Geometry of rank-one tensors and typical (sub)ranks of order-three tensors

Geometry of rank-one tensors and typical (sub)ranks of order-three tensors Dissertation zur Erlangung des Grades Doktor der Naturwissenschaften (Dr. rer. nat.) des Fachbereichs Mathematik/Informatik/Physik der Universität Osnabrück vorgelegt von Sarah Eggleston Betreuer Prof. Dr. rer. nat. Paul Breiding Osnabrück, 2025 Contents Introduction 5 1 Preliminaries 9 1.1 Tensors ...................................... 9 1.1.1 Rankandsubrank ............................ 11 1.2 Metric algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.1 Concepts from differential geometry . . . . . . . . . . . . . . . . . . 17 2 Reach of Segre-Veronese Manifolds 21 2.1 Introduction.................................... 21 2.2 Tensor products of Riemannian manifolds . . . . . . . . . . . . . . . . . . . 23 2.2.1 The second fundamental form of a tensor product manifold . . . . . 24 2.3 Geodesics and the second fundamental form of Segre-Veronese manifolds . . 25 2.3.1 Veronese manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2 Segre-Veronese manifold . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Reach of the Segre-Veronese manifold . . . . . . . . . . . . . . . . . . . . . . 32 2.4.1 Extremal curvature of the Segre-Veronese manifold . . . . . . . . . . 32 2.4.2 Bottlenecks of the Segre-Veronese manifold . . . . . . . . . . . . . . 34 2.5 Volume of the tubular neighborhood . . . . . . . . . . . . . . . . . . . . . . 34 2.5.1 Perfect matchings in graphs and random determinants . . . . . . . . 36 3 Typical ranks of order-three tensors 41 3.1 Introduction.................................... 41 3.2 Typicalranks................................... 42 3.3 Geometric perspective on typical ranks . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Random linear subspaces . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.2 A probabilistic version of Friedland’s theorem . . . . . . . . . . . . . 46 3.3.3 Typical ranks of tall tensors . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Random 3×3×5tensors and cubic surfaces . . . . . . . . . . . . . . . . . 49 3.5 Asymptotics and heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4 Typical subranks of order-three tensors 55 4.1 Introduction.................................... 55 4.2 Real and complex subrank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 Typical subranks are consecutive . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4 Subrank for specific order-three formats . . . . . . . . . . . . . . . . . . . . 62 4.4.1 Typical subranks of 2×m×ntensors................. 62 4.4.2 A geometric method to bound the subrank from above . . . . . . . . 64 4.4.3 Typical subranks of 3×3×5tensors.................. 64 4.4.4 Typical subranks of 4×4×4tensors.................. 66 4.5 Subrank of direct sums of real division algebras . . . . . . . . . . . . . . . . 67 Bibliography 71 3 Introduction As generalizations of matrices to higher dimensions, tensors are ubiquitous in a wide range of fields such as data science, psychometrics, and various topics in mathematics. In particular, the rank of a tensor T, or the minimum number of simple tensors v1⊗···⊗vdneeded to sum to T, is an important property that quantifies in a sense the degree of independence of the data stored in the d-dimensional table of numbers represented by an order-dtensor. The study of principal component analysis is a powerful tool in data analysis used to estimate the most important trends in a given dataset; mathematically, this is performed by approximating the data by a low-rank matrix. The study of geometric properties of rank-one tensors, known as the Segre variety, and ranks of random order-three tensors is the focus of this thesis. If we think of an order-dtensor as storing the probabilities of different events based on d independent random input variables, the probabilities will be multiplicative, and hence the tensor will have rank one. In this setting, higher rank indicates interdependencies among the variables, which is very common in real-world applications. In addition, data inevitably contains noise, leading to a random component in the entries of the tensor. Even if the variables were to be completely independent, this noise would cause the rank of the tensor to be higher; indeed, the typical ranks of a given tensor size describe the possible ranks that a random tensor could take, and there are even small formats of tensors for which the typical ranks are only known from numerical simulations. The study of tensor rank and decomposition is not new; since the definition of tensor rank by Hitchcock in 1927 [35] as the number of outer products of vectors required to sum to the tensor, much progress has been made in this field, but there are still many open questions. Tensor rank is important in the field of complexity theory, for example; improving the efficiency of algorithms for performing calculations has been important to mathematicians for centuries, and with the development of computers in the mid-twentieth century, interest in complexity theory has only grown. For example, multiplication of two 2×2tensors using the usual algorithm requires eight multiplications; Strassen’s proof that only seven are required [55] provides a more efficient algorithm that is still implemented in computing software today. In the language of tensor rank: the multiplication of two 2×2 tensors can be encoded in a 4×4×4tensor with precisely eight non-zero entries that is hence of rank at most 8. Strassen showed that these eight terms can be indeed written as the sum of only seven rank-one tensors, thus demonstrating that the rank is actually at most 7; shortly thereafter, Winograd proved that the rank is exactly equal to 7 [66]. Matrix rank can be defined in many equivalent ways, such as the maximum number of ones on the diagonal of the matrix after performing Gaussian elimination to the rows and columns; the number of linearly independent rows or columns of the matrix; or the number of non-zero singular values of the matrix. For higher order tensors, however, these definitions no longer coincide, and generalizations of different definitions have appeared in the literature. In addition to Hitchcock’s definition, subrank [57], slice rank [60] and geometric rank [41] are also being studied, among others. For example, the subrank of a tensor T is the maximum number of ones on the diagonal of the tensor after performing Gaussian 5 Introduction elimination along all axes; in a sense, this is dual to the notion of rank. Relationships between the different notions of rank for a given tensors have been studied; for example, it can be seen from the definitions that the subrank of a tensor Tis always at most equal to the rank of T. However, although the same basic questions can be asked in many cases, such as what the rank/subrank/geometric rank/slice rank of a generic n×n×ntensor is, the strategies for proofs can be significantly different depending on the type of rank that is of interest. In this thesis, we focus on both rank and subrank of tensors and show that while there is a direct correspondence between the two notions of rank for some formats of tensors, this does not hold in general. While both tensor rank and subrank are defined algebraically, both have geometric aspects, just as in the case of matrix rank. In this thesis, we use tools both from algebra and geometry to answer questions about tensor rank. In particular, we relate the rank of a tensor to the number of intersection points of two manifolds: the Segre variety, for which the degree is known [e.g. 32, Lecture 18], and a linear space related to the tensor. This allows us to use tools from geometry and intersection theory that date back to the eighteenth century with Bézout’s Theorem. Intersection theory is a rich field of its own, though we will not rely too heavily on this branch of algebraic geometry here. In many cases, particularly in data science, the exact rank of a given tensor may be of less interest than a close low-rank approximation of the tensor. An optimal rank-one approximation of a tensor Tis defined as the “closest” rank-one tensor to T, i.e. the shortest distance from Tto the set of rank-one tensors. In Chapter 2, we turn to the study of the geometry of the set of rank-one partially symmetric tensors. While the geometric foundations of this chapter date back to the eighteenth and nineteenth centuries with results from Gauss and Riemann on the curvature of manifolds, the heart of this chapter is based on metric algebraic geometry, a field that applies tools from classical geometry to the objects of interest in algebraic geometry: varieties. Tensors can be defined over any field; however, while those coming from applications, such as data science are typically defined over R, many aspects of tensor rank are more easily proved over the complex numbers because the Cis algebraically closed. In this thesis we focus on tensors over the real numbers, where there are still many open questions. To investigate low-rank approximations of tensors, it is first necessary to define a loss function to minimize, such as a metric providing a distance function within the tensor space. However, this alone does not guarantee the existence of a unique solution; depending on the metric and the geometry of the set of rank-one tensors, there may be a tensor for which two distinct rank-one tensors minimizes the distance to the given tensor. To determine which points in the tensor space have unique rank-one approximations, we can calculate the reach of the set of rank-one tensors, which is the focus of Chapter 2. Although finding a rank-one (or otherwise low-rank) approximation to a given tensor is advantageous in a field such as data science, where the given data are most likely influenced by measurement error including noise, it may also be of interest to compute the exact rank of a given tensor. In some cases, a decomposition of the tensor as a sum of rank-one tensors may also be desirable. For small tensor formats, computation of the rank of a tensor is a purely algebraic problem and can be solved, accordingly, exclusively using tools from algebra. For example, given a generic 2×n×ntensor Tfor n≥2, it is possible to determine whether Thas rank nor not, as proved by [8]: slice Talong the first direction to obtain two n×nmatrices T1and T2and consider the degree-npolynomial p(λ) = det(T1−λT2). The rank of Tis nprecisely when p(λ)has nroots. For an algebraically closed field, such as the complex numbers, this is always the case; however, this is not true over every field. 6 Regarding other tensor formats, [62] showed that the rank of a generic 3×3×5tensor is related to the number of roots of a degree-seven polynomial whose coefficients are functions of the entries of the tensor, and other authors have made progress in this area using purely algebraic results for larger tensor formats [51]. Here, we address the question of tensor rank of generic tensors using approaches from algebraic geometry. Thus, instead of relying on the algebraic definition of tensor rank, we use an equivalent geometric definition, allowing us to express the condition that the rank is equal to a certain number in terms of the number of intersection points of two manifolds. In Chapter 4, we turn to typical subranks of real tensors. While the generic subrank, which is also the largest typical subrank, is a known function of the dimensions of the vector spaces, we explore some small formats and prove that there are multiple typical subranks for certain formats. To do this, we develop a geometric strategy, again related to the number of intersection points of two spaces. 7 Introduction Acknowledgments I am very grateful to many people for their help and support over the last three years; without them, this thesis would not have been possible. My advisor, Paul Breiding, took a chance on accepting me with my rather unconventional background. I continue to be impressed by his ability to always provide enough help while also challenging his students. I am so happy to have had this opportunity and the chance to learn so much and meet so many other fantastic mathematicians through Paul. I am deeply indebted to Nick Vannieuwenhoven for his useful feedback on various chapters of this thesis during two trips to Leuven. Many thanks to Jan Draisma for encouraging me to continue in math after my master’s degree and for the many inspirational discussions. Although he only co-authored one of the papers that this dissertation is based on, he provided many ideas used in each of the other chapters. It was a great pleasure to finally have the opportunity to collaborate with Benjamin Biaggi. I have learned so much from Beni not only about math but also bouldering and am very grateful to have had a place to stay while visiting Bern! Working together with Andrea Rosana was a wonderful experience. Andrea taught me so much about algebraic geometry and was a fantastic tour guide in Trieste. I have yet to co-author a paper with Tim Seynnaeve, but he has been extremely helpful throughout my mathematical career. Tim’s knowledge and interests in math are so broad; I have learned so much from him, and much of the material in this thesis benefited from conversations with him. Lianne de Jonge, Bernhard Hafer, Justus Bruckamp, Daniel Köhne, Tarek Emmrich, Mathias Hockmann, Kinga Nagy, Harsha and everyone else at the math institute made the last three years much more enjoyable. A very special thanks to Tarek Emmrich for proofreading parts of this thesis. Lennart Kaiser, Mario Dyczka and David Rolfes generously provided me with a second office at the Ringlokschuppen. Many thanks for your hospitality, the many cookies, and the technical support! The members of our group, Viktoriia Borovik, Pierpaola Santarsiero, Elima Shehu, Erin Connelly and Leo Mathis, provided very useful feedback to the various projects in this thesis. I am especially grateful to Erin for thoroughly proofreading my thesis, and to Pierpaola for many helpful tips and quick answers to any questions I might have about tensors. Many thanks for the group meetings and the wonderful time in Osnabrück and together at conferences! Petra Schoppenhauer has been instrumental in providing organizational support from the very beginning; while German bureaucracy does not make anything particularly easy, Petra always made sure it was done right. When I left a comfortable position in Switzerland to do a PhD in math, I’m not sure many of my friends or family understood why; but nevertheless, they and my newfound friends in Osnabrück supported me so much over the past three years, for which I am extremely grateful. A PhD is an adventure in many ways, and this one would not have been nearly as much fun without Thomas Hänel at my side. I am deeply indebted to Thomas for his infinite patience, encouragement and support. 8 1 Preliminaries In this thesis, we use algebraic geometry to study several properties of tensors. We will generally use bold-faced capital letters to refer to matrices and higher-order tensors Tand bold-faced lower-case letters to refer to vectors v. However, note that because tensors are also used to refer to multi-linear maps, we will sometimes refrain from using bold-faced notation in this context. We work both in affine and projective space and will denote projective space by Pn(or RPnor CPnwhen referring to real or complex projective space, respectively). The projectivization of a linear space Lwill generally be denoted by LP, but we will sometimes omit the subscript and use Lfor both the affine and projective subspace; when necessary, it will be specified. We will use ⟨,⟩to denote an inner product, and span{v1,...,vn}for the linear span of the vectors vi. The set {1, . . . , r}will also be written as [r]. When considering the probability of an event A, we will write P{A}, and the expectation value of a random variable Xwill be denoted as E(X). 1.1 Tensors While it will generally suffice to consider a tensor to be a d-dimensional table of numbers, we also make use of several properties of tensors, such as the correspondence between tensors and multilinear maps. Formally, a tensor is an element of a tensor product: Definition 1.1.1 (Tensor product).Given a field Kand K-vector spaces U, V , the tensor product of Uand Vis given by U⊗V=F(U×V)/N, where F(U×V)is the free vector space over the Cartesian product of Uand V, and Nis the linear space spanned by elements of the form (λu,v)−λ(u,v) (u, λv)−λ(u,v) (u1+u2,v)−(u1,v)−(u2,v) (u,v1+v2)−(u,v1)−(u,v2) for λ∈K,u,u1,u2∈Uand v,v1,v2∈V. The tensor product can naturally be extended to d(or infinitely many) vector spaces of arbitrary dimensions; for K-vector spaces V1, . . . , Vd, we call an element T∈V1⊗···⊗Vd an order-dor d-dimensional tensor. A tensor of order zero is simply a number; a onedimensional tensor is a vector; an order-two tensor is a matrix. In the following chapters, we will mostly focus on order-three tensors; and in any case, only tensors of finite order (d < ∞). Furthermore, while tensors may be defined over any product of linear spaces, we will typically restrict ourselves to coordinate linear spaces, sometimes called “Cartesian tensors”, as an n-dimensional linear space over the field Kis isomorphic to Kn. We will use different notation to describe tensors, depending on the context and application. 9 1 Preliminaries While this is not yet completely solved, it is known that this holds in the case where the vector spaces are of small dimension. Theorem 1.1.21 ([19]).Let K=Ror Cand let A1, A2, B1, B2, C1, C2be vector spaces over K. Assume T1∈A1⊗B1⊗C1,T2∈A2⊗B2⊗C2and T:= T1⊕T2. Furthermore, assume that dim B2≤3and dim C2≤3. Then rank(T) = rank(T1) + rank(T2). Note that this theorem is not directly relevant for computing typical ranks of tensors, because given two generic tensors T1,T2, the direct sum T1⊕T2will not be generic. A very simple counterexample to the additivity of generic ranks is seen by setting A1=A2= 1, B1=B2=C1=C2=nfor any n≥1. The only typical rank of 1×n×ntensors (in this case, n×nmatrices) is n; however, the space of 2×2n×2ntensors has typical ranks 2n and 2n+ 1. In the case of subrank, it has also been shown that additivity of subrank over direct sums of tensors may only hold for small dimensions: Theorem 1.1.22 ([25]).There are tensors T1,T2∈Kn×n×nwith Q(T1), Q(T2)≤√3n−2 and Q(T1⊕T2)≥n. For n= 12, this theorem guarantees us the existence of tensors T1and T2with subrank at most 5 while T1⊕T2has subrank at least 12 >5·2. Nevertheless, we will use the additivity of subrank over tensor spaces of small dimension to investigate subranks of some specific tensors, which will be important to our investigation of the subrank of multiplication tensors of division algebras at the end of Chapter 4. 1.2 Metric algebraic geometry Given points in a Euclidean space, a natural question is how far they are from each other. This necessitates the choice of a norm to measure this distance. With this choice, we can also describe characteristics of a set of points in the space: broadly speaking, the shape of the set. Given a point x, a set Sin an embedded Riemannian manifold M⊆RNand the Riemannian metric dinherited from RN, we are interested in determining the shortest distance between xand S, i.e. the solution to the following optimization problem: inf{d(x, y)|y∈S}. This motivates our definition of the distance from a point to a set: Definition 1.2.1. Given a set Sin a Riemannian manifold Mwith a Riemannian metric dand a point x∈M, we define the distance of xto Sas d(x, S) := inf{d(x, y)|y∈S}. While a general point on Mwill have a unique closest point on S, a certain set of points, called the medial axis, has more than one closest point. Definition 1.2.2 (Medial axis).Given a set Sin a Riemannian manifold Mwith distance function dgiven by the Riemannian metric, the medial axis of Sis given by Med(S) := {x∈M| ∃ y1, y2∈S, y1=y2:d(x, y1) = d(x, y2) = d(x, S)}. 16 1.2 Metric algebraic geometry The medial axis of Sper definition is disjoint from the set Sitself; nevertheless, it is useful for characterizing the geometry of S. For example, the distance of the medial axis to the set, or the reach of S, will be the main focus of study in Chapter 2. Definition 1.2.3 (Reach).Given a set Sand a distance function d, the reach of Sis given by τ(S) := inf y∈S{d(y, Med(S))}. This notion was first introduced by Federer [27]. Specifically, in this thesis, we will focus on the variety of partially symmetric rank-one tensors. Given a partially symmetric tensor, the closest point on the corresponding SegreVeronese variety represents the best rank-one approximation of this tensor. We consider a tubular neighborhood around the variety, or the set of points at most a given distance from the variety. Any point in the tubular neighborhood with radius less than the reach of the Segre-Veronese variety is then guaranteed to have a unique best rank-one approximation. 1.2.1 Concepts from differential geometry As the reach is a geometric property of a variety, we will rely on many results from differential geometry in Chapter 2. In order to be able to discuss distances, we must be in a Riemannian manifold. Definition 1.2.4 (Manifold).Let n, N ∈Nwith n≤N. Then M⊆RNis an ndimensional differentiable manifold of RNif there exists a family of open sets Ui⊆Rnand a family of injective mappings φi:Ui→Msuch that ⋃︁iφi(Ui) = Mand for any pair i, j, if V:= φi(Ui)∩φj(Uj)=∅, then φ−1 i(V)and φ−1 j(V)are open in Rnand the map φ−1 j◦φiis differentiable. Each pair (Ui, φi)is a chart, and the family of charts (Ui, φi)iis an atlas. A non-singular variety, such as the projective Veronese variety, is a manifold. We will use this to show that the Segre-Veronese is also a manifold in Proposition 2.2.1. A useful description of a manifold is the tangent space, as this is, for each point in the manifold, a linear space which can therefore easily be parameterized. Definition 1.2.5 (Tangent space).Let M⊆RNbe an n-dimensional manifold and the point p∈M; furthermore, let (U, φ)be a chart of Mwith p∈φ(U). Given two curves γ1:I1→Mand γ2:I2→M, where I1, I2are intervals containing 0and γ1(0) = p= γ2(0), we write that γ1∼γ2if (φ◦γ1)′(0) = (φ◦γ2)′(0). The tangent space to Mat p, TpM, is then given by the set of equivalence classes of this equivalence relation. Perpendicular to the tangent space, we define the normal space of the manifold at a point: Definition 1.2.6 (Normal space).Let M⊆RNbe an n-dimensional manifold, the point p∈M, and TpMthe tangent space of Mat p. Let ⟨·,·⟩ be an inner product on RN. Then the normal space to Mat pis the linear subspace given by NpM={v∈RN|v⊥TpM}, where the orthogonality of vand TpMis respect to the given inner product. 17 1 Preliminaries When considering the distance between a manifold Mand a point x∈ M, the distance in Definition 1.2.1 is minimized by a point y∈Mwhere the vector vfrom xto yis orthogonal to Mat y; in other words, v∈NyM. In calculation the reach of M, we are interested in points in the ambient space for which the distance to the manifold is minimized by two (or more) distinct points on M; hence, we will seek vectors vin the normal space at two (or more) distinct points of M:v∈Ny1M∩Ny2M. The reach is also a function of the curvature of the manifold: consider an osculating circle to the manifold at a point of high curvature. While the center of circle may not be equidistant to multiple points on the manifold, if the manifold is not the circle itself, the radius of the osculating circle is nevertheless an upper bound for the reach, as for any ε > 0a point with distance at most εfrom the center will have this property. Because we are interested in the maximum curvature, we can rely on curves on the surface of the manifold. Specifically, we consider geodesics: Definition 1.2.7 (Geodesic).Let M⊆RNbe a manifold and γ:I→Mbe a curve along M. Then γis a geodesic if γ′′(t)∈Nγ(t)Mfor all t∈I. We say that γis parametrized by arc length if ∥γ′(t)∥= 1 for all t∈I. Here, the distance function is induced by the Riemannian metric on Minherited from RN. Hence, a curve γalong Mis a geodesic if it locally minimizes the distances between two points. In Euclidean space, geodesics are straight lines; on the sphere, geodesics are great circles. Here, we are interested in the volume of partially symmetric tensors of a given format within a certain distance – the reach – of the Segre-Veronese variety. This computation requires us to first determine the Weingarten map and the second fundamental form, as these quantify the principal curvatures of the manifold. Definition 1.2.8 (Second fundamental form, Weingarten map).Given a Riemannian manifold M⊆RNand a point x∈M, the second fundamental form of Mat xis given by the map IIx:TxM×TxM×NxM→R,(v,w,a)↦→ ⟨︃∂v(u) ∂wu=x,a⟩︃. Here, v(u)represents a local smooth tangent field of Mwith v(x) = v. The Weingarten map of Min x∈Mand a∈NxMis then given by La:TxM→TxM, such that IIx(v,w,a) = ⟨v, La(w)⟩. Finally, Weyl’s tube formula [65] will allow us to compute the volume of a tubular neighborhood of the Segre-Veronese variety. In our application in Chapter 2, we are interested in the special case of a manifold on the sphere SN⊂RN+1. Theorem 1.2.9 ([65]).Let M⊆SNbe an n-dimensional manifold. Then the volume of a tubular neighborhood of radius εaround Mis given by V(ε) = ∑︂ 0≤2i≤n κiJi(ε), where Ji(ε) = ∫︂ε 0 (sin ϕ)N−n+2i−1·(cos ϕ)n−2idϕ=∫︂tan ε t=0 tN−n+2i−1 (1 + t2)N+1 2 dt 18 1.2 Metric algebraic geometry and κi=∫︂G∈M(︄∫︂a∈NGM:∥a∥=1 m2i(La) da)︄dG. Here, the coefficients m2i(La)represent the sum of the 2i-principal minors of the Weingarten map Lain the normal direction a. 19 2 Reach of Segre-Veronese Manifolds This chapter is based on the publication [15]. 2.1 Introduction In this chapter we study the metric geometry of rank-one tensors. More specifically, we compute the reach and the volume of a tubular neighborhood of the Segre-Veronese variety. Since rank-one tensors form a cone, we intersect the Segre-Veronese variety with the unit sphere, thus obtaining the spherical Segre-Veronese manifold; the proof that this is a manifold is provided below. We first describe the setting. Let Hn,d denote the vector space of homogeneous polynomials in n+1 variables x0, . . . , xn of degree d. As discussed in Chapter 1, this is isomorphic to the projective space of symmetric tensors Symd(Kn). We consider the Bombieri-Weyl inner product ⟨,⟩on Hn,d: this is the inner product corresponding to the orthogonal basis vectors mα:= √︂(︁d α)︁xα, where (︁d α)︁is the multinomial coefficient for α= (α0, . . . , αn). The reason for this choice is that the Bombieri-Weyl inner product is invariant under an orthogonal change of coordinates; this was proved by Kostlan [42, 43]. The norm of a polynomial f∈Hn,d is ∥f∥=√︁⟨f,f⟩, and the sphere is S(Hn,d) := {f∈Hn,d | ∥f∥= 1}. For n, d ≥1, we denote the real variety of powers of linear forms in Hn,d by ˆ︁ Vn,d := {±ℓd|ℓis a linear form in x0, . . . , xn}. As discussed in Chapter 1, ˆ︁ Vn,d is isomorphic to the projective Veronese variety. The hat indicates that ˆ︁ Vn,d is the cone over a spherical variety V◦ n,d := ˆ︁ Vn,d ∩S(Hn,d), which we call the spherical Veronese variety. Its dimension is dim V◦ n,d =n. We fix r-tuples of positive integers d= (d1, . . . , dr)and n= (n1, . . . , nr)and write Hn,d:= Hn1,d1⊗···⊗Hnr,dr. The elements in Hn,dare partially symmetric tensors. They are multihomogeneous forms in rsets of variables. The number d=d1+···+dris called the total degree of the tensors in Hn,d. For a tensor F=∑︁α1,...,αrFα1,...,αrmα1⊗···⊗mαr∈Hn,d, we use the short form F= (Fα1,...,αr). The following defines an inner product on Hn,d: ⟨F,G⟩:= ∑︂ α1,...,αr Fα1,...,αr·Gα1,...,αr,where F= (Fα1,...,αr),G= (Gα1,...,αr)∈Hn,d. With this, Hn,dbecomes a Euclidean space, and we can measure volumes and distances in Hn,d. The norm of a tensor F∈Hn,dis ∥F∥:= √︁⟨F,F⟩, and the angular distance is dS(F,G) := arccos⟨F,G⟩for F,Gin the unit sphere S(Hn,d)⊂Hn,d. 21 2 Reach of Segre-Veronese Manifolds The (spherical) Segre-Veronese variety in Hn,dis Xn,d:= {︂f1⊗···⊗fr|fi∈ˆ︁ Vni,di}︂∩S(Hn,d).(2.1) This is the variety of products of powers of linear forms in Hn,dthat have unit norm. We prove in Proposition 2.2.1 that Xn,dis an embedded smooth submanifold of S(Hn,d) of dimension dim Xn,d=n1+···+nr; hence, we call Xn,da (spherical) Segre-Veronese manifold. The main focus of this chapter is the reach and the volume of a tubular neighborhood of Xn,d. In our first main theorem we calculate the reach of the (spherical) Segre-Veronese manifold. Theorem 2.1.1 (The reach of (spherical) Segre-Veronese manifolds).Let d= (d1, . . . , dr) and n= (n1, . . . , nr)be r-tuples of positive integers, and let d:= d1+···+dr≥2be the total degree. The reach of the (spherical) Segre-Veronese manifold is τ(Xn,d) = ⎧ ⎨ ⎩ π 4, d ≤5 √︂d 2(d−1), d > 5. In particular, the reach only depends on the total degree dand not on the dimensions of the Veronese varieties Vni,di. This extends a theorem by Cazzaniga, Lerario and Rosana [21], who proved this formula for the Veronese variety, which is the special case r= 1. Another special case worth mentioning is d1=··· =dr= 1, which corresponds to the Segre manifold. Since Xn,dis a smooth submanifold of the sphere, its reach is the minimum of the inverse of its maximal curvature and its smallest bottleneck. We also compute these. The next theorem explains which curves in Xn,dhave maximal and minimal curvature; this is proved in Section 2.4.1. Theorem 2.1.2 (Extremal curvature of curves in Segre-Veronese manifolds).Let the total degree of the (spherical) Segre-Veronese manifold Xn,dbe d=d1+···+dr≥2. Consider a geodesic γ(t) = γ1(t)⊗···⊗γr(t)∈Xn,d. 1. The maximum curvature of γ(t)is √︂2(d−1) d.It is attained by geodesics where the γi(t) are geodesics in V◦ ni,diof constant speed ∥γ′ i(t)∥=√︂di d. 2. The minimal curvature is √︂2(dℓ−1) dℓ, where dℓ= min{d1, . . . , dr}. It is attained by geodesics where γℓ(t)is a geodesic parametrized by arc length in V◦ nℓ,dℓand the other γi(t)are constant. Our third main result concerns the volume of the tubular neighborhood U(ε) := {F∈S(Hn,d)dS(F,Xn,d)< ε}. In Section 2.5 we compute this volume in terms of complete matchings in a weighted graph. For a tuple (v1, . . . , vr)of nonnegative integers let G= (V, E)be the complete graph on 22 2.2 Tensor products of Riemannian manifolds v:= v1+···+vrvertices. Recall that the tuple of degrees is d= (d1, . . . , dr). We define weights on Eas follows: the vertices of Gare partitioned into rgroups V=I1⊔···⊔Ir of cardinalities |Ik|=vk. The weight w(e)of an edge ebetween vertices in group Ikis w(e) = dk−1 dk, and the weight of an edge across groups is 1. Given a perfect matching C⊂Ewe define its weight to be w(C) := ∏︁e∈Cw(e).This defines the function Dd(v1, . . . , vr) := (−1)v 2∑︂ C⊂Eperfect matching w(C).(2.2) We now have the following result. Theorem 2.1.3 (Volume of a tubular neighborhood).In the following, let n= dim Xn,d, N= dim(S(Hn,d)) and c=N−n. Define Ji(ε) = ∫︁ε 0(sin ϕ)N−n+2i−1·(cos ϕ)n−2idϕ and write θi=Γ(c 2) 2iΓ(i+c 2)∑︂ v1,...,vr∈N:vi≤ni v1+···+vr=2i Dd(v1, . . . , vr). Then for ε<τ(Xn,d), we have vol(U(ε)) = √︁(d1···dr)n 2r−1·vol(Sn1)···vol(Snr)·vol(Sc−1)·∑︂ 0≤2i≤n θi·Ji(ε). The proof of this theorem is based on computing the Weingarten map of Xn,d, which we do in Theorem 2.3.8. We show that the Weingarten map of Xn,dadmits a block structure where the diagonal blocks are the Weingarten maps of the Veronese factors. At the end of Section 2.5.1 we compute the coefficients θifrom Theorem 2.1.3 for the Segre manifold Xn,dwhere n=1r:= (1,1,...,1) and d=1r. This Segre manifold is the image of S1×···×S1under the Segre embedding. Organization In Section 2.2 we discuss the differential geometry and curvature of manifolds defined by tensor products of vectors. We then apply the results from Section 2.2 in Section 2.3 to study the curvature of the (spherical) Segre-Veronese manifold Xn,d. In particular, we work out the Weingarten map of Xn,d. In Section 2.4 we compute the reach and prove Theorems 2.1.1 and 2.1.2. Finally, in Section 2.5, we compute the volume of the tubular neighborhood and prove Theorem 2.1.3. 2.2 Tensor products of Riemannian manifolds The tensor space Rm1+1 ⊗···⊗Rmr+1 is a Euclidean space for the inner product defined by ⟨x1⊗ ··· ⊗ xr,y1⊗ ··· ⊗ yr⟩=⟨x1,y1⟩···⟨xr,yr⟩, where ⟨x,y⟩=xTy. Write N:= (m1+1) ···(mr+ 1)−1; then SNis the sphere in Rm1+1 ⊗···⊗Rmr+1. We consider for 1≤i≤ra smooth embedded submanifold Miof the sphere Smi⊂Rmi+1. We define the tensor product of these manifolds to be M1⊗···⊗Mr:= {x1⊗···⊗xr|x1∈M1,...,xr∈Mr}. Proposition 2.2.1. For 1≤i≤rlet Mibe a smooth Riemannian submanifold of Smiof dimension ni, and denote M:= M1⊗ ··· ⊗ Mr.Furthermore, denote the tensor product map by ψ:M1×···×Mr→M,(x1,...,xr)↦→ x1⊗···⊗xr. Then: 23 2 Reach of Segre-Veronese Manifolds 1. Mis a Riemannian submanifold of SNof dimension n1+···+nr. 2. The tangent space of Mat x=x1⊗···⊗xris TxM=Tx1M1⊗x2⊗···⊗xr+···+x1⊗x2⊗···⊗TxrMr. 3. ψis a local isometry. Proof. For 1≤i≤rlet Ai= (Ui,j, φi,j)jbe an atlas for Misuch that u∈Ui,j implies that the antipodal point −u∈ Ui,j. Such an atlas exists since 0∈ Mi. Define the open sets Uj1,...,jr:= ψ(U1,j1×···×Ur,jr); then ψ|U1,j1×···×Ur,jris an isomorphism, so we have an atlas for Mwith charts Uj1,...,jrand maps (φ1,j1×. . .×φr,jr)◦(ψ|U1,j1×···×Ur,jr)−1. This also shows that we have dim M= dim(M1×···×Mr) = n1+···+nr. The Riemannian structure on the ambient space SNinduces a Riemannian structure on M. For the second statement, we use that T(x1,...,xr)(M1×···×Mr) = Tx1M1×···×TxrMr. For 1≤i≤rlet v∈TxiMi. By multilinearity, the derivative of ψat (x1,...,xr)maps D(x1,...,xr)ψ(0,...,0,v,0,...,0) = x1⊗···⊗xi−1⊗v⊗xi+1 ⊗···⊗xr. This proves the second statement, since TxMis the image of D(x1,...,xr)ψ. Finally, for v∈TxiMiand w∈TxjMjwe have ⟨x1⊗···⊗v⊗···⊗xr,x1⊗···⊗w⊗···⊗xr⟩={︄⟨v,w⟩, i =j ⟨v,xi⟩⟨w,xj⟩, i =j. Since ⟨v,xi⟩=⟨w,xj⟩= 0, this shows that the inner product between the images of (0,...,0,v,0,...,0) and (0,...,0,w,0,...,0) under D(x1,...,xd)is ⟨v,w⟩, if i=j, and 0 otherwise. This shows that the derivative D(x1,...,xr)ψpreserves inner products on a basis of T(x1,...,xr)(M1× ··· × Mr)and hence is an orthogonal map. This proves the third statement. Using the notation of Proposition 2.2.1 we can now write Xn,d={︄Vn1,d1⊗···⊗Vnr,dr,if one is diodd; (Vn1,d1⊗···⊗Vnr,dr)∪ −(Vn1,d1⊗···⊗Vnr,dr),if all diare even. (2.3) Furthermore, Proposition 2.2.1 implies that Xn,dis a smooth submanifold of the sphere of dimension dim Xn,d=n1+···+nr. Therefore, we will henceforth call it the (spherical) Segre-Veronese manifold. 2.2.1 The second fundamental form of a tensor product manifold The next proposition provides the Weingarten map for a tensor product of manifolds. Proposition 2.2.2. Let M1,...,Mrbe as in Proposition 2.2.1 and M=M1⊗···⊗Mr. Consider a point x=x1⊗ ··· ⊗ xr∈Mand a normal vector a∈NxM. A matrix representation of the Weingarten map of Mat xin direction arelative to orthonormal coordinates is La=⎡ ⎢ ⎢ ⎢ ⎣ L1L1,2··· L1,r (L1,2)TL2··· L1,r−1 ... (L1,r)T(L1,r−1)T··· Lr ⎤ ⎥ ⎥ ⎥ ⎦, 24 2.3 Geodesics and the second fundamental form of Segre-Veronese manifolds where the matrices Li,j and Liare defined as follows: let v(i) 1,...,v(i) nibe an orthonormal basis for the tangent space TxiMi. 1. The off-diagonal blocks are Li,j := [︂⟨x1⊗···⊗v(i) k⊗···⊗v(j) ℓ⊗···⊗xr,a⟩]︂1≤k≤ni,1≤ℓ≤nj∈Rni×nj. 2. Write Ri:= x1⊗ ··· ⊗ xi−1⊗NxiMi⊗xi+1 ⊗ ··· ⊗ xr,and let the orthogonal projection of aonto Ribe x1⊗···⊗xi−1⊗ai⊗xi+1 ⊗···⊗xr. Then ai∈NxiMi, and Li∈Rni×niis a matrix representation of the Weingarten map Laiof Miat xi in direction aiwith respect to the orthonormal basis v(i) 1,...,v(i) ni. Proof. By Proposition 2.2.1, x1⊗ ··· ⊗ v(i) k⊗ ··· ⊗ xrfor 1≤i≤rand 1≤k≤ni is an orthonormal basis of TxM. Fix tangent vectors v=x1⊗ ··· ⊗v(i) k⊗··· ⊗xrand w:= x1⊗···⊗v(j) ℓ⊗···⊗xr. Furthermore, let v(i) k(ui)be a local smooth tangent field of Miwith v(i) k(xi) = v(i) k. Then we obtain a local smooth tangent field of Mwith v(x) = v by setting v(u1⊗···⊗ur) := u1⊗···⊗v(i) k(ui)⊗···⊗ur. By multilinearity, ∂v(u) ∂w=⎧ ⎪ ⎨ ⎪ ⎩ x1⊗···⊗ ∂v(i) k(ui) ∂v(i) ℓ⊗···⊗xr,if i=j; x1⊗···⊗v(i) k⊗···⊗v(j) ℓ⊗···⊗xr,if i=j. This shows that the off-diagonal blocks of Laare the matrices Li,j. For the diagonal blocks (i=j)we observe that x1⊗···⊗ ∂v(i) k(ui) ∂v(i) ℓ⊗···⊗xr∈Ri, so ⟨v, La(w)⟩= IIx(v,w,a) = ⟨︃x1⊗···⊗ ∂v(i) k(ui) ∂v(i) ℓ⊗···⊗xr,a⟩︃ =⟨︃∂v(i) k(ui) ∂v(i) ℓ ,ai⟩︃=⟨v(i) ℓ, Lai(v(i) k)⟩. This settles the case i=j. 2.3 Geodesics and the second fundamental form of Segre-Veronese manifolds We now use the results from the previous section to compute the second fundamental form and the Weingarten map for a (spherical) Segre-Veronese manifold Xn,d.The first step towards this goal is considering the Veronese manifold (r= 1). 2.3.1 Veronese manifolds The Bombieri-Weyl inner product on the space of homogeneous polynomials Hn,d has the property that ⟨f,ℓd⟩=f(ℓ0, . . . , ℓn),where ℓ(x) = ℓ0x0+···+ℓnxn;(2.4) 25 2 Reach of Segre-Veronese Manifolds 2.4 Reach of the Segre-Veronese manifold We compute the reach τ(Xn,d)of the Segre-Veronese manifold. We adapt the strategy from [21] and calculate the reach as the minimum of two quantities: τ(Xn,d) = min{ρ1, ρ2}, where ρ1is the inverse of the maximal curvature of a curve in Xn,dthat is parametrized by arc length: 1 ρ1 = sup {︁∥PE(γ′′(0))∥ | γis a geodesic in Xn,dparametrized by arc length}︁; here PEdenotes the orthogonal projection onto TES(Hn,d). The second quantity, ρ2,is the width of the smallest bottleneck: ρ2= min {︁1 2dS(F,E)F∈Xn,d,F=Eand F−E∈R·E⊕NEXn,d}︁. The goal of this section is to prove the following proposition, giving formulas for both ρ1 and ρ2. Proposition 2.4.1. Let d= (d1, . . . , dr)and n= (n1, . . . , nr)be r-tuples of positive integers, and let d:= d1+···+dr≥2. For the (spherical) Segre-Veronese manifold Xn,d of total degree d, we have 1. ρ1=√︂d 2(d−1). 2. ρ2=π 4. We prove Proposition 2.4.1 (1) in Section 2.4.1 and (2) in Section 2.4.2. Because the reach is the minimum of ρ1and ρ2, this proves Theorem 2.1.1. 2.4.1 Extremal curvature of the Segre-Veronese manifold Let γ(t)be a geodesic in Xn,dparametrized by arc length. By orthogonal invariance (Lemma 2.3.5) we can assume that γ(0) = E. As shown in Lemma 2.3.11, geodesics in Xn,dparametrized by arc length that pass through Ecan, without loss of generality, be written as γ(t) = γ1(t)⊗···⊗γr(t) where γi(t) = (cos(d−1/2 iait)x0+ sin(d−1/2 iait)x1)diand ai,1≤i≤r, are real numbers with a2 1+···+a2 r= 1. By (2.11), we have PE(γ′′(0)) = W+G∈ W⊕G, where the latter are defined in (2.12). As W ⊥ G and the mαform orthonormal bases, the magnitude 32 2.4 Reach of the Segre-Veronese manifold of PE(γ′′(0)) is ∥PE(γ′′(0))∥2=∥W∥2+∥G∥2 = r ∑︂ i=1 a4 i·2(di−1) di + 4 ∑︂ 1≤i<j≤r a2 ia2 j = r ∑︂ i=1 a4 i·2(di−1) di + 2 r ∑︂ i=1 a2 i∑︂ j=i a2 j = r ∑︂ i=1 (︃a4 i·2(di−1) di + 2a2 i(1 −a2 i))︃,(because r ∑︂ i=1 a2 i= 1) = 2 r ∑︂ i=1 (︃a2 i−a4 i di)︃. To maximize this expression under the constraint ∑︁r i=1 a2 i= 1 we consider the Lagrange function L(a1, . . . , ar, λ) := r ∑︂ i=1 (︃a2 i−a4 i di)︃−λ(︄1− r ∑︂ i=1 a2 i)︄. Setting the derivatives of Lto zero, we have 0 = ∂L ∂ai = 2ai−4 di a3 i+ 2λai=⇒ai=√︃di(1 + λ) 2or ai= 0. Let us first consider the case when none of the aiare equal to zero. In this case, the equation ∑︁r i=1 a2 i= 1 implies 1 = r ∑︂ i=1 a2 i= r ∑︂ i=1 di(1 + λ) 2=d(1 + λ) 2, where d=d1+···+dris the total degree. This shows λ=2 d−1, so that ai=√︃di d. Thus, in this case ∥PE(γ′′(0))∥=⌜   ⎷2 r ∑︂ i=1 di d−di d2=√︃2(d−1) d For the other critical values of (a1, . . . , ar)we get √︂2(d′−1) d′, where d′=∑︁i∈Idiis the total degree of a subset I⊂ {1, . . . , r}of factors. Since x↦→ √︂2(x−1) xis an increasing function for x≥1, this shows that √︂2(d−1) dis indeed the maximal curvature. It also shows that √︂2(dℓ−1) dℓis the minimal curvature, where dℓ= min{d1, . . . , dr}. Proof of Theorem 2.1.2. We have shown above that √︂2(d−1) dis the maximal curvature, and that √︂2(dℓ−1) dℓ, where dℓ= min{d1, . . . , dr}, is the minimal curvature. The geodesics that attain these curvatures are given by the critical values ai, and, as shown by [21], γi(t) = (︁cos(d−1/2 iait)x0+ sin(d−1/2 iait)x1)︁diis a geodesic in Vni,di. 33 2 Reach of Segre-Veronese Manifolds 2.4.2 Bottlenecks of the Segre-Veronese manifold We compute ρ2, the width of the smallest bottleneck of the Segre-Veronese manifold. Recall that ρ2is the minimum over the distances 1 2dS(F,E)where F∈Xn,dwith F=Eand F−E∈R·E⊕NEXn,d. The latter is equivalent to ⟨F−E,G⟩= 0 for all G∈TEXn,d. We have TEXn,d=Txd1 0 Vn1,d1⊗xd2 0⊗ ··· ⊗ xdr 0+··· +xd1 0⊗xd2 0⊗ ··· ⊗ Txdr 0 Vnr,dr by Proposition 2.2.1 (2). We check that F−Eis orthogonal to each summand in this decomposition: let us write F=ℓd1 1⊗···⊗ℓdr r where the ℓiare linear forms and consider the inner product of F−Ewith elements from the first summand in the decomposition of TEXn,dabove. By Lemma 2.3.1 the monomials xd1−1 0xk, for 1≤k≤n1, span the tangent space Txd1 0 Vn1,d1. Consider G= (xd1−1 0xk)⊗xd2 0⊗···⊗xdr 0∈Txd1 0 Vn1,d1⊗xd2 0⊗···⊗xdr 0. We have that ⟨︁F−E,G⟩︁=⟨︁F,(xd1−1 0xk)⊗xd2 0⊗···⊗xdr 0⟩︁by (2.6) =⟨ℓd1 1, xd1−1 0xk⟩ r ∏︂ i=2⟨ℓdi i, xdi 0⟩ by (2.4) =⟨ℓ1, x0⟩d1−1⟨ℓ1, xk⟩ r ∏︂ i=2⟨ℓi, x0⟩di. This inner product is zero for every 1≤k≤n1if either ℓ1=x0or ⟨ℓ1, x0⟩= 0. We proceed similarly for the other summands in the decomposition of TEXn,d. Ultimately, we find that ⟨F−E,G⟩= 0 for all G∈TEXn,dif and only if either ℓ1=··· =ℓr=x0or there is at least one ℓiwith ⟨ℓi, x0⟩= 0, in which case ⟨F,E⟩= 0 by (2.6). Since F=E, it must be that the latter holds. Therefore, the bottlenecks of Xn,dall have width arccos 0 = π 2, so ρ2=1 2·π 2=π 4. 2.5 Volume of the tubular neighborhood Recall from Theorem 2.1.1 that the reach of the (spherical) Segre-Veronese manifold is τ(Xn,d) = π 4, if d≤5, and τ(Xn,d) = √︂d 2(d−1), if d > 5. In this section we prove Theorem 2.1.3 by computing the volume of the tubular neighborhood for ε<τ(Xn,d) U(ε) := {︂F∈S(Hn,d)dS(F,Xn,d)< ε}︂. The proof will be completed in Section 2.5.1 below. For the computation we use Weyl’s tube formula [65]. We denote n= dim Xn,d=n1+···+nr, N = dim(S(Hn,d)), and Ji(ε) = ∫︂ε 0 (sin ϕ)N−n+2i−1·(cos ϕ)n−2idϕ=∫︂tan ε t=0 tN−n+2i−1 (1 + t2)N+1 2 dt. Then Weyl’s tube formula implies that the volume of U(ε)is given as the following linear combination of the functions Ji: vol(U(ε)) = ∑︂ 0≤2i≤n κi·Ji(ε),(2.13) 34 2.5 Volume of the tubular neighborhood with coefficients given by κi=∫︂G∈Xn,d(︄∫︂F∈NGXn,d:∥F∥=1 m2i(LF) dF)︄dG, where m2i(LF)denotes the sum of the 2i-principal minors of the Weingarten map LF in the normal direction F. The coefficients κiare called curvature coefficients; they are isometric invariants of Xn,d. It follows from Lemma 2.3.5 that the integral in the formula for κiis independent of G, so that κi= vol(Xn,d)∫︂F∈NEXn,d:∥F∥=1 m2i(LF) dF,(2.14) where now the inner integral is over the sphere in the normal space of Xn,dat the point E=xd1 0⊗···⊗xdr 0. The volume of the (spherical) Segre-Veronese manifold is computed next. Lemma 2.5.1. vol(Xn,d) = √︁(d1···dr)n 2r−1·vol(Sn1)···vol(Snr). Proof. We consider the map ψfrom Proposition 2.2.1 in the case of Xn,d. If there is at least one odd di, the map ψis 2r−1: 1. Proposition 2.2.1 (3) therefore implies vol(Xn,d)by (2.3) = vol(Vn1,d1⊗···⊗Vnr,dr) = 1 2r−1·vol(Vn1,d1)···vol(Vnr,dr). On the other hand, if all diare even, ψis 2r: 1 and we have vol(Xn,d)by (2.3) = 2 ·vol(Vn1,d1⊗···⊗Vnr,dr)=2·1 2r·vol(Vn1,d1)···vol(Vnr,dr). Finally, vol(V◦ n,d) = √dn·vol(Sn)(see, e.g., [26]). Remark 2.5.2. The volume of the k-sphere is vol(Sk) = 2πk+1 2 Γ(k+1 2). The main task in computing the volume of U(ε)therefore is integrating the principal minors of the Weingarten map LFover the normal space. For this we pass from the uniform distribution on the sphere to the Gaussian distribution. Since Lλ·F=λ·LFfor F∈NEXn,dand λ∈R, we have m2i(Lλ·F) = λ2i·m2i(LF).(2.15) Suppose that Fis a Gaussian vector in the normal space, that is, a random tensor in NEXn,dwith probability distribution (2π)−c 2exp(−1 2∥F∥2). Then the two random variables ∥F∥and F/∥F∥are independent. We define the scalars λi:= E F∈NEXn,dGaussian ∥F∥2i. Using (2.15) we can then pass between the uniform distribution on the sphere and the Gaussian distribution as follows: E F∈NEXn,dGaussian m2i(LF) = λi·E F∈NEXn,duniform in the sphere m2i(LF). 35 2 Reach of Segre-Veronese Manifolds Since ∥F∥2has a χ2 c-distribution with c= dim NEXn,ddegrees of freedom, λiis the i-th moment of χ2 c; i.e., λi= 2iΓ(i+c 2) Γ(c 2)[53]. We have thus proved the following reformulation of (2.14). Lemma 2.5.3. Let c= dim NEXn,d. Then κi= vol(Xn,d)·vol(Sc−1)·θi,where θi:= Γ(c 2) 2iΓ(i+c 2)·E F∈NEXn,dGaussian m2i(LF). For computing the expectation of m2i(LF)we can rely on Corollary 2.3.9. Recall that this corollary implies that if Fis Gaussian, LFis a random symmetric matrix with independent blocks LF∼⎡ ⎢ ⎣ L1··· L1,r ... (L1,r)T··· Lr ⎤ ⎥ ⎦,Lk∼√︂2(dk−1) dkGOE(nk), Li,j ∼N(0, Ini⊗Inj). (2.16) In general it is difficult to evaluate the expected value of the minors of this random matrix. We make an attempt using graph theory in the next subsection. 2.5.1 Perfect matchings in graphs and random determinants In this section we give a formula for Em2i(LF)when Fis Gaussian using concepts from graph theory. In the following, the degrees d= (d1, . . . , dr)are fixed. Define the following random symmetric matrix with independent blocks: Ld(v1, . . . , vr) := ⎡ ⎢ ⎣ L1··· L1,r ... (L1,r)T··· Lr ⎤ ⎥ ⎦,Lk∼√︂2(dk−1) dkGOE(vk), Li,j ∼N(0, Ivi⊗Ivj). This differs from (2.16) in that we allow the sizes of the blocks to be arbitrary, not necessarily given by the dimension ni= dim Vni,di. We can write the expected principal minors of LFas E F∈NEXn,dGaussian m2i(LF) = ∑︂ v1,...,vr∈N:vi≤ni v1+···+vr=2i Edet Ld(v1, . . . , vr). Recall the definition of Dd(v1, . . . , vr)from (2.2): for a tuple (v1, . . . , vr)of nonnegative integers let G= (V, E)be the complete graph on v:= v1+··· +vrvertices where the vertices are partitioned into rgroups V=I1⊔ ··· ⊔ Irof cardinalities |Ik|=vk. The weight w(e)of an edge between vertices in group Ikis w(e) = dk−1 dk. The weight of an edge across groups is 1. Given a perfect matching C⊂Eits weight is w(C) := ∏︁e∈Cw(e). Then, Dd(v1, . . . , vr)=(−1)v 2∑︂ C⊂Eperfect matching w(C). The main goal of this section is to prove the following characterization of the function Dd. In combination with (2.13), Lemma 2.5.1 and Lemma 2.5.3 the next proposition completes the proof of Theorem 2.1.3. 36 2.5 Volume of the tubular neighborhood Proposition 2.5.4. Let (v1, . . . , vr)be nonnegative integers. Then Dd(v1, . . . , vr) = Edet(Ld(v1, . . . , vr)). Example 2.5.5. Recall from Example 2.3.10 that the random matrix for the Segre manifold Xn,d=S2⊗S2⊗S1⊗S1(i.e., n1=n2= 2 and n3=n4= 1) is L1(2,2,1,1) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 F1100 F1200 F1010 F1001 0 0 F2100 F2200 F0201 F2001 F1100 F2100 0 0 F0110 F0101 F1200 F2200 0 0 F0210 F0201 F1010 F2010 F0110 F0210 0F0011 F1001 F2001 F0101 F0201 F0011 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , where the entries are all i.i.d. standard Gaussian. We compute the expected determinant of this matrix using Theorem 2.5.4. The corresponding graph has n1+n2+n3+n4= 6 vertices in four groups I1={1,2},I2={3,4},I3={5},I4={6}: 1 3 46 2 5 The edges within groups all have weight zero (they can be deleted). All other edges have weight one, so D1(2,2,1,1) = Edet L1(2,2,1,1) is given by the negative of the number of perfect matchings in this graph. We can match {1,2}with {3,4}. There are two possible such matches. Or we can match 1with either 5or 6, in which case we have to match 2 with either 3or 4. There are 4 such matches. Finally, we can also match 2with either 5or 6, and by symmetry there are again 4 such matches. In total these are 10 matches, which shows that D1(2,2,1,1) = −10. Proof of Proposition 2.5.4. Let v:= v1+···+vrand write Ld(v1, . . . , vr)=(ℓi,j)1≤i,j≤v. Since the expectation is linear, Laplace expansion of the determinant yields Edet Ld(v1, . . . , vr) = ∑︂ π∈Sv sgn(π)E v ∏︂ i=1 ℓi,π(i), where Svis the symmetric group on velements. The ℓi,π(i)are all Gaussian with mean Eℓi,π(i)= 0 and independent. This implies that the only terms whose expectation is not zero are those where all ℓi,π(i)appear as a square. In other words, only those expectations are not zero where π∈Svhas the property that π(i)=ifor all iand π(i) = jimplies π(j) = i. Such π∈Svonly exist when vis even. 37 2 Reach of Segre-Veronese Manifolds Figure 2.2: The empirical distribution of det L1(2,2,1,1) for 105sample points. The empirical mean of this sample is −9.9995. We show in Example 2.5.5 that the actual mean value is −10. If vis odd, we therefore have Edet Ld(v1, . . . , vr) = 0. Since for vodd, no perfect matchings can exist, we also have Dd(v1, . . . , vr)=0. If vis even, on the other hand, the π∈Svwith the above property are precisely products of v 2transpositions, so that Edet Ld(v1, . . . , vr) = (−1)v 2∑︂ π∈Sv: πis product of v 2transpositions E v ∏︂ i=1 ℓi,π(i). There is a 1:1correspondence between products of v 2transpositions and perfect matchings C⊂E, where Eis the set of edges in the complete graph G= (V, E)on vvertices. Let C={(i1, i2),(i3, i4),...,(iv−1, iv)}be the matching corresponding to π; i.e., for jodd, π(ij) = ij+1. Then, using independence, we obtain E v ∏︂ i=1 ℓi,π(i)=E(ℓ2 i1,i2···ℓ2 iv−1,iv) = Eℓ2 i1,i2···Eℓ2 iv−1,iv=σ2 i1,i2···σ2 iv−1,iv, where σ2 ij,ij+1 is the variance of ℓij,ij+1 . By the definition of Ld(v1, . . . , vr), the variance of the off-diagonal entries in the diagonal blocks is dk−1 dk, while the variance of the entries in the off-diagonal blocks of Ld(v1, . . . , vr)is 1. That is: σ2 ij,ij+1 ={︄dk−1 dk, ij, ij+1 ∈ Ik 1, ij, ij+1 are in different groups of vertices , which shows that E∏︁v i=1 ℓi,π(i)=w(C), so Dd(v1, . . . , vr) = Edet Ld(v1, . . . , vr). The last example of this thesis is the computation of the curvature coefficients κiin Weyl’s tube formula (2.13) for the Segre manifold Xn,d=S1⊗···⊗S1⊂S2r−1. 38 2.5 Volume of the tubular neighborhood Example 2.5.6. For the special case of the Segre manifold Xn,d=S1⊗···⊗S1we have d=1r= (1,...,1) and n=1r. In this case, Lemma 2.5.1 yields vol(Xn,d) = 1 2r−1·vol(S1)···vol(S1) = 2πr. Furthermore, the codimension of Xn,dis c= 2r−1−r. This implies that κi= 2πr·vol(Sc−1)·θi,where θiis defined as in Theorem 2.1.3 and Lemma 2.5.3. We compute the θi. By Theorem 2.1.3, we have θi=Γ(c 2) 2iΓ(i+c 2)∑︂Dd(v1, . . . , vr), where the sum is over all tuples (v1, . . . , vr)∈ {0,1}rwith and v1+···+vr= 2i. There are (︁r 2i)︁such tuples. Fix a tuple (v1, . . . , vr); this corresponds to the complete graph with 2ivertices where all edges have weight 1, and there are (2i−1)!! perfect matchings on this graph. Therefore, by (2.2) we have D12i(12i)=(−1)i(2i−1)!!. It follows that θi= (−1)i·Γ(c 2) 2iΓ(i+c 2)·(︃r 2i)︃·(2i−1)!!; hence, κi= (−1)i·2πr·vol(Sc−1)·Γ(c 2) 2iΓ(i+c 2)·(︃r 2i)︃·(2i−1)!!. This completes the computation of the curvature coefficients of this Segre manifold. 39 3 Typical ranks of order-three tensors This chapter is based on the publication [16]. 3.1 Introduction The determination of tensor rank has broad applications in many fields, including mathematics, psychometrics, and data science [40]. This chapter focuses on typical ranks. A typical rank of a given format is any number rsuch that the set of tensors of rank ris a full-dimensional semialgebraic set [29, §7]. While over the complex numbers (in fact, over any algebraically closed field) there is only one typical rank, much less is known about typical ranks of real tensors. Here, we focus on the family of real order-three tensors of the format n1×n2×n3where 2≤n1≤n2≤n3. We take the perspective of probability and define typical ranks of tensors as follows. Definition 3.1.1. We call T= (ti,j,k)∈Rn1×n2×n3aGaussian tensor if its entries ti,j,k are i.i.d. standard Gaussian. An integer r > 0is called a typical rank for the format n1×n2×n3if for a Gaussian tensor Twe have P{rank(T) = r}>0. As discussed in Chapter 1, this is equivalent to the definition of typical rank given in Definition 1.1.17, which is independent of the choice of probability distribution. Our first main theorem is the following. The proof comes in Subsection 3.3.2. Theorem 3.1.2. Suppose that (n1−1)(n2−1) + 1 ≤n3≤n1n2. Then, the typical ranks of real n1×n2×n3tensors are contained in {n3, n3+ 1}. Moreover, n3is always a typical rank. In [58, Theorem 8.1] the authors give an algebraic proof of Theorem 3.1.2 assuming 3≤n1. Our proof, on the other hand, is geometric and is based on the General Position Theorem for general linear sections of algebraic varieties (see Proposition 3.3.2). The lower bound of n3= (n1−1)(n2−1)+1 in Theorem 3.1.3 is the codimension of the variety of rank-one matrices in Rn1×n2, called the Segre variety. This is a real projective variety that we denote by Σn1,n2⊂Pn1n2−1. Our next theorem links rank probabilities to the probabilities of having a certain number of intersections points of Σn1,n2with a uniform subspace L⊂Pn1n2−1. This random subspace is the projectivization of a uniform random variable in the Grassmannian G(n3, n1n2)of n3-dimensional linear spaces in Rn1×n2(see Subsection 3.3.1 for more details). We give a proof of the theorem in Subsection 3.3.2. Theorem 3.1.3. Let T∈Rn1×n2×n3. If n3= (n1−1)(n2−1) + 1, we have P T∈Rn1×n2×n3 Gaussian {rank(T) = n3}= P L∈G(n3,n1n2) uniform {#(L∩Σn1,n2)≥n3}. For (n1−1)(n2−1) + 1 < n3≤n1n2we have P T∈Rn1×n2×n3 Gaussian {rank(T) = n3}= P L∈G(n3,n1n2) uniform {L∩Σn1,n2=∅}. 41 3 Typical ranks of order-three tensors Another consequence of Proposition 3.3.4 is that, if 2≤n1≤n2≤n3<(n1−1)(n2−1) + 1, then with probability one, the projectivization of the linear span of the m×nslices of a Gaussian tensor T∈Rn1×n2×n3will have no intersection with the Segre variety Σn1,n2, since its dimension is lower than the codimension of Σn1,n2. Therefore the probability that the condition given in Proposition 3.3.4 to have rank(T)≤n3is satisfied is 0. This means that typical ranks are all greater than or equal to n3+ 1. Deciding whether or not n3+1 is a typical rank is more subtle. In view of Theorem 3.2.1, we can see that having rank n3+ 1 is equivalent to the existence of an (n3+ 1)-plane containing the linear span of the slices of T, such that its projectivization intersects Σn1,n2 in at least n3+ 1 linearly independent real points. The subtlety is that we are not asking for a general (n3+ 1)-plane containing the span of the slices to have this property, but we only require the existence of a single such plane. Therefore, dimension counting arguments, which only provide general results, cannot be used. 3.3.3 Typical ranks of tall tensors We conclude this section by giving another proof of a result of ten Berge. In [61] for 2≤n1≤n2≤n3, an n1×n2×n3tensor is defined to be tall if (n1−1)n2< n3< n1n2, and it is shown that the only typical rank for such tensors is always n3. We now will use Proposition 3.3.4 once again to give a different proof of this result. Let T∈Rn1×n2×n3be a tall Gaussian tensor and denote by T1,...,Tn3its n1×n2slices. Let k:= n1n2−n3. With probability one, the dimension of span{T1,...,Tn3}is n3and therefore there exist A1,...,Ak∈Rn1×n2such that span{T1,...,Tn3}={M∈Pn1n2−1|Trace(MTA1) = ··· = Trace(MTAk)=0} (see (3.2)). For a rank-one matrix x⊗ywith x∈Rn1and y∈Rn2, the condition x⊗y∈span{T1,...,Tn3}is equivalent to the system of equations xTA1y=··· =xTAky= 0, which is in turn equivalent to xT·(A1y. . . Aky)=0,(3.5) i.e. xTis in the left kernel of the n1×kmatrix having columns Aiyfor i= 1, . . . , k. For every i= 1, . . . , n2define Bi∈Rn1×kas the matrix containing the i-th columns of the matrices A1,...,Ak, i.e. the j-th column of Biis the i-th column of Aj. Then we have (A1y,...,Aky)=(B1y1+···+Bn2yn2) where y= (y1, . . . , yn2). Equation (3.5) becomes xT·(B1y1+···+Bn2yn2) = 0.(3.6) Since Tis tall, we have k < n1and equation (3.6) always has non-trivial solutions. This means that for every y∈Rn2we can find x∈Rn1such that x⊗ybelongs to span{T1,...,Tn3}. By Proposition 3.3.4 it follows that rank(T)≤n3. By Theorem 3.1.2, n3is also the smallest typical rank, which implies that n3is the only typical rank. 48 3.4 Random 3×3×5tensors and cubic surfaces 3.4 Random 3×3×5tensors and cubic surfaces In this section, we focus on the special case of Gaussian tensors of the format 3×3×5. ten Berge [62] showed that tensors in R3×3×5admit two typical ranks: 5and 6. In this section, we express the probability that a Gaussian tensor Thas rank 5as the probability that the 27 lines on a certain random cubic surface are all real. Recall that blowing up six points in CP2that are in general position (no three points are collinear and not all lie on a conic) produces a smooth cubic surface, and indeed every smooth cubic surface can be generated in this way [22]. Lemma 3.3.3 implies that a 3×3×5 Gaussian tensor Tinduces a uniform subspace L⊂P8of dimension 4. By Theorem 3.1.3, with probability one, if Thas rank 5, it intersects Σ3,3in 6 points x1⊗y1,...,x6⊗y6. We can therefore associate a random cubic surface to Tthat is the blow up of P2in x1,...,x6 or y1,...,y6. Our results are based on the following theorem. Note that codim ΣC 3,3= 4 and deg ΣC 3,3= 6, so a general 4-dimensional subspace of CP8intersects ΣC 3,3in 6points. Theorem 3.4.1. Let Lbe a four-dimensional general linear subspace of P8. Then L intersects ΣC 3,3in the 6points x1⊗y1,...,x6⊗y6∈CP8if and only if there exists a cubic surface Sthat results both from blowing up CP2in x1,...,x6as well as from blowing up CP2in y1,...,y6, such that S={[z0:··· :z3]∈P3|det(z0M0+···+z3M3)=0}, where span{M0,...,M3}=L⊥is the orthogonal complement (see (3.3)). Proof. The theorem is proved in [23, Theorem 6.9] for a general linear subspace of CP8. Since the real Grassmannian is Zariski dense in the complex Grassmannian, it also holds for a general subspace in P8. Remark 3.4.2. [23, Theorem 6.9] also proves that the exceptional lines {h1, . . . , h6}from blowing up in {x1,...,x6}and the exceptional lines {h′ 1, . . . , h′ 6}from blowing up in {y1,...,y6}form a Schläfli double six on S. Remark 3.4.3. Remarkably, [23] is a paper that studies a problem in computer vision. The connection to ranks of random tensors is surprising. Example 3.4.4. Consider the real rank-one matrices xi⊗yifor i= 1,...,6where x1= (1,0,0) x2= (0,1,0) x3= (0,0,1) x4= (1,1,1) x5= (3,5,1) y1= (1,0,0) y2= (0,1,0) y3= (0,0,1) y4= (1,1,1) y5= (8,2,1) and x6= (−1 3,7 5,3 17)and y6= (−4 3,2 5,−1 17). The first five rank-one matrices are linearly independent, but the sixth is linearly dependent on the first five. Therefore, the six rankone matrices span a subspace L∈G(5,9). The orthogonal complement is L⊥=⎧ ⎨ ⎩⎡ ⎣ 0 3z03z2 −37z0−2z1−5z2+z30 3z3 34z0−z1+ 2z2−4z33z10⎤ ⎦z0, . . . , z3∈R⎫ ⎬ ⎭. The cubic surface Sconstructed in Theorem 3.4.1 is thus given by S={z∈P3|z0z3(34z0−z1+ 2z2−4z3) + z1z2(−37z0−2z1−5z2+z3) = 0}. It contains 27 real lines. 49 3 Typical ranks of order-three tensors Figure 3.1: The cubic surface Sin Example 3.4.4 intersected with the hyperplane z0= 1 seen from two different viewpoints. The pictures were created using Surfer. In 1849, Cayley and Salmon proved that every smooth cubic surface contains 27 lines [20]. Since their classical proof, many different proofs have been produced throughout the years. It was shown in 1858 by [52] that a real cubic surface can only have 3,7,15, or 27 real lines, leading to a classification of cubic surfaces. We will need the following result due to Polo-Blanco and Top. Proposition 3.4.5. [50, Proposition 2.1] Let Sbe a smooth, real cubic surface. If Sis obtained as the blow-up of 1. 6real points in the plane, it contains 27 real lines; 2. 4real points and a pair of complex conjugate points in the plane, it contains 15 real lines; 3. 2real points and 2pairs of complex conjugate points in the plane, it contains 7real lines 4. 3pairs of complex conjugate points in the plane, it contains 3real lines. Remark 3.4.6. There is a fifth case in [50, Proposition 2.1] in which a cubic surface that is obtained as the blow-up of 6 real points contains only 3 real lines. However, in this case, the sixth point must fulfill a specific condition with respect to the other five points, which does not occur in the general setting, so we do not include it. Corollary 3.4.7. Let M0,M1,M2,M3∈R3×3be independent Gaussian matrices and consider the random cubic surface S={[z∈P3|det(z0M0+··· +z3M3)=0}.Let L∈G(5,9) be a uniform subspace and denote also by Lits projectivization. Denote by pi:= P{#(L∩Σ3,3) = i}and qj:= P{Scontains jreal lines}. Then, p0=q3, p2=q7, p4=q15, p6=q27. Proof. Since deg ΣC 3,3= 6, a general real projective linear subspace Lintersects Σ3,3in either 0,2,4or 6real points. Therefore, p0+p2+p4+p6= 1. On the other hand, a smooth cubic surface contains either 3,7,15 or 27 real lines. Since Sis smooth with probability one, q3+q7+q15 +q27 = 1. By Lemma 3.3.1, L∈G(5,9) is uniform if and only if L⊥∈G(4,9) is uniform. Moreover, by Lemma 3.3.3 span{M0,...,M3} ∈ G(4,9) is also uniformly distributed. It follows from 50 3.5 Asymptotics and heuristics Theorem 3.4.1 that piis the probability that Sis obtained as the blow up of ireal points and (6 −i)/2pairs of complex conjugate points in the plane. Proposition 3.4.5 implies that p6≤q27,p4≤q15,p2≤q7and p0≤q3. If one of those was a strict inequality, then q3+q7+q15 +q27 < p0+p2+p4+p6= 1, which is a contradiction. Hence, all are equalities. The stage is now set to relate the probability that a random real 3×3×5tensor has rank 5 to the probability that all 27 lines on a random cubic surface are real. Proof of Theorem 3.1.4. As before, we write p6= P{#(L∩Σ3,3)=6}. By Theorem 3.1.3, P{rank T= 5}is the probability that for a uniform subspace L∈G(5,9), its projectivization Lintersects Σ3,3in at least 5points. Since Lintersects Σ3,3in either 0,2,4or 6 points with probability one, we have P{rank T= 5}=p6. The statement follows from Corollary 3.4.7. We conclude this section by studying the expected number of real lines on our random cubic. Denote by E:= Enumber of real lines on S, where S={det(z0M0+···+z3M3) = 0}with Miindependent and Gaussian. We show in Example 3.5.1 below that the expected number of points in L∩Σ3,3for uniform Lis 3. Together with Corollary 3.4.7 we obtain the following system of equalities and inequalities: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0≤p0, p2, p4, p6≤1 p0+p2+p4+p6= 1 2p2+ 4p4+ 6p6= 3 3p0+ 7p2+ 15p4+ 27p6=E Treating p0, p2, p4, p6and also Eas variables, the solution set of this system is a twodimensional polytope. Projected to the (E, p6)-plane this polytope is given as the convex hull of the four points (11,0),(12,0),(12,1 4),(15,1 2)From this we obtain the following nontrivial bound for E. Corollary 3.4.8. We have 11 ≤E≤15. The expected number of lines for other models of random cubic surfaces has been computed in [2, 3, 7]. Comparing Corollary 3.4.8 to these results, Eseems to be comparably large. 3.5 Asymptotics and heuristics In Theorem 3.1.3 we proved that the rank of a random tensor T∈Rn1×n2×n3with n3:= (n1−1)(n2−1) + 1 depends on whether the corresponding random linear subspace spanned by its slices projectively intersects the Segre variety in enough real points. In this section, we will compute the average number of real points in such an intersection and compute its asymptotic for n1fixed as n2→+∞. Comparing the growth of this expectation with the growth of the required number of real points, we obtain some insight 51 3 Typical ranks of order-three tensors p6 E • (11,0) •(12,0) •(15,1 2) • (12,1 4) Figure 3.2: The polygon of possible values for (E, p6), where Eis the expected number of real lines on the random cubic, and p6the probability that it has 27 real lines. into how the probability of having rank n3behaves. We start by stating the real projective version of Howard’s kinematic formula [37]. Recall that given a Riemannian manifold (M, g), a submanifold Nι ↪→Minherits an induced Riemannian metric by pullback through ι, denoted as ι∗g. By vol(N)we mean the total volume of Nwith respect to the induced metric (if dim(N)<dim(M)its volume with respect to gis 0). If Nis compact, vol(N)will always be finite. In the case M=Pn1n2−1 with the Fubini-Study metric and N= Σn1,n2, Howard’s kinematic formula yields E#(L∩Σn1,n2) = vol(Σn1,n2) vol(Pn1+n2−2), where Lcorresponds to a uniform subspace in G(n3, n1n2). Moreover, vol(Σn1,n2) = vol(Pn1−1)·vol(Pn2−1)and vol(Pk−1) = πk 2 Γ(k 2), leading to E(#{Σn1,n2∩L}) = √π·Γ(n1+n2−1 2) Γ(n1 2)Γ(n2 2).(3.7) Example 3.5.1. For n1= 3, Equation (3.7) becomes E(#{Σ3,n2∩L}) = n2. Therefore, for a random tensor T∈R3×n2×(2n2−1) with the usual notation, Theorem 3.1.3 combined with Markov’s inequality yields P{rank(T)=2n2−1}= P{#{Σ3,n2∩L} ≥ 2n2−1} ≤ n2 2n2−1<1. This gives another proof of the fact that there are two typical ranks when n1= 3 and n3= (n1−1)(n2−1) + 1, since it implies that P{rank(T)>2n2−1}is positive. This is also an alternative proof to [59, Theorem 5.5] for the case n1= 3. We are now interested in the asymptotics of (3.7) when n1is fixed and n2→+∞. In order to compute this, we consider two cases, depending on the parity of n1. Proposition 3.5.2. Let Lbe a uniform projective (n1−1)(n2−1)-dimensional subspace. For n1fixed, as n2→+∞, we have the asymptotic expression E(#{Σn1,n2∩L}) = c·n n1−1 2 2(︃1 + O(︃1 n2)︃)︃, where c=1 (n1−2)!! for n1odd and c=√︁π 2 1 (n1−2)!! for n1even. 52 3.5 Asymptotics and heuristics Proof. We first consider the case when n1is odd. Let n1= 2k+ 1 for some k≥1. In this case, (3.7) reads E(#{Σ2k+1,n ∩L}) = √π·Γ(n2 2+k) Γ(k+1 2)Γ(n2 2). By the multiplicative property of the Gamma function Γ(x+ 1) = xΓ(x), we obtain Γ(︂n2 2+k)︂= Γ (︂n2 2)︂k−1 ∏︂ i=0 (︂i+n2 2)︂,and Γ(︃1 2+k)︃=√π k−1 ∏︂ i=0 (︃i+1 2)︃, leading to E(#{Σ2k+1,n2∩L}) = k−1 ∏︂ i=0 (︁n2 2+i)︁ (︁1 2+i)︁.(3.8) This is a polynomial of degree k=n1−1 2in n2, with leading coefficient given by (1 2)k ∏︁k−1 i=0 2i+1 2 =1 ∏︁k−1 i=0 (1 + 2i)=1 (2k−1)!!, where (2k−1)!! = (2k−1) ·(2k−3) ·····3·1is the double factorial. Thus we obtain the asymptotic expression E(#{Σn1,n2∩L}) = 1 (n1−2)!! n n1−1 2 2(︃1 + O(︃1 n2)︃)︃ for n1odd.(3.9) We now consider the case when n1is even. Let n1= 2kfor some k≥2. In this case, (3.7) reads E(#{Σ2k,n2∩L}) = √πΓ(︁n2−1 2+k)︁ Γ(k)Γ (︁n2 2)︁. Again by the multiplicative property of the Gamma function we have Γ(︃n2−1 2+k)︃= Γ (︃n2+ 1 2)︃k−1 ∏︂ i=1 (︃n2−1 2+i)︃. Using above identity and Γ(k)=(k−1)!, we obtain E(#{Σ2k,n2∩L}) = π·∏︁k−1 i=1 (︁n2−1 2+i)︁ (k−1)! ·Γ(︁n2+1 2)︁ Γ(︁n2 2)︁Γ(︁1 2)︁=π·∏︁k−1 i=1 (︁n2−1 2+i)︁ (k−1)! ·1 B(︁1 2,n2 2)︁, where B(x, y) = Γ(x)Γ(y) Γ(x+y)is the Beta function. Using the asymptotic B(︁1 2,n2 2)︁∼√︂2π n2(see [54, Equation 43:9:3]) and the identity 2k−1(k−1)! = (2k−2)!! = (n1−2)!!, we have E(#{Σn1,n2∩L}) = √︃π 2 1 (n1−2)!!n n1−1 2 2(︃1 + O(︃1 n2)︃)︃ for n1even.(3.10) 53 3 Typical ranks of order-three tensors We can now compare the results in Proposition 3.5.2 with Theorem 3.1.3. When we have n3= (n1−1)(n2−1)+ 1, Theorem 3.1.3 states that for a random n1×n2×n3tensor having rank n3is, with probability one, equivalent to #{Σn1,n2∩L} ≥ n3. While n3grows linearly in n2for n1fixed, the expectation in (3.7) grows as n n1−1 2 2. For n1= 3, both quantities grow linearly, but while n3is given by 2n−1, the expectation is n2. Qualitatively, this means that the number of linearly independent points required to be real in order to have rank n3= 2n2−1grows faster, albeit with the same order, compared to the expected value of such points. This suggests that the probability that the rank is 2n2−1should go to 0as n2→+∞. For n1≥4, while n3grows linearly, the expected value of real points grows as a higher power of n2. Qualitatively, this tells us that as n2increases, the number of points we require to be real becomes smaller and smaller relative to the expected value. Based on this, we expect that the probability of having rank n3approaches 1as n2→+∞. 54 4 Typical subranks of order-three tensors This chapter is based on the preprint [11]. 4.1 Introduction Given a K-vector space V, an order-three tensor T∈V∗⊗V∗⊗Vdefines a bilinear map V×V→Valso denoted T. The subrank is the largest rsuch that there exist linear maps φ1:Kr→V, φ2:Kr→Vand φ3:V→Krwith φ3(T(φ1(a), φ2(b))) = Ir(a,b)for all a,b∈Kr, where Iris the bilinear map given by rindependent scalar multiplications: Ir:Kr×Kr→Kr; ((a1, . . . , ar),(b1, . . . , br)) ↦→ (a1b1, . . . , arbr). The rank of Tis dually the smallest ℓsuch that there exist linear maps ψ1:V→Kℓ, ψ2:V→Kℓand ψ3:Kℓ→Vwith T(v,w) = ψ3(Iℓ(ψ1(v), ψ2(w))) for all v,w∈V. Thus the rank corresponds to the minimal number of linearly independent scalar multiplications required to evaluate T, or the “cost” of the tensor, and the subrank is the maximal number of linearly independent scalar multiplications that can be embedded into T, the “value” of the tensor. In this chapter, we focus on the case where K=R. Hence, we discuss different aspects of the following natural question: Given a real bilinear map, how many linearly independent Rscalar multiplication can be embedded into the bilinear map? Recent work by Derksen-Makam-Zuiddam [25] and Pielasa-Šafránek-Shatsila [49] solved this question for sufficiently general tensors over algebraically closed fields; in particular, they showed that the generic subrank of T∈Kn⊗Kn⊗Knis ⌊√3n−2⌋. However, over R, there can be multiple typical subranks for certain tensor formats. Similarly, while over an algebraically closed field there is a single generic rank for a given tensor format, over Rthere can be multiple typical ranks. In addition to several general results, we will investigate many small formats of order-three tensors. In general, it is difficult to find upper bounds on the subrank of a given tensor. The slice rank of T∈V1⊗. . . ⊗Vdis the minimal sum ∑︁d i=1 dim(Ui)where Uiare linear subspaces of Visuch that T∈ d ∑︂ i=1 V1⊗···⊗Vi−1⊗Ui⊗Vi+1 ⊗···⊗Vd; see [60], where it is also shown that the slice rank ≤rlocus is closed. Using the fact that slice rank can only drop under linear transformations and that the slice rank of Irequals r, we find that the slice rank is an upper bound to the subrank. However, this bound is by no means tight for sufficiently general tensors: the generic subrank is strictly smaller than the generic slice rank: the latter equals nwhen all Vihave dimension n, while the former is O(n1/(d−1))[25]. 55 4 Typical subranks of order-three tensors In the remainder of this chapter, we will restrict ourselves to the subrank of real tensors T∈Rn1⊗···⊗Rnd, so all the linear maps in the definition are R-linear: Q(T) = max{r| ∃R-linear maps φi:Rni→Rrwith (φ1⊗···⊗φd)T=Ir}. We will sometimes consider the complex subrank, which we will denote by QC(T). For real matrices A∈Rn1⊗Rn2we know that the subrank equals the matrix rank of A, and this is the same over all possible field extensions, so Q(A) = QC(A). However, for higher-order tensors, the complex subrank only provides an upper bound on the real subrank. Let T∈Rn⊗Rn⊗Rn. For tensor rank, it is known that the rank of a real order-d tensor is bounded by dtimes its complex rank [5], so it is natural to expect a similar result for subrank. In Section 4.2, we establish such a result, as follows: Theorem 4.1.1 (Theorem 4.2.2 below).For any order-three tensor Twe have Q(T)≥ ⌊√︁QC(T)⌋. We do not think, however, that this bound is close to sharp; in particular, we do not have examples where Q(T)grows more slowly than linearly in QC(T). Over the complex numbers, the locus of tensors with subrank rin Cn1⊗Cn2⊗Cn3is a constructible set. Therefore, there exists a unique rfor which the locus of tensors with subrank ris dense. This ris called the generic subrank of tensors in Cn1⊗Cn2⊗Cn3[25]. Over the real numbers, as in the case of rank, there does not have to be a dense subset in the Euclidean topology where the subrank is constant. For tensor rank, we know that any rank between the smallest typical rank (which is the generic rank) and the largest typical rank is also typical [10]. In Section 4.3, we prove that the same is true for typical subranks: Theorem 4.1.2. Let rand sbe typical subranks of tensors in Rn1⊗Rn2⊗Rn3with r≤s. Then all integers ℓwith r≤ℓ≤sare also typical subranks. As discussed in Chapter 1, both 2 and 3 are typical ranks of real 2×2×2tensors [45]; combining this fact with Remark 1.1.16, this gives us that 1 and 2 are both typical subranks of tensors in R2⊗R2⊗R2. In Section 4.4, we look at more examples of n1×n2×n3tensor spaces and describe the typical subranks. We summarize our findings in the following theorem. Theorem 4.1.3. Let Rn1⊗Rn2⊗Rn3the space of real order-three tensors. 1. For (n1, n2, n3) = (2,2,2), the typical subranks are 1 and 2. 2. For n1= 2,n2≥2and n3>2, the only typical subrank is 2. 3. For (n1, n2, n3)∈ {(3,3,3),(3,3,4)}, the only typical subrank is 2. 4. For (n1, n2, n3) = (3,3,5), the typical subranks are 2 and 3. 5. For (n1, n2, n3)∈ {(3,4,4),(4,4,4)}, the typical subranks are 2 and 3. 56 4.2 Real and complex subrank To give an upper bound for the subrank of some concrete tensors T∈Rn1⊗Rn2⊗Rn3, we use a simple geometric condition for the subrank of Tto be at least some given integer r, namely, that the linear space spanned by the slices of the tensor along one axis contains at least rlinearly independent rank-one matrices. In the final section of this paper, we turn to real division algebras and prove the following theorem. Theorem 4.1.4 (Theorem 4.5.3 and Corollary 4.5.4 below).Let Dbe a division algebra over Rof dimension ≥2. Let fnbe the componentwise multiplication map fn:Dn×Dn→Dn; (a,b) = ((a1,...an),(b1, . . . , bn)) ↦→ (a1b1, . . . , anbn) =: a∗b. Regarding fnas an R-bilinear map, we have Q(fn)≤nd, where d=1 2dimRD. In particular, for D=Cwe have Q(fn) = nand for D=H, the quaternions, we have Q(fn)=2n. We stress that this result for arbitrary ndoes not trivially follow from the case where n= 1, since the subrank of tensors can be strictly superadditive [25]. 4.2 Real and complex subrank Given the complex subrank of a real tensor, we are interested in finding bounds on the real subrank. Let T∈Rn1⊗Rn2⊗Rn3be a tensor. As discussed above, we have the trivial upper bound Q(T)≤QC(T). In the following, we first consider an order-three tensor Tof any format n1×n2×n3 and provide a lower bound of Q(T)as a function of the complex subrank of T. We then show that this bound can be improved in the case that QC(T) = n1=n2=n3. For these results, we use the following lemma. Lemma 4.2.1. Let T∈Rr⊗Rs⊗Rsbe a tensor where the rslices T1,...,Tr∈Rs⊗Rs along the first axis are a spanning set of the space of s×smatrices. Then Q(T) = s. Proof. For all i∈[s] := {1, . . . , s}, there exist coefficients ci1, . . . , cir ∈Rsuch that ei⊗ei=Eii =ci1T1+. . . +cirTr. Let φ1:Rr→Rsbe the linear map defined by the matrix ⎛ ⎜ ⎝ c11 ··· c1r . . .. . . cs1··· csr ⎞ ⎟ ⎠. Then (φ1⊗id ⊗id)T= (φ1⊗id ⊗id) r ∑︂ i=1 ei⊗Ti= s ∑︂ i=1 ei⊗(ci1T1+. . . +cirTr) = s ∑︂ i=1 ei⊗ei⊗ei=Ir. This shows Q(T)≥s, and equality holds, as the subrank is bounded from above by the dimension of Rs. Theorem 4.2.2. Let T∈Rn1⊗Rn2⊗Rn3be a real tensor. Then Q(T)≥ ⌊√︁QC(T)⌋. 57 4 Typical subranks of order-three tensors 4.4.2 A geometric method to bound the subrank from above There is no established method to give an upper bound for the subrank of a given tensor; hence, showing that a given integer that is strictly smaller than the generic subrank is a typical rank is difficult. In the following, we give a necessary condition for a tensor to have subrank at least r, which we use to give upper bounds on the subrank of 3×3×5and 4×4×4tensors. We prove that in those cases, there exist multiple typical subranks. Let T∈Rn1⊗Rn2⊗Rn3be a tensor and denote by T1,...,Tn1∈Rn2⊗Rn3the slices of Talong the first axis. Assume that the subrank of Tis ≥r. Then there exist surjective linear maps π2:Rn2→Rrand π3:Rn3→Rrand an r-dimensional linear subspace W⊂span{˜ T1,..., ˜ Tn1}, where ˜ Ti:= (π2⊗π3)(Ti), which up to left and right multiplication is of the form ⎧ ⎪ ⎨ ⎪ ⎩⎛ ⎜ ⎝ x10 0 0...0 0 0 xr ⎞ ⎟ ⎠|x1, . . . , xr∈R⎫ ⎪ ⎬ ⎪ ⎭. This is a necessary and sufficient condition for Q(T)≥r. We formulate the existence of this subspace using the Segre variety: for R-vector spaces U, V we denote by ΣU,V := {[A]∈(U⊗V)P|rank(A)=1}, the projective variety corresponding to the affine cone of rank-one tensors of order 2. For U=Kn1and V=Kn2we also just write Σn1,n2. By the discussion above, we can now state the following necessary condition for subrank ≥r. Lemma 4.4.7. Let T∈Rn1⊗Rn2⊗Rn3be a tensor with subrank at least r. Then there exist linear maps π2:Rn2→Rr,π3:Rn3→Rrsuch that the slices ˜ T1,..., ˜ Tn1∈Rr⊗Rr of (id ⊗π2⊗π3)Tsatisfy: #(span{˜ T1,..., ˜ Tn1}P∩Σ(U×V)) ≥r. Remark 4.4.8. The converse of the above is false. For example, the tensor e1⊗e1⊗e1+ e1⊗e2⊗e2∈R2⊗R2⊗R2has slice rank one, and hence also subrank one. However, #(span{e1⊗e1,e2⊗e2}P∩Σ2,2) = 2. 4.4.3 Typical subranks of 3×3×5tensors In this section, we use ideas from Chapter 3, where we used a geometric approach to show that the typical ranks of 3×3×5tensors are 5 and 6. Combining this with the construction in Section 4.4.2gives us the typical ranks of 3×3×5tensors. Given a sufficiently general tensor T∈R3⊗R3⊗R5, denote by T1,...,T5∈R3⊗R3 the slices of this tensor along the third axis. Let V:= span{T1,...,T5}P⊆(R3⊗R3)P be the projectivization of the space spanned by these 5 matrices. For a typical tensor, the slices are linearly independent and so Vhas projective dimension 4. If QR(T)=3, there exists a linear subspace W⊆Vwhich has the form ⎧ ⎨ ⎩⎛ ⎝ x0 0 0y0 0 0 z⎞ ⎠|x, y, z ∈R⎫ ⎬ ⎭ 64 4.4 Subrank for specific order-three formats up to conjugation. We thank Tim Seynnaeve for the construction of Tin the second part of the following proof. Theorem 4.4.9. The typical subranks of real 3×3×5tensors are 2and 3. Proof. The generic rank is min{3,3,5,⌊√3+3+5−2⌋} = 3, so 3is a typical subrank. Also, note that 2 is the only typical subrank of 3×3×3tensors, so 2 is a lower bound on the possible typical subranks. It is left to show that 2 is a typical subrank of 3×3×5 tensors. Using Lemma 4.4.7, it suffices to show that there is a an open subset U⊂R3⊗R3⊗R5 of full dimension such that the intersection span{T1,...,T5}P∩Σ3,3 has at most two (real) points for all T∈U, where T1,...,T5are the slices of Talong the third axis. Indeed, for T∈Uwe then have Q(T)<3. To show this, consider the tensor T∈R3⊗R3⊗R5with slices given by T1=⎛ ⎝ 1 0 0 0 0 0 0 0 −1⎞ ⎠,T2=⎛ ⎝ 0 0 0 0 1 0 0 0 −1⎞ ⎠,T3=⎛ ⎝ 010 100 000⎞ ⎠, T4=⎛ ⎝ 0 0 1 0 0 0 1 0 0⎞ ⎠,T5=⎛ ⎝ 000 001 010⎞ ⎠. The linear space L:= span{T1,...,T5}equals ⎧ ⎨ ⎩⎛ ⎝ x a b a y c b c −x−y⎞ ⎠|a, b, c, x, y ∈R⎫ ⎬ ⎭. Any matrix Ain Lis symmetric and real, hence has real eigenvalues, which sum up to the trace, namely 0. This implies that if Ais nonzero, then it has rank at least 2. So #(LP)∩Σ3,3= 0. This then also holds for tensors in a neighborhood of T; hence, 2is a typical subrank. Remark 4.4.10. In the context of Theorem 4.4.9, we showed in Chapter 3 that, with probability one, rank(T) = 5 if and only if #(span{T1,...,T5}P∩Σ3,3) = 6. There exist four cases with nonzero probability: #(span{T1,...,T5}P∩Σ3,3)∈ {0,2,4,6}. If the number of intersection points is 6, the rank is 5. Thus, there exists a rank decomposition T=∑︁5 i=1 v1i⊗v2i⊗v3i, where v1i,v2i∈R3and v3i∈R5. With probability one, {v3i}5 i=1 are linearly independent. We define the linear map π3:R5→R3that projects coordinatewise onto the first three coordinates. Then (id ⊗id ⊗π3)T∈R3⊗R3⊗R3is concise and of rank 3. By Remark 1.1.16, Q((id ⊗id ⊗π3)T)=3, and hence Q(T)≥3. Because Twas taken to be typical, the subrank must be 3. If the number of intersection points is ≤2, the subrank is at most 2. In particular, if the entries of Tare taken to be i.i.d. Gaussians, the probability that there are at most 2 points in this intersection was shown to be equal to the probability that a random determinantal cubic surface contains 3 or 7 real lines, which is indeed strictly positive. In the remaining case of four intersection points, the argument of Section 4.4.2 does not show whether the subrank is 2 or 3. 65 4 Typical subranks of order-three tensors 4.4.4 Typical subranks of 4×4×4tensors In this section, we show that 2 and 3 are both typical subranks of 4×4×4tensors. To do so, we show that the tensor defined by the bilinear product of the quaternions has subrank 2. Recall that the quaternions Hare a four dimensional division algebra over R, with basis {1, i, j, k}and multiplication 1x=x1 = xfor all x∈H, i2=j2=k2=−1, ij =k, jk =i, ki =j, ji =−k, kj =−i, ik =−j. The bilinear product H×H→Hdefines a tensor T∈(R4)⊗3whose slices along the first axis are given by T1=⎛ ⎜ ⎜ ⎝ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎞ ⎟ ⎟ ⎠,T2=⎛ ⎜ ⎜ ⎝ 0−1 0 0 1000 000−1 0010 ⎞ ⎟ ⎟ ⎠, T3=⎛ ⎜ ⎜ ⎝ 0 0 −1 0 0 0 0 1 1 0 0 0 0−100 ⎞ ⎟ ⎟ ⎠,T4=⎛ ⎜ ⎜ ⎝ 0 0 0 −1 0 0 −1 0 0 1 0 0 1 0 0 0 ⎞ ⎟ ⎟ ⎠. Let a∈H; then left multiplication defines a linear map La:H→H, b ↦→ a·b, where a·bdenotes the image of the bilinear product. For a=c11 + c2i+c3j+c4k∈H, the matrix of Lais c1T1+c2T2+c3T3+c4T4. Since the quaternions are a division algebra, this matrix has full rank 4for all a∈H\{0}. Lemma 4.4.11. The real subrank of the tensor T∈(R4)⊗3defined by the quaternions is 2 and there is a neighborhood of Twhere the real subrank is 2. Proof. We start by showing that Q(T)≤2. Let T1,...,T4be the slices of Talong the first axis. As described above, all nonzero matrices in the linear space span{T1,...,T4} ⊆ R4⊗R4are invertible. Applying two linear maps π2, π3:R4→R3to a rank-four matrix can reduce the rank at most by 2, so that it still has rank >1. Hence by Lemma 4.4.7, Q(T)≤2. Next, we show that there is a neighborhood in the Euclidean topology of Twhere all tensors have real subrank at most 2. For all v,w∈R4with ∥v∥=∥w∥= 1, there exist ε, δ > 0with the following property. For all S∈Bδ(T)and all ˜ v∈Bε(v)⊆S3,˜ w∈ Bε(w)⊆S3, the intersection span{˜ S1,..., ˜ S4}P∩Σ˜ u⊥,˜ w⊥ is empty, where ˜ S1,..., ˜ S4are the four slices of ˜ S:= (id ⊗π˜ u⊥⊗π˜ w⊥)(S)along the first axis, π˜ u⊥:R4→˜ u⊥is the orthogonal projection and π˜ w⊥is defined analogously. The open balls Bε(v)×Bε(w)form an open cover of S3×S3, so using compactness, there exists 66 4.5 Subrank of direct sums of real division algebras a finite subcover. Let δ0be the minimal δof these coverings. Then for all S∈Bδ0(T), we have # span{˜ S1,..., ˜ S4}P∩Σu⊥,w⊥= 0 for all u,w∈S3, which shows that Q(S)≤2. For completeness, to show that the subrank of Tis 2, one can check that (A⊗B⊗C)T= I2for A=(︃1 0 0 0 0 1 0 0)︃, B =(︃1 0 0 0 0 0 1 0)︃, C =(︃1 0 0 0 0 0 0 1)︃. Lastly, the fact that (A⊗B⊗C)T=I2implies that the Cayley hyperdeterminant ∆((A⊗B⊗C)Tε)>0for Tεin a small enough neighborhood of T. This shows that in a neighborhood of T, all tensors have subrank 2. Theorem 4.4.12. For 4×4×4tensors, the typical subranks are 2 and 3. Proof. We know that 3is the largest typical subrank, as it is the generic subrank for tenors in C4⊗C4⊗C4. As 2 is the typical subrank of 2×2×3tensors, we have that the typical subranks are at least 2. So it is left to show that 2is a typical subrank. In Lemma 4.4.11, we show that the tensor of the quaternions has a neighborhood with subrank 2. Hence, 2 is a typical subrank. Corollary 4.4.13. For real tensors of the format 3×4×4, the typical subranks are 2 and 3. Proof. We know that 3 = ⌊√3+4+4−2⌋is a typical subrank. If 2 were not typical, this would imply that 2 is also not a typical subrank for tensors in R4⊗R4⊗R4. Remark 4.4.14. The same argument shows that the tensor defined by the complex numbers, seen as a two dimensional real algebra, has subrank 1. For the octonions, one can construct projections onto five-dimensional subspaces and show that the intersection with the Segre variety is still empty. This shows that the subrank of the octonions is at most 4, which equals the generic subrank of 8×8×8tensors. We conjecture something stronger, however. Conjecture 4.4.15. The real subrank of the 8×8×8multiplication tensor of the octonions equals 3. 4.5 Subrank of direct sums of real division algebras In this section, we examine the subrank of multiplication tensors of division algebras. Let Dbe a real division algebra, i.e., a finite-dimensional vector space equipped with a bilinear map D×D→D, (a, b)↦→ ab such that for each nonzero athe left multiplication La: b↦→ ab is invertible. No further conditions are imposed; in particular, the multiplication may be non-associative or even non-alternative. Of course, by the celebrated Hopf-BottMilnor-Kervaire theorem [36, 14, 39], it follows that dimRD∈ {1,2,4,8}. The bilinear map f:D×D→Dhas subrank 1 if D=C(see Example 1.1.2) and 2 if D=H(see Section 4.4.4). We now turn to direct sums of the tensors corresponding to multiplication in those division algebras; i.e. componentwise multiplication over Dn. 67 4 Typical subranks of order-three tensors Proposition 4.5.1. Let Rn1,Rn2,Rn3be finite-dimensional R-vector spaces, and let f:Rn1×Rn2→Rn3 be a bilinear map and write f(u,v) = u∗v. Then Q(f)is the maximal rfor which there exist u1,...,ur∈Rn1and v1,...,vr∈Rn2 such that u1∗v1,...,ur∗vrare linearly independent modulo the space span{{ui∗vj|1≤i, j ≤r, i =j}}R. Proof. Set r:= Q(f). Then there exist linear maps φ1:Rr→Rn1,φ2:Rr→Rn2, and φ3:Rn3→Rrsuch that the composition φ3◦f◦(φ1×φ2) = Ir:Rr×Rr→Rr is the component-wise multiplication. Denote the standard basis of Rrby e1,...,erand set ui:= φ1(ei)and vj:= φ2(ej)for i, j = 1, . . . , r. Then φ3(ui∗vi) = eiand φ3(ui∗vj) = 0 for i=j. Hence the ui∈Rn1and the vj∈Rn2have the required property. Conversely, if the uiand the vihave the property in the proposition, then define φ1(ei) := ui,φ2(ej) := vj, and note that there exists a linear map φ3:Rn3→Rrthat maps ui∗vito eiand all ui∗vjwith i=jto zero. Then φ3◦f◦(φ1×φ2) = Ir:Rr×Rr→Rr is the componentwise scalar multiplication, so Q(f)≥r. Lemma 4.5.2. For a bilinear map f:Rn1×Rn2→Rn3,(u,v)↦→ u∗vand a q∈Nthe following two statements are equivalent: 1. Q(f)≤q; and 2. given any integer r≥qand any u1,...,ur∈Rn1and v1,...,vr∈Rn2, define the space of linear relations R:= {α∈R[r]×[r]|∑︂ i,j αijui∗vj= 0} and the projection to the diagonal π:R[r]×[r]→Rr,α↦→ (α11, . . . , αrr). Then dim π(R)≥r−q. Proof. Suppose that the second statement holds; let r > q, and let u1,...,ur∈Rn1and v1,...,vr∈Rn2. By the second statement, the ui∗viare not linearly independent modulo the ui∗vjwith i=j. Hence by Proposition 4.5.1, Q(f)≤q. Now suppose that Q(f)≤qand let r≥qand u1,...,ur∈Rn1and v1,...,vr∈Rn2. Let sbe the dimension of the image of span{u1∗v1,...,ur∗vr}Runder projection modulo the space Wspanned by the ui∗vjwith i=j. By Proposition 4.5.1, s≤Q(f), and hence s≤q. Without loss of generality, u1∗v1,...,us∗vsare linearly independent modulo W. Then u1∗v1,...,us∗vs,ui∗viare not, for any i=s+ 1, . . . , r. This gives rise to r−s≥r−qelements in Rwhose images under πare linearly independent. We are now ready to prove the main result of this section. 68 4.5 Subrank of direct sums of real division algebras Theorem 4.5.3. Let Dbe a division algebra over Rof dimension ≥2. Let fnbe the componentwise multiplication map fn:Dn×Dn→Dn; (a,b) = ((a1,...an),(b1, . . . , bn)) ↦→ (a1b1, . . . , anbn) =: a∗b. Regarding fnas an R-bilinear map, we have Q(fn)≤nd, where d=1 2dimRD. Proof. We proceed by induction on n. For n= 0 the statement is obvious; so we assume that n > 0and that the statement is true for all strictly smaller values of n. By Proposition 4.5.1, we need to show that if u1,...,ur∈Dnand v1,...,vr∈Dnhave the property that the ui∗viare R-linearly independent modulo the space Wspanned over Rby the ui∗vjwith i=j, then r≤nd. It follows immediately that u1,...,urare linearly independent over R, and so are v1,...,vr: indeed, if ui=∑︁j=iγjujfor certain γj∈R, then ui∗vilies in W, a contradiction to the assumption that the ui∗viare linearly independent modulo W. Suppose, for a contradiction, that r > nd. After permuting coordinates, we may assume that ur∈ {0}n−k×(D\{0})kwith k > 0. We define the left multiplication map by uras Lur:Dn→ {0}n−k×Dk,v↦→ ur∗v. We restrict this map to span{v1,...,vr}Rand define r1:= dimRLur(span{vj|j= 1, . . . , r}R) = dimRspan{ur∗vj|j= 1, . . . , r}R. Let r2=r−r1be the dimension of the kernel of said restriction of Lur. Since Dis a division algebra, r2= dimRspan{v1,...,vr}R∩(Dn−k×{0}k). Define V:= {γ∈Rr| r ∑︂ i=1 γivi∈Dn−k×{0}k}, so that dimR(V) = r2. Let I⊆[r]be a subset of cardinality r2such that the projection Rr→RIrestricts to an isomorphism on V. Then for all j∈Iwe have ˜︁ vj:= vj+∑︂ l∈I γjlvl∈Dn−k×{0}k for suitable coefficients γjl ∈R. We claim that the vectors ui∗˜︁ viwith i∈Iare linearly independent modulo the space ˜︂ Wspanned by the ui∗˜︁ vjwith i, j ∈Iand i=j. Indeed, suppose that ∑︂ i,j∈I αijui∗˜︁ vj= 0 for certain scalars αij ∈R, where the αii are not all zero. Substituting the ˜︁ vj, we obtain 0 = ∑︂ i,j∈I αij(ui∗vj+∑︂ l∈I γjlui∗vl) = ∑︂ i,j∈I αijui∗vj+∑︂ i∈I∑︂ l∈I (∑︂ j∈I αijγjl)ui∗vl, which is a linear relation among the ui∗vjwith i∈I, j ∈[r]in which the coefficients of the ui∗viare not all zero, a contradiction. This proves the claim. Note further that the claim remains intact if we replace, in each uiwith i∈I, the last kentries by zeros—after 69 4 Typical subranks of order-three tensors all, this does not affect the products ui∗˜︁ vj. Hence, by the induction hypothesis applied to n−k, we find that r2≤(n−k)d, and therefore r1=r−r2≥r−(n−k)d > kd, where the last inequality follows from the assumption that r > nd. Next let ψ:Dn→Dn−kbe the projection onto the first n−kcoordinates. Define u′ i:= ψ(ui)and v′ j:= ψ(vj)for i, j = 1,...r−1. Let Rbe the space of relations among their products: R:= {α∈R[r−1]×[r−1] |∑︂ i,j αiju′ i∗v′ j= 0}, and let π:R[r−1]×[r−1] →Rr−1be the projection to the diagonal. By Lemma 4.5.2 and the induction hypothesis applied to these vectors, the projection π(R)has dimension at least (r−1) −(n−k)d≥kd (here we use that r−1≥nd, which follows from r > nd and the fact that dis an integer, i.e., that dim(D)≥2). Let R′⊆Rbe a kd-dimensional subspace on which πis injective. Now consider the R-linear map ψ:R′→ {0}n−k×Dk⊆Dndefined by ψ(α) = ∑︂ i,j αijui∗vj. Any nonzero αin the kernel of ψwould give a linear relation among the ui∗vjwith i, j = 1, . . . , r−1in which the coefficients of the ui∗viare not all zero, a contradiction. Hence ψ is injective. Therefore dimR(Im(ψ)) = kd. But now Im(ψ)and span{ur∗v1,...,ur∗vr}R are R-subspaces of Dk× {0}n−kof dimensions kd and r1> kd, respectively. Since d= 1 2dimR(D), we have dimR({0}n−k×Dk) = 2kd, so these spaces intersect nontrivially: say ψ(α) = β1ur∗v1+···+βrur∗vr= 0 for suitable α∈R′and β1, . . . , βr∈R. But then ∑︂ i,j∈[r−1] αijui∗vj− r ∑︂ j=1 βjur∗vj= 0 and not all αii are zero, a contradiction. We conclude that r≤nd, as desired. Corollary 4.5.4. Let Dbe the division algebra Cor Hover R. Let fnbe the componentwise multiplication map fn:Dn×Dn→Dn; (a,b) = ((a1,...an),(b1, . . . , bn)) ↦→ (a1b1, . . . , anbn) =: a∗b. Regarding fnas an R-bilinear map, the subrank is Q(fn) = nq, where qis the subrank of f1. In other words, the subrank is additive for direct sums of the division algebra with itself. Proof. The direction Q(fn)≥nq always holds for direct sums of tensors. The other direction follows from Theorem 4.5.3 above using that for D=Cor D=Hwe have Q(f1) = dimR(D)/2. Remark 4.5.5. For the octonions, the above upper bound is also valid. 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