ORIGINAL RESEARCH
published: 06 June 2019
doi: 10.3389/fams.2019.00026
Frontiers in Applied Mathematics and Statistics | www .frontiersin.org 1 June 2019 | V olume 5 | Article 26
Edited by:
Jean-Luc Bouchot,
Beijing Institute of T echnology , China
Reviewed by:
Nguyen Quang T ran,
Aalto University , Finland
Shao-Bo Lin,
Wenzhou University , China
Martin Lotz,
University of Warwick,
United Kingdom
*Correspondence:
Peter Jung
peter [email protected]
Richard Kueng
[email protected]
Specialty section:
This article was submitted to
Mathematics of Computation and
Data Science,
a section of the journal
Frontiers in Applied Mathematics and
Statistics
Received: 23 September 2018
Accepted: 01 May 2019
Published: 06 June 2019
Citation:
Jung P , Kueng R and Mixon DG
(2019) Derandomizing Compressed
Sensing With Combinatorial Design.
Front. Appl. Math. Stat. 5:26.
doi: 10.3389/fams.2019.00026
Derandomizing Compr essed Sensing
With Combinatorial Design
Peter Jung 1 * , Richard Kueng 2 * and Dustin G. Mixon 3
1 Communications and Information Theory Group, T echnische Universität Berlin, Berlin, Germany , 2 Department of Computing
and Mathematical Sciences, Institute for Quantum Information and Matter , California Institute of T echnology , Pasadena, CA,
United States, 3 Department of Mathematics, Ohio State University , Columbus, OH, United States
Compr essed sensing is the art of effectively r econstructing structur ed n -dimensional
vectors from substantially fewer measur ements than naively anticipated. A plethora of
analytical reconstruction guarantees support this cr edo. The strongest among them
ar e based on deep results fr om large-dime nsional pr obability theory and r equire a
considerable amount of randomness in the measurement design. Her e, we demonstrate
that derandomization techniques allow for a considerable reduction in the randomness
r equir ed for such proof strategies. Mor e pr ecisely , we establish uniform s -sparse
r econstruction guarantees for Cs log ( n ) measurements that ar e chosen independently
fr om strength-4 orthogonal arrays and maximal sets of mutually unbiased bases,
r espectively . These are highly structur ed families of ˜
Cn 2 vectors that imitate signed
Ber noulli and standard Gaus sian vectors in a (partially) derandomized fashion.
Keywords: compr essed sensing, k -wise independence, orthogonal arrays, spherical design, derandomization
1. INTRODUCTION AND MAIN RESUL TS
1.1. Motivation
Compressed sensing is the art of effectively rec onstructing structured signals from substantially
fewer measurements than would naively be required for st andard techniques like least squares.
Although not entirely novel, rigorous treat ments of this obser vation [ 1 , 2 ] spurred considerable
scientific attention from 2006 on (see e.g., [ 3 , 4 ]) and references therein. While deterministic
results do exist, the strongest theoretic reconstruction guarantees still rely on randomness. Each
measurement corresponds to the obser ved inner product of the unknown vector , with a vector
chosen randomly from a fixed measurement ensemble. Broadly, these ensembles can be grouped
into two families:
(i) Generic measurements such as independent Gaussian, or Bernoulli vectors. Such an abundance
of randomness allows establishing very strong results by following comparatively simple and
instructive proof techniques. The downside is that concrete implementations do require a lot of
randomness. In fact, they might be too random to be useful for certain applications.
(ii) Structured measurements such as random rows of a Fourier , or Hadamard matrix. In contrast
to generic measurements, these feature a lot of structure t hat is geared toward applications.
Moreover , selecting random rows from a fixed matrix does require very little randomness.
For example, log 2 ( n ) random bits are sufficient to select a row of the DFT uniformly at
random. In contrast, generating an i.i.d. Bernoulli vector would consume n bits of randomness.
Structure and comparatively little randomness ha ve a downside, though. Theoretic convergence
guarantees tend to be weaker than their generic counterparts. It should also not come as a
surprise that the necessary proof techniques become considerably more involved.

Jung et al. Partially Derandomizing Compressed Sensing
Typically, results for type (i) precede the results for type (ii).
Phase retrieval via PhaseLift is a concrete example for such a
development. Generic convergence guarantees [ 5 , 6 ] preceded
(partially) de-randomized results [ 7 , 8 ]. Compressed sensing
is special in this regard. The two seminal works [ 1 , 2 ] from
2006 provided both results almost simultaneously. This had
an interesting consequence. Despite considerable effort, to date
there still seems to be a gap between both proof te chniques.
Here, we try to close this gap by applying a method
that is very well established in theoretical computer s cience:
par tial der andomiza tion . We start with a proof technique of
type (i) and considerably reduce the amount of randomness
required for it to work. While doing so, we keep careful
track of the amount of randomness that is still necessary.
Finally, we replace the original (generic) random measurements
with pseudo-random ones that mimic them in a sufficiently
accurate fashion. Our results highlight that this te chnique almost
allows bridging the gap between existing proof techniques for
generic and structured measurements: the results are still strong
but require slightly more randomness than choosing vectors
uniformly from a bounded orthogonal system, such as Fourier
or H adamard vectors.
There is also a didactic angle to this work: wit hin the realm of
signal processing, partial-derandomization techniques ha ve been
successfully applied to matrix reconstruction [ 8 , 9 ] and phase
retrieval via PhaseLift [ 7 , 10 , 11 ]. Although similar in spirit, t he
more involved nature of these problems may obscure the key
ideas, intuition and tricks behind such an approach. However ,
the same techniques have not yet been applied to the original
problem of compressed sensing. Here, we fill this gap and, in
doing so, provide an introduction to partial derandomization
techniques by example. To preser ve this did actic angle, we try to
keep the presentation as simple and self-conta ined as possible.
Finally, one may argue that compressed sensing has not fully
lived up to the high expect ations of the community yet (see
e.g., Tropp [ 12 ]). Arguably, one of the most glaring problems
for applications is the requirement of choosing individual
measurements at random 1 . While we are not able to fully
overcome this drawback here, the methods described in this
work do limit the amount of randomness required to generate
individual structured measurements. We belie ve that this may
help to reduce the discrepancy between “what c an be proved ” and
“what can be done ” in a variety of concrete applications.
1.2. Pr eliminaries on Compr essed Sensing
Compressed sensing aims at reconstructing s -sparse vectors x ∈
C n from m ≪ n linear measurements:
y = Ax ∈ C m .
Since m ≪ n , the matrix A is singular and there are infinitely
many solutions to this equation. A convex penalizing function
is used to promote sparsity among these solutions. Typically, this
1 Existing deterministic constructions (see e.g., Bandeira et al. [ 13 ]), do not (yet)
yield comparable statements.
penalizing function is the ℓ 1 -norm k z k ℓ 1 = P n
i = 1 | z i | :
minimize
z ∈ C n k z k ℓ 1 (1)
subject to Az = y
Strong mathematical proofs for correct re covery have been
established. By and large , these statements require randomness
in the sense that each row a i ∈ C n of A is an independent copy of
a random vector a ∈ R n . Prominent examples include
(1) m = Cs log( n / s ) standard complex Gaussian measurements:
a g ∼ N ( 0 , I / √ 2) + i N ( 0 , I / √ 2),
(2) m = Cs log( n / s ) signed Bernoulli (R ademacher)
measurements: a sb ∼ { ± 1 } n ,
(3) m = Cs log 4 ( n ) random rows of a DFT matrix: a f ∼
{ f 1 , ... , f n } ,
(4) for n = 2 d : m = Cs log 4 ( n ) random rows of a H adamard
matrix: a h ∼ { h 1 , ... , h n } .
A rigorous treatment of all th ese cases can be found in Foucart
and R auhut [ 3 ]. Here, and throughout this work, C > 0
denotes an absolute constant whose exact value depends on the
context but is always independent of the problem parameters n , s ,
and m . It is instructive to compare the amount of randomness
that is required to generate one instance of t he random vectors
in question. A random signed Bernoulli vector a sb ∈ R n
requires n random bits (one for each coordinate), while a
total of log 2 ( n ) random bits suffice to select a random row
a h ∈ R n of a H adamard matrix. A comparison between
complex standard Gaussian vectors a g ∈ C n and random
Fourier vectors a f ∈ C n indicates a similar discrepancy. In
summary: highly structured random vectors, like a f , a h require
exponentially fewer random bits to generate than generic random
vectors, like a g , a sb . Importantly, this transition from generic
measurements to highly structured ones comes at a price. The
number of measurements required in cases (3) and (4) scales
poly-logarithmically in n . More sophisticated approaches allow
for converting this offset into a poly-logarithmic s caling in s
rather than n [ 14 , 15 ]. Another , arguably even higher price, is
hidden in the proof techniques behind these results. They are
considerably more involved.
The following two subsections are de voted to introducing
formalisms that allow for partially de-randomizing
signed Bernoulli vectors and complex standard Gaussian
vectors, respectively.
1.3. Partially De-randomizing Signed
Ber noulli V ectors
Throughout this work, we endow C n with the standard i nner
product h x , y i = P n
i = 1 ¯ x i y i . We denote the associated (Euclidean)
norm by k z k 2
ℓ 2 = h z , z i . Let a sb = ( ǫ 1 , ... , ǫ n ) T be a
signed Bernoulli vector with coefficients ǫ i ∼ { ± 1 } chosen
independently at random (Ra demacher random varia bles). Then,
E  ǫ i ¯ ǫ j  = E  ǫ i ǫ j  = δ ij . (2)
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Jung et al. Partially Derandomizing Compressed Sensing
This feature is equivalent to isotropy of signed Bernoulli vectors:
E  | h z , a sb i | 2  =
n
X
i , j = 1
E  ǫ i ¯ ǫ j  ¯ z i z j = h z , z i = k z k 2
ℓ 2 . (3)
for all z ∈ C n .
Independent sign entries are sufficient, but not necessary for
this feature. Indeed, suppose that n = 2 d is a power of two.
Then, the rows of a Sylvester Hadamard matrix correspond to a
particular subset of n sign vectors that are also proportional to an
orthonormal basis. The orthonormal b asis property ensures that
a randomly selected Hadamard row a h ∈ R n is also isotropic.
In turn, the coordinates a i ∈ { ± 1 } of a h also obey (2), despite
not being independent instances of random signs. This fe ature is
called pairwise independence and naturally generalizes to k ≥ 2:
Definition 1. ( k -wise independence.) Fix k ≥ 2 and let ǫ i denote
independent inst ances of a s igned Bernoulli rand om variable. We
call a r andom sign vector a ∈ { ± 1 } n k- wise independent , if its
components a 1 , ... , a n obey
E 

k
Y
i = 1
a i k 
 = E 

k
Y
i = 1
ǫ i k 
 (4)
for all k-tuples of indice s 1 ≤ i 1 , ... , i k ≤ n .
For k = 2, this defining property reduces to Equation (2)
and is equivalent to demanding isotropy of the distribution.
R andom rows of H adamard matrices form a concrete example
for pairwise independence.
What is more, explicit constructions for k -wise independent
vectors in R n are known for any k and n . In this work we focus on
particular constructions that rely on generalizing the following
instructive example. Fix n = 3 and consider the M = 4 rows of
the following matrix:




111
1 − 1 − 1
− 1 1 − 1
− 1 − 1 1




The first two columns summarize all possible tuple s ( k =
2 combinations) of ± 1. The coefficients of the third column
correspond to their entry-wise product. Hence, the third column
is completely characterized by the first two and therefore three
components of a selected row are not mutually independent.
Nonetheless, each subset of two coordinates does mimic
independent beha vior : all possible length-two combinations of
± 1 occur exactly once in these two columns. This ensures that
a randomly selected row obeys Equation (4) for k = 2 (pairwise
independence). The concept of or thogonal arr ays [ 16 ] generalizes
this simple example.
Definition 2. (Orthogonal array) A (binary) or thogonal arr ay of
strength k with M rows and n columns is a M × n sign ma tr ix such
tha t e very selection of k columns conta ins all elements of { ± 1 } k an
equal number of times.
The example from above is an 4 × 3 ort hogonal array of
strength 2. Strength- k orthogonal arrays are closely related to
the concept of k -wise independence. The following implication
is straightforward.
Lemma 1. Selecting a row of an M × n or thogonal arr ay of
strength k uniformly a t rando m produces k-wise independent
r andom vectors in n dimens ions.
This correspondence identifies orthogonal arrays as general-
purpose seeds for pseudo-random beha vior. What is more,
explicit constructions of orthogonal arrays are known for any k
and any n . In contrast to the “full” array that lists all 2 n possible
elements of { ± 1 } n as its rows, these constructions typically only
require M ≥ O  n k / 2  rows. This fundamental restriction is
called Rao’ s bound and only scales polynomially in the dimension
n . Choosing a random row from such an array only requires
log 2 ( M ) = O ( k log 2 ( n )) random bits and produces a random
vector that is k -wise independent.
1.4. Partially Derandomizing Complex
Standar d Gaussian V ectors
Let us now discuss another general-purpose tool for (partial)
de-randomization. Concentration of measure implies that n -
dimensional standard complex Gaussian vectors concentrate
sharply around the complex sphere of radius √ n . Hence, they
beha ve very similarly to vectors a s ∈ C n chosen uniformly from
this sphere. For any k ∈ N , such spherical random vectors obey
E h |h z , a s i| 2 k i = n k Z w ∈ S n − 1 |h z , w i| 2 k d w = n k  n + k − 1
k  − 1
k z k 2 k
ℓ 2 .
for all z ∈ C n .
Here, d w denotes the uniform measure on the complex unit
sphere S n − 1 =  x ∈ C n : k x k ℓ 2 = 1  . This formula characterizes
even moments of the uniform distribution 2 on S n − 1 . The concept
of k -designs [ 17 ] uses this moment formula as a starting point for
partial de-randomization. Roughly speaking, a k -design is a finite
subset of √ n -length vectors such that the uniform distribution
over these vectors reproduces the uniform measure on √ n S n − 1
up to k -th moments. More precisely:
Definition 3. A set of N vectors { w i } N
i = 1 ⊂ √ n S n − 1 is called a
(complex projective) k -design if, for an y l ∈  k  =  1, ... , k  , a
r andomly selected element a ( k ) obeys
E h |h z , a ( k ) i| 2 l i = n l  n + l − 1
l  − 1
k z k 2 l
ℓ 2 for all z ∈ C n .
(Spherical) k -designs were originally developed as cubature
formulas for the real-valued unit sphere [ 17 ]. The concept has
since been extended to other sets. A generalization to the complex
projective space C P n − 1 gives rise to Definition 3. Complex
projective k -designs are known to exist for any k and any
2 For comparison, a complex standard Gaussian vector obeys E h |h z , a g i| 2 k i =
k ! k z k 2 k
ℓ 2 instead.
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Jung et al. Partially Derandomizing Compressed Sensing
dimension n (see e.g., [ 18 – 20 ]). However , explicit constructions
for k ≥ 3 are notoriously difficult to find. In contrast, se veral
explicit families of 2-designs ha ve been identified. Here, we will
focus on one such family. Two orthonormal bases { b i } n
i = 1 and
{ c i } n
i = 1 of C n are called mutually unbiased if
  h b i , c j i  
2 = 1
n for all i , j ∈ [ n ] . (5)
A prominent example for such a basis pair is the st andard basis
and the Fourier , or H adamard, basis, respectively. One can show
that at most n + 1 different orthonormal bases may exist t hat ha ve
this property in a pairwise fashion [ 21 , Theorem 3.5]. Such a set
of n + 1 bases is called a maximal set of mutually unbiased bases
(MMUB). For instance, in n = 2 the stand ard basis together with
1
√ 2  1
1  , 1
√ 2  1
− 1  , 1
√ 2  1
i  , 1
√ 2  1
− i 
forms a MMUB. Importantly, MMUBs are always (proportional
to) 2-designs [ 22 ]. Explicit constructions exist for any prime
power dimension n and one can ensure that the st andard basis
is always included. Here we point out one construction that is
particularly simple if the dimension n ≥ 5 is prime [ 23 ]: The
standard basis ve ctors e 1 , ... , e n ∈ C n together with all vectors
whose entry-wise coefficients correspond to
 b α , λ  k = 1
√ n ω ( k + α ) 3 + λ ( k + α )
n (6)
form a MMUB. Here ω n = exp  2 π i
n  is a n -th root of unity.
The parameter α ∈ [ n ] singles out one of the n different
bases, while λ ∈ [ n ] labels the n corresponding basis vectors.
Excluding the standard basis, t his set of n 2 vectors corresponds
to all time-frequency shifts of a discrete Alltop sequence [ f ] k =
ω k 3
n [ 24 ].
1.5. Main Results
In the following ˜ c > 0, like C , denotes an absolute constant whose
precise value may depend on the context.
Theorem 1. (CS from orthogonal array measurements.) Suppose
tha t a matrix A conta ins m ≥ Cs log(2 n ) rows tha t are
chosen independently from s trength-4 or thogonal arr ay. Then,
with proba bility at le ast 1 − 2e −˜ cm , any s-sparse x ∈ C n c an
be recovered from y = Ax by solving the convex optimiza tion
problem (1).
Theorem 2. (CS from time-frequency shifted Alltop sequences.)
Let n ≥ 5 be prime and suppose tha t A contains m ≥ Cs log(2 n )
rows tha t correspond to r andom time-frequency s h ifts of the n-
dimens ional Alltop sequence (6). Then, with proba bility at le ast
1 − 2e − ˜ cm , any s-spar se x ∈ R n can be recovered from y = Ax
by solving (1).
This result readily generalizes to measurements that are
sampled from a maximal set of mutually unbiased bases
(excluding the standard basis). Time-frequency shifts of the
Alltop sequence are one concrete construction that applies to
prime dimensions only.
Note that the cardinality of all Alltop shifts is n 2 . Hence,
2 log 2 ( n ) random bits suffice to select a random time-frequency
shift. In turn, a total of
2 log 2 ( n ) m ≃ 2 Cs log 2 ( n ) (7)
random bits are required for sampling a complete measurement
matrix A . This number is exponentially smaller than the number
of random bits required to generate a matrix with independent
complex Gaussian entries. A similar comparison holds true for
random signed Bernoulli matrices and columns sampled from a
strength-4 orthogonal array.
Highly structured families of vectors – such as rows of a
Fourier , or Hadamard matrix – require even less randomness to
sample from: only log 2 ( n ) bits are required to select such a row
uniformly at random. However , existing recovery guarantees are
weaker than the main results presented here. They require an
order of Cs polylog( s ) log( n ) random measurements to establish
comparable results. Thus, the total number of random bits
required for such a procedure scales like Cs polylog( s ) log 2 ( n ).
Equation (7) still establishes a logarithmic improvement in terms
of sparsity.
The recovery guarantees in Theorem 1 and Theorem 2 can
be readily extended to ensure st ability with respect to noise
corruption in the measurements and robustness with respect
to violations of the model assumption of sparsity. We refer to
section 3 for details.
We also emphasize that there are results in the lit erature
that establish compressed sensing guarantees with comparable ,
or even less, randomness. Obviously, deterministic constructions
are the extreme case in this regard. E arly deterministic
results suffer from a “quadratic bottleneck.” The number of
measurements must scale quadratic ally in the sparsity: m ≃ s 2 .
Although this obst acle was overcome , existing progress is still
comparatively mild. Bourgain et al. [ 25 ], Mixon [ 26 ], Bandeira
et al. [ 27 ] establish deterministic convergence guarantees for
m ≃ s 2 − ǫ , where ǫ > 0 is a (very) small constant.
Closer in spirit to this work is Bandeira et al. [ 2 8 ]. There, the
authors employ the Legendre symbol – which is well known for
its pseudorandom beha vior – to partially derandomize a signed
Bernoulli matrix. In doing so, they establish uniform s -sparse
recovery from m ≥ Cs log 2 ( s ) log( n ) measurements that require
an order of s log( s ) log( n ) random bits to generate. Compared
to the main results presented here, this result gets by with
less randomness, but requires more measurements. The proof
technique is also very different.
To date, the strongest de-randomized reconstruction
guarantees hail from a close connection between s -sparse
recovery and Johnson-Lindenstrauss embeddings [ 29 , 30 ].
These ha ve a wide range of applications in modern data
science. Kane and Nelson [ 31 ] established a very strong
partial de-randomization for such embeddings. This result
may be used to establish uniform s -sparse recovery for
m = Cs log( n / s ) measurements that require an order of
s log  s log( n / s ) log( n / s )  random bits. This result surpasses
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Jung et al. Partially Derandomizing Compressed Sensing
the main results presented here in both sampling rate and
randomness required.
However , this strong result follows from “reducing ” the
problem of s -sparse recovery to a (seemingly) very different
problem: find Johnson-Lindenstrauss embeddings. Such a
reduction typically does not preser ve problem-spe cific structure.
In contrast, the approach presented here addresses the problem
of sparse recovery directly and relies on tools from signal
processing. In doing so, we maintain structural properties that are
common in several applications of s -sparse recovery. Ort hogonal
array measurements, for instance, have ± 1-entries. This is
well-suited for the single pixel camera [ 32 ]. Alltop sequence
constructions, on the other hand, have succes sfully been applied
to stylized radar problems [ 33 ]. Both types of me asurements also
ha ve the property that every entry has unit modulus. This is an
important feature for communication appli cations like CDMA
[ 34 ]. H a ving pointed out these high-le vel connections, we want
to emphasize that careful, problem specific adapt ations may be
required to rigorously exploit them. The framework developed
here may ser ve as a guideline on how to achieve this goal in
concrete scenarios.
2. PROOFS
2.1. T extbook-W orthy Pr oof for Real-V alued
Compr essed Sensing With Gaussian
Measur ements
This section is devoted to summarizing an elegant argument,
originally by Rudelson and Vershynin [ 14 ], see also [ 35 – 37 ] for
arguments that are similar in spirit. This argument only applies
to s -sparse recovery of real-valued signals. We will generalize a
similar idea to the complex case later on.
In this work we are concerned with uniform re construction
guarantees: with high probability, a single realization of the
measurement matrix A allows for reconstructing any s -sparse
vector x by means of ℓ 1 -regularization (1). A ne cessary
pre-requisite for uniform recovery is t he demand that the
kernel, or nullspace , of A must not contain any s -sparse
vectors. This condition is captured by the nullspace proper ty
(NSP). Define,
T s =  z ∈ S n − 1 : k z k ℓ 1 ≥ 2 σ s ( z )  , (8)
where σ s ( x ) = inf k z k 0 ≤ s k x − z k ℓ 1 is the approximation error
(measured in ℓ 1 -norm) one incurs when approximating x by
s -sparse vectors.
A matrix A obeys the NSP of order s ( s -NSP) if,
inf
z ∈ T s k Az k ℓ 2 > 0. (9)
The set T s is a subset of the unit sphere that contains all
normalized s -sparse vectors. This justifies the informal definition
of the NSP: no s -sparse vector is an element of the nullspace
of A . Importantly, the NSP is not only necessary, but also
sufficient for uniform recovery (see e.g., Foucart and R auhut
[ 3 , Theorem 4.5]). Hence, universal recovery of s -sparse signals
readily follows from establishing Rel. (9). The nullspace property
and its relation to s -sparse recovery has long been somewhat
folklore. We refer to Foucart and R auhut [ 3 , Notes on se ction 4]
for a discussion of its origin.
The following powerful statement allows for exploiting
generic randomness in order to establish nullspace properties.
It is originally due to Gordon [ 38 ], but we utilize a
more modern reformulation, see Foucart and R auhut
[ 3 , Theorem 9.21].
Theorem 3. (Gordon’ s esc ape through a mesh.) Let A be a real-
valued m × n stand ard Gauss ian matrix 3 and let E ⊆ S n − 1 ⊂ R n
be a subset of t he real-valued unit sph ere. Define the Gaussian
width ℓ ( E ) = E sup z ∈ E h a g , z i , where the expecta tion is over
realiza tions a g ∼ N (0, I ) of a s tandard Gaus sian r andom vector.
Then, for t ≥ 0 the bound
inf
z ∈ E k Az k ℓ 2 ≥ √ m − 1 − ℓ ( E ) − t
is true with proba bility at le ast 1 − e − t 2 / 2 .
This is a deep statement that connec ts random matrix theory
to geometry: the Gaussian width is a rough me asure of the size
of the set E ⊆ S n − 1 . Setting E = T s allows us to conclude that a
matrix A encompassing m independent Gaussian measurements
is very likely to obey the s -NSP (9), provided that ( m − 1) exceeds
ℓ ( T s ) 2 . In order to derive an upper bound on ℓ ( T s ), we may use
the following inclusion
T s ⊂ 2conv ( 6 s ) ,
see e.g., Ka banava and Rauhut [ 35 , Lemma 3] and Rudelson and
Vershynin [ 14 , Lemma 4.5]. Here, 6 s ⊆ S n − 1 denotes the set of
all s -sparse vectors with unit length. In turn,
ℓ ( T s ) ≤ 2 E sup
z ∈ conv( 6 s ) h a g , z i = 2 E sup
z ∈ 6 s h a g , z i , (10)
because the line ar function z 7→ h a g , z i achieves its maximum
value at the boundary 6 s of the convex set conv ( 6 s ) . The
right-hand side of (10) is the expected supremum of a Gaussian
process indexed by z ∈ 6 s . Dudley’ s inequality [ 39 ], see also [ 3 ,
Theorem 8.23], states
E sup
z ∈ 6 s h a g , z i ≤ 4 √ 2 Z 1
0 q ln  C  6 s , k · k ℓ 2 , u  d u ,
where C ( 6 s , k · k ℓ 2 , u ) are covering numbers associated with the
set 6 s . They are defined as the smallest cardinality of an u -
covering net with respect to the Euclide an distance. A volumetric
counting argument yields C ( 6 s , k · k ℓ 2 , u ) ≤  e n
s  s  1 + 2
u  s and
Dudley’ s inequality therefore implies
ℓ ( T s ) ≤ c p s log ( e n / s ) ,
3 This is a m × n matrix where each entry is sampled independently from the
standard normal distribution N (0, 1).
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Jung et al. Partially Derandomizing Compressed Sensing
where c is an absolute constant. This readily yields the following
assertion (choose t = √ 2 ˜ cm for ˜ c sufficiently small).
Theorem 4. (NSP for Gaussian measurements.) A number
of m ≥ c 2 s log(e n / s ) independent real-valued Gaus sian
measurements obeys the (re al-valued) s-NSP with proba bility at
least 1 − e −˜ cm .
This argument is exemplary for generic proof techniques:
strong results from probability theory allow for establishing
close-to-optimal recovery guarantees in a relatively
succinct fashion.
2.2. Extending the Scope to Subgaussian
Measur ements
The extended arguments presented here are largely due to
Dirksen, Lecue and R auhut [ 36 ]. Again, we will focus on the
real-valued case.
Gordon’ s esc ape through a mesh is only valid for Gaussian
random matrices A . Novel methods are required to extend
this proof technique beyond this idealized c ase. Comparatively
recently, Mendelson provided such a generalization [ 40 , 41 ].
Theorem 5. (Mendelson’ s small ball meth od, Tropp’ s
formulation [ 37 ]). Suppose tha t A is a r andom m × n ma trix
whose rows correspond to m independent re aliza tions of a r andom
vector a ∈ R n . F ix a set E ⊆ R n , and define,
Q ξ ( a , E ) = inf
z ∈ E Pr [ | h z , a i | ≥ ξ ] for ξ > 0,
W m ( a , E ) = E sup
z ∈ E h z , h i where h = 1
√ m
m
X
i = 1
ǫ i a i ∈ R n ,
is the empirical a ver a ge over m independent copies of a weigh ted
by uniformly r andom signs ǫ i ∼ { ± 1 } . Then, for any t , ξ > 0
inf
z ∈ E k Az k ℓ 2 ≥ ξ √ mQ 2 ξ ( a , E ) − 2 W m ( a , E ) − ξ t
with proba bility at le ast 1 − 2e − t 2 / 2 .
It is worthwhile to point out that for real-valued Gaussian
vectors this result recovers Theorem 3 up to constants. Fix ξ > 0
of appropriate size. Then, E ⊆ S n − 1 ensures that ξ Q 2 ξ ( a g , E )
is constant. Moreover , W m ( a g , E ) reduces to the usual Gaussian
width ℓ ( E ).
Mendelson’ s small b all method can be used t o establish the
nullspace property for independent random measurements a ∈
R n that exhibit subgauss ian beha vior:
E exp ( θ h z , a i ) ≤ exp  θ 2
2 k z k 2
ℓ 2 
for all z ∈ R n , θ > 0. (11)
Signed Bernoulli vectors are a concrete example. Such random
vectors are isotropic (3) and direct computation also re veals
E  h z , a sb i 4  =
n
X
i , j , k , l = 1
E  ǫ i ǫ j ǫ k ǫ l  z i z j z k z l =
n
X
i = 1
E  ǫ 4
i  z 4
i
+ 3 X
i 6= j
E  ǫ 2
i  E h ǫ 2
j i z 2
i z 2
j
=
n
X
i = 1
z 4
i + 3 X
i 6= j
z 2
i z 2
j = 3 k z k 4
ℓ 2 − 2 k z k 4
ℓ 4 ≤ 3 k z k 4
ℓ 2 ,
(12)
because there are 3 possible pairings of four indices. Next, set E =
T s ⊂ S n − 1 . An application of the Paley-Zygmund inequality then
allows for bounding the parameter Q 2 ξ ( a sb , T s ) in Mendelson’ s
small ball method from below:
Q 2 ξ ( a sb , T s ) ≥ inf
z ∈ S n − 1 Pr [ |h z , a sb i| ≥ 2 ξ ] = inf
z ∈ S n − 1
Pr  h z , a sb i 2 ≥ 4 ξ 2 E  h z , a sb i 2 
≥ inf
z ∈ S n − 1  1 − 4 ξ 2  2 E  h z , a sb i 2  2
E  h z , a sb i 4  ≥ (1 − 4 ξ 2 ) 2
3 .
This lower bound is constant for any ξ ∈ (0, 1 / 2).
Next, note that X z = h z , h i is a stochastic process that is
indexed by z ∈ R n . This process is centered ( E X z = 0)
and Equation (11) implies that it is also subgaussian. Moreover ,
E  | X z − X y | 2  = k z − y k 2
ℓ 2 readily follows from isotropy (3).
Unlike Gordon’ s esc ape through a mesh, Dudley’ s inequality
does remain valid for such stochastic processes with subgaussian
marginals. We can now repeat the width analysis from the
previous section t o obtain
W m ( a sb , T s ) ≤ 2 E sup
z ∈ 6 s h z , h i ≤ c p s log(e n / s ).
Fixing ξ > 0 sufficiently small, setting t = √ 2 ˜ cm and inserting
these bounds into Equation (5) yields the following result.
Theorem 6. (NSP for signed Bernoulli measurements.) A matrix
A encompass ing m ≥ Cs log(e n / s ) r andom signed Bernoulli
measurements obeys the (re al-valued) s-NSP with proba b ility a t
least 1 − e −˜ cm .
A similar result remains valid for other classes of independent
measurements with subgaussian marginals (11).
2.3. Generalization to Complex-V alued
Signals and Partial De-Randomization
The nullspace property, as well as its connection to uniform
s -sparse recovery readily generalizes to complex-valued s -sparse
vectors (see e.g., Foucart and R auhut [ 3 ], section 4). In turn,
Mendelson’ s small ball met hod may also be generalized to the
complex-valued case:
Theorem 7. (Mendelson’ s small ball met hod for complex vector
spaces.) Suppose th a t the rows of A corres pond to m independent
copies of a r andom vector a ∈ C n . F ix a set E ⊆ C n and define
Q ξ ( a , E ) = inf
z ∈ E Pr [ |h z , a i| ≥ ξ ] for ξ > 0,
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Jung et al. Partially Derandomizing Compressed Sensing
W m ( a , E ) = E sup
z ∈ E | h z , h i | where h = 1
√ m
m
X
i = 1
ǫ i a i .
Then, for any t , ξ > 0
inf
z ∈ E k Az k ℓ 2 ≥ √ 2  ξ √ mQ 2 3 / 2 ξ / 2 − 2 W m ( E , a ) − ξ t 
with proba bility at le ast 1 − 2e − t 2 / 2 .
Such a generalization was conjectured by Tropp [ 37 ], but we
are not aware of any rigorous proof in the literature. We provide
one in subsection 5.1 and belie ve that this generalization may be
of independent interest. In particular , this extension allows for
generalizing the arguments from the previous subse ction to the
complex-valued case.
Let us now turn to the main s cope of this work: partial
de-randomization. Effectively, Mendelson’ s small ball method
reduces the task of establishing nullspace properties to bounding
the two parameters Q 2 3 / 2 ξ ( a , T s ) and W m ( a , T s ) in an appropriate
fashion. A lower bound on the former readily follows from the
P aley-Zygmund inequality, provided that the following relations
hold for any z ∈ C n :
E  |h a , z i| 2  = k z k 2
ℓ 2 (isotropy),
E  |h a , z i| 4  ≤ C 4 k z k 4
ℓ 2 (4h moment bound).
Here, C 4 > 0 is constant.
Indeed, inserting these bounds into the Paley-Zygmund
inequality yields
Q 2 3 / 2 ξ ( a , T s ) ≥ C − 1
4  1 − 8 ξ 2  2 for any ξ ∈  0, 2 − 3 / 2  .
(13)
In contrast, establishing an upper bound on W m ( a , T s )
via Dudley’ s inequality requires subgaussian marginals (11)
(that must not depend on the ambient dimension). This
implicitly imposes stringent constraints on all moments of a
simultaneously. An additional assumption allows to considerably
weaken these demands:
max
1 ≤ k ≤ n |h e k , a i| 2 = 1 almost surely (incoherence). (14)
Here, e 1 , ... , e n denotes the standard basis of C n . Incoherence
has long been identified as a key ingredient for developing
s -sparse recovery guarantees. Here, we utilize it to establish
an upper bound on W m ( A , T s ) that does not rely on
subgaussian marginals.
Lemma 2. Let a ∈ C n be a r andom vector th a t is isotropic and
incoherent. Let T s ⊂ C n be the complex-v alued generaliza tion of
the set de fined in Equa tion (8) and assume m ≥ log(2 n ) . Then,
W m ( a , T s ) ≤ 4 p 2 s log(2 n ). (15)
This bound only requires an appropriate scaling of t he first two
moments (isotropy) but comes at a price. The bound sc ales
logarithmically in n rather than n / s . We defer a proof of this
statement to subsection 5.2 below. I nserting the bounds (13) and
(15) into the assertion of Theorem 7 readily yields the main
technical result of this work:
Theorem 8. Suppose tha t a ∈ C n is a random vector th a t obeys
incoherence, isotropy and the 4t h moment bound. Then, choos ing
m ≥ Cs log( n )
inst ances of a uniformly a t r andom results in a mea surement
ma trix A tha t obeys t he complex-valued nullspace property of order
s with proba bility at le ast 1 − 2e −˜ cm .
In complete analogy to the real-valued case, the complex
nullspace property ensures uniform recovery of s -sparse vectors
x ∈ C n from y = Ax via solving the convex optimization
problem (1).
2.4. Recovery Guarantee for Str ength-4
Orthogonal Arrays
Suppose that a oa ∈ { ± 1 } n is chosen uniformly from a
strength-4 orthogonal array. By definition, e ach component
a i of a has unit modulus. This ensures incoherence . Moreover ,
E  a i a j  = E  ǫ i ǫ j  = δ ij , because 4-wise independence
necessarily implies 2-wise independence. Isotropy then
readily follows from (3). Finally, 4-wise independence
suffices to establish the 4th moment bound. By assumption
E  a i a j ¯ a k ¯ a l  = E  ǫ i ǫ j ǫ k ǫ l  and we may t hus infer
E h |h z , a oa i| 4 i =
n
X
i , j , k , l = 1
E  ǫ i ǫ j ǫ k ǫ l  ¯ z i ¯ z j z k z l
=
n
X
i = 1
E h ǫ 4
i i | z i | 4 + X
i 6= j
E h ǫ 2
i i E h ǫ 2
j i  ¯ z 2
i z 2
j + 2 | z i | 2 | z j | 2 
= 2 k z k 4
ℓ 2 + X
i 6= j ¯ z 2
i z 2
j − k z k 4
ℓ 4 ≤ 3 k z k 4
ℓ 2 .
In summary: a oa meets all the requirements of Theorem 8.
Theorem 1 then follows from the fact that the complex
nullspace property ensures uniform recovery of all s -sparse
signals simultaneously.
2.5. Recovery Guarantee for Mutually
Unbiased Bases
Suppose that a mub ∈ C n is chosen uniformly from a maximal
set of n mutually unbiased bases (excluding the standard basis)
whose elements are re-normalized to length √ n . R andom time-
frequency shift of the Alltop sequence (6) are a concrete example
for such a sampling procedure, provided that the dimension n ≥
5 is prime.
The vector a mub is chosen from a union of n bases that are
all mutually unbiased with respect to the st andard basis, see
Equation (5). Together with normalization to lengt h √ n , this
readily establishes incoherence: max 1 ≤ k ≤ n |h e k , a i| 2 = n
n =
1 with probability one. Next, by assumption a mub is chosen
uniformly from a union of n re-scaled orthonormal b ases. Let us
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Jung et al. Partially Derandomizing Compressed Sensing
denote each of them by b ( l )
1 , ... , b ( l )
n , where 1 ≤ l ≤ n la bels the
different basis. Then,
E  |h a mub , z i| 2  = 1
n 2
n
X
l = 1
n
X
i = 1 | √ n h b ( l )
i , z i| 2 = 1
n
n
X
l = 1 k z k 2
ℓ 2
= k z k 2
ℓ 2 for all z ∈ C n
which implies isotropy. Finally, a maximal set of ( n + 1) mutually
unbiased bases – including the standard b asis which we denote
by b ( n + 1)
k = e k – forms a 2-design according to Definition 3. For
any z ∈ C n this property ensures
E  |h a mub , z i| 4  =
n + 1
X
l = 1
n
X
i = 1 |h b ( l )
i , z i| 4 −
n
X
k = 1 | h e k , z i | 4
= 2 k z k 4
ℓ 2 − k z k 4
ℓ 4 ≤ 2 k z k 4
ℓ 2
which establishes the 4th moment bound. In summary, the
random vector a mub ∈ C n meets the requirements of Theorem 8.
Theorem 2 then readily follows form t he implications of the
nullspace property for s -sparse recovery.
3. EXTENSION TO NOISY
MEASUREMENTS
The nullspace property may be generalized to address two
imperfections in s -sparse recovery simulta neously: (i) the vector
x ∈ C d may only be approximately sparse in the sense that it is
well-approximated by an s -sparse vector , (ii) the measurements
may be corrupted by additive noise: y = Ax + s with s ∈ C m .
To state this generalization, we need s ome additional notation.
For z ∈ C n and s ∈ [ n ] , let z s ∈ C n be the vector that only
contains the s largest entries in modulus. All other entries are set
to zero. Likewise, we write z ¯ s = z − z s to denote t he remainder.
In particular , σ s ( z ) = k z ¯ s k ℓ 1 . An m × n matrix A obeys the
robust nulls pace proper ty of order s with parameters ρ ∈ (0, 1)
and τ > 0 if
k z s k ℓ 2 ≤ ρ
√ s k z ¯ s k ℓ 1 + τ k Az k ℓ 2 for all z ∈ S n − 1 ,
see e.g., Foucart and R auhut [ 3 , Definition 4.21]. This extension
of the nullspace property is closely related to stable s -sparse
recovery via basis pursuit denoising:
minimize k z k ℓ 1
subject to k Az − y k ℓ 2 ≤ η . (16)
Here, η > 0 denotes an upper bound on t he strength of the noise
corruption: k s k ℓ 2 ≤ η . [ 3 , Theorem 4.22] draws the following
precise connection: suppose t hat A obeys the robust nullspace
property with parameters ρ and τ . Then, the solution z ♯ ∈ C n
to (16) is guaranteed to obey
k z ♯ − x k ℓ 2 ≤ D 1
√ s σ s ( x ) + D 2 η , (17)
where D 1 = (1 + ρ ) 2 / (1 − ρ ) and D 2 = (3 + ρ ) τ / (1 − ρ ). The
first term on the r.h.s. vanishes if x is exactly s -sparse and remains
small if x is well approximated by a s -sparse vector. The second
term scales linearly in t he noise bound η ≥ k s k ℓ 2 and vanishes in
the absence of noise corruption.
In the previous se ction, we have esta blished the classical
nullspace property for measurements that are chosen
independently from a vector distribution that is isotropic,
incoherent and obeys a bound on the 4th moments. This
argument may readily be extended to establish the robust
nullspace property with relatively little extra effort. To this end,
define the set
T ρ , s =  z ∈ S n − 1 : k z s k ℓ 2 > ρ
√ s k z ¯ s k ℓ 1  .
A moment of thought re veals that t he matrix A obeys the robust
nullspace property with parameters ρ , τ if
inf
z ∈ T ρ , s k Az k ℓ 2 ≥ 1
τ . (18)
What is more, the following inclusion formula is also valid:
T ρ , s ⊂ 3
ρ conv ( 6 s ) ,
see Ka banava and Rauhut [ 35 , Lemma 3] and Rudelson and
Vershynin [ 14 , Lemma 4.5]. This ensures that the bounds on
the parameters in Mendelson’ s small ba ll method generalize in
a rather straightforward fashion. Isotropy, incoherence and t he
4th moment bound ensure
Q 2 ξ ( a , T ρ , s ) ≥ (1 − 2 ξ 2 ) 2
C 4
and W m ( a , T ρ , s ) ≤ 12
ρ p 2 s log(2 n ).
Now, suppose that A subsumes m ≥ C ρ − 2 s log(2 n ) independent
copies of the random vector a ∈ C n . Then, Theorem 7 readily
asserts that with probability at least 1 − 2e −˜ cm
inf
z ∈ T ρ , s k Az k ℓ 2 ≥ c
ρ √ m , (19)
where c > 0 is another constant. Pre viously, we employed
Mendelson’ s small b all method to merely assert that a similar
infimum is not equal to zero. Equation (19) provides a strictly
positive lower bound with comparable effort. Comparing this
relation to Equation (18) highlights that this is enough to
establish the robust nullspace property with parameters ρ and
τ = ρ
c √ m . In turn, a stable generalization of the main recovery
guarantee follows from Equation (17).
Theorem 9. F ix ρ ∈ (0, 1) and s ∈ N . Suppose tha t we
sample m ≥ C ρ − 2 s log( n ) independent copies of an isotropic,
incoherent r andom vector a ∈ C n tha t also obeys the 4th moment
bound. Then, with proba bility at le ast 1 − 2e −˜ cm , the result ing
measurement ma trix A allows for sta ble, uniform recovery of
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Jung et al. Partially Derandomizing Compressed Sensing
(approxima tely) s-s parse vectors. More precisely, the solut ion z ♯ to
(16) is guaran teed to obey
k x − z ♯ k ℓ 2 ≤ D 1
√ s σ s ( x ) + D 2
η
√ m ,
where D 1 , D 2 > 0 depend only on ρ .
4. NUMERICAL EXPERIMENTS
In this part we demonstrate the per formance which can be
achieved with our proposed derandomized constructions and we
compare this to generic measurement matrices (Gaussian, signed
Bernoulli). However , since the orthogonal array construction is
more involved we first provide additional details relevant for
numerical experiments.
Details on orthogonal arrays: An orthogonal array
OA( λσ k , n , σ , k ) of strength k , with n factors and σ le vels
are an λσ k × n array of σ different symbols such that in any k
columns every ordered σ k -tuple occurs in exactly λ rows. Arrays
with λ = 1 are called simple. A comprehensive treatment can
be found in the book [ 16 ]. Known arrays are listed in se veral
libraries 4 . Often the symbol alphabet is not relevant, but we use
the set Z σ = { 0, ... , σ − 1 } for concreteness. Such arrays can
be represented as a matrix in Z λσ k × n
σ . For σ = q p with q prime
the simple orthogonal array OA( σ k , n , σ , k ) is linear if the q pt
rows of the matrix form a vector space over F q . The runs of a n
orthogonal array (the rows of the corresponding matrix) c an
also be interpreted as codewords of a code and vice versa. The
array is linear if and only if the corresponding code is linear [ 16 ,
Chapter 4]. This relationship allows to employ classical code
constructions to construct orthogonal arrays.
In this work we propose to generate sampling matrices A ∈
Z m × n
σ by selecting m ≤ M = λσ k rows at random from an
orthogonal array OA( λσ 4 , n , σ , 4) of strength k = 4 and with
n factors. Intuitively, m log 2 ( M ) bits are then required to specify
such a matrix A . For k = 4, a classical lower bound due to R ao
[ 42 ] demands
M = λσ 4 ≥ 1 + n +  n
2  =  ( n 2 ). (20)
Arrays that saturate this bound are c alled tight (or complete). In
summary, an order of s log 2 ( n ) bits are required to sample a m × n
matrix A with m ≥ Cs log( n ) rows according to this procedure.
Strength- 4 Constructions: For compressed sensing applications,
we want arrays with a large number of factors n since this
corresponds to the ambient dimension of the sparse vectors
to recover. On the ot her hand the run size M should scale
“moderately” to describe the random matrices only wit h few
bits. Most constructions use an existing orthogonal array as
a seed to construct larger arrays. Known binary arrays of
strength 4 are for example t he simple array OA(16, 5, 2, 4), or
OA(80, 6, 2, 4). P at [ 43 ] proposes an algorithm that uses a linear
orthogonal array OA( N , n , σ , k ) as a seed to construct a linear
4 For example http://neilsloane.com/oadir/ or http://pietereendebak.nl/oapage/
orthogonal array OA( N 2 , n 2 + 2 n , σ , k ). This procedure may then
be iterated.
Numerical results for orthogonal arrays: Figure 1 summarizes
the empirical performance of b asis pursuit (1) from independent
orthogonal array measurements. We consider real-valued signals
x and quantify the performance in terms of the normalized
ℓ 2 -recovery error (NMSE) k z ♯ − x k ℓ 2 / k x k ℓ 2 where z ♯ is the
solution of (1). To construct the orthogonal array, algorit hm
[ 43 ] is applied twice OA(16, 5, 2, 4) → OA(256, 35, 2, 4) →
OA(65536, 1295, 2, 4). The 323 rows are uniformly samples from
this array, i.e., the sa mpling matrix A has ± 1 entries (instead
of binary entries) and size 323 × 1295. But note that, in the
case of non-negative sparse vectors, the corresponding binary
0/1-matrices may be used instead directly to re cover with non-
negative least-squares [ 44 ]. The sparsity of the unknown vector
has been varied between 1 ... 180. For e ach sparsity many
experiments are performed to compute NMSE. In each run, the
support of the unknown vector has been chosen uniformly at
random and the values are independent instances of a st andard
Gaussian random variable. For comparison, we ha ve also
included the corresponding performances of a generic sampling
matrix (signed Bernoulli) of the same size. Numeric ally, the
partially derandomized orthogonal array construction achieves
essentially the same performance as its generic counterpart.
Numerical results for the Alltop design: Figure 1 shows
the NMSE achieved for measurement matrices b ased on
subsampling from an Alltop-design (6). The data is obtained
in the same way as above, but the sparse vectors are
generated as iid. complex-normal distributed on the support.
For comparison the results for a (complex) standard Gaussian
sampling matrix are included as well. Again, the performance
of random Alltop-design measurements essentially matches its
generic (Gaussian) counterpart.
5. ADDITIONAL PROOFS
5.1. Pr oof of Theor em 7
The proof is based on rather straig htforward modifications of
Tropp’ s proof for Mendelson’ s small ball method [ 37 ]. Let a ∈ C n
be a complex-valued random vector. Suppose that a 1 , ... , a m ∈
C n are independent copies of a and let A be the m × n matrix
whose i -th row is given by a i . The goal is to obtain a lower bound
on inf z ∈ E k Az k ℓ 2 , where E ⊂ C n is an arbitrary set. First, note
that ℓ 1 and ℓ 2 norms on R 2 m are related via k v k ℓ 1 ≤ √ 2 m k v k ℓ 2 .
For any z ∈ E this relation ensures
k Az k ℓ 2 = v
u
u
t
m
X
i = 1 |h a i , z i| 2 = v
u
u
t
m
X
i = 1
Re ( h a i , z i ) 2 +
m
X
i = 1
Im ( h a i , z i ) 2
≥ 1
√ 2 m m
X
i = 1 | Re( h a i , z i ) | +
m
X
i = 1 | Im( h a i , z i ) | !
= 1
√ 2 m
m
X
i = 1  | Re( h a i , z i ) | + | Im( h a i , z i ) |  .
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Jung et al. Partially Derandomizing Compressed Sensing
FIGURE 1 | ( Left ) Performance of basis pursuit for m = 323 and n = 1295 (real-valued signals) from random orthogonal array measur ements and their generic
counterpart (signed Bernoulli). ( Right ) Performance of basis pursuit for m = 255 and n = 1021 (complex-valued s ignals) from random time-fr equency shifted Alltop
sequences and their generic counterpart (standard complex Gaussian vectors).
Next, fix ξ > 0 and introduce the indicator function I { x ≥ ξ }
which obeys x ≥ ξ I { x ≥ ξ } for all x ≥ 0. Consequently,
k Az k ℓ 2 ≥ ξ
√ 2 m
m
X
i = 1  I  | Re( h a i , z i ) | ≥ ξ  + I  | Im( h a i , z i ) | ≥ ξ  .
(21)
Also, note that the expect ation value of each summand obeys
E  I  | Re( h a i , z i ) | ≥ ξ  + E  I  | Im( h a i , z i ) | ≥ ξ 
= Pr  | Re( h a i , z i ) | ≥ ξ  + Pr  | Im( h a i , z i ) | ≥ ξ 
≥ Pr  | Re( h a i , z i ) | ≥ ξ ∨ | Im( h a i , z i ) | ≥ ξ 
≥ Pr h |h a i , z i| ≥ √ 2 ξ i ,
according to the union bound. The last line follows from a simple
obser vation. Let z = a + ib be a complex number. Then, | z | =
√ a 2 + b 2 ≥ √ 2 ξ necessarily implies either | a | ≥ ξ , or | b | ≥ ξ
(or both). Next, define
Q 2 ξ ( z ) = Pr  | Re( h a i , z i ) | ≥ 2 ξ  + Pr  | Im( h a i , z i ) | ≥ 2 ξ 
and note that the estimate from above ensures
inf
z ∈ E Q 2 ξ ( z ) ≥ inf
z ∈ E Pr  |h a , z i| ≥ 2 3 / 2 ξ  = Q 2 3 / 2 ξ ( a , E ). (22)
Adding and subtracting ξ √ m / 2 Q 2 ξ ( z ) to Equation (21) and
taking the infimum yields
inf
z ∈ E k Az k ℓ 2
≥ inf
z ∈ E  ξ r m
2 Q 2 ξ ( z ) − ξ r m
2 Q 2 ξ ( z )
+ ξ
√ 2 m
m
X
i = 1  I  | Re( h a i , z i ) | ≥ ξ  + I  | Im( h a i , z i ) | ≥ ξ  !
≥ ξ r m
2 Q 2 3 / 2 ξ ( a , E ) − ξ
√ 2 m sup
z ∈ E mQ 2 ξ ( z )
−
m
X
i = 1
( I  | Re( h a i , z i ) | ≥ ξ  + I  | Im( h a i , z i ) | ≥ ξ  ) ! . (23)
Here we have applied Equation (22) to bound the contribution of
the first term. Since Q 2 ξ ( z ) features both a real and imaginary part
and we can split up the remaining supremum accordingly. The
suprema over real and complex parts individually correspond to
R ( E , a ) = sup
z ∈ E
m
X
i = 1  Pr  | Re( h a i , z i ) | ≥ 2 ξ  − I  | Re( h a i , z i ) | ≥ ξ  ,
I ( E , a ) = sup
z ∈ E
m
X
i = 1  Pr  | Im( h a i , z i ) | ≥ 2 ξ  − I  | Im( h a i , z i ) | ≥ ξ  ,
and the vectors a 1 , ... , a m are independent copies of a single
random vector a ∈ C n . The bounded difference inequality [ 45 ,
section 6.1] asserts that both expressions concentrate around
their expectation. More pre cisely, for any t > 0
Pr  R ( E , a ) ≥ E R ( E , a ) + t √ m  ≤ e − t 2 / 2 and
Pr  I ( E , a ) ≥ E I ( E , a ) + t √ m  ≤ e − t 2 / 2 .
Therefore, the union bound grants a transition from R ( E , a ) +
I ( E , a ) to E R ( E , a ) + E I ( E , a ) + 2 √ mt with probability at least
1 − 2e − t 2 / 2 . These expectation values can be furt her simplified.
Define the soft indicator function
ψ ξ ( s ) = 




0 | s | ≤ ξ ,
( | s | − ξ ) /ξ ξ ≤ | s | ≤ 2 ξ ,
1 | s | ≥ 2 ξ
which admits the following bounds: I { | s | ≥ 2 ξ } ≤ ψ ξ ( s ) ≤
I { | s | ≥ ξ } for all s ∈ R . Moreover , ξ ψ ξ ( s ) is a contraction, i.e., a
Frontiers in Applied Mathematics and Statistics | www .frontiersin.org 10 June 2019 | V olume 5 | Article 26

Jung et al. Partially Derandomizing Compressed Sensing
real-valued function with Lips chitz constant one that also obeys
ξ ψ ξ (0) = 0. Rademacher symmetrization [ 3 , Lemma 8.4] and the
R ademacher comparison principle [ 46 , Equation (4.20)] yield
E R ( E , a ) = E sup
z ∈ E
m
X
i = 1  EI  | Re( a i , z ) | ≥ 2 ξ  − I  | Re( h a i , z i ) | ≥ ξ 
≤ E sup
z ∈ E
m
X
i = 1  E ψ ξ (Re( h a i , z i )) − ψ ξ (Re( h a i , z i ) 
≤ 2 E
m
X
i = 1
ǫ i ψ ξ (Re( h a i , z i ) ≤ 2
ξ
E sup
z ∈ E
m
X
i = 1
ǫ i Re( h a i , z i )
≤ 2 √ m
ξ
E sup
z ∈ E | h z , h i | ,
where h = 1
√ m P m
i = 1 ǫ i a i ∈ C n . A completely analogous
bound holds true for E I ( E , a ). Inserting both bounds into
Equation (23) establishes
inf
z ∈ E k Az k ℓ 2 ≥ ξ r m
2 Q 2 3 / 2 ξ − ξ
√ 2 m  4 √ m
ξ
E sup
z ∈ E |h z , h i| + 2 √ mt 
= ξ r m
2 Q 2 3 / 2 ξ − 2 3 / 2 E sup
z ∈ E |h z , h i| − √ 2 ξ t
with probability at least 1 − 2e − t 2 / 2 . Setting W m ( E , z ) =
E sup z ∈ E |h z , h i| est ablishes the claim.
5.2. Pr oof of Lemma 2
The inclusion T s ⊂ 2conv( 6 s ) remains valid in the complex case.
Moreover , e very z ∈ conv( 6 s ) necessarily obeys
max
z ∈ conv( 6 s ) k z k ℓ 1 ≤ max
z ∈ 6 s k z k ℓ 1 = √ s ,
because the maximum value of a convex function is achieved at
the boundary. Hoelder’ s inequality therefore implies
W m ( a , T s ) = E sup
z ∈ T s |h z , h i| ≤ 2 E sup
z ∈ conv( 6 s ) k z k ℓ 1 k h k ℓ ∞
≤ 2 √ s E k h k ℓ ∞ , (24)
where h = 1
√ m P m
i = 1 ǫ i a i ∈ C n . Moreover ,
E k h k ℓ ∞ = E max
1 ≤ k ≤ n |h e k , h i| ≤ E max
1 ≤ k ≤ n   Re( h e k , h i )  
+ E max
1 ≤ k ≤ n | Im( h e k , h i|
and we may bound both expressions on the r.h.s. independently.
For the first term, fix θ > 0 and use Jensen’ s inequality (the
logarithm is a concave function) to obtain
E max
1 ≤ k ≤ n | Re( h e k , h i ) | = E max
1 ≤ k ≤ n max
σ =± σ Re( h e k , h i )
≤ 1
θ log  E exp  max
1 ≤ k ≤ n max
σ =± θ σ Re ( h e k , h i )  .
Monotonicity and non-negativity of the exponential function
then imply
E exp  max
1 ≤ k ≤ n max
σ =± θ σ Re ( h e k , h i ) 
≤
n
X
k = 1 X
σ =±
E exp ( θ σ Re ( h e k , h i ))
=
n
X
k = 1 X
σ =±
m
Y
i = 1
E exp  θ σ
√ m ǫ i Re ( h e k , a i i )  ,
where we have also used that all ǫ i ’ s and a i ’ s are independent.
The remaining moment generating functions can be bounded
individually. Fix 1 ≤ k ≤ n , σ ∈ { ± 1 } and
1 ≤ i ≤ m and exploit the R ademacher randomness
to infer
E exp  θ σ
√ m ǫ i Re ( h e k , a i i )  = E cosh  θ σ
√ m Re ( h e k , a i i ) 
≤ E exp  θ 2 σ 2
2 m Re ( h e k , a i i ) 2 
= E exp  θ 2
2 m Re ( h e k , a i i ) 2  ,
because σ 2 = 1. Incoherence moreover ensures (Re( h e k , a i i ) 2 ≤
|h e k , a i i| 2 ≤ 1. This ensures that the remaining expect ation
value is upper-bounded by exp  θ 2
2 m  . Inserting these individual
bounds into the original expression yields
E max
1 ≤ k ≤ n   Re( h e k , h i )   ≤ 1
θ log 

n
X
k = 1 X
σ =±
m
Y
i = 1
E exp  θ σ
√ m ǫ i Re  h e k , a i i   

≤ 1
θ log 

n
X
k = 1 X
σ =±
m
Y
i = 1
exp θ 2
2 m ! 

= 1
θ log 2 n exp θ 2
2 !! = log(2 n )
θ + θ
2
for any 0 < θ ≤ √ 2 m . Choosing θ = p 2 log(2 n )
minimizes this upper bound and is feasible, by assumption. A
completely analogous bound can be derived for the expe cted
maximum absolute value of the imaginary part. Combining
both yields
E k h k ℓ ∞ ≤ p 2 log(2 n ) + p 2 log(2 n ) = 2 p 2 log(2 n )
and inserting this bound into (24) ensures
W m ( a , T s ) ≤ 4 p 2 s log(2 n ).
AUTHOR CONTRIBUTIONS
RK developed the te chnical aspects of the paper. PJ is
responsible for the numerical aspects and t he discussion of
Frontiers in Applied Mathematics and Statistics | www .frontiersin.org 11 June 2019 | V olume 5 | Article 26

Jung et al. Partially Derandomizing Compressed Sensing
orthogonal arrays. Some of the main conceptual ide as are due
to DM. These in particular include the idea of employing
orthogonal arrays to partially derandomize signed Bernoulli
matrices and the insight that a partial de-randomization of
Gordon’ s esc ape through a mesh is essential for achieving
the results. All authors contributed to the introduction and
general presentation.
FUNDING
PJ is supported by the DFG grant JU 2795/3 and D AAD
grant 57417688. RK was in part supported by Joel A.
Tropp under the ONR Award No. N00014-17-12146 and also
acknowledges funding provided by the Institute of Quantum
Information and Matter , an NSF Physics Frontiers Center (NSF
Grant PHY -1733907). DM was partially supported by AFOSR
F A9550-18-1-0107, NSF DMS 1829955, and the Simons Institute
of the Theory of Computing. We acknowledge support by the
German Research Foundation and the Open A ccess Publication
Fund of TU Berlin.
ACKNOWLEDGMENTS
The authors want to thank David Gross for providing the original
ideas that ultimately led to this public ation.
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Conflict of Interest Statement: The authors declare t hat the research was
conducted in the absence of any commercial or financia l relationships that could
be construed as a potential conflict of interest.
Copyright © 2019 Jung, Kueng and M ixon. This is an open-access article distributed
under the terms of the Cre a tive Commons A ttr ibution License (CC BY). The
use, distribution or reproduction in ot her forums is permitted, provided the
original author(s) and the copyright owner(s) are cred ited and tha t the original
publica tion in th is journal is cited, in accordance wit h accepted academic pr actice.
No use, distribution or reproduction is permitted wh ich doe s not comply with the se
terms.
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