McShane-Whitney extensions for fuzzy Lipschitz maps

McShane-Whitney extensions for fuzzy Lipschitz maps E. Jim´enez Fern´andez Departamento de Econom´ıa Universitat Jaume I, Campus del Riu Sec, s/n 12071 Castell´o de la Plana. Spain J. Rodr´ıguez-L´opez Instituto Universitario de Matem´atica Pura y Aplicada Universitat Polit`ecnica de Val`encia, Camino de Vera s/n 46022 Valencia. Spain E. A. S´anchez-P´erez Instituto Universitario de Matem´atica Pura y Aplicada Universitat Polit`ecnica de Val`encia, Camino de Vera s/n 46022 Valencia. Spain. Tel.0034-963877660, Fax. 0034-963877669 Abstract We present a McShane-Whitney extension theorem for real-valued fuzzy Lipschitz maps defined between fuzzy metric spaces. Motivated by the potential applications of the obtained results, we generalize the mathematical theory of extensions of Lipschitz maps to the fuzzy context. We develop the problem in its full generality, explaining the similarities and differences with the classical case of extensions on metric spaces. Keywords: Fuzzy metric, fuzzy Lipschitz map, extension, McShane-Whitney Theorem 2010 MSC: 26A16, 54E70, 54C20. ∗Corresponding author Email addresses: [email protected] (E. Jim´enez Fern´andez), [email protected] (J. Rodr´ıguez-L´opez), [email protected] (E. A. S´anchez-P´erez) Preprint submitted to Fuzzy sets and systems May 29, 2020 1. Introduction and basic definitions The so called McShane-Whitney Theorem is a classical result which establishes that, given a real valued function on a subspace of a metric space, it can always be extended to the whole space preserving the Lipschitz constant. This theoretical result can be easily proved using a constructive procedure: the extension is given by concrete known formulas involving a supremum— the McShane formula— or an infimum—the Whitney formula—. The aim of the present paper is to obtain a similar result for the case of fuzzy Lipschitz maps on fuzzy metric spaces as a contribution to the general theory of these spaces, that could find applications in different fields. In the case of finite spaces, the formulas cited above provide effective computational tools for getting these Lipschitz extensions, and allow to develop useful applications. Among them, we are particularly interested in the construction of algorithms for artificial intelligence. A classic and simple model for some machine learning developments is based on controlled extensions of real valued functions that act in subspaces of metric spaces. For example, suppose we have a metric model for a given problem —i.e., the individuals in the model are elements of a subspace (S0, d) of a metric space (X, d)—, and assume that the learning tool is a real function I. A basic reinforcement learning scheme is then provided by an increasing subset of metric subspaces S0⊆S1⊆S2· · · ⊆ Xand the corresponding extensions of the index I0, I1, I2,· · · ,which improve as the process progresses, since we incorporate new information about the system at each step. Similar arguments provide standard tools in distance learning (see for example [8]). This and other issues on the relation among machine learning and Lipschitz functions are of current interest; the reader can find concrete information about in [1, 8, 9, 19, 32, 24] and the references therein. Concretely, the authors of the present paper are involved in the development of algorithms for reinforcement learning based on the basic idea explained above. Although this is still an open project, some results on this research have been already published and can be found in [5]. On the other hand, nowadays it seems to be an objective of the scientific community to enrich the set of mathematical tools for machine learning by introducing increasingly sophisticated fuzzy methods. The nature of the topic itself and the multiple applications of the different areas of the machine learning motivated an early introduction of techniques of fuzzy mathematics in these areas, both to justify theoretical developments and for concrete 2 applications (see for example [18, 36]). Motivated in part by this situation, the second objective of this work is to merge the theoretical contexts of Lipschitz maps and fuzzy metrics to facilitate the construction of new tools in machine learning. In particular, our aim is to adapt a central theorem of the theory of Lipschitz functions —the McShane-Whitney extension theorem for real valued Lipschitz functions— to the framework of fuzzy metric spaces (see [2, 4, 31, 34]). Let us recall some fundamental definitions. A map f: (X, d)→(Y, q) between metric spaces is said to be Lipschitz if there is a constant K > 0 such that q(f(x), f(y)) ≤Kd(x, y), x, y ∈X. The Lipschitz constant of fis the infimum of all the constants Ksatisfying the inequality. It is well known that Lipschitz real functions acting in metric subspaces can always be extended to the whole metric space preserving the Lipschitz constant. Indeed, the so called McShane-Whitney Theorem establishes that if Sis a subspace of a metric space (X, d) and f:S→Ris a Lipschitz function with Lipschitz constant K, there exists an extension to X with the same Lipschitz constant. In fact, a lot of extensions are available. For example, the following function fM(x) := sup s∈S {f(s)−K d(s, x)}, x ∈X, that is called the McShane extension of f, gives one of them, and also the Whitney extension defined as fW(x) := inf s∈S{f(s) + K d(s, x)}, x ∈X. The aim of the present paper is to extend this result to the setting of real-valued fuzzy Lipschitz functions f: (X, M, ∗)→(R, N, ~) between fuzzy metric spaces as introduced in [37]: fis fuzzy Lipschitz if for every t > 0,the supremum sup x6=y 1−N(f(x), f(y), t) 1−M(x, y, t) is finite. However, in order to extend the result in a coherent way, we will first consider functions f:X×(0,∞)→R,that is, we will consider explicitly the dependence of the original map on the parameter t. Our main theoretical result —Corollary 22— gives an extension theorem for fuzzy Lipschitz maps. 3 If (X, M, ∗) is a fuzzy metric space and (R, N, ~) is what we call a Euclidean fuzzy metric space, suppose S⊆Xand f: (S, M, ∗)×(0,∞)→(R, N, ~) is a fuzzy Lipschitz map. Then, under some requirements on M, f can be extended to a fuzzy Lipschitz map f:X×(0,∞)→R. Although this result seems to be the most adequate generalization after understanding the nature of fuzzy Lipschitz maps, from the point of view of the applications seems to be relevant the case of extending fuzzy Lipschitz functions non-depending on tacting in Sto fuzzy Lipschitz functions nondepending on tacting in X. We will show that this is also possible using our construction. We intend in this way to provide new tools for opening the door to a new methodological approach to machine learning, including fuzzy notions in the design of algorithms of artificial intelligence that use Lipschitz extensions. Following the principles underlying the fuzzy philosophy, our technique allows to modulate, using the “t” parameter, the level of uncertainty under which the function fulfills the Lipschitz inequality. This idea is automatically translated into the context of machine learning, thus allowing to build in future research work a technique to introduce probabilistic interpretations into forecasting methods based on reinforcement learning such as those that can be found in [5]. The structure of the paper is as follows. We begin recalling in Section 2 some necessary concepts about fuzzy metric spaces. Our technique for extending a fuzzy Lipschitz map is based on a method for constructing a family of metrics from a fuzzy metric. The problem of obtaining a compatible metric from a fuzzy metric has already treated in the literature [6, 12, 26, 27]. We summarize some results in Section 3 and provide a new method in Lemma 7. Section 4 is devoted to introduce the concept of fuzzy Lipschitz function as considered in [37]. For completeness, we provide in this section some results about the relationship of this concept with other notions of Lipschitzness in the fuzzy framework coming from fuzzy contractiveness notions. Finally, in Section 5 we obtain a McShane-Whitney extension theorem for fuzzy Lipschitz maps taking values in what we call a Euclidean fuzzy metric space. 2. Fuzzy metric spaces The origins of fuzzy metric spaces are due mainly to Menger [21] (see also [29]) who introduced the concept of probabilistic metric space which gives a probabilistic interpretation of the distance between two points by assigning 4 a distribution function with every pair of elements. This concept has evolved in the last decades to various concepts of fuzzy metrics. One of the most widespread notion is that due to George and Veeramani [10] (see also [11]) and this will be the notion that we will work with. Let us recall its definition. Definition 1 ([10]).A triple (X, M, ∗)is called a fuzzy metric space if Xis a nonempty set, ∗is a continuous t-norm, and Mis a fuzzy set on X2×(0,∞) such that for each x, y, z ∈Xand t, s > 0, (1) M(x, y, t)>0, (2) M(x, y, t)=1if and only if x=y, (3) M(x, y, t) = M(y, x, t), (4) M(x, y, t)∗M(y, z, s)≤M(x, z, t +s),and (5) M(x, y, ·) : (0,∞)→[0,1],is continuous. The pair (M, ∗)is said to be a fuzzy metric on X. Example 2 ([10]).Given a metric space (X, d), let Mdbe the fuzzy set on X2×(0,∞)defined by Md(x, y, t) = t t+d(x, y). For every continuous t-norm ∗,(Md,∗)is a fuzzy metric on Xwhich is called the standard fuzzy metric induced by d. The class of stationary fuzzy metric spaces was introduced in [15] when the authors were studying completions of fuzzy metric spaces. We extend this definition as follows. Definition 3 (see [15]).A fuzzy metric space (X, M, ∗)is said to be eventually stationary (or (M, ∗)is a eventually stationary fuzzy metric on X) if we can find t0>0such that the function M(x, y, ·) : [t0,+∞)→[0,1] is constant for every x, y ∈X. (M, ∗)is said to be stationary if M(x, y, ·)is constant for every x, y ∈X. Another class of fuzzy metric spaces which includes the stationary fuzzy metric spaces are the so-called strong fuzzy metric spaces introduced and studied in [12]. 5 Definition 4 ([12]).A fuzzy metric space (X, M, ∗)is said to be strong if M(x, y, t)∗M(y, z, t)≤M(x, z, t) for all x, y, z ∈Xand all t > 0. It is obvious that every stationary fuzzy metric space is strong. Furthermore, (M, ∗) is strong if and only if {(Mt,∗) : t > 0}is a family of stationary fuzzy metrics on Xassociated to M, where Mt:X×X×(0,+∞)→[0,1] is given by Mt(x, y, s) = M(x, y, t),for all x, y ∈Xand all s > 0. 3. Metrics from fuzzy metrics George and Veeramani [10] showed that every fuzzy metric (M, ∗) on a nonempty set Xgenerates a Hausdorff topology τ(M) on X. Moreover, this topology is metrizable [14]. This naturally leads to the problem of constructing in an easy way a metric don Xcompatible with the topology τ(M) such that we can infer results for the fuzzy metric (M, ∗) from classic results about metrics. In this way Radu constructed in [26] such a metric dwhich has been successfully applied to prove fixed point theorems for complete fuzzy metric spaces from classic results in the context of metric spaces. This construction was improved and modified in [6, 27] which allowed the authors to prove several fixed point theorems for different types of contractions in the context of fuzzy metric spaces. Their main tool is the construction of a metric from a fuzzy metric such that it preserves a certain contraction notion. In a different way, Gregori, Morillas and Sapena [12] developed a method for constructing a metric from a strong fuzzy metric as follows. Proposition 5 ([12]).Let (X, M, ∗)be a strong fuzzy metric space such that ∗≥∗ L, where ∗ L is the Lukasiewicz t-norm. Let {Mt:t > 0}be the family of stationary fuzzy metrics associated to (M, ∗). Then (i) {dM t:t > 0}is a family of metrics on Xwhere dM t(x, y) = 1−M(x, y, t) for all x, y ∈Xand all t > 0. (ii) d= supt>0dM tis a metric on Xsuch that τ(Mt)⊆τ(d). We notice that part (i) of Proposition 5 provides the following characterization of strong fuzzy metrics with respect to the Lukasiewicz t-norm ∗ L. 6 Proposition 6. A fuzzy metric space (X, M, ∗ L)is strong if and only if {dM t:t > 0}is a family of metrics on Xwhere dM t(x, y)=1−M(x, y, t)for all x, y ∈Xand all t > 0. Proof. Necessity follows from Proposition 5. For the converse, fix t > 0. Then given x, y, z ∈Xwe have that dM t(x, y) + dM t(y, z)≥dM t(x, z) 1−M(x, y, t)+1−M(y, z, t)≥1−M(x, z, t) 1−M(x, y, t)−M(y, z, t)≥ −M(x, z, t) M(x, y, t) + M(y, z, t)−1≤M(x, z, t) max{M(x, y, t) + M(y, z, t)−1,0} ≤ M(x, z, t) M(x, y, t)∗ L M(y, z, t)≤M(x, z, t). Since tis arbitrary then (M, ∗ L) is strong. Observe that the method provided by Proposition 5 of constructing a metric dfrom a fuzzy metric (M, ∗) is only valid when (M, ∗) is strong. Moreover ddoes not preserve important properties of Msince, for example, τ(d)6=τ(M) in general. Next we present a new method for constructing a family of metrics from a fuzzy metric which is better behaved for our purposes. This method is based on an standard “convexification” process inspired by classical metrization theorems [35, Theorem 23.4] and the obtention of a subadditive function by means of the inf-convolution [7] (it is a particular case of the metrics dε considered in [20] when ε=∞). Lemma 7. Let (X, M, ∗)be a fuzzy metric space, and let ϕ: [0,1) →[0,1) be an increasing function such that ϕ−1(0) = {0}. Fix t > 0and consider the function pt:X×X→R+defined by pt(x, y) := inf nn X i=1 ϕ1−M(xi, xi+1, t):x1=x, xn+1 =y, xi∈Xo for every x, y ∈X. Then (i) ptis a pseudo metric on X, and 7 (ii) if (x, y)7→ ϕ1−M(x, y, t)satisfies the triangular inequality, then pt(·,·) = ϕ1−M(·,·, t), and it defines a metric. Proof. (i) The function is clearly symmetric, due to the symmetry of M. On the other hand, a direct calculation using the properties of the infimum gives the triangular inequality. (ii) Note that, if ϕ1−M(·,·, t)satisfies the triangular inequality then it is a pseudo metric. Moreover given x, y ∈Xwe have that for every x1, . . . , xn+1 ∈Xsuch that x1=x, xn+1 =ythen ϕ1−M(x, y, t)≤ n X i=1 ϕ1−M(xi, xi+1, t), and so the infimum pt(x, y) coincides with ϕ1−M(x, y, t).On the other hand, we have that ϕ1−M(x, y, t)= 0 if and only if 1 −M(x, y, t) = 0, and by the definition of fuzzy metric this happens if and only if x=y. This gives (ii). In the literature we can find a lot of examples of fuzzy metrics but many of them are constructed starting from a classic metric [13, 28]. We next show that the above Lemma allows to recover the metric from the fuzzy metric in some cases. Example 8 (cf. [12, Example 25]).Let (X, d)be a metric space and let us consider, following [13, Example 5], the fuzzy metric (M, ·)on Xgiven by M(x, y, t) := e(−d(x,y)/g(t)), x, y ∈X, t > 0, where g:R+→R+is an increasing continuous function. Let us consider ϕ: [0,1) →[0,1) given by ϕ(x) = −log(1 −x). It is obvious that ϕ−1(0) = {0}and that ϕis subadditive since it is concave. Following Lemma 7, given t > 0we can construct the following metric pton X: pt(x, y) := ϕ(1 −M(x, y, t)) = −log(1 −1 + e−d(x,y)/g(t)) = d(x, y) g(t). 8 Observe that if (X, M, ∗) is a strong fuzzy metric space, then by Proposition 5 {dM t:t > 0}is a family of metrics on X. Thus, if ϕis a metric preserving function [7] then ϕ(1 −M(·,·, t)) is also a metric on Xfor all t > 0 so we obtain (ii) of the previous result. We next present an example borrowed from [17], where Lemma 7 can be applied but Proposition 5 cannot. Example 9 ([17, Example 2]).Let X={a, b, c}and M:X×X×(0,+∞)→ [0,1] given by M(x, y, t) = M(y, x, t) =          1if x=y 2t+ 1 2t+ 2 if x=aor x=band y=c t t+ 2 if x=a, y =b . It was proved in [17] that (M, ∗ L)is a fuzzy metric on Xwhich is not strong. Notice that by Proposition 6, there has to be t > 0such that dM tis not a metric. In fact, dM tis not a metric for every t > 0,since dM t(a, b) = 1 −M(a, b, t) = 2 t+ 2 6≤ dM t(a, b) + dM t(b, c) = 1 −M(a, b, t)+1−M(b, c, t) = 2 2t+ 2. Hence, we cannot apply Proposition 5 to construct a metric from (M, ∗ L). Nevertheless, Lemma 7 allows this construction. If we consider the function ϕ: [0,1) →[0,1) given by ϕ(x) = xthen this lemma provides a family of metrics {pt:t > 0}where pt(x, y) =    2 2t+ 2 if x6=y 0if x=y for every t > 0. 4. Fuzzy Lipschitz maps Since our aim is to obtain a McShane-Whitney extension theorem in the context of fuzzy metric spaces, we must look for an appropriate notion of Lipschitz function in this context. Hence it is natural to analyze the different notions of fuzzy Lipschitz maps that have been proposed in the literature. We collect some of them in the following definition. 9 Example 20 (cf. [13, Example 6]).Let us consider two functions φ, g : R+→R+such that φis strictly increasing, subadditive, bounded by k > 0 and φ−1(0) = {0},meanwhile gis decreasing, bounded by 1 kand greater than 0. Then it is easy to see that (M, ∗ L)is a Euclidean fuzzy metric on Rwhere M(x, y, t) = 1 −φ(|x−y|)g(t). We only check the triangle inequality. Let x, y, z ∈Rand t, s > 0. Since φ is subadditive and strictly increasing then φ(|x−y|)≤φ(|x−y|+|y−z|)≤ φ(|x−z|) + φ(|z−y|). Using that gis decreasing we have that φ(|x−y|)g(t+s)≤φ(|x−z|)g(t+s) + φ(|z−y|)g(t+s) ≤φ(|x−z|)g(t) + φ(|z−y|)g(s). That is, 1−φ(|x−y|)g(t+s)≥1−φ(|x−z|)g(t)−φ(|z−y|)g(s), what gives M(x, y, t +s)≥M(x, z, t)∗ L M(z, y, s). Let (X, M, ∗) be a fuzzy metric space and consider a Euclidean fuzzy metric space (R, Nφ,g,~). If f: (X, M, ∗)×(0,∞)→(R, Nφ,g,~) is a fuzzy Lipschitz map then for each t > 0 there is a positive real number K(t) such that 1−Nφ,g(f(x, t), f(y, t), t)≤K(t)1−M(x, y, t) φ(|f(x, t)−f(y, t)|)g(t)≤K(t)1−M(x, y, t) for all x, y ∈X. By composing with the increasing function φ−1we obtain |f(x, t)−f(y, t)| ≤ φ−1K(t) g(t)1−M(x, y, t), x, y ∈X. Due to the fact that | · | is a norm on R,we automatically get from the inequality above that for each t > 0,the inequality |f(x, t)−f(y, t)| 16 ≤inf nnn X i=1 φ−1K(t) g(t)1−M(xi, xi+1, t), x1=x, x2, ..., xn, xn+1 =y, xi∈Xo, where x, y ∈S. We will write dt,f (x, y) for the second term of this inequality, that is clearly a pseudometric on Xfor each t > 0.Therefore, we obtain in this way a typical Lipschitz inequality for the function f(·, t) : X→R, where the spaces Xand Rare assumed to be pseudometric spaces with the explained pseudometrics. This construction gives the proof of the following extension theorem. Note that we have to take the elements xiin all the set Xin the definition of dt,f and not only the ones of S. Theorem 21. Let (X, M, ∗)be a fuzzy metric space and (R, Nφ,g,~)be a Euclidean fuzzy metric space. Let S⊆Xand suppose that f: (S, M, ∗)× (0,∞)→(R, Nφ,g,~)is a fuzzy Lipschitz map. Then the following statements are equivalent. (i) f(·, t):(S, dt,f )→Ris nonexpansive for all t > 0. (ii) The function fcan be extended as a fuzzy Lipschitz map to X×(0,∞). Proof. (i) ⇒(ii) Fix t > 0. By assumption, the function f(·, t) : (X, dt,f )→ Ris Lipschitz so, for example, the McShane formula provides an extension ˆ f(·, t) of f(·, t) to all the set X. Therefore, we have in particular that for each x, y ∈X, |ˆ f(x, t)−ˆ f(y, t)| ≤ dt,f (x, y)≤φ−1K(t) g(t)1−M(x, y, t), and so 1−Nφ,g(ˆ f(x, t),ˆ f(y, t), t) = φ(|f(x, t)−f(y, t)|)g(t)≤K(t)1−M(x, y, t). Since t > 0 is arbitrary, ˆ fis a fuzzy Lipschitz map. (ii) ⇒(i) Suppose that fallows an extension ˆ fto the whole Xwhich is fuzzy Lipschitz. Then given t > 0 we have that 1−Nφ,g(ˆ f(x, t),ˆ f(y, t), t)≤K(t)(1 −M(x, y, t)) φ(|ˆ f(x, t)−ˆ f(y, t)|)g(t)≤K(t)1−M(x, y, t) |ˆ f(x, t)−ˆ f(y, t)| ≤ φ−1K(t) g(t)1−M(x, y, t)(1) 17 for all x, y ∈X. Fix now x, y ∈S. Given n∈Nand {x1, . . . , xn+1} ⊆ Xsuch that x1=xand xn+1 =ywe can use inequality (1) to obtain |f(x, t)−f(y, t)|=|ˆ f(x, t)−ˆ f(y, t)| ≤ n X i=1 |ˆ f(xi, t)−ˆ f(xi+1, t)| ≤ n X i=1 φ−1K(t) g(t)1−M(xi, xi+1, t). Since this holds for all n∈Rand for all xi0s, we obtain that for x, y ∈S, |f(x, t)−f(y, t)| ≤ dt,f (x, y). Therefore, (i) holds. The following result will be the main tool for working in concrete applications. Corollary 22. Let (X, M, ∗)be a fuzzy metric space and (R, Nφ,g,~)be a Euclidean fuzzy metric space. Let S⊆Xand suppose that f: (S, M, ∗)× (0,∞)→(R, Nφ,g,~)is a fuzzy Lipschitz map with extended dilation K(t). Assume also that for every t > 0the map ρt:X×X→R+given by ρt,f (x, y) := φ−1K(t) g(t)1−M(x, y, t), x, y ∈X, is a metric on X. Then the function fcan be extended as a fuzzy Lipschitz map to X×(0,∞). Proof. If the above defined map is a metric on X, then we clearly have that it coincides with dt,f .Then we directly get (i) in Theorem 21, and the result holds. Remark 23. Observe that under the hypotheses of the previous corollary, we can provide directly an extension of fby using: •the following modified McShane formula fM(x, t) = sup s∈Sf(s)−φ−1K(t) g(t)(1 −M(x, s, t)) for all x∈Xand all t > 0. 18 •the following modified Whitney formula fW(x, t) = inf s∈Sf(s) + φ−1K(t) g(t)(1 −M(x, s, t)) for all x∈Xand all t > 0. Remark 24. The main problem to define the extension of a fuzzy Lipschitz function is that the Lipschitz extension obtained by our method depends on the parameter t. However, under some assumptions, it can be proved that we can obtain an extension not depending on t. For example, let (X, M, ∗),(R, Nφ,g,~)be two stationary fuzzy metric spaces such that (Nφ,g,~)is a Euclidean fuzzy metric. Given S⊆Xand a fuzzy Lipschitz function f: (S, M, ∗)→(Y, Nφ,g,~),then fis a stationary fuzzy Lipschitz function. Furthermore, if φ−1is subadditive then it is easy to see [7] that the function ρtconsidered in the previous corollary is a metric on Xwhich does not depend on t, so the function fcan be extended to the whole Xwithout depending on t(see also [3]). Nevertheless, we can improve a little bit this result as follows. Proposition 25. Let (X, M, ∗)be a eventually stationary fuzzy metric space with ∗ ≥ ∗ L and (R, Nφ,g,~)be a Euclidean stationary fuzzy metric space such that φis strictly increasing and φ−1is subadditive. Let S⊆Xand suppose that f: (S, M, ∗)→(R, Nφ,g,~)is fuzzy Lipschitz. Then fcan be extended as a fuzzy Lipschitz map to X. Proof. Although we could use Corollary 22 to prove this proposition, for completeness, we present a direct proof by using a suitable McShane formula. Since (Nφ,g,~) is stationary then gmust be constant. Otherwise, if g(t1)6=g(t2) for some t1, t2>0 then Nφ,g(0,1, t1) = 1 −φ(|1−0|)g(t1)6= 1−φ(|1−0|)g(t2) = Nφ,g(0,1, t2) which contradicts stationarity of (Nφ,g,~). Furthermore, since (X, M, ∗) is eventually stationary we can find t0>0 such that M(x, y, t) = M(x, y, s) for all x, y ∈Xand all t, s ≥t0.By assumption we can find K(t0)>0 such that 1−Nφ,g(f(s), f(s0), t0)≤K(t0)(1 −M(s, s0, t0)) for all s, s0∈S. Observe that dt0:X×X→[0,+∞) given by dt0(x, y) = K(t0) g(t0)(1 −M(x, y, t0)) is a metric on X. 19 Consider ˆ f:X→Rdefined by the following modified McShane formula: ˆ f(x) = sup s∈Sf(s)−φ−1K(t0) g(t0)(1 −M(x, s, t0)) for all x∈X. Let us check that ˆ fis fuzzy Lipschitz. Given x, y ∈Xwe have that 1−Nφ,g(ˆ f(x),ˆ f(y), t) = φ(|ˆ f(x)−ˆ f(y)|)g(t0) =φsup s∈Sf(s)−φ−1K(t0) g(t0)(1 −M(x, s, t0)) −sup s∈Sf(s)−φ−1K(t0) g(t0)(1 −M(y, s, t0))g(t0) ≤φsup s∈Snf(s)−φ−1K(t0) g(t0)(1 −M(x, s, t0)) −f(s) + φ−1K(t0) g(t0)(1 −M(y, s, t0))og(t0) =φ sup s∈Sφ−1K(t0) g(t0)(1 −M(y, s, t0))−φ−1K(t0) g(t0)(1 −M(x, s, t0)) =φ( sup s∈Sφ−1(dt0(y, s)) −φ−1(dt0(x, s))g(t0) ≤φφ−1(dt0(x, y))g(t0) = K(t0)(1 −M(x, y, t0)) ≤K(t0)(1 −M(x, y, t)) where in the last inequality we have used that M(x, y, t) = M(x, y, t0) whenever t≥t0and M(x, y, t)≤M(x, y, t0) whenever t < t0.Consequently, ˆ fis fuzzy Lipschitz. Moreover, ˆ fextends fto X. In fact, if s0∈Sthen f(s0) = f(s0)−φ−1K(t0) g(t0)(1 −M(s0, s0, t0))≤ˆ f(s0). On the other hand, since fis fuzzy Lipschitz for every s∈Swe have that 1−Nφ,g(f(s), f(s0), t0)≤K(t0)(1 −M(s, s0, t0)), φ(|f(s)−f(s0)|)g(t0)≤K(t0)(1 −M(s, s0, t0)), |f(s)−f(s0)| ≤ φ−1K(t0) g(t0)(1 −M(s, s0, t0)), 20 and then f(s)−φ−1K(t0) g(t0)(1 −M(s, s0, t0))≤f(s0). Hence ˆ f(s0)≤f(s0) so ˆ f(s0) = f(s0). Remark 26. Notice that under the conditions of the previous proposition, we can provide directly an extension of fnot depending on tby using •the following modified McShane formula (as used in the proof) fM(x) = sup s∈Sf(s)−φ−1K(t0) g(t0)(1 −M(x, s, t0)) for all x∈X, or •the following modified Whitney formula fW(x) = inf s∈Sf(s) + φ−1K(t0) g(t0)(1 −M(x, s, t0)) for all x∈X. 6. Applications: the extension formulas for two fuzzy metric spaces To finish the paper, let us explain how to obtain explicit formulas providing parameterized families of extensions. These families can be used to build machine learning tools as the ones in [5], but integrating new fuzzy elements in them. This would provide more flexible algorithms, in the sense of not depending so heavily on the Lipschitz constant, which could easily increase a lot as a result of outliers among the data, producing as a consequence imprecise extensions. Although the same construction that we present here can be done for a broader class of fuzzy metric spaces, we will center our attention in two standard cases for the aim of clarity, and also because these particular cases can be easily implemented in the context of [5]. Let us consider a strong fuzzy metric space (X, M, ∗) with ∗≥∗ L and the Euclidean fuzzy metric space (R, NE,∗ L) (see Example 20), given by NE(x, y, t) = 1 −min{|x−y|,1}g(t), x, y ∈R, where g: [0,+∞)→(0,1] is a decreasing function. 21 Let S⊆Xand I: (S, M, ∗)→(R, NE,∗ L) be a fuzzy Lipschitz map. Since (M, ∗) is a strong fuzzy metric, by using Proposition 5 we can obtain that the hypotheses of Corollary 22 are satisfied. Thus, we know (see Remark 23) that there are two canonical extensions of I, —provided by the McShane and Whitney formulas—, IM, IW:X×(0,+∞)→R.Parameter dependent interpolations of these functions can be considered as optimal extensions of I, and would be given by Iα(x, t) := α(t)IM(x, t) + 1−α(t)IW(x, t), x ∈X, t > 0. Here, α: (0,+∞)→[0,1]. •The metric model depending on a parameter. Take a metric space (X, d) and construct the associated strong fuzzy metric space (X, Mk,∗ L) (cf. [13, Example 6]) defined by Mk(x, y, t)=1−min{d(x, y), k} h(t), x, y ∈X, where k > 0 and h: (0,+∞)→(k, +∞) is an increasing continuous function. Let S⊆X. Suppose that the function I: (S, Mk,∗ L)→ (R, NE,∗ L) is a fuzzy Lipschitz map. Its corresponding Lipschitz inequality is 1−NE(I(x), I(y), t)≤K(t)1−Mk(x, y, t), x, y ∈S, for all x, y ∈Xand all t > 0,which can be rewritten as min{|I(x)−I(y)|,1} ≤ K(t) g(t)h(t)min{d(x, y), k}. That is, there is a function Q:R+→R+such that min{|I(x)−I(y)|,1} ≤ Q(t) min{d(x, y), k} for all x, y ∈Sand all t > 0.Then the McShane and Whitney extensions of Ito X×(0,∞) are given by IM(x, t) := sup s∈S {I(s)−Q(t) min{d(s, x), k}}, x ∈X, 22 and IW(x, t) := inf s∈S{I(s) + Q(t) min{d(s, x), k}}, x ∈X. Thus, a possible family of extensions would be given by functions as Iα(t)(x, t) =α(t)IM(x, t) + (1 −α(t)) IW(x, t) =α(t) sup s∈S {I(s)−Q(t) min{d(s, x), k}} + (1 −α(t)) inf s∈S{I(s) + Q(t) min{d(s, x), k}}, x∈X, t > 0.An adequate function α(t) could be given for example by an optimization procedure, in order to define a machine learning method incorporating fuzzy elements. •The exponential fuzzy model. In this case, we consider the stationary fuzzy metric (M1,·) given by [13, Example 5], M1(x, y, t) = e−d(x,y), x, y ∈X, where (X, d) is a metric space. As above, let S⊆Xand I: (S, M1,·)→ (R, NE,∗ L) be a fuzzy Lipschitz function. Then, for each t > 0 we can find K(t)>0 such that 1−NE(I(x), I(y), t)≤K(t)1−M1(x, y, t), x, y ∈S, for all x, y ∈Xand all t > 0,which can be rewritten as min{|I(x)−I(y)|,1} ≤ K(t) g(t)(1 −e−d(x,y)). Notice that since ·≥∗ L we have by Proposition 5 that 1 −e−d(x,y)is a metric on X. Using the same arguments than in the previous example, we obtain that the family of extensions would be given by functions as Iα(t)(x, t) = α(t)IM(x, t) + (1 −α(t)) IW(x, t), x ∈X, t > 0, where IM(x, t) = sup s∈SnI(s)−K(t) g(t)1−e−d(s,x)o, x ∈X, t > 0, 23 and IW(x, t) = inf s∈SnI(s) + K(t) g(t)1−e−d(s,x)o, x ∈X, t > 0. Acknowledgements. The first and the last authors gratefully acknowledge the support of the Ministerio de Ciencia, Innovaci´on y Universidades, Agencial Estatal de Investigaciones and FEDER under grant MTM2016-77054C2-1-P. The second author acknowledges financial support from FEDER/Ministerio de Ciencia, Innovaci´on y Universidades-Agencia Estatal de Investigaci´on-Proyecto PGC2018-095709-B-C21. [1] K. Asadi, M. Dipendra and M. L. Littman, Lipschitz continuity in model-based reinforcement learning. arXiv preprint arXiv:1804.07193 (2018). [2] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000. [3] J. Bors´ık and J. Doboˇs, On metric preserving functions. Real Anal. Exchange 13 (1987/88), 285–293. [4] A. Brudnyi and Y. Brudnyi, Methods of geometric analysis in extension and trace problems, Volume 1, Monographs in Mathematics, 102, Birkh¨auser/Springer Basel AG, Basel, 2012. [5] J. M. Calabuig, H. Falciani and E. A. S´anchez-P´erez, Dreaming machine learning: Lipschitz extensions for reinforcement learning on financial markets, Neurocomputing 398 (2020), 172–184. [6] F. Castro-Company, S. Romaguera and P. Tirado, On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings. Fixed Point Theory Appl. 2015, 226 (2015). [7] J. Doboˇs, Metric preserving functions, ˇ Stroffek, Koˇsice, 1998. [8] M. Dong, Metric Learning with Lipschitz Continuous Functions, Doctoral dissertation, University College London (2019). 24 [9] M. Dong, X. Yang, Y. Wu and J.-H. Xue, Metric learning via maximizing the Lipschitz margin ratio. arXiv:1802.03464v1 (2018) [10] A. George and P.Veeramai, On some result in fuzzy metric space, Fuzzy Sets and Systems 64 (1994), 395–399. [11] A. George and P. Veeramani, On some result of analysis for fuzzy metric spaces, Fuzzy Sets Syst. 90 (1997), 365–368. [12] V. Gregori, S. Morillas and A. Sapena, On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161 (2010), 2193–2205. [13] V. Gregori, S. Morillas and A. Sapena, Examples of fuzzy metrics and applications, Fuzzy Sets and Systems 170 (2011), 95–111. [14] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems 115 (2000), 485–489. [15] V. Gregori and S. Romaguera, Characterizing completable fuzzy metric spaces, Fuzzy Sets and Systems 144 (2004), 411–420. [16] V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002), 245–252. [17] J. Guti´errez-Garc´ıa and S. Romaguera, Examples of non-strong fuzzy metrics, Fuzzy Sets and Systems 162 (2011), 91–93. [18] E. H¨ullermeier, Fuzzy methods in machine learning and data mining: Status and prospects, Fuzzy Sets and Systems 156, no. 3 (2005), 387–406. [19] R. Kyng, A. Rao, S. Sachdeva and D. A. Spielman, Algorithms for Lipschitz learning on graphs. In: Proceedings of the Conference on Learning Theory 2015, (2015), 1190–1223. [20] D. H. Leung and W.-K. Tang, Functions that are Lipschitz in the small, Rev. Mat. Complut. 30 (2017), 25–34. [21] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942), 535–537. [22] D. Mihet¸, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems 144 (2004), 431–439. 25