Luis M Briceño-Arias

Patrick L Combettes

Francisco J Silva
email: francisco.silva@unilim.fr

Proximity Operators of Perspective Functions with Nonlinear Scaling

A perspective function is a construction which combines a base function defined on a given space with a nonlinear scaling function defined on another space and which yields a lower semicontinuous convex function on the product space. Since perspective functions are typically nonsmooth, their use in first-order algorithms necessitates the computation of their proximity operator. This paper establishes closed-form expressions for the proximity operator of a perspective function defined on a Hilbert space in terms of a proximity operator involving its base function and one involving its scaling function.

Introduction

Throughout, H and G are real Hilbert spaces and Γ 0 pHq is the class of proper lower semicontinuous convex functions from H to s´8, `8s. The focus of this paper is on the following construction, which arises in application such as control, mean field games, optimal transportation, information theory, and the calculus of variations; see, e.g., [START_REF] Achdou | Mean field games: Numerical methods for the planning problem[END_REF][START_REF] Achdou | Mean field type control with congestion[END_REF][START_REF] Achdou | Mean field type control with congestion (II): An augmented Lagrangian method[END_REF][START_REF] Avalos-Fernandez | Representation learning of compositional data[END_REF][START_REF] Bercher | Some properties of generalized Fisher information in the context of nonextensive thermostatistics[END_REF][START_REF] Brenier | Optimal multiphase transportation with prescribed momentum[END_REF][START_REF] Cardaliaguet | Geodesics for a class of distances in the space of probability measures[END_REF][START_REF] Carlen | Some trace inequalities for exponential and logarithmic functions[END_REF][START_REF] Carrillo | Boltzmann to Landau from the gradient flow perspective[END_REF][START_REF] Dolbeault | A new class of transport distances between measures[END_REF][START_REF] Effros | A matrix convexity approach to some celebrated quantum inequalities[END_REF][START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF][START_REF] Maas | Gradient flows of the entropy for finite Markov chains[END_REF][START_REF] Zellner | Specification and estimation of Cobb-Douglas production function models[END_REF][START_REF] Zozor | φ-informational measures: Some results and interrelations[END_REF] (alternative constructions of nonlinearly scaled perspective functions in certain settings have been studied in [START_REF]On a functional operation generating convex functions, Part 1: Duality[END_REF][START_REF]On a functional operation generating convex functions, Part 2: Algebraic properties[END_REF][START_REF] Zȃlinescu | On the second conjugate of several convex functions in general normed vector spaces[END_REF]; see [START_REF] Briceño-Arias | Perspective functions with nonlinear scaling[END_REF] for a discussion). Definition 1.1 [START_REF] Briceño-Arias | Perspective functions with nonlinear scaling[END_REF] The preperspective of a base function ϕ : H Ñ r´8, `8s with respect to a scaling function s : G Ñ r´8, `8s is

ϕ ˙s : H ˆG Ñ r´8, `8s px, yq Þ Ñ $ & % spyqϕ ˆx spyq ˙, if 0 ă spyq ă `8; `8, if ´8 ď spyq ď 0 or spyq " `8, (1.1)
and the perspective of ϕ with respect to s is the largest lower semicontinuous convex function ϕ § s minorizing ϕ ˙s.

The classical notion of a perspective function, first studied in [START_REF] Rockafellar | Level sets and continuity of conjugate convex functions[END_REF] and further discussed in [START_REF] Combettes | Perspective functions: Properties, constructions, and examples[END_REF][START_REF] Rockafellar | Convex Analysis[END_REF], corresponds to the case of a linear scaling function. Thus, if ϕ P Γ 0 pHq, G " R, and s : y Þ Ñ y in Definition 1.1, it follows from [START_REF] Rockafellar | Level sets and continuity of conjugate convex functions[END_REF]Theorem 3.E] that ϕ § s in (1.1)

where rec ϕ denotes the recession function of ϕ. A key tool in Hilbertian convex analysis to study variational problems and design solution algorithms for them is Moreau's proximity operator [START_REF] Moreau | Fonctions convexes duales et points proximaux dans un espace hilbertien[END_REF][START_REF] Moreau | Proximité et dualité dans un espace hilbertien[END_REF]. Recall that, given f P Γ 0 pHq and x P H, prox f x is the unique minimizer over H of the function y Þ Ñ f pyq `1 2 }x ´y} 2 .

(1.3)

This process defines the proximity operator prox f : H Ñ H of f , which is extensively discussed in [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]. Formulas for the proximity operator of the classical perspective function r ϕ of (1.2) were derived in [START_REF] Combettes | Perspective functions: Proximal calculus and applications in highdimensional statistics[END_REF][START_REF] Combettes | Perspective maximum likelihood-type estimation via proximal decomposition[END_REF] and they have been employed to solve minimization problems arising in areas such as statistical biosciences [START_REF] Combettes | Regression models for compositional data: General log-contrast formulations, proximal optimization, and microbiome data applications[END_REF], information theory [START_REF] Gheche | Proximity operators of discrete information divergences[END_REF], signal recovery [START_REF] Kuroda | Block-sparse recovery with optimal block partition[END_REF], and machine learning [START_REF] Yamada | Hierarchical convex optimization by the hybrid steepest descent method with proximal splitting operators -Enhancements of SVM and Lasso[END_REF]. Likewise, first order methods for solving variational problems involving the perspective functions of Definition 1.1, which are typically nonsmooth, require their proximity operators; see, e.g., [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF][START_REF] Chambolle | An introduction to continuous optimization for imaging[END_REF][START_REF] Combettes | Fixed point strategies in data science[END_REF][START_REF] Condat | Proximal splitting algorithms for convex optimization: A tour of recent advances, with new twists[END_REF] and the references therein. It is the objective of the present paper to derive these proximity operators. The closed-form expressions we obtain for prox ϕ § s are formulated in terms of a proximity operator involving the base function ϕ and one involving its scaling function s.

In Section 2, we define our notation and provide the background necessary to our investigation. Section 3 is devoted to preliminary results. Closed-form expressions of prox ϕ § s are established in Section 4. Examples are provided in Section 5.

Notation and background

The scalar product of a Hilbert space is denoted by x¨| ¨y and the associated norm by } ¨}. The closed ball with center x P H and radius ρ P s0, `8r is denoted by Bpx; ρq. The Hilbert direct sum of H and G is denoted by H ' G. Let f : H Ñ r´8, `8s. Then dom f " x P H ˇˇf pxq ă `8( is the domain of f , epi f " px, ξq P H ˆR ˇˇf pxq ď ξ ( is the epigraph of f ,

f ˚: H Ñ r´8, `8s : x ˚Þ Ñ sup xPH `xx | x ˚y ´f pxq ˘, (2.1)
is the conjugate of f , and Bf is the subdifferential of f . We declare

f convex if epi f is convex, lower semicontinuous if epi f is closed, and proper if ´8 R f pHq ‰ t`8u. The recession of f P Γ 0 pHq is rec f : H Ñ r´8, `8s : x Þ Ñ lim 0ăλÑ`8 ˆf pz `λxq ´f pzq λ ˙, (2.2)
where z P dom f is arbitrary. Let C be a subset of H. Then ι C is the indicator function of C and σ C " ι C is the support function of C; if C is nonempty, closed, and convex, then proj C " prox ι C is the projection operator onto C. See [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF] for background on Hilbertian convex analysis and [START_REF] Rockafellar | Convex Analysis[END_REF] for the Euclidean setting.

Definition 2.1 Let f : H Ñ r´8, `8s. Then p@ξ P r0, `8rq ξdf "

# ι dom f , if ξ " 0; ξf, if ξ ą 0. (2.3)
In addition,

f _ : H Ñ s´8, `8s : x Þ Ñ # f pxq, if ´8 ă f pxq ă 0; `8, otherwise (2.4)
and the İ envelope of f is f İ " f _˚˚. Furthermore,

f ^: H Ñ s´8, `8s : x Þ Ñ # f pxq, if 0 ă f pxq ă `8; `8, otherwise (2.5)
and the IJ envelope of

f is f IJ " f ^˚˚.
Let us record a few facts. Lemma 2.3 [START_REF] Rockafellar | Level sets and continuity of conjugate convex functions[END_REF] Let f P Γ 0 pHq and γ P r0, `8r. Then the following hold:

(i) γdf P Γ 0 pHq.

(ii) r r f p¨, γqs ˚" γdf ˚and pγdf q ˚" Ă f ˚p¨, γq.

Lemma 2.4 [START_REF] Briceño-Arias | Perspective functions with nonlinear scaling[END_REF] Let f P Γ 0 pHq be such that f ´1ps´8, 0rq ‰ ∅. Then the following hold:

(i) f İ P Γ 0 pHq.

(ii) dom f İ " f ´1ps´8, 0rq " f ´1ps´8, 0sq.

(iii) Let x P H be such that f pxq P s´8, 0s. Then f İ pxq " f pxq.

Lemma 2.5 [START_REF] Briceño-Arias | Perspective functions with nonlinear scaling[END_REF] Let f P Γ 0 pHq be such that f ´1ps0, `8rq ‰ ∅. Then the following hold:

(i) f IJ P Γ 0 pHq.
(ii) dom f IJ " conv f ´1ps0, `8rq.

(iii) f IJ pdom f IJ q Ă r0, `8r.

(iv) Let x P H be such that f pxq P s0, `8r. Then f IJ pxq " f pxq.

Preliminary results

We establish results on which the derivations of Section 4 will rest.

Lemma 3.1 Let f P Γ 0 pHq, x P H, p P H, and γ P r0, `8r. Then the following hold:

(i) prox γdf " # proj dom f , if γ " 0; prox γf , if γ P s0, ` 8r .
(ii) ran prox γdf Ă dom pγdf q Ă dom f .

(iii) p " prox γdf x ô p@y P Hq xy ´p | x ´py `pγdf qppq ď pγdf qpyq.

(iv) p " prox γdf x ô pγdf qppq `pγdf q ˚px ´pq " xp | x ´py.

(v) Suppose that γ ą 0. Then p " prox γf x ô f ppq `f ˚ppx ´pq{γq " xp | x ´py{γ.

(vi) Suppose that γ ą 0. Then x " prox γf x `γ prox f ˚{γ px{γq.

Proof. Recall from Lemma 2.3(i) that γdf P Γ 0 pHq.

(i): This follows from (2.3).

(ii): This follows from (1.3) and (2.3).

(iii): In view of (2.3), for γ " 0, this is the characterization of the projection of x onto the nonempty closed convex set dom f [7, Theorem 3.16] while, for γ ą 0, this is [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Proposition 12.26].

(iv): By virtue of (ii), dom pγdf q Ă dom f . Hence, Lemma 2. Lemma 3.2 Let γ P r0, `8r, let φ P Γ 0 pRq be even and such that 0 P int dom φ, set ϕ " φ ˝} ¨}, and let x P H. Then ϕ P Γ 0 pHq and the following hold:

(i) prox γdϕ x " $ & % prox γdφ }x} }x} x, if x ‰ 0; 0, if x " 0.
(ii) ϕpprox γdϕ xq " φpprox γdφ }x}q.

Proof. Since (ii) follows from (i), we prove the latter. We have ϕ P Γ 0 pHq. In addition, by [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Propositions 16.17(ii) and 16.27], Bφp0q is a symmetric compact interval, say Bφp0q " r´τ, τ s, where τ P r0, `8r. We also note that there exists ρ P s0, `8s such that dom φ "

# r´ρ, ρs, if ρ ă `8; R, if ρ " `8
and dom ϕ "

# Bp0; ρq, if ρ ă `8; H, if ρ " `8. (3.3)
If ρ ă `8, we derive from (3.3) and [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Example 3.18] that

proj dom ϕ x " ρx maxt}x}, ρu " $ & % ρ}x} maxt}x}, ρu}x} x, if x ‰ 0; 0, if x " 0 " $ ' & ' % proj dom φ }x} }x} x, if x ‰ 0; 0, if x " 0 (3.4)
whereas, if ρ " `8, it is clear that proj dom ϕ x coincides with the last term above. In view of Lemma 3.1(i), this establishes the claim for γ " 0. Now suppose that γ ą 0. Then it follows from [12, Proposition 2.1] that

prox γϕ x " $ & % prox γφ }x} }x} x, if }x} ą γτ ; 0, if }x} ď γτ. (3.5)
Moreover, since, in view of (1.3), }x} ď γτ ô }x} P γBφp0q ô prox γφ }x} " 0, (3.5) reduces to

prox γϕ x " $ & % prox γφ }x} }x} x, if x ‰ 0; 0, if x " 0, (3.6)
as required.

Lemma 3.3 Let f P Γ 0 pHq, let x P H, and set φ : r0, `8r Ñ s´8, `8s : γ Þ Ñ f pprox γdf xq. Then the following hold:

(i) Let µ P r0, `8r and γ P sµ, `8r. Then φpγq ď φpµq ´}prox µdf x ´prox γf x} 2 {pγ ´µq.

(ii) φ is decreasing on r0, `8r.

(iii) φ is continuous.

Proof. First note that Lemma 2.3(i) guarantees that prox γdf and, therefore φ, are well defined.

(i): Set p " prox µdf x and q " prox γf x, and note that (1.3) implies that q P dom f . If µ " 0, we assume that p " proj dom f x P dom f since, otherwise, φpµq " `8 and the inequality holds trivially. By which is equivalent to the announced inequality.

(ii): Clear from (i).

(iii): Set T : r0, `8r Ñ H : γ Þ Ñ prox γdf x. It follows from [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Proposition 23.31(iii)] applied to the maximally monotone operator Bf that T is continuous on s0, `8r and from [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Theorem 23.48] that it is right-continuous at 0. Now suppose that pγ n q nPN be a sequence in s0, `8r such that γ n Ñ µ P r0, `8r. Then T pγ n q Ñ T pµq. If µ " 0, by invoking the lower semicontinuity of f and (ii), we get φp0q " f `T pµq ˘ď lim f `T pγ n q ˘" lim φpγ n q ď lim φpγ n q ď φp0q (3.8)

and therefore φpγ n q Ñ φp0q. If µ ą 0, the continuity of φ at µ is established in [START_REF] Attouch | Variational Convergence for Functions and Operators[END_REF]Lemma 3.27].

The following proposition provides explicit expressions for the perspective function of Definition 1.1. Proposition 3.4 [START_REF] Briceño-Arias | Perspective functions with nonlinear scaling[END_REF] Let ϕ P Γ 0 pHq and let s : G Ñ r´8, `8s be such that S " s ´1ps0, `8rq ‰ ∅. Let x P H and y P G. Then the following hold:

(i) Suppose that ϕ ˚pHq Ă r0, `8s, pϕ ˚q´1 ps0, `8rq ‰ ∅, and ´s P Γ 0 pGq. Then

pϕ § sqpx, yq " $ ' ' ' & ' ' ' % spyqϕ ˆx spyq ˙, if 0 ă spyq ă `8;
prec ϕqpxq, if spyq " 0;

`8, otherwise.

(

(ii) Suppose that ϕ ˚pHq Ă t0, `8u. Then pϕ § sqpx, yq " ϕpxq `ιconv S pyq.

(iii) Suppose that ϕ ˚pHq Ă s´8, 0s Y t`8u, pϕ ˚q´1 ps´8, 0rq ‰ ∅, and s P Γ 0 pGq. Then

pϕ § sqpx, yq " $ ' ' ' & ' ' ' % spyqϕ ˆx spyq ˙, if 0 ă spyq ă `8;
prec ϕqpxq, if y P conv S and spyq ď 0;

`8, otherwise.

(3.10)

Additionally, in each case, ϕ § s P Γ 0 pH ' Gq.

Our derivation of the proximity operator of perspective functions will rely on the following properties. Proposition 3.5 [START_REF] Briceño-Arias | Perspective functions with nonlinear scaling[END_REF] Let ϕ : H Ñ s´8, `8s be proper, let s : G Ñ r´8, `8s be such that S " s ´1ps0, `8rq ‰ ∅, let x ˚P H, and let y ˚P G. Then the following hold:

(i) Suppose that ϕ ˚pHq Ă r0, `8s and pϕ ˚q´1 ps0, `8rq ‰ ∅.

Computation of the proximity operator

We are now ready to present our main result.

Theorem 4.1 Let ϕ P Γ 0 pHq and let s : G Ñ r´8, `8s be such that S " s ´1ps0, `8rq ‰ ∅. Let x P H, y P G, and γ P s0, `8r. Then the following hold:

(i) Suppose that ϕ ˚pHq Ă r0, `8s, pϕ ˚q´1 ps0, `8rq ‰ ∅, and ´s P Γ 0 pGq. Then there exists a unique η P r0, `8r such that

p´sq İ ˆprox γϕ ˚`prox η γ dϕ ˚`x γ ˘˘dp´sq İ y ˙`η " 0. (4.1)
Furthermore,

prox γpϕ § sq px, yq " ˆx ´γ prox η γ dϕ ˚´x γ ¯, prox γϕ ˚`prox η γ dϕ ˚`x γ ˘˘dp´sq İ y ˙. (4.2)
(ii) Suppose that ϕ ˚pHq Ă t0, `8u. Then prox γpϕ § sq px, yq " pprox γϕ x, proj conv S yq.

(iii) Suppose that ϕ ˚pHq Ă s´8, 0s Y t`8u, pϕ ˚q´1 ps´8, 0rq ‰ ∅, and s P Γ 0 pGq. Then there exists a unique η P r0, `8r such that On the other hand, (4.9) yields q P dom s, while (4.7) and Proposition 3.5(i) imply that px ´pq{γ P pϕ ˚q´1 pr0, `8rq. Therefore, since (4.9) and Lemma 2.4(iii) yield Consequently, we deduce from (4.14) that η " spqq P r0, `8r solves (4.1), from which (4.2) follows.

ϕ ˚ˆprox 1 γ s IJ pprox γηds IJ yqdϕ ˚´x γ ¯˙`η " 0. ( 4
0 ď spqq " ´p´sq İ pqq, ( 4
To establish the uniqueness of the solution to (4.1), let us define Therefore, since 0 ď s IJ P Γ 0 pGq by Lemma 2.5(i)&(iii), ps IJ q ´1ps0, `8rq " S ‰ ∅ by Lemma 2.5(iv), and ´p´ϕ ˚qİ " ´p´ϕ ˚q " ϕ ˚P Γ 0 pHq, it follows from (i) that pt, rq "

$ & % φ 1 : r0, `8r Ñ s´8, 0s : µ Þ Ñ p´sq İ `prox γµdp´sq İ y φ2 : r0, `8r Ñ r0, `8r : η Þ Ñ ϕ ˚`prox η γ dϕ ˚px{γq ˘. ( 4
ˆy γ ´1 γ prox γηds IJ y, prox 1 γ s IJ pprox γηds IJ yqdϕ ˚ˆx γ ˙˙, (4.27)
where η is the unique solution in r0, `8r to (4.3). The conclusion then comes from (4.23).

Next, we provide explicit formulas for prox γpϕ § sq px, yq in items (i) and (iii) of Theorem 4.1 (item (ii) is already explicit).

Proposition 4.2 Consider the assumptions and notation of Theorem 4.1(i), and set

$ ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' % Ω 1 " " pu
* Ω 4 " pH ˆGq pΩ 1 Y Ω 2 Y Ω 3 q. (4.28)
Then exactly one of the following holds:

(i) px, yq P Ω 1 , η " 0, and prox γpϕ § sq px, yq " ´x ´γ proj dom ϕ ˚px{γq, proj S y ¯.

(ii) px, yq P Ω 2 , η " 0, and prox γpϕ § sq px, yq " ´x ´γ proj dom ϕ ˚px{γq, prox γϕ ˚`proj dom ϕ ˚px{γq ˘p´sq İ y ¯.

(iii) px, yq P Ω 3 , η " spproj S yq ą 0, and prox γpϕ § sq px, yq " ´x ´γ prox spproj S yq γ ϕ ˚px{γq, proj S y ¯.

(iv) px, yq P Ω 4 , η ą 0 solves

η " s ˆprox γϕ ˚`prox η γ ϕ ˚`x γ ˘˘p´sq İ y ˙, (4.29)
and Hence, if ϕ ˚pprox η γ ϕ ˚px{γqq " 0, we deduce from (4.31) that 0 ă η " spproj S yq, which yields px, yq P Ω 3 . However, since px, yq P Ω 4 , we have ϕ ˚pprox η γ ϕ ˚px{γqq ą 0. Consequently, the claim follows from Lemma 3.1(i).

prox γpϕ § sq px, yq " ˆx ´γ prox η γ ϕ ˚´x γ ¯, prox γϕ ˚`prox η γ ϕ ˚`x γ ˘˘p´sq İ y ˙. ( 4
Finally, it is clear from (4.28) that Ω 1 X Ω 2 " ∅ and Ω 1 X Ω 3 " ∅. Moreover, we infer from (ii) and (iii) that Ω 2 X Ω 3 " ∅. Altogether, pΩ i q 1ďiď4 is a partition of H ˆG and the proof is complete.

γ ¯˙ă 0 * Ξ 4 " pH ˆGq pΞ 1 Y Ξ 2 Y Ξ 3 q. (4.40)
Then exactly one of the following holds:

(i) px, yq P Ξ 1 , η " 0, and prox γpϕ § sq px, yq " ´x ´γ proj dom ϕ ˚px{γq, proj conv S y ¯.

(ii) px, yq P Ξ 2 , η " 0, and prox γpϕ § sq px, yq " ´x ´γ prox 1 γ s IJ pproj conv S yqϕ ˚px{γ q, proj conv S y ¯.

(iii) px, yq P Ξ Hence, if s IJ pprox γηs IJ yq " 0, 0 ą ´η " ϕ ˚pproj dom ϕ ˚px{γqq, which yields px, yq P Ξ 3 . However, since px, yq P Ξ 4 , we have s IJ pprox γηs IJyq ą 0. Consequently, the claim follows from Lemma 3.1(i). Finally, it is clear from (4.40) that Ξ 1 X Ξ 2 " ∅ and Ξ 1 X Ξ 3 " ∅. Moreover, we infer from (ii) and (iii) that Ξ 2 X Ξ 3 " ∅. Altogether, pΞ i q 1ďiď4 is a partition of H ˆG and the proof is complete. Remark 4. [START_REF] Antoniadis | Comments on: 1 -penalization for mixture regression models[END_REF] In cases (i)-(iii) of Proposition 4.2, the computation of prox γpϕ § sq px, yq requires only the ability to compute the projection operators onto dom ϕ ˚and S, as well as the proximity operators of ϕ ˚and p´sq İ . Examples of explicit formulas for these operators can be found in [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF][START_REF] Chierchia | The proximity operator repository[END_REF]. The case (iv) requires additionally the solution η P s0, `8r to (4.29). To determine η, let us define φ 1 and φ 2 as in (4.22) and note that it is the root of T " φ 1 ˝φ2 `Id. Since Lemma 3.3 implies that T is strictly increasing and continuous on s0, `8r, η can be found via standard one-dimensional rootfinding routines [START_REF] Press | Numerical Recipes -The Art of Scientific Computing[END_REF]Chapter 9]. A similar observation can be made for Proposition 4.3.

  2 and (2.1) yield xp | x ´py ď pγdf qppq `pγdf q ˚px ´pq " sup yPdom f `xy | x ´py `pγdf qppq ´pγdf qpyq ˘. (3.1) On the other hand, we derive from (iii) that p " prox γdf x ô sup yPdom f `xy | x ´py `pγdf qppq ´pγdf qpyq ˘ď xp | x ´py. (3.2) Combining (3.1) and (3.2) furnishes the desired characterization. (v): This follows from (iv) and Lemma 2.3(ii). (vi): See [7, Proposition 14.3(ii)].

  Lemma 3.1(iii), xq ´p | x ´py ď µpf pqq ´f ppqq and xp ´q | x ´qy ď γpf ppq ´f pqqq. Adding these inequalities yields }p ´q} 2 ď pγ ´µq `f ppq ´f pqq ˘" pγ ´µq `φpµq ´φpγq ˘, (3.7)

Lemma 2.2 [

  7, Proposition 13.15] Let f : H Ñ s´8, `8s be proper, let x P H, and let x ˚P H. Then f pxq `f ˚px ˚q ě xx | x ˚y.

  Thenpϕ § sq ˚px ˚, y ˚q " Suppose that ϕ ˚pHq Ă t0, `8u. Then `ϕ § s ˘˚px ˚, y ˚q " ι pϕ ˚q´1 pt0uq px ˚q `σconv S py ˚q. Suppose that ϕ ˚pHq Ă s´8, 0s Y t`8u and pϕ ˚q´1 ps´8, 0rq ‰ ∅. Then `ϕ § s ˘˚px ˚, y ˚q "

	(ii) (3.12)
	(iii) $ ' ' ' ' ' & ´ϕ˚p x ˚qs IJ˚˜y	φ˚p x	˚q ¸, if ´8 ă ϕ ˚px ˚q ă 0;
	' σ conv S py ˚q,		if ϕ ˚px ˚q " 0;	(3.13)
	'
	'
	'
	' % `8,		if ϕ ˚px ˚q " `8.
	$ ' ' ' ' ' &	ϕ ˚px ˚qp´sq İ˚˜y	φ˚p x
				(3.11)
	'
	'
	'
	'
	'
	%

˚q ¸, if 0 ă ϕ ˚px ˚q ă `8; σ conv S py ˚q, if ϕ ˚px ˚q " 0; `8, if ϕ ˚px ˚q " `8.

  .22) Lemma 3.3(ii) asserts that φ 1 and φ 2 are decreasing, which renders φ 1 ˝φ2 increasing. In turn, φ 1 ˝φ2 `Id is strictly increasing and therefore the equation pφ 1 ˝φ2 qpηq `η " 0, which is precisely (4.1), has at most one solution in r0, `8r. Since Lemma 2.5(i)-(ii) yields conv S " dom s IJ " dom s IJ ˚˚, we invoke[START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF] Proposition 13.49] to get rec `sIJ˚˘" σ dom s IJ " σ conv S . Now set pr, tq " prox pϕ § sq ˚{γ px{γ, y{γq. Then we derive from (4.25) and [7, Proposition 24.8(iv)] that pt, rq " prox ps IJ˚ § p´ϕ ˚qq{γ ˆy γ

					(4.24)
	In turn, by virtue of Proposition 3.5(iii) and Proposition 3.4(i), we obtain
	pϕ § sq ˚: px ˚, y ˚q Þ Ñ	$ ' & ´ϕ˚p x ˚qs IJ˚˜y	φ˚p x	˚q ¸, if ´8 ă ϕ ˚px ˚q ă 0;
		' % σ conv S py ˚q,	if ϕ ˚px ˚q " 0
	" `sIJ˚ § p´ϕ ˚q˘p y ˚, x ˚q.	(4.25)
		,	x γ	˙.	(4.26)
	(ii): This follows from Proposition 3.4(ii) and [7, Proposition 24.11].
	(iii): A noteworthy consequence of Lemma 3.1(vi) is that
				,	y γ	˙.	(4.23)

pp, qq " px, yq ´γ prox pϕ § sq ˚{γ ˆx γ

  Therefore, if ϕ ˚pproj dom ϕ ˚px{γqq " 0, then (4.37) yields 0 " spproj S yq, which implies that px, yq P Ω 1 . On the other hand, if ϕ ˚pproj dom ϕ ˚px{γqq ą 0, then (4.37) yields and thus px, yq P Ω 2 . However, since px, yq P Ω 4 , we have px, yq R Ω 1 Y Ω 2 and obtain a contradiction. This shows that η ą 0. In turn, (4.33) reduces to

	η " s ˆprox γϕ ˚`prox η γ ϕ ˚`x γ ˘˘dp´sq İ y ˙.	(4.39)
					.30)
	Proof. Lemma 2.4(ii) yields
	dom p´sq İ " p´sq ´1ps´8, 0rq " S " s ´1pr0, `8rq.	(4.31)
	Hence, it follows from Lemma 3.1(ii) that
	p@µ P r0, `8rq prox µdp´sq İ y P s ´1pr0, `8rq.	(4.32)
	Therefore, Lemma 2.4(iii) implies that (4.1) is equivalent to
	η " s ˆprox γϕ ˚`prox η γ dϕ	˚`x γ ˘˘dp´sq İ y ˙.	(4.33)
	(i): Since Lemma 3.1(i) and (4.31) yield
	s ˆprox γϕ ˚`proj dom ϕ ˚`x			(4.34)
	we deduce from (4.33) and Lemma 3.1(i) that η " 0. The claim therefore follows from (4.2).
	(ii): Lemma 3.1(i) yields
	s ˆprox γϕ ˚`proj dom ϕ ˚`x γ ˘˘dp´sq İ y ˙" s ˆprox γϕ ˚`proj dom ϕ ˚`x γ ˘˘p´sq İ y ˙" 0,	(4.35)
	and we deduce from (4.33) and Lemma 3.1(i) that η " 0. Therefore, the claim follows from (4.2).
	(iii): Since spproj S yq ą 0, Lemma 3.1(i) and (4.31) yield
	s ˆprox γϕ ˚´prox spproj S yq γ	dϕ ˚`x γ ˘¯dp´sq İ y ˙" s ˆprox γϕ ˚´prox spproj S yq γ	ϕ ˚`x γ ˘¯dp´sq İ y	"
				s `proj S y ˘,	(4.36)
	and we deduce from (4.33) that η " spproj S yq ą 0. Therefore, the claim follows from (4.2).
	(iv): Suppose that η " 0. Then it follows from (4.33) that
	0 " s ˆprox	γϕ ˚`proj dom ϕ ˚`x γ ˘˘dp´sq İ y ˙.	(4.37)
	0 " s ´prox γϕ ˚`proj dom ϕ ˚`x

γ ˘˘dp´sq İ y ˙" s `proj S y ˘" 0, γ ˘˘p´sq İ y ¯(4.38)

  ˆG ˇˇs IJ pproj conv S vq " 0 and ϕ ˚ˆproj dom ϕ ˆG ˇˇs IJ pproj conv S vq P s0, `8r and ϕ ˚ˆprox 1γ s IJ pproj conv S vqϕ

	Proposition 4.3 Consider the assumptions and notation of Theorem 4.1(iii), and set
	$	"		*
	' ' ' ' ' Ξ 1 "	pu, vq P H	˚´u γ	¯˙" 0
	'
	' ' ' ' ' & Ξ 2 "	" pu, vq P H		˚´u γ	¯˙" 0
	'
	'
	'
	'
	'
	'
	'
	'
	'
	'
	'
	%

* Ξ 3 " " pu, vq P H ˆG ˇˇs IJ ´prox γp´ϕ ˚pproj dom ϕ ˚p u γ qqqs IJv ¯" 0 and ϕ ˚ˆproj dom ϕ ˚´u

  s IJ pprox γp´ϕ ˚pproj dom ϕ and we deduce from (4.3) that η " ´ϕ˚p proj dom ϕ ˚px{γqq ą 0. Therefore, the claim follows from (4.4). Therefore, if s IJ pproj conv S yq " 0, then (4.48) yields 0 " ϕ ˚pproj dom ϕ ˚px{γqq, which implies that px, yq P Ξ 1 . On the other hand, if s IJ pproj conv S yq ą 0, then (4.48) yields 0 " ϕ ˚ˆprox 1 γ s IJ pproj conv S yqϕ and thus px, yq P Ξ 2 . At the same time, since px, yq P Ξ 4 , we have px, yq R Ξ 1 Y Ξ 2 . This contradiction shows that η ą 0. In turn, (4.3) reduces to ´η " ϕ ˚ˆprox 1 γ s IJ pprox γηs IJ yqdϕ

	(i): Since Lemma 3.1(i) and (4.44) yield
	ϕ ˚ˆprox 1 γ s IJ pproj conv S yqdϕ ˚´x γ	¯˙" ϕ ˚ˆproj dom ϕ	˚´x γ	¯˙" 0,	(4.45)
	we deduce from (4.3) and Lemma 3.1(i) that η " 0. The claim therefore follows from (4.4).
	(ii): Lemma 3.1(i) yields
	ϕ ˚ˆprox 1 γ s IJ pproj conv S yqdϕ ˚´x γ	¯˙" ϕ ˚ˆprox 1 γ s IJ pproj conv S yqϕ	˚´x γ	¯˙" 0,	(4.46)
	and we deduce from (4.3) and Lemma 3.1(i) that η " 0. Therefore, the claim follows from (4.4).
	(iii): Since ϕ ˚pproj dom ϕ ˚p x γ qq ă 0, Lemma 3.1(i) yields
							¯"
	ϕ ˚ˆprox 1 γ s IJ pprox γp´ϕ ˚pproj dom ϕ	˚p x γ qqqds IJ yqdϕ	˚´x γ
									¯"
				ϕ ˚ˆprox 1			˚p x γ qqqs IJ yqdϕ ˚´x γ
								¯˙(4.47)
				ϕ ˚ˆproj dom ϕ ˚´x γ
			¯˙(4.49)
	˚´x γ
			˚´x γ	¯˙.	˚´x γ	¯, prox `´γϕ ˚`proj dom ϕ	˚`x γ	˘˘˘s	IJ y ¯.	(4.41) (4.50)
	(iv) px, yq P Ξ 4 , η ą 0 solves
	ϕ ˚ˆprox 1 γ s IJ pprox γηs IJ yqϕ ˚´x γ	¯˙`η " 0			(4.42)
	and
	prox γpϕ § sq px, yq " ˆx ´γ prox 1 γ s IJ pprox γηs IJ yqϕ ˚´x γ	¯, prox γηs IJy ˙.	(4.43)
	Proof. Lemma 2.5(ii) yields
	dom s IJ " conv S.								(4.44)

3 , η " ´ϕ˚`p roj dom ϕ ˚`x{γ ˘˘ą 0, and prox γpϕ § sq px, yq " ´x ´γ proj dom ϕ γ (iv): Suppose that η " 0. Then it follows from (4.3) that

0 " ϕ ˚ˆprox 1 γ s IJ pproj conv S yqdϕ ˚´x γ ¯˙.

(4.48)

Examples

We provide four examples of computation of the proximity operator of perspective functions.

Example 5.1 Let φ P Γ 0 pRq be an even coercive function such that φ ˚pRq Ă r0, `8s and pφ ˚q´1 ps0, `8rq ‰ ∅, and set ϕ " φ ˝} ¨}. Then, φ ˚P Γ 0 pRq is even, 0 P int dom φ ˚by [7, Proposition 14.16], ϕ P Γ 0 pHq, and [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Example 13.8] implies that ϕ ˚" φ ˚˝}¨}. In turn, ϕ ˚pHq Ă r0, `8s and pϕ ˚q´1 ps0, `8rq ‰ ∅. Furthermore, let ´s P Γ 0 pGq, let px, yq P H ˆG, and note that Proposition 3. [START_REF] Antoniadis | Comments on: 1 -penalization for mixture regression models[END_REF]

form a partition of H ˆG, which brings up four cases for consideration:

• px, yq P Ω 1 : We derive from Lemma 3.1(i), Lemma 3.2(i), and Proposition 4.2(i) that prox γpϕ § sq px, yq "

(5.3)

• px, yq P Ω 2 : We derive from Lemma 3.1(i), Lemma 3.2(i), and Proposition 4.2(ii) that

(5.4)

• px, yq P Ω 3 : We derive from Lemma 3. `0, prox p´γφ ˚p0qqs IJy ˘, if x " 0.

(5.14)

Starting with the work [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF], convex optimization problems involving the perspective function with linear scaling (1.2) appear in optimal transport theory and in mean field games [START_REF] Achdou | Mean field games: Numerical methods for the planning problem[END_REF][START_REF] Benamou | Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations[END_REF][START_REF] Lasry | Mean field games[END_REF][START_REF] Villani | Topics in Optimal Transportation[END_REF]. In this context, numerical methods employing its proximity operator are investigated in [START_REF] Briceño-Arias | Proximal methods for stationary mean field games with local couplings[END_REF]. Extensions to variational models with qth root scaling functions have been proposed to address optimal control of McKean-Vlasov systems with congestion [START_REF] Achdou | Mean field type control with congestion[END_REF][START_REF] Achdou | Mean field type control with congestion (II): An augmented Lagrangian method[END_REF], as well as optimal transport with nonlinear mobilities [START_REF] Dolbeault | A new class of transport distances between measures[END_REF]. In the following example, we compute the proximity operator of perspective functions with such scaling functions and incorporate a scale constraint which can be used, in particular, to model density constraints [START_REF] Cardaliaguet | First order mean field games with density constraints: Pressure equals price[END_REF][START_REF] Daudin | Optimal control of diffusion processes with terminal constraint in law[END_REF][START_REF] Daudin | Optimal control of the Fokker-Planck equation under state constraints in the Wasserstein space[END_REF][START_REF] Mészáros | A variational approach to second order mean field games with density constraints: The stationary case[END_REF].

Example 5.3 Let p P s1, `8r and q P s0, 1r. Set p ˚" p{pp ´1q, and φ " | ¨|p {p. Let I Ă r0, `8r be a closed interval such that 0 P I and define

´8, if y ă 0, and set s " ψ ´ιI .

(

Note that, in view of (5.19), (5.18) can be written as

In the case when x ‰ 0, let us point out that, given ξ P s0, `8r, [7, Example 24.38] asserts that ρpξq " prox ξ γ φ ˚p}x}{γq is the unique solution to }x} " ργ `ξρ p ˚´1 .

(

On the other hand, for every µ P s0, `8r, in view of Lemma 3.1(iii), zpµq " prox γµp´ψq y P s0, `8r is the unique solution to y " z ´qγµz q´1 .

(

Therefore, finding η P s0, `8r such that (5.19) holds amounts to solving η " |proj I pzpρpηq p ˚{p ˚qq| q , that is,

which can be handled by one-dimensional root-finding methods.

A fundamental objective in statistical inference is the estimation of both location x (i.e., the regression vector) and scale y (e.g., the standard deviation of the noise or some other parameter) of the statistical model from the data. In robust statistics, the maximum likelihood-type estimator (Mestimator) for location with concomitant scale [32, p. 179] couples both parameters via a convex objective function in the form of a standard perspective (1.2); see [START_REF] Antoniadis | Comments on: 1 -penalization for mixture regression models[END_REF][START_REF] Combettes | Perspective maximum likelihood-type estimation via proximal decomposition[END_REF][START_REF] Combettes | Regression models for compositional data: General log-contrast formulations, proximal optimization, and microbiome data applications[END_REF][START_REF] Lambert-Lacroix | The adaptive BerHu penalty in robust regression[END_REF][START_REF] Owen | A robust hybrid of lasso and ridge regression[END_REF] for further instances of this model. Most of these models involve the perspective function of the Huber function with a scalar scale. Our analysis allows us to extend it to nonlinear scales. An illustration of Example 5.2 in this context is provided in the following example, where the proximity operator of the resulting function, a central piece in algorithms for solving concomitant estimation problems [START_REF] Combettes | Perspective functions: Proximal calculus and applications in highdimensional statistics[END_REF][START_REF] Combettes | Perspective maximum likelihood-type estimation via proximal decomposition[END_REF][START_REF] Combettes | Regression models for compositional data: General log-contrast formulations, proximal optimization, and microbiome data applications[END_REF], is computed.

Example 5.4 Let φ is the Huber function with parameter α P s0, `8r, that is,

(