Capra-Convexity: Support & Rank Recovery

• Capra-Convexity is a generalized convexity framework that normalizes data along primal rays to analyze support-based functions.
• It enables exact biconjugacy for the ℓ0 pseudonorm under orthant-strict monotonicity, overcoming the limitations of classical Fenchel conjugacy.
• The framework extends to matrix rank recovery and introduces variational formulations and generalized top-k support norms for robust convex analysis.

Capra-convexity is a form of generalized convexity built from a coupling that is constant along primal rays, so that the primal variable is analyzed through its normalized direction rather than its magnitude. In the vector setting, the Capra coupling replaces the standard bilinear pairing by $\langle x,y\rangle/\|x\|$ for $x\neq 0$, which makes it adapted to $0$-homogeneous objects such as functions of the support and, in particular, the $\ell_0$ pseudonorm. Within this framework, several results that fail under classical Fenchel conjugacy become exact: nondecreasing finite-valued functions of the support are Capra-convex under orthant-strict monotonicity assumptions, $\ell_0$ equals its Capra-biconjugate for $\ell_p$ source norms with $1<p<\infty$, and an analogous construction yields rank-based conjugacies for matrices, with exact recovery of the rank function in the Frobenius case (Chancelier et al., 2019, Chancelier et al., 2020, Chancelier et al., 2020, Barbier et al., 2021, Franc et al., 8 Sep 2025).

1. Capra coupling and generalized Fenchel–Moreau structure

The defining object is a source norm $\|\cdot\|$ on $\mathbb{R}^d$, together with the normalization mapping

$$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$

Depending on the source, the same map is also written $x\neq 0$0 or $x\neq 0$1. The associated Capra coupling is

$x\neq 0$2

Its characteristic property is

$x\neq 0$3

which is the origin of the phrase constant along primal rays (Chancelier et al., 2019, Chancelier et al., 2020, Chancelier et al., 2020).

For a function $x\neq 0$4, the Capra-Fenchel–Moreau conjugate and biconjugate are

$x\neq 0$5

and satisfy the generalized Fenchel–Young inequality $x\neq 0$6. A function is Capra-convex precisely when equality holds, i.e. $x\neq 0$7 (Franc et al., 2021, Franc et al., 8 Sep 2025).

A structural characterization parallels ordinary convex analysis but is expressed through normalization. In the one-sided linear coupling formalism, $x\neq 0$8, a function $x\neq 0$9 is $0$0-convex if and only if there exists a closed convex function $0$1 such that $0$2. Specializing to the Capra normalization map yields the factorization property

$0$3

so Capra-convexity is convexity after collapsing each positive ray to a point on the unit sphere (Chancelier et al., 2019, Chancelier et al., 2020, Franc et al., 8 Sep 2025).

2. Why Capra-convexity is adapted to support-based functions

The primary motivation is the support mapping

$0$4

which is $0$5-homogeneous: $0$6 Any function of the support is a composition $0$7, where $0$8. Such functions are constant along rays and therefore lie outside the natural scope of classical Fenchel conjugacy (Chancelier et al., 2020).

The degeneration of the ordinary Fenchel framework is explicit. For support level sets, the Fenchel conjugate of the indicator may collapse to $0$9, and the Fenchel biconjugate may collapse to $\ell_0$0. More generally, the papers show that Fenchel biconjugates of functions of the support are degenerate, so the standard convex hull construction loses the combinatorial support information (Chancelier et al., 2020, Chancelier et al., 2019).

Capra conjugacy avoids that collapse by inserting normalization directly into the coupling. Under the assumption that both the source norm and its dual norm are orthant-strictly monotonic, any nondecreasing finite-valued set function $\ell_0$1 satisfies

$\ell_0$2

so every such function of the support is Capra-convex (Chancelier et al., 2020).

Orthant-strict monotonicity is defined through entrywise comparisons within a common orthant: $\ell_0$3 The theory also gives an equivalent condition involving support-matched dual vectors: $\ell_0$4 This criterion is central in proving nonemptiness of Capra-subdifferentials and hence exact biconjugacy (Chancelier et al., 2020).

3. The $\ell_0$5 pseudonorm: exact biconjugacy, hidden convexity, and Capra-subdifferentials

The canonical example is

$\ell_0$6

which counts nonzero coordinates and is $\ell_0$7-homogeneous. Under classical Fenchel conjugacy, $\ell_0$8. Under Capra conjugacy, the situation changes completely: for source norms $\ell_0$9 with $\ell_0$0, one has

$\ell_0$1

and $\ell_0$2 is Capra-subdifferentiable at every point (Chancelier et al., 2019, Chancelier et al., 2020, Franc et al., 2021).

The Capra conjugate of $\ell_0$3 can be written in several equivalent forms. In the Euclidean E-Capra setting,

$\ell_0$4

where $\ell_0$5 is the top-$\ell_0$6 norm (Chancelier et al., 2019). For $\ell_0$7 source norms with dual exponent $\ell_0$8, the explicit formula becomes

$\ell_0$9

where

$\ell_p$0

for a permutation $\ell_p$1 ordering coordinates by decreasing absolute value (Franc et al., 2021).

A major consequence is the existence of a proper convex lower semicontinuous function agreeing with $\ell_p$2 on the unit sphere. In the Euclidean case,

$\ell_p$3

and

$\ell_p$4

The later norm-general formulation writes, for a nondecreasing $\ell_p$5 with $\ell_p$6,

$\ell_p$7

with $\ell_p$8 proper, convex, and lower semicontinuous (Chancelier et al., 2019, Chancelier et al., 2020).

The Capra-subdifferential is defined by

$\ell_p$9

and, unlike the classical convex subdifferential of $1<p<\infty$0, it is nontrivial. For all $1<p<\infty$1,

$1<p<\infty$2

For $1<p<\infty$3, $1<p<\infty$4, $1<p<\infty$5, and $1<p<\infty$6, the paper gives an explicit characterization involving three ingredients: $1<p<\infty$7 must lie in the normal cone to the $1<p<\infty$8-norm unit ball at $1<p<\infty$9; off-support coordinates satisfy $\|\cdot\|$0; and a family of rank-ordered top-norm inequalities constrains the order statistics of $\|\cdot\|$1 (Franc et al., 2021).

The extreme norms are exceptional. For $\|\cdot\|$2,

$\|\cdot\|$3

and for $\|\cdot\|$4,

$\|\cdot\|$5

Accordingly, $\|\cdot\|$6 is not Capra-convex in these edge cases (Franc et al., 2021).

4. Generalized top-$\|\cdot\|$7 and local-$\|\cdot\|$8-support norms, convex factorization, and variational formulations

The concrete convex-analytic machinery behind Capra-convexity is expressed through norm families indexed by support size or support set. In the vector case, the generalized top-$\|\cdot\|$9 dual norm is

$\mathbb{R}^d$0

and its dual is the generalized $\mathbb{R}^d$1-support dual norm

$\mathbb{R}^d$2

Under orthant-monotonicity, these coincide with the coordinate-$\mathbb{R}^d$3 constructions (Chancelier et al., 2020).

For nonnegative nondecreasing $\mathbb{R}^d$4 with $\mathbb{R}^d$5, the generalized convex factor $\mathbb{R}^d$6 admits several exact variational descriptions. One form is

$\mathbb{R}^d$7

and, for $\mathbb{R}^d$8,

$\mathbb{R}^d$9

The minimum is attained by the decomposition that selects $$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$0 for $$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$1 and $$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$2 otherwise (Chancelier et al., 2020).

The support-function generalization replaces the integer parameter $$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$3 by arbitrary subsets $$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$4. The papers define generalized local-top-$$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$5 dual norms and local-$$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$6-support dual norms, and establish exact normalized variational formulations for nondecreasing finite-valued functions $$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$7. When $$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$8, one obtains

$$\rho(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\ 0, & x=0. \end{cases}$$9

under the corresponding decomposition and norm constraints recorded in the paper. At a point $x\neq 0$00 with support $x\neq 0$01, the minimum is achieved by $x\neq 0$02 and $x\neq 0$03 for $x\neq 0$04 (Chancelier et al., 2020).

These formulas supply the precise meaning of hidden convexity. A support-based function may remain nonconvex as a function on $x\neq 0$05, yet after normalization to the unit sphere it becomes the restriction of a proper convex lower semicontinuous function. The Capra framework therefore does not replace combinatorial sparsity by an approximation; under the stated assumptions it gives an exact convex factorization of a ray-invariant quantity (Chancelier et al., 2019, Chancelier et al., 2020, Chancelier et al., 2020).

5. Matrix extension: rank-based norms and Capra-convexity of rank

The same construction extends from vectors to matrices. On $x\neq 0$06, with trace inner product $x\neq 0$07 and source matrix norm $x\neq 0$08, the Capra coupling is

$x\neq 0$09

It is again constant along primal rays (Barbier et al., 2021).

For each $x\neq 0$10, $x\neq 0$11, the generalized dual $x\neq 0$12-rank norm is defined by

$x\neq 0$13

and the generalized $x\neq 0$14-rank norm is its dual: $x\neq 0$15 The sequence $x\neq 0$16 is nondecreasing in $x\neq 0$17, whereas $x\neq 0$18 is nonincreasing, with

$x\neq 0$19

When the source norm is unitarily invariant, these norms factor through singular values via a symmetric gauge $x\neq 0$20: $x\neq 0$21 For the Frobenius norm, $x\neq 0$22 is the $x\neq 0$23 top-$x\neq 0$24 norm of the singular values: $x\neq 0$25 (Barbier et al., 2021).

Since

$x\neq 0$26

the rank function inherits a Capra-conjugacy parallel to that of $x\neq 0$27. For $x\neq 0$28,

$x\neq 0$29

If $x\neq 0$30 with $x\neq 0$31, then for $x\neq 0$32,

$x\neq 0$33

Specializing to $x\neq 0$34 gives a variational lower bound for rank, valid for any source norm (Barbier et al., 2021).

The Frobenius case is exact. The paper proves

$x\neq 0$35

hence the rank function is Capra-convex relative to the Frobenius source norm. For other source norms, the same expression remains a lower bound, but equality is not established in general (Barbier et al., 2021).

6. Capra-convex sets, geometric closure on the sphere, and a distinct second-order usage

A set $x\neq 0$36 is called Capra-convex when its indicator $x\neq 0$37 is Capra-convex. The set-theoretic theory shows that

$x\neq 0$38

and therefore

$x\neq 0$39

Geometrically, Capra-convex sets are precisely those cones whose direction set on $x\neq 0$40 is closed and convex (Franc et al., 8 Sep 2025).

Several consequences follow. Any Capra-convex set is a cone, $x\neq 0$41 is closed, and if the unit ball of the source norm is rotund, these properties become part of a sufficiency criterion. Every closed convex cone is Capra-convex, and if it is pointed, then $x\neq 0$42 is also Capra-convex. Moreover, sublevel sets of Capra-convex functions are Capra-convex sets (Franc et al., 8 Sep 2025).

This geometric perspective clarifies the optimization reduction behind Capra-convexity. If $x\neq 0$43 and $x\neq 0$44 are Capra-convex, then $x\neq 0$45 factorizes as $x\neq 0$46 for a proper lower semicontinuous convex $x\neq 0$47. Optimization can then be transferred from $x\neq 0$48 to the sphere of normalized directions, where standard convex structure reappears (Franc et al., 8 Sep 2025).

The theory is nevertheless norm-sensitive. Exact vector biconjugacy for support-based functions requires orthant-strict monotonicity of both the source norm and its dual; the standard examples are $x\neq 0$49 norms with $x\neq 0$50, while $x\neq 0$51 and $x\neq 0$52 are exceptional (Chancelier et al., 2020, Franc et al., 2021). In the matrix case, exact recovery of rank is proved for the Frobenius norm, whereas other source norms presently yield lower bounds (Barbier et al., 2021). For sets, rotundness of the source norm’s unit ball plays a decisive role in sufficiency statements (Franc et al., 8 Sep 2025).

A distinct usage of the name appears in a separate line of work on symbolic convexity certification from Hessians. There, a graph-based second-order calculus proves positive semidefiniteness of Hessian computational graphs by using PSD-preserving rules, domain-to-positivity propagation, and variance-form templates, and it is shown to be at least as powerful as disciplined convex programming for twice-differentiable functions and strictly more powerful than the state-of-the-art CVX implementation for differentiable functions (Klaus et al., 2022). This suggests a broader, graph-centric use of “Capra-Convexity” as a computational convexity calculus, but that usage is conceptually separate from the coupling-based generalized convexity theory centered on support, $x\neq 0$53, and rank (Klaus et al., 2022).