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Matrix $\Phi$-Entropic Measures of Dependence

Chenyu Wang, Amin Gohari

arXiv:2610.29576v1cs.ITmath.IT

TL;DR

Dependence measures with tensorization have applications including distribution simulation, coding, and simulation of nonlocal correlations. This paper extends scalar Φ-ribbons to the matrix case and establishes their structural and SDPI properties. It also gives independence-structure bounds that are sharp for Φ(t) = t log t.

  • Problem

    Dependence measures with tensorization have applications in noninteractive distribution simulation, distributed source and channel coding, and simulation of nonlocal correlations via wirings.

    Source: §I · p. 1

  • Method

    The paper extends scalar Φ-ribbons to matrix-valued functions and proves their data-processing and tensorization properties, alongside scalar and matrix Φ-entropy inequalities.

    Source: §I · p. 2

  • Results

    Scalar and matrix Φ-ribbons characterize their corresponding Φ-SDPI constants, and the convex-hull inner bound from prescribed independence structures is sharp for Φ(t) = t log t.

    Source: §I · p. 2

  • Takeaways & Limitations

    For Φ(t) = t log t, the independence-based inner bound exactly identifies the matrix Φ-ribbon guarantee under the stated independence structure.

    Source: §I · p. 2

Abstract

from arXiv · show

We introduce matrix $\Phi$-ribbons, extending the scalar $\Phi$-ribbons to the matrix case. For $\Phi$ in the subadditivity class, we establish data processing and tensorization properties of matrix $\Phi$-ribbons. We establish scalar and matrix $\Phi$-entropy inequalities and use them to show that both $\Phi$-ribbons characterize the corresponding $\Phi$-strong data processing inequality (SDPI) constants. We obtain exact evaluations of the SDPI constants for several sources. We provide examples showing that the matrix $\Phi$-SDPI constant can exceed the scalar counterpart. Finally, we derive inner bounds for $\Phi$-ribbons from prescribed independence structures and prove that these bounds are optimal when $\Phi(t)=t\log t$.

I. INTRODUCTION

The paper extends scalar dependence measures with tensorization to matrix Φ-ribbons and establishes structural and SDPI characterizations. It also derives ribbon bounds from prescribed independence structures, with sharp bounds for Φ(t) = t log t.

I. INTRODUCTION

  • For Φ in the subadditivity class, matrix Φ-ribbons satisfy data processing and tensorization, and scalar and matrix ribbons characterize their corresponding Φ-SDPI constants.The characterization extends to matrix Φ-entropy and removes the earlier compact-domain assumption on Φ.Source: §I · p. 2
  • The convex hull of incidence vectors for a downward-closed hypergraph of prescribed independent subsets lies inside the matrix Φ-ribbon, and this bound is sharp for Φ(t) = t log t.Every point outside that hull is excluded by some finite joint distribution satisfying the prescribed independence relations.Source: §I · p. 2

A. Organization

  • The paper’s organization assigns matrix-ribbon properties to Section III, matrix SDPI constants and evaluations to Section IV, and wiring monotonicity to Section V.Section VI treats independence-derived inner bounds and their optimality for Φ(t) = t log t.Source: A. Organization · p. 2

B. Notations

  • The paper assumes finite-valued random variables and sets notation for indexed variable families, distributions, and parameter vectors.Source: B. Notations · p. 2, §I · p. 2

II. SCALAR Φ-RIBBONS, Φ-SDPI CONSTANTS AND A NEW Φ-ENTROPIC INEQUALITY

  • New scalar and matrix Φ-entropy inequalities yield ribbon characterizations of the corresponding SDPI constants and imply mutual-information inequalities described as new.These inequalities remove the compactness assumption used in the earlier scalar characterization.Source: §II · p. 3

A. Scalar Φ-ribbons and Φ-SDPI constants

B. New Φ-entropic inequalities

This section establishes scalar Φ-entropy inequalities and uses them to derive related information inequalities; the scalar proofs also introduce techniques used later for the matrix case.

B. New Φ-entropic inequalities

  • The scalar proofs introduce techniques used later for the matrix Φ-entropy inequalities.The authors recommend reading the scalar proofs before the corresponding matrix proofs.Source: B. New Φ-entropic inequalities · p. 5
  • Theorem 2 and Theorem 4 each assume a finite joint distribution pXY for (X,Y), define μ as its smallest positive probability, and consider any function f:X×Y→dom(Φ).Source: B. New Φ-entropic inequalities · pp. 5, 8
  • Theorem 2’s proof uses a quadratic estimate, and Lemma 3 supplies the estimate used in that proof.The conditional form uses f_Y and its related conditional expectation, as defined in the section.Source: B. New Φ-entropic inequalities · pp. 5–6
  • Corollaries 5–7 derive further inequalities from Theorem 2 and Theorem 4, including inequalities involving a copied output variable eY.For Shannon mutual information, the section interprets eY as a copy of Y and notes that μ enters the resulting information inequality.Source: B. New Φ-entropic inequalities · pp. 8–9

C. Proof of Theorem 1

The proof establishes that the SDPI constant η equals the ribbon-derived quantity α by proving inequalities in both directions. It extends the known result [8, Theorem 23] to the matrix setting and proves the desired inclusion using Proposition 2 and Theorem 14.

C. Proof of Theorem 1

  • For each positive-entropy f(X), every ribbon point with λ2 > 0 yields the bound used to show η ≤ α.The proof takes the supremum over f(X) and then the infimum over ribbon points.Source: C. Proof of Theorem 1 · pp. 9–10
  • η = α, as the proof establishes both η ≤ α and α ≤ η.The reverse bound is first shown as α ≤ η + δ for arbitrary δ > 0, then δ is taken arbitrarily small.Source: C. Proof of Theorem 1 · p. 10
  • For the reverse bound, the proof applies Theorem 4 to conditional expectations and obtains α ≤ η + δ for arbitrary δ > 0.It chooses n > max{1, η + δ, Kδ}; the final step uses the chain rule.Source: C. Proof of Theorem 1 · p. 10
  • For the reverse inclusion, the proof uses real-valued, mean-zero matrix coordinates and shifts each by a constant to be nonnegative without changing conditional variance.Finite alphabets ensure such a constant exists for each coordinate.Source: A. Preliminaries on matrix functions · pp. 16–17
  • The proof extends the known result [8, Theorem 23] to the matrix version.The passage identifies [8, Theorem 23] as the known result, then states that it is extended to matrices.Source: A. Preliminaries on matrix functions · p. 17
  • The proof invokes Proposition 2 and Theorem 14, then combines the resulting steps to obtain the desired inclusion.Source: A. Preliminaries on matrix functions · p. 17

IV. STRONG DATA PROCESSING INEQUALITY CONSTANTS FOR SCALAR AND MATRIX ENTROPIES

This section extends scalar strong data processing inequality results to the matrix case. It studies SDPI constants for scalar and matrix entropies.

IV. STRONG DATA PROCESSING INEQUALITY CONSTANTS FOR SCALAR AND MATRIX ENTROPIES

  • The section generalizes scalar strong data processing inequality results to the matrix case.The authors describe scalar SDPI results as well studied and frame the section as extending many of them to matrices.Source: §IV · p. 18
  • The section’s scope includes SDPI constants for both scalar and matrix entropies.This is stated in the section title.Source: §IV · p. 18
  • The stated aim is to generalize many scalar results, rather than to claim that every scalar result is extended.The passage says “many scalar results.”Source: §IV · p. 18

A. Basic properties

The matrix Φ-ribbon has data processing and tensorization properties. The section also relates matrix SDPI constants to scalar results, including a lower bound by squared maximal correlation in the scalar setting.

A. Basic properties

  • In the scalar case, every Φ-SDPI constant for Φ ∈ F is at least the squared maximal correlation; the paper says it obtains similar results for matrices.The scalar inequality is ηΦ(X;Y) ≥ ρ²(X;Y).Source: A. Basic properties · p. 18
  • The matrix Φ-ribbon satisfies data processing: the theorem states the property for a Markov chain X → Y → Z.The supplied passage introduces the Markov-chain condition but omits the displayed relation.Source: A. Basic properties · p. 18
  • For independent pairs, the matrix Φ-ribbon of the paired variables is given by the intersection of the two individual ribbons.The displayed equality is supplied as a fragment, so no further conditions are inferred here.Source: A. Basic properties · p. 18

B. Matrix Φ-entropy inequalities

The section extends the entropy bounds used to characterize scalar Φ-ribbons to matrix-valued functions, yielding the matrix Φ-SDPI characterization. It also treats the doubly symmetric binary source, extending the scalar analysis to the matrix setting.

B. Matrix Φ-entropy inequalities

  • Matrix Φ-entropy bounds are the key ingredient in proving that the matrix Φ-SDPI constant equals the boundary slope of the matrix Φ-ribbon.The stated result applies in dimension d.Source: B. Matrix Φ-entropy inequalities · p. 19
  • The matrix entropy inequality in Theorem 18 uses a constant Kδ that depends on the minimum positive joint probability μ.The supplied theorem statement identifies finite alphabets and admissible matrix-valued functions; Kδ is given as 1 + π²/(4δμ²).Source: B. Matrix Φ-entropy inequalities · p. 19

C. Doubly symmetric binary source

  • The paper extends the scalar DSBS(λ) result ηΦ(X; Y) = λ² to the matrix case.The supplied result states the scalar value for Φ in F and says the proof is extended to matrices; it does not expose the matrix theorem’s formula.Source: C. Doubly symmetric binary source · p. 20

D. Sums of i.i.d. random variables

The section gives an exact matrix result for sums of i.i.d. variables, develops ribbon properties for wirings of no-signaling boxes, and bounds ribbons under prescribed independence structures. The independence-structure bounds are tight for Φ(t)=t log t, while a dimension-dependence question remains open.

D. Sums of i.i.d. random variables

  • For 1 ≤ m ≤ n, the matrix version of the sum-of-i.i.d. result gives the corresponding SDPI constant m/n.The variables are nondegenerate, finite-valued, real-valued, and i.i.d.Source: D. Sums of i.i.d. random variables · p. 21

V. WIRINGS OF NO-SIGNALING BOXES

  • The matrix Φ-ribbon for no-signaling boxes satisfies data processing and tensorization under wirings.The theorem considers a target box generated by wiring n no-signaling boxes.Source: §V · p. 22

A. General independence structures

  • For prescribed hypergraph independence structures, the paper derives inner bounds on matrix Φ-ribbons, and the convex-hull characterization is tight when Φ(t) = t log t.The stated tightness is specific to Φ(t)=t log t.Source: A. General independence structures · p. 23

VII. FUTURE WORK

  • Whether matrix SDPI constants can strictly increase with dimension when Φ remains in the subadditivity class is left open.The open question concerns some pair of matrix dimensions and Φ in the corresponding subadditivity class.Source: §VII · p. 24

APPENDIX A PROOF OF THE PROPERTIES OF THE MATRIX Φ-RIBBON

The appendices supply proof details for structural properties of matrix Φ-ribbons and for tensorization under wirings of no-signaling boxes. The wiring proof formulates each party’s actions using its own transcript and potentially different box order.

APPENDIX A PROOF OF THE PROPERTIES OF THE MATRIX Φ-RIBBON

  • For product pairs, the proof combines conditional matrix Φ-entropy inequalities and the chain rule to establish ribbon tensorization.The derivation applies the component inequalities to the two independent pairs and then combines them.Source: Appendix A · p. 26
  • The proof establishes the data-processing property for the matrix Φ-ribbon using Markov-chain and conditional-entropy steps.The appendix concludes that the ribbon for (X1, Y1) is contained in the ribbon for (X2, Y2).Source: Appendix A · pp. 26–27

APPENDIX B PROOF OF THEOREM 23

  • The wiring proof tracks Alice’s and Bob’s local transcripts separately, allowing their box choices and action orders to differ.Each party selects a box and input using only its own local transcript and private randomness.Source: Appendix B · pp. 27–28, Table I · p. 28

APPENDIX C PROOF OF THEOREM 16

The appendix proves tensorization of the matrix Φ-SDPI constant across independent pairs. Its argument applies the SDPI to conditioned functions, then uses independence and convexity to combine the bounds.

APPENDIX C PROOF OF THEOREM 16

  • The proof reduces tensorization to two independent pairs, with the general case following by induction.It applies the SDPI to functions conditioned on the other pair’s output.Source: Appendix C · pp. 33–34
  • Convexity of the matrix Φ-entropy functional supports averaging the conditional bounds across the independent pair.The appendix invokes convexity for functions of X1 and the independent pair (X2, Y2).Source: Appendix C · p. 34
  • Independence lets the proof identify conditional expectations of functions of one variable pair using only that pair’s output.For a function of Xi, conditioning on Y1 and Y2 gives the same expectation as conditioning on Yi.Source: Appendix C · p. 34

APPENDIX D PROOF OF THEOREM 17

The proof constructs a matrix-valued example whose matrix Φ-SDPI constant depends strictly on dimension. It uses a smooth strictly convex Φ with concave 1/Φ′′.

APPENDIX D PROOF OF THEOREM 17

  • The construction starts with independent Bernoulli(1/2) variables U, V, and J, then defines X=(U,V) and Y=(J,Z).It also defines a 2 × 2 positive definite matrix-valued function F(U,V).Source: Appendix D · pp. 35–36
  • The example’s Φ is smooth and strictly convex, and its reciprocal second derivative is concave.The proof uses this concavity to place Φ in the stated function class.Source: Appendix D · p. 35
  • The matrix Φ-SDPI constant can depend strictly on matrix dimension, even for a smooth strictly convex Φ with concave 1/Φ′′.The example evaluates matrices that are not simultaneously diagonalizable.Source: Appendix D · p. 36

APPENDIX E MATRIX Φ-ENTROPY INEQUALITIES

Theorem 18 establishes two matrix Φ-entropy inequalities for functions on finite joint distributions. The proof uses convexity and Jensen’s inequality, then extends the inequalities to boundary-valued functions by perturbation and continuity.

APPENDIX E MATRIX Φ-ENTROPY INEQUALITIES

  • Theorem 18 gives two matrix Φ-entropy inequalities for a function on a finite joint distribution, with the smallest positive joint probability denoted by µ.The function’s spectra are required to lie in dom(Φ).Source: Appendix E · pp. 36, 40
  • The proof first treats functions whose spectra lie in int(dom(Φ)), then extends both inequalities to the boundary by perturbation and continuity.The probability space is finite, so continuity of Φ suffices for the limiting step.Source: Appendix E · pp. 37, 40
  • The proof applies Jensen’s inequality to a jointly convex matrix Φ-entropy expression after conditioning on Y.It uses conditional expectations F^X, F^Y, and E[F^X|Y].Source: Appendix E · pp. 37–38

APPENDIX F PROOF OF PROPOSITION 5

The proof reduces the desired matrix inequality to convexity along a symmetric pair of values around their mean. It establishes that convexity and handles boundary spectra by perturbation and continuity.

APPENDIX F PROOF OF PROPOSITION 5

  • It suffices to treat functions with spectra in int(dom(Φ)); perturbation and continuity cover the boundary case.The proof also uses the defining convexity property of the matrix function class.Source: Appendix F · pp. 40–41
  • For values M−Z and M+Z around their mean M, the proof establishes a trace inequality comparing the endpoint pair with the pair scaled by λ.The endpoint expression is bounded below by the expression at M±λZ, with the latter weighted by λ².Source: Appendix F · p. 40
  • The proof reduces the comparison to showing that ξ(s) is convex on [0,1].It first establishes convexity of ξ′(s) using the second Fréchet derivative.Source: Appendix F · pp. 40–41

APPENDIX G PROOF OF THEOREM 25

The proof constructs finite-field random variables whose mutual independence matches a prescribed downward-closed family of subsets. It uses these constructions to complete the scalar HC-ribbon proof and sets up the matrix version through Φ-entropy.

APPENDIX G PROOF OF THEOREM 25

  • For a sufficiently large prime q, linear maps can be chosen so that their concatenation has full rank for every subset in E.The proof gives a probabilistic argument for existence, with an explicit Vandermonde-matrix construction also noted.Source: Appendix G · p. 41
  • The construction defines X_i=(Y_i,C_i) so that X_S are mutually independent exactly for S in E.For S outside E, a minimal forbidden set contained in S supplies dependent C variables.Source: Appendix G · p. 43
  • The finite-field construction completes the scalar HC-ribbon proof.The proof then treats the matrix version by expressing mutual information in Φ-entropy form and taking F(x^n)=f_u(x^n)I_d.Source: Appendix G · p. 43
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