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Symmetrizing Bregman Divergence on the Cone of Positive Definite Matrices: Which Mean to Use and Why

Tushar Sial, Abhishek Halder

Abstract

This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

Symmetrizing Bregman Divergence on the Cone of Positive Definite Matrices: Which Mean to Use and Why

Abstract

This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

Paper Structure

This paper contains 9 sections, 4 theorems, 34 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. Arithmetic Mean is Canonical for Forward Symmetrization
  4. Arithmetic Mean in the Dual Space is Canonical for Reverse Symmetrization
  5. Examples
  6. The Case $\psi(X)=\|X\|_{\mathrm{F}}^2$
  7. The Case $\psi(X)={\mathrm{trace}}(X\log X - X)$
  8. The Case $\psi(X)=-\log{\mathrm{det}} X$
  9. Conclusion

Key Result

Theorem 1

Consider the forward Bregman symmetrization defSymmetrizedBregmanDiv with fixed $X,Y\in\mathbb{S}^{n}_{++}$. The unique minimizer in McanForward is

Theorems & Definitions (15)

  • Definition 1: Bregman divergence on $\mathbb{S}^{n}_{++}$
  • Remark 1
  • Definition 2: Mean on $\mathbb{S}^{n}_{++}$
  • Remark 2: Geometric mean
  • Remark 3: Logarithmic mean
  • Remark 4: Log-Euclidean mean
  • Definition 3: ${\mathrm{GL}}(n)$ and ${\mathrm{O}}(n)$ invariant means
  • Theorem 1
  • proof
  • Corollary 2
  • ...and 5 more