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arxiv: 2605.00396 · v1 · submitted 2026-05-01 · 🧮 math.OC

Riemannian Optimization over Symmetric Positive Definite Matrices with the Alpha-Procrustes Geometry

Derun Zhou , Keisuke Yano , Mahito Sugiyama This is my paper

Pith reviewed 2026-05-09 19:18 UTC · model grok-4.3

classification 🧮 math.OC
keywords Riemannian optimizationsymmetric positive definite matricesAlpha-Procrustes geometryRiemannian Hessianmetric geometryoptimization on manifoldsill-conditioned matrices
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The pith

For alpha equal to 1 the Alpha-Procrustes geometry on symmetric positive definite matrices keeps Riemannian Hessian eigenvalues uniformly bounded independent of the matrix.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the Alpha-Procrustes geometry as a one-parameter family of Riemannian metrics on the manifold of symmetric positive definite matrices. This family includes the log-Euclidean metric when alpha is zero and the Bures-Wasserstein metric when alpha is one half. At alpha equal to one the eigenvalues of the induced Riemannian metric operator remain bounded by constants that do not depend on the underlying matrix. Therefore, whenever the Euclidean Hessian of the objective satisfies uniform spectral bounds, the corresponding Riemannian Hessian also satisfies those same bounds. The result supplies a geometric setting in which Riemannian optimization algorithms retain good conditioning even when the matrices become severely ill-conditioned.

Core claim

When the parameter alpha is set to one, every eigenvalue of the Riemannian metric operator induced by the Alpha-Procrustes geometry is bounded above and below by positive constants that are independent of the symmetric positive definite matrix. Consequently, if the Euclidean Hessian satisfies uniform spectral bounds on the manifold, then all eigenvalues of the Riemannian Hessian are likewise bounded independently of the matrix. This property distinguishes the alpha-one case from the affine-invariant and Bures-Wasserstein metrics, which can become arbitrarily ill-conditioned as the matrix condition number grows.

What carries the argument

The Riemannian metric operator induced by the Alpha-Procrustes geometry at alpha equal to one, whose eigenvalues remain uniformly bounded independent of the symmetric positive definite matrix and thereby control the conditioning of the Riemannian Hessian.

If this is right

  • Riemannian first- and second-order optimization methods achieve asymptotic convergence rates that do not deteriorate when the underlying symmetric positive definite matrix becomes ill-conditioned.
  • The geometry supplies a single robust choice of metric for any optimization problem whose objective has a uniformly well-conditioned Euclidean Hessian.
  • Numerical performance remains stable across applications that routinely encounter near-singular or high-condition-number covariance or kernel matrices.
  • The alpha-one case unifies several existing metrics while adding a uniform-conditioning guarantee that the log-Euclidean and Bures-Wasserstein choices lack.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Users can adopt alpha equal to one as a default metric without separate preconditioning or matrix regularization steps.
  • The uniform bound may extend to other matrix manifolds that inherit similar eigenvalue-control properties from the Alpha-Procrustes construction.
  • Convergence analyses that previously required explicit dependence on matrix condition numbers can now be stated in a parameter-free manner for the alpha-one geometry.

Load-bearing premise

The Euclidean Hessian satisfies uniform spectral bounds across the manifold.

What would settle it

An explicit counter-example in which the Euclidean Hessian eigenvalues stay bounded yet the Riemannian Hessian eigenvalues for alpha equal to one grow without bound as the condition number of the symmetric positive definite matrix increases would falsify the claim.

read the original abstract

In Riemannian optimization, it is well known that the condition number of the Riemannian Hessian at an optimum strongly influences the asymptotic convergence behavior of optimization algorithms. On the manifold of symmetric positive definite (SPD) matrices, several commonly used metrics for optimization, such as the Affine-Invariant (AI) and Bures--Wasserstein (BW) metrics, tend to become ill-conditioned as the underlying SPD matrix becomes ill-conditioned. As a result, even when the Euclidean Hessian remains uniformly well-conditioned on the SPD manifold, optimization may still become difficult near an optimum associated with an ill-conditioned SPD matrix. In this paper, we address this issue through the Alpha-Procrustes (AP) geometry on the SPD manifold. This geometry generalizes several well-known metrics, including the Log-Euclidean (LE) metric for \(\alpha=0\) and the BW metric for \(\alpha=1/2\). We first show that, when \(\alpha=1\), all eigenvalues of the Riemannian metric operator induced by the AP geometry are uniformly bounded independently of the underlying SPD matrix. Therefore, under the assumption that the Euclidean Hessian satisfies the uniform spectral bounds, all the eigenvalues of the corresponding Riemannian Hessian are uniformly bounded independently of the underlying SPD matrix. Consequently, the case \(\alpha=1\) provides a robust geometric framework for several Riemannian optimization problems involving ill-conditioned SPD matrices. Finally, we validate our theoretical findings through extensive numerical experiments across a range of applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces the Alpha-Procrustes (AP) geometry on the SPD manifold, generalizing the Log-Euclidean metric (α=0) and Bures-Wasserstein metric (α=1/2). It establishes that for α=1 the eigenvalues of the Riemannian metric operator are uniformly bounded independently of the underlying SPD matrix. Under the additional assumption that the Euclidean Hessian satisfies uniform spectral bounds, the corresponding Riemannian Hessian eigenvalues are likewise uniformly bounded independently of the SPD matrix. This is positioned as providing a robust geometric setting for Riemannian optimization on ill-conditioned SPD matrices, with the claims supported by theoretical analysis and numerical experiments across applications.

Significance. If the central bound holds, the AP geometry at α=1 supplies a concrete improvement over the Affine-Invariant and Bures-Wasserstein metrics by eliminating the dependence of the Riemannian Hessian condition number on the conditioning of the SPD matrix itself. The result is parameter-free for the chosen α and directly addresses a recognized practical limitation in SPD Riemannian optimization. The generalization of existing metrics and the explicit numerical validation strengthen the contribution.

minor comments (3)
  1. The abstract and introduction would benefit from a brief statement of the explicit form of the AP metric operator (or its eigenvalue expression) at α=1 so that the uniform-bound claim can be traced without immediately consulting the full derivation.
  2. In the numerical section, the reported condition-number plots should include the precise definition of the Euclidean Hessian used in each experiment so that the assumption of uniform spectral bounds can be checked against the displayed data.
  3. A short remark on the range of α values for which the uniform bound fails would help readers understand the special role of α=1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript on the Alpha-Procrustes geometry, as well as for the significance assessment and recommendation of minor revision. We are pleased that the central contribution regarding uniform bounds on the Riemannian Hessian eigenvalues at α=1 is recognized as addressing a practical limitation in SPD Riemannian optimization.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the uniform eigenvalue bounds on the Riemannian metric operator for α=1 directly from the definition of the Alpha-Procrustes geometry and its induced operator. It then states a conditional consequence for the Riemannian Hessian eigenvalues under an explicit external assumption on the Euclidean Hessian. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central claim remains an independent mathematical derivation from the geometry's properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard Riemannian geometry definitions and the new Alpha-Procrustes construction; no free parameters are fitted to data, and the only invented element is the geometry family itself.

axioms (1)
  • standard math Standard definitions and properties of Riemannian metrics, Hessians, and the SPD manifold
    Invoked to relate the Riemannian Hessian to the Euclidean one and to define the metric operator.
invented entities (1)
  • Alpha-Procrustes geometry no independent evidence
    purpose: Parameterized family of metrics on the SPD manifold that generalizes Log-Euclidean and Bures-Wasserstein
    Introduced in the paper as the central new object; no independent evidence outside the definition is provided.

pith-pipeline@v0.9.0 · 5563 in / 1312 out tokens · 37959 ms · 2026-05-09T19:18:06.312986+00:00 · methodology

discussion (0)

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