arXiv subjects

Derun Zhou

Publications and source records attributed to Derun Zhou.

4 recordsLinked to original sources

arXiv2026

Centered Weak Discrete Riemannian Gradients: A Unified Framework for Riemannian Optimization

We introduce the centered weak discrete Riemannian gradient (c-wDRG) framework for the unified analysis of optimization methods on Riemannian manifolds. The framework uses a center point to represent the relevant logarithmic differences in a common tangent space and covers Riemannian steepest descent, proximal point, proximal gradient, implicit midpoint, geodesic average-vector-field, Gonzalez, and Itoh--Abe methods. We derive c-wDRG certificates for these schemes and establish curvature-aware convergence results using explicit metric distortion bounds. The framework yields sublinear and linear convergence for nonaccelerated methods in the geodesically convex and strongly convex settings, respectively. We further develop accelerated c-wDRG schemes, obtaining accelerated sublinear convergence in the convex case and accelerated linear convergence in the strongly convex case. Our results provide explicit curvature-dependent stepsize conditions and convergence rates within a common framework for Riemannian first-order and discrete-gradient methods.

math.OC↗

arXiv2026

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

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.

math.OC↗

arXiv2025

Dual Riemannian Newton Method on Statistical Manifolds

In probabilistic modeling, parameter estimation is commonly formulated as a minimization problem on a parameter manifold. Optimization in such spaces requires geometry-aware methods that respect the underlying information structure. While the natural gradient leverages the Fisher information metric as a form of Riemannian gradient descent, it remains a first-order method and often exhibits slow convergence near optimal solutions. Existing second-order manifold algorithms typically rely on the Levi-Civita connection, thus overlooking the dual-connection structure that is central to information geometry. We propose the dual Riemannian Newton method, a Newton-type optimization algorithm on manifolds endowed with a metric and a pair of dual affine connections. The dual Riemannian Newton method explicates how duality shapes second-order updates: when the retraction (a local surrogate of the exponential map) is defined by one connection, the associated Newton equation is posed with its dual. We establish local quadratic convergence and validate the theory with experiments on representative statistical models. Thus, the dual Riemannian Newton method thus delivers second-order efficiency while remaining compatible with the dual structures that underlie modern information-geometric learning and inference.

stat.CO↗

arXiv2023

Time-varying effect in the competing risks based on restricted mean time lost

Patients with breast cancer tend to die from other diseases, so for studies that focus on breast cancer, a competing risks model is more appropriate. Considering subdistribution hazard ratio, which is used often, limited to model assumptions and clinical interpretation, we aimed to quantify the effects of prognostic factors by an absolute indicator, the difference in restricted mean time lost (RMTL), which is more intuitive. Additionally, prognostic factors may have dynamic effects (time-varying effects) in long-term follow-up. However, existing competing risks regression models only provide a static view of covariate effects, leading to a distorted assessment of the prognostic factor. To address this issue, we proposed a dynamic effect RMTL regression that can explore the between-group cumulative difference in mean life lost over a period of time and obtain the real-time effect by the speed of accumulation, as well as personalized predictions on a time scale. Through Monte Carlo simulation, we validated the dynamic effects estimated by the proposed regression having low bias and a coverage rate of around 95%. Applying this model to an elderly early-stage breast cancer cohort, we found that most factors had different patterns of dynamic effects, revealing meaningful physiological mechanisms underlying diseases. Moreover, from the perspective of prediction, the mean C-index in external validation reached 0.78. Dynamic effect RMTL regression can analyze both dynamic cumulative effects and real-time effects of covariates, providing a more comprehensive prognosis and better prediction when competing risks exist.

stat.ME↗