arXiv subjects

Yamin Zhou

Publications and source records attributed to Yamin Zhou.

3 recordsLinked to original sources

arXiv2026

Proximal DCA for Fréchet Regression on Riemannian Manifolds with Bounded Curvature

Fréchet regression generalizes linear regression to metric-space-valued responses by defining fitted values as minimizers of weighted Fréchet functionals. Since these weights may have mixed signs, the resulting objective is a signed barycenter problem rather than a standard convex barycenter problem. On Riemannian manifolds, this is further complicated by the lack of global geodesic convexity and possible nonsmoothness of squared distances near cut loci. We study signed Fréchet regression on complete manifolds with two-sided bounded sectional curvature. By restricting optimization to a strongly convex normal ball containing the response support, we use local smoothness, Hessian comparison, and Jacobi-field estimates to formulate the problem as a locally controlled Riemannian proximal DC problem. This leads to FRIDA (Fréchet Regression via Riemannian Iterative DC Algorithm), an exact and inexact proximal DC algorithm for computing regression fits. We prove existence and interiority of minimizers under explicit signed-weight conditions, establish curvature-dependent strong convexity of the proximal subproblems, and show descent and convergence of the iterates to stationary points. We also derive sublinear complexity estimates and, under real-analyticity, obtain full-sequence convergence with KL-type local rates. These results provide a rigorous optimization framework for signed Fréchet regression on manifolds with bounded curvature.

math.OC↗

arXiv2024

ReLUs Are Sufficient for Learning Implicit Neural Representations

Motivated by the growing theoretical understanding of neural networks that employ the Rectified Linear Unit (ReLU) as their activation function, we revisit the use of ReLU activation functions for learning implicit neural representations (INRs). Inspired by second order B-spline wavelets, we incorporate a set of simple constraints to the ReLU neurons in each layer of a deep neural network (DNN) to remedy the spectral bias. This in turn enables its use for various INR tasks. Empirically, we demonstrate that, contrary to popular belief, one can learn state-of-the-art INRs based on a DNN composed of only ReLU neurons. Next, by leveraging recent theoretical works which characterize the kinds of functions ReLU neural networks learn, we provide a way to quantify the regularity of the learned function. This offers a principled approach to selecting the hyperparameters in INR architectures. We substantiate our claims through experiments in signal representation, super resolution, and computed tomography, demonstrating the versatility and effectiveness of our method. The code for all experiments can be found at https://github.com/joeshenouda/relu-inrs.

eess.IV↗

arXiv2024

Which shapes can appear in a Curve Shortening Flow Singularity?

We study possible tangles that can occur in singularities of solutions to plane Curve Shortening Flow. We exhibit solutions in which more complicated tangles with more than one self-intersection disappear into a singular point. It seems that there are many examples of this kind and that a complete classification presents a problem similar to the problem of classifying all knots in $\mathbb R^3$. As a particular example, we introduce the so-called $n$-loop curves, which generalize Matt Grayson's Figure-Eight curve, and we conjecture a generalization of the Coiculescu-Schwarz asymptotic bow-tie result, namely, a vanishing $n$-loop, when rescaled anisotropically to fit a square bounding box, converges to a "squeezed bow-tie," i.e. the curve $\{(x, y) : |x|\leq 1, y=\pm x^{n-1}\}\cup\{(\pm 1, y) : |y|\leq 1\}$. As evidence in support of the conjecture, we provide a formal asymptotic analysis on one hand, and a numerical simulation for the cases $n=3$ and $n=4$ on the other.

math.DG↗