arXiv subjects

Jiyuan Tao

Publications and source records attributed to Jiyuan Tao.

4 recordsLinked to original sources

arXiv2026

New Bounds for Exact Penalized Cardinality-Constrained Optimization with Pseudonormality Conditions

Cardinality-constrained optimization (CCO) is a popular topic in sparse learning and signal recovery, yet remains challenging due to the inherent nonconvexity and discontinuity of cardinality constraints. This paper investigates the exact penalty theory for CCO problems with general equality and inequality constraints. In particular, we extend the pseudonormality condition to the cardinality-constrained framework and establish the local exact penalization without imposing Lipschitz continuity on the objective function. We further analyze both the projected subgradient method and its stochastic variant with convergence guarantees for the derived exact penalty formulation. Compared with the existing results, we give some more precise bounds of the iterate sequence and the objective function value.

math.OC↗

arXiv2026

Heaviside Low-Rank Support Matrix Machine

Support matrix machine (SMM) is an emerging classification framework that directly handles matrix-structured observations, thereby avoiding the spatial correlations destroyed by vectorization. However, most existing SMM variants rely on convex or nonconvex surrogate loss functions, which may lead to high sensitivity to noise. To address this issue, we propose a novel Heaviside low-rank SMM model called HL-SMM, which leverages the Heaviside loss instead of the common hinge or ramp losses for robustness. Moreover, the low-rank constraint is adopted to accurately characterize the inherent global structure. In theory, we analyze the Karush-Kuhn-Tucker (KKT) points and rigorously prove the sufficient and necessary conditions. In algorithms, we develop an effective proximal alternating minimization (PAM) scheme, where all subproblems have closed-form solutions. Extensive experiments on benchmark datasets validate that the proposed HL-SMM achieves superior classification accuracy and robustness compared to state-of-the-art methods.

cs.LG↗

arXiv2020

Some log and weak majorization inequalities in Euclidean Jordan algebras

Motivated by Horn's log-majorization (singular value) inequality $s(AB)\underset{log}{\prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)\underset{w}{\prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $λ(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} λ(A)*λ(B)$ for positive semidefinite matrices and $λ(|A\circ B|) \underset{w}{\prec} λ(|A|)*λ(|B|)$ for general (Hermitian) matrices, where $A\circ B$ denotes the Jordan product of $A$ and $B$ and $*$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $λ\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} λ(a)*λ(b)$ for $a,b\geq 0$ and $λ\big (|a\circ b|\big )\underset{w}{\prec} λ(|a|)*λ(|b|)$ for all $a$ and $b$, where $P_u$ and $λ(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $λ(|A\bullet b|)\underset{w}{\prec} λ({\mathrm{diag}}(A))*λ(|b|)$, where $A$ is a real symmetric positive semidefinite matrix and $A\,\bullet\, b$ is the Schur product of $A$ and $b$. In the form of an application, we prove the generalized Hölder type inequality $||a\circ b||_p\leq ||a||_r\,||b||_s$, where $||x||_p:=||λ(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,r\in [1,\infty]$ with $\frac{1}{p}=\frac{1}{r}+\frac{1}{s}$. We also give precise values of the norms of the Lyapunov transformation $L_a$ and $P_a$ relative to two spectral $p$-norms.

math.FA↗

arXiv2017

The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra

A weighted complementarity problem (wCP) is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem (wHLCP) in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of R_0, R, and P properties and discuss the solvability of wHLCPs under nonzero (topological) degree conditions. A uniqueness result is stated in the setting of R^n. We show how our results naturally lead to interior point systems.

math.OC↗