Enliang Hu | Searcharxiv

arXiv2026

Exact Rank-Space KL Projection for Shared-Marginal Low-Rank Factors: Application to Doubly Stochastic Clustering

We study exact Kullback--Leibler (KL) projection for low-rank factorizations whose two nonnegative factors have prescribed row marginals and a shared, learned column marginal. For arbitrary positive row marginals of equal total mass, the joint KL projection reduces exactly to a strictly convex gauge-fixed dual with only $r-1$ effective variables; its Hessian is a sum of categorical covariance terms and admits $O((n+m)r)$ matrix-free Hessian--vector products. The projection theorem is objective-independent. We then specialize this geometry to doubly stochastic (DS) graph learning through $W=U\operatorname{Diag}(g)^{-1}V^\top$, where row-simplex factors with a common column mass induce an exactly DS graph without materializing an $n\times n$ optimization variable. Combined with observed-edge sparse fitting, a stochastic anchor-reduced manifold regularizer, and Bregman backtracking, the resulting mirror-descent method preserves exact feasibility at every accepted step. Under a nonvanishing latent-mass condition, it satisfies sufficient decrease and an $O(1/N)$ mirror-stationarity bound, while strictly positive accumulation points are KKT stationary. Matched clustering experiments show competitive accuracy, feasibility residuals near numerical precision, and favorable anytime behavior without a dense learned graph.

cs.LG↗

arXiv2018

Provable Exactness for Asymmetric Low-Rank SDP Learning

Low-rank factorization is a standard way to make structured optimization problems in machine learning more tractable by replacing matrix variables with compact factors. For positive semidefinite (PSD) variables, the symmetric Burer--Monteiro factorization (sBMF) writes $Z=XX^\top$ with a single low-rank factor $X$. A recent asymmetric alternative (aBMF) writes $Z=XY^\top$ and adds a quadratic penalty $(\gamma/2)\|X-Y\|_F^2$ to encourage symmetry. This split is attractive because it yields a biconvex objective with alternating convex subproblems, but its practical value depends strongly on how the penalty parameter $\gamma$ is chosen. We study a unified regularized aBMF framework and derive an explicit lower bound on $\gamma$ that guarantees exactness: under mild assumptions, any $\gamma$ above this threshold makes aBMF and sBMF share the same critical points. This gives a principled way to use the asymmetric formulation without altering the critical-point structure of the symmetric problem. In particular, it answers the open question of whether an exact penalty exists for asymmetric relaxation.

cs.LG↗