提出精确的低秩因子共享列边际投影方法，实现双随机图学习的高效无误差优化。
Exact Rank-Space KL Projection for Shared-Marginal Low-Rank Factors: Application to Doubly Stochastic Clustering
- 通过严格凸对偶简化为 r-1 个有效变量，支持 O((n+m)r) 快速计算。
- 在非零潜在质量条件下保证收敛性，可行性残差接近数值精度。
- 适合需要精确双随机约束的图学习与聚类任务，尤其适用于稀疏观测场景。
我们研究具有指定行边际和共享学习列边际的低秩因子分解的精确熵散度（KL）投影问题。对于总质量相等的任意正行边际，联合KL投影可精确化为仅含 r-1 个有效变量的严格凸规范对偶，其海森矩阵由类别协方差项之和构成，支持 O((n+m)r) 的无矩阵海森-向量乘积。该投影定理与目标函数无关。随后将其应用于双随机（DS）图学习：通过 W = U Diag(g)^{-1} V^T，使行单纯形因子共享列质量，从而在不显式构造 n×n 优化变量的情况下生成精确双随机图。结合观测边稀疏拟合、随机锚点约简流形正则项及Bregman回溯，所提出的镜面下降法在每一步接受后均保持精确可行性。在非零潜在质量假设下，算法满足充分下降性与 O(1/N) 镜面平稳性界，严格正积累点为KKT驻点。匹配聚类实验表明其具备竞争性准确率、近数值精度的可行性残差，以及无需稠密学习图的优越即时行为。
原文摘要 · Abstract (English)
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.
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