AI 中文总结
该研究提出共享边缘低秩因子的精确秩空间KL投影方法，将其应用于双随机图学习，结合稀疏拟合等技术实现高效聚类，取得良好性能。
AI 中文摘要
我们研究低秩分解的精确Kullback-Leibler（KL）投影，该分解具有两个非负因子，这些因子具有规定的行边缘和共享的、学习得到的列边缘。对于总质量相等的任意正行边缘，联合KL投影可精确简化为具有r-1个有效变量的严格凸规范固定对偶；其Hessian是分类协方差项的和，允许O((n+m)r)的无矩阵Hessian-向量乘积。该投影定理与目标无关。随后，我们将此几何结构专门应用于双随机（DS）图学习，通过W=U Diag(g)⁻¹Vᵀ实现，其中具有公共列质量的行单纯形因子可生成精确的双随机图，无需实例化n×n的优化变量。结合观测边的稀疏拟合、随机锚定降维流形正则化器和Bregman回溯，所得的镜像下降方法在每个接受的步骤都保持精确可行性。在非消失潜在质量条件下，该方法满足充分下降和O(1/N)的镜像平稳性界，而严格正的累积点为KKT平稳点。匹配的聚类实验显示出具有竞争力的准确率、接近数值精度的可行性残差，以及无需学习密集图的良好任意时刻性能。
英文摘要
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.
