半正定因子分解的Lee-Seung算法的非交换推广
A Non-commutative Extension of Lee-Seung's Algorithm for Positive Semidefinite Factorizations
摘要：
给定具有非负项的矩阵$X\in \mathbb{R}_+^{m\times n}$，$X\in \mathbb{R}_+^{m\times n}$的半正定（PSD）分解是$r \times r$维PSD矩阵$\{A_i\}$和$\{B_j\}$的集合，其对于所有$\ i\in [m],\ j\in [n]$满足$X_{ij}= \mathrm{tr}(A_i B_j)$. PSD因子分解从根本上与理解半定程序的表达性以及信息论中量子资源的能力和局限性有关。PSD因式分解任务推广了非负矩阵因式分解（NMF）问题，其中我们寻求满足$X_{ij}= a_i^\top b_j$的$r$维非负向量$\{a_i\}$和$\{b_j\}$的集合，对于所有$i\in [m],\ j\in [n]$-可以通过将PSD因式分解中的矩阵选择为对角线来恢复后一个问题。计算矩阵的NMF的最广泛使用的算法是由Lee和Seung开发的乘法更新算法，其中更新的非负性通过用正对角矩阵缩放来保持。在本文中，我们描述了一个非交换的扩展李Seung的算法，我们称之为矩阵乘法更新（MMU）算法，计算PSD分解。MMU算法通过与适当PSD矩阵的矩阵几何平均值进行同余缩放来确保更新保持PSD，并且它保留了Lee-Seung算法所享有的实现的简单性。基于优化最小化框架，我们证明了在我们的更新方案下，平方损失目标是非增的，不动点对应于临界点。分析依赖于Lieb的凹定理。除了PSD因子分解，我们使用MMU算法作为计算块对角PSD因子分解和张量PSD因子分解的原语。我们证明了我们的方法的实用性与真实和合成数据的实验。
英文摘要：
Given a matrix $X\in \mathbb{R}_+^{m\times n}$ with nonnegative entries, a Positive Semidefinite (PSD) factorization of $X$ is a collection of $r \times r$-dimensional PSD matrices $\{A_i\}$ and $\{B_j\}$ satisfying $X_{ij}= \mathrm{tr}(A_i B_j)$ for all $\ i\in [m],\ j\in [n]$. PSD factorizations are fundamentally linked to understanding the expressiveness of semidefinite programs as well as the power and limitations of quantum resources in information theory. The PSD factorization task generalizes the Non-negative Matrix Factorization (NMF) problem where we seek a collection of $r$-dimensional nonnegative vectors $\{a_i\}$ and $\{b_j\}$ satisfying $X_{ij}= a_i^\top b_j$, for all $i\in [m],\ j\in [n]$ -- one can recover the latter problem by choosing matrices in the PSD factorization to be diagonal. The most widely used algorithm for computing NMFs of a matrix is the Multiplicative Update algorithm developed by Lee and Seung, in which nonnegativity of the updates is preserved by scaling with positive diagonal matrices. In this paper, we describe a non-commutative extension of Lee-Seung's algorithm, which we call the Matrix Multiplicative Update (MMU) algorithm, for computing PSD factorizations. The MMU algorithm ensures that updates remain PSD by congruence scaling with the matrix geometric mean of appropriate PSD matrices, and it retains the simplicity of implementation that Lee-Seung's algorithm enjoys. Building on the Majorization-Minimization framework, we show that under our update scheme the squared loss objective is non-increasing and fixed points correspond to critical points. The analysis relies on Lieb's Concavity Theorem. Beyond PSD factorizations, we use the MMU algorithm as a primitive to calculate block-diagonal PSD factorizations and tensor PSD factorizations. We demonstrate the utility of our method with experiments on real and synthetic data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
