发表机构
Frontier Research Center, Toyota Motor Corporation（丰田汽车公司前沿研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Monarch矩阵逼近发展了几何理论，刻画了其Bruhat型胞腔结构，证明等块情形的Monarch-Schur定理，并引入四次正则化器恢复梯度流全局收敛性，为结构化分解层中的权重衰减提供几何解释。
AI 中文摘要
Helmke和Shayman证明了，对于一般目标，到秩为r的矩阵流形的最小二乘距离是一个非退化的Morse函数，其指标由Bruhat胞腔的余维数决定。我们为Monarch矩阵发展了类似的理论，Monarch矩阵是一个固定置换矩阵与两个块对角因子的乘积。参数化的规范维度以及相关Bruhat型胞腔的余维数和闭包关系，由一个共同的传递矩阵不变量决定，尽管这些量通常是不同的。在等块情形下，我们证明了Monarch-Schur定理，刻画了使规范自由度最小化并使流形维度最大化的置换，并推导了任意划分的一般下界。我们进一步建立了二阶非退化性、一个Morse指标公式以及一个显式的零测度焦点集。最后，我们证明无正则化的梯度流因无界平坦方向而缺乏强制性。一个四次正则化器恢复了全局收敛性，在存在精确解时选择规范的最小范数精确解，否则产生定义良好的最佳逼近图像。该正则化器保留了Monarch架构，并为结构化分解层中的权重衰减提供了几何解释。
英文摘要
Helmke and Shayman showed that, for generic targets, least-squares distance to the manifold of rank-r matrices is a nondegenerate Morse function whose indices are governed by Bruhat-cell codimensions. We develop the analogous theory for Monarch matrices, products of a fixed permutation matrix with two block-diagonal factors. The parametrization's gauge dimension, as well as the codimensions and closure relations of the associated Bruhat-type cells, is determined by a common transfer-matrix invariant, although these quantities are generally distinct. In the equal-block case, we prove the Monarch-Schur theorem, characterizing the permutations that minimize gauge freedom and maximize manifold dimension, and derive a general lower bound for arbitrary partitions. We further establish generic second-order nondegeneracy, a Morse-index formula, and an explicit focal set of measure zero. Finally, we show that the unregularized gradient flow lacks coercivity because of unbounded flat directions. A quartic regularizer restores global convergence, selects a canonical minimum-norm exact solution when one exists, and otherwise yields a well-defined best-approximation image. The regularizer preserves the Monarch architecture and provides a geometric interpretation of weight decay in structured factorized layers.
Comments28 pages
