We study the geometry of the algebraic set of tuples of composable matrices which multiply to a fixed matrix, using tools from the theory of quiver representations. In particular, we determine its codimension $C$ and the number $\theta$ of its top-dimensional irreducible components. Our solution is presented in three forms: a Poincar\'e series in equivariant cohomology, a quadratic integer program, and an explicit formula. In the course of the proof, we establish a surprising property: $C$ and $\theta$ are invariant under arbitrary permutations of the dimension vector. We also show that the real log-canonical threshold of the function taking a tuple to the square Frobenius norm of its product is $C/2$. These results are motivated by the study of deep linear neural networks in machine learning and Bayesian statistics (singular learning theory) and show that deep linear networks are in a certain sense ``mildly singular".

翻译：我们利用箭图表示论的工具，研究可组合矩阵元组在乘积为固定矩阵条件下的代数集几何结构。特别地，我们确定了该代数集的余维数 $C$ 及其最高维不可约分支的数量 $\theta$。我们的解以三种形式呈现：等变上同调中的庞加莱级数、二次整数规划以及显式公式。在证明过程中，我们揭示了一个令人惊讶的性质：$C$ 和 $\theta$ 在维数向量的任意置换下保持不变量。我们还证明了将元组映射至其乘积的平方Frobenius范数的函数的实对数典范阈值为 $C/2$。这些研究动机源于机器学习与贝叶斯统计学（奇异学习理论）中对深度线性神经网络的探讨，并表明深度线性网络在某种意义上是"轻度奇异"的。