通过随机Chebyshev展开实现大规模对数行列式计算
Large-scale Log-determinant Computation through Stochastic Chebyshev Expansions
AI总结：
本文提出一种线性时间随机算法，利用随机迹近似和Chebyshev多项式展开，高效计算大规模正定及一般非奇异矩阵的对数行列式，具有高精度和速度优势。
摘要：
对数行列式在机器学习中广泛应用，如高斯图模型、高斯过程模型、离散图模型的配分函数、最小体积椭球、度量学习和核学习。对数行列式计算通常涉及Cholesky分解，其计算复杂度为变量数的三次方，不适用于大规模应用。本文提出一种线性时间随机算法，通过随机迹近似（Hutchinson方法）和Chebyshev多项式展开，结合高效的矩阵向量乘法，实现对大规模正定和一般非奇异矩阵的对数行列式近似计算。我们建立了依赖输入矩阵条件数的严格加法和乘法近似误差界。实验表明，该算法在速度上比Cholesky分解和Schur补快多个数量级，并能计算包含数百万变量的矩阵的对数行列式。
英文摘要：
Logarithms of determinants of large positive definite matrices appear ubiquitously in machine learning applications including Gaussian graphical and Gaussian process models, partition functions of discrete graphical models, minimum-volume ellipsoids, metric learning and kernel learning. Log-determinant computation involves the Cholesky decomposition at the cost cubic in the number of variables, i.e., the matrix dimension, which makes it prohibitive for large-scale applications. We propose a linear-time randomized algorithm to approximate log-determinants for very large-scale positive definite and general non-singular matrices using a stochastic trace approximation, called the Hutchinson method, coupled with Chebyshev polynomial expansions that both rely on efficient matrix-vector multiplications. We establish rigorous additive and multiplicative approximation error bounds depending on the condition number of the input matrix. In our experiments, the proposed algorithm can provide very high accuracy solutions at orders of magnitude faster time than the Cholesky decomposition and Schur completion, and enables us to compute log-determinants of matrices involving tens of millions of variables.
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