用神经网络学习矩阵奇异值

Derek Xu, William Shiao, Jia Chen, E. Papalexakis
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引用次数: 0

摘要

奇异值分解(SVD)将一个矩阵分解为三个独立的矩阵:两个(半)酉矩阵,其列是左/右奇异向量,一个对角矩阵,其对角项是奇异值。通常,在大矩阵上执行SVD是很费力的,因为它在维度的三次顺序上的计算复杂性。随着深度学习技术在广泛应用中的进步和快速发展,一个基本问题出现了:深度神经网络能学习矩阵的奇异值吗?为了回答这个问题,我们提出了一种新的算法,即sv - learn,通过利用神经网络的进步来预测给定输入矩阵的奇异值。数值结果表明,在使用实际数据集进行奇异值预测时,我们提出的方法在实现更低的归一化均方误差方面优于竞争方案。此外,预测的奇异值与输入数据的奇异向量相结合,使我们能够重建具有良好性能的输入矩阵。
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SV-Learn: Learning Matrix Singular Values with Neural Networks
The singular value decomposition (SVD) factors a matrix into three separate matrices: two (semi-)unitary matrices whose columns are left/right singular vectors and one diagonal matrix whose diagonal entries are singular values. Typically, performing SVD on big matrices is taxing due to its compu-tational complexity in the cubic order of its dimensions. With the advances and rapid growth of deep learning techniques in a broad spectrum of applications, a fundamental question arises: can deep neural networks learn the singular values of a matrix? To answer this question, we propose a novel algorithm, namely SV-Iearn, to predict the singular values of a given input matrix by leveraging the advances of neural networks. Numerical results demonstrate that our proposed method outperforms the competing alternatives in terms of achieving lower normalized mean square error on singular value prediction when using real-world datasets. Further, the predicted singular values combined with singular vectors of an input data allow us to reconstruct the input matrices with promising performance.
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