Computing eigenvalue decomposition (EVD) of a given linear operator, or finding its leading eigenvalues and eigenfunctions, is a fundamental task in many machine learning and scientific computing problems. For high-dimensional eigenvalue problems, training neural networks to parameterize the eigenfunctions is considered as a promising alternative to the classical numerical linear algebra techniques. This paper proposes a new optimization framework based on the low-rank approximation characterization of a truncated singular value decomposition, accompanied by new techniques called nesting for learning the top-$L$ singular values and singular functions in the correct order. The proposed method promotes the desired orthogonality in the learned functions implicitly and efficiently via an unconstrained optimization formulation, which is easy to solve with off-the-shelf gradient-based optimization algorithms. We demonstrate the effectiveness of the proposed optimization framework for use cases in computational physics and machine learning.

翻译：计算给定线性算子的特征值分解（EVD），或求其主导特征值与特征函数，是众多机器学习与科学计算问题中的基础任务。针对高维特征值问题，采用神经网络参数化特征函数被视为经典数值线性代数技术的一种有前景的替代方案。本文提出一种基于截断奇异值分解低秩逼近特性的新型优化框架，并引入名为"嵌套"的新技术，以按正确顺序学习前$L$个奇异值及奇异函数。所提方法通过无约束优化公式隐式且高效地促进学习函数所需的正交性，该公式易于利用现成的梯度优化算法求解。我们通过计算物理与机器学习领域的应用案例，验证了所提优化框架的有效性。