Low-Rank Plus Sparse Decomposition of Covariance Matrices Using Neural Network Parametrization

Area of Science:

• Linear Algebra
• Machine Learning
• Optimization

Background:

• Matrix decomposition is crucial in various fields, including finance.
• Existing methods for low-rank and sparse decomposition have limitations.

Purpose of the Study:

• To develop a novel deep learning approach for decomposing positive semidefinite matrices into low-rank and sparse components.
• To apply this decomposition to portfolio optimization problems.

Main Methods:

• Representing the low-rank component as M*M^T, where M is parameterized by a deep neural network.
• Employing a gradient descent algorithm to minimize a loss function over network parameters.
• Analyzing convergence rates based on Lipschitz smoothness of the loss function.

Main Results:

• The deep neural network approach effectively decomposes matrices into low-rank and sparse parts.
• Convergence rate to a local optimum is shown to grow polynomially with input-output dimensions and hidden layer sizes.

Conclusions:

• The proposed deep learning framework offers an efficient method for matrix decomposition.
• This technique has direct applications in portfolio optimization by analyzing asset covariance matrices.