Low-Rank plus Sparse Decomposition of Covariance Matrices using Neural Network Parametrization
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This paper revisits the problem of decomposing a positive semidefinite matrix as a sum of a matrix with a given rank plus a sparse matrix. An immediate application can be found in portfolio optimization, when the matrix to be decomposed is the covariance between the different assets in the portfolio. Our approach consists in representing the low-rank part of the solution as the product MM^T, where M is a rectangular matrix of appropriate size, parameterized by the coefficients of a deep neural network. We then use a gradient descent algorithm to minimize an appropriate loss function over the parameters of the network. We deduce its convergence speed to a local optimum from the Lipschitz smoothness of our loss function. We show that the rate of convergence grows polynomially in the dimensions of the input, output, and each of the hidden layers and hence conclude that our algorithm does not suffer from the curse of dimensionality.
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