Guaranteed Nonconvex Low-Rank Tensor Estimation via Scaled Gradient Descent

• URL: http://arxiv.org/abs/2501.01696v1
• Date: Fri, 03 Jan 2025 08:26:01 GMT
• Title: Guaranteed Nonconvex Low-Rank Tensor Estimation via Scaled Gradient Descent
• Authors: Tong Wu,
• Abstract summary: This paper develops a scaled gradient descent (ScaledGD) algorithm to directly estimate the tensor factors. In theory, ScaledGD achieves linear convergence at a constant rate that is independent of the condition number of the ground truth low-rank tensor. numerical examples are provided to demonstrate the efficacy of ScaledGD in accelerating the convergence rate of ill-conditioned low-rank tensor estimation.
• Score: 4.123899820318987
• License:
• Abstract: Tensors, which give a faithful and effective representation to deliver the intrinsic structure of multi-dimensional data, play a crucial role in an increasing number of signal processing and machine learning problems. However, tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. A fundamental challenge is to reliably extract the meaningful information from corrupted tensor data in a statistically and computationally efficient manner. This paper develops a scaled gradient descent (ScaledGD) algorithm to directly estimate the tensor factors with tailored spectral initializations under the tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) framework. In theory, ScaledGD achieves linear convergence at a constant rate that is independent of the condition number of the ground truth low-rank tensor for two canonical problems -- tensor robust principal component analysis and tensor completion -- as long as the level of corruptions is not too large and the sample size is sufficiently large, while maintaining the low per-iteration cost of gradient descent. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties for low-rank tensor estimation with the t-SVD decomposition. Finally, numerical examples are provided to demonstrate the efficacy of ScaledGD in accelerating the convergence rate of ill-conditioned low-rank tensor estimation in these two applications.

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