Title: Local and Global Convergence of General Burer-Monteiro Tensor Optimizations
Abstract: Tensors, a multi-dimensional generalization of vectors and matrices, provide natural representations for multi-way datasets and find numerous applications in signal processing and machine learning. In this work, we consider tensor optimization with a convex and well-conditioned objective function and reformulate it into a non-convex optimization using the Burer-Monteiro type parameterization. We analyze the local convergence of applying vanilla gradient descent to the factored formulation and establish a local regularity condition under mild assumptions. We also provide a linear convergence analysis of the gradient descent algorithm started in a neighborhood of the true tensor factors. Complementary to the local analysis, this work also characterizes the global geometry of a best rank-one tensor approximation problem and demonstrates that for orthogonally decomposable tensors the problem has no spurious local minima and all saddle points are strict except for the one at zero which is a third-order saddle point.
Bio: Shuang Li received the Ph.D. degree in Electrical Engineering from the Colorado School of Mines, Golden, CO, USA, in 2020. She is currently an Assistant Professor in the Department of Electrical and Computer Engineering at Iowa State University. She was a Hedrick Assistant Adjunct Professor with the Department of Mathematics, University of California, Los Angeles, CA, USA, from 2020 to 2023. Her research interests include developing optimization-based techniques with optimality guarantees for fundamental problems in signal processing, machine learning and data science. She was the recipient of the DARPA Riser Award in 2022, and CPAL Rising Stars Award in 2024.
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