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<p>Low-rank tensor approximations with closed and convex constraints, e.g., nonnegative entries or Lorentz cones, have become increasingly important and ubiquitous in the era of data analysis. Proposing a unified approach to solve this type-dependent problem is challenging. We, in this paper, present a spectral projective gradient method to find a structured tensor factorization over closed convex sets. This factorization is a non-convex optimization problem, and even finding an exact nonnegative tensor approximation of dimension 2 is NP-hard. In addition to the classical choice of steplength, an available spectral projected gradient direction is incorporated to alleviate further trial projections during the search process. Convergence properties for this updated procedure are given. In numerical results, we see that this approach exhibits a unified strategy and remarkable performance in various well-known problems.</p>

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A modified spectral projected gradient method for tensor approximations over closed convex sets

  • 期刊论文

  • 发表日期：

    2025年1月11日

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  • Matthew M. Lin,
  • Chieu Thanh Nguyen

摘要

Low-rank tensor approximations with closed and convex constraints, e.g., nonnegative entries or Lorentz cones, have become increasingly important and ubiquitous in the era of data analysis. Proposing a unified approach to solve this type-dependent problem is challenging. We, in this paper, present a spectral projective gradient method to find a structured tensor factorization over closed convex sets. This factorization is a non-convex optimization problem, and even finding an exact nonnegative tensor approximation of dimension 2 is NP-hard. In addition to the classical choice of steplength, an available spectral projected gradient direction is incorporated to alleviate further trial projections during the search process. Convergence properties for this updated procedure are given. In numerical results, we see that this approach exhibits a unified strategy and remarkable performance in various well-known problems.