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On decomposability and subdifferential of the tensor nuclear norm

Jiewen Guan, Bo Jiang, Zhening Li

TL;DR

The paper advances the theory of tensor nuclear norms by establishing full decomposability over carefully chosen subspaces for tensors of any order and by deriving new subdifferential inclusions that expand beyond prior results. It also proves dual-norm based decomposability results for the tensor spectral norm and applies these insights to analyze tensor robust PCA, providing exact recovery guarantees for tensors of arbitrary order. The work presents a detailed subspace analysis of subgradients, introduces improved decomposability via $U^I$-type subspaces, and develops a golfing-scheme construction for dual certificates. Collectively, these contributions deepen the understanding of low-rank tensor optimization and yield a first general-order statistical result for tensor robust PCA. The results have potential implications for tensor completion, regression, and related convex-tensor optimization problems, and point to future directions on tightening constants and removing logarithmic factors in sample-size requirements.

Abstract

We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a full decomposability over specific subspaces and determine the largest possible subspaces that allow the full decomposability. We derive novel inclusions of the subdifferential of the tensor nuclear norm and study its subgradients in a variety of subspaces of interest. All the results hold for tensors of an arbitrary order. As an immediate application, we establish the statistical performance of the tensor robust principal component analysis, the first such result for tensors of an arbitrary order.

On decomposability and subdifferential of the tensor nuclear norm

TL;DR

The paper advances the theory of tensor nuclear norms by establishing full decomposability over carefully chosen subspaces for tensors of any order and by deriving new subdifferential inclusions that expand beyond prior results. It also proves dual-norm based decomposability results for the tensor spectral norm and applies these insights to analyze tensor robust PCA, providing exact recovery guarantees for tensors of arbitrary order. The work presents a detailed subspace analysis of subgradients, introduces improved decomposability via -type subspaces, and develops a golfing-scheme construction for dual certificates. Collectively, these contributions deepen the understanding of low-rank tensor optimization and yield a first general-order statistical result for tensor robust PCA. The results have potential implications for tensor completion, regression, and related convex-tensor optimization problems, and point to future directions on tightening constants and removing logarithmic factors in sample-size requirements.

Abstract

We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a full decomposability over specific subspaces and determine the largest possible subspaces that allow the full decomposability. We derive novel inclusions of the subdifferential of the tensor nuclear norm and study its subgradients in a variety of subspaces of interest. All the results hold for tensors of an arbitrary order. As an immediate application, we establish the statistical performance of the tensor robust principal component analysis, the first such result for tensors of an arbitrary order.

Paper Structure

This paper contains 27 sections, 42 theorems, 188 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preparations
  3. Basic notations
  4. Tensor operations
  5. Tensor spectral norm and nuclear norm
  6. Tensors subspaces
  7. Subspaces
  8. Orthogonal transformations and projections
  9. Tensor spectral and nuclear norms over subspaces
  10. Decomposability of the tensor nuclear norm
  11. A natural decomposability
  12. An improved decomposability
  13. Subdifferential of the tensor nuclear norm
  14. Decomposability and subdifferential
  15. Subgradients in subspaces
  16. ...and 12 more sections

Key Result

Lemma 2.1

If two tensors $\mathbf{T},\mathbf{S}\in\mathbb{R}^{n_1\times n_2\times\dots\times n_d}$, then Moreover,

Figures (3)

  • Figure 1: The four basic subspaces of $\mathbb{R}^{n_1\times n_2}$.
  • Figure 2: Subspaces of $\mathbb{R}^{n_1\times n_2\times n_3}$ presented by shaded blocks where each block represents a basic subspace of $\mathbb{R}^{n_1\times n_2\times n_3}$ and the union of all eight blocks represents $\mathbb{R}^{n_1\times n_2\times n_3}$ in each subfigure.
  • Figure 3: Subspaces of $\mathbb{R}^{n_1\times n_2\times n_3}$ presented by shaded blocks where the union of all blocks represents $\mathbb{R}^{n_1\times n_2\times n_3}$ in each subfigure.

Theorems & Definitions (83)

  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Example 4.1
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 73 more