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Tensor train completion: local recovery guarantees via Riemannian optimization

Stanislav Budzinskiy, Nikolai Zamarashkin

TL;DR

The paper addresses TT completion from limited observations by formulating it as a Riemannian optimization problem on the TT manifold $\mathcal{M}_{\bm{r}}$ and derives local convergence guarantees for RGD under RIP-type conditions. A curvature bound based on the harmonic mean of the TT unfoldings' smallest singular values is established, along with a core-coherence measure $\mu_C$ to quantify TT incoherence. The authors prove that a tangent-space RIP holds with high probability for random samples and derive explicit sample-size bounds for TT recovery and TT completion, including extensions to auxiliary subspace information that reduce the required samples. These results quantify when TT-based methods can reliably recover or complete high-dimensional tensors from sparse observations and provide guidance for sample complexity in large-scale settings.

Abstract

In this work, we estimate the number of randomly selected elements of a tensor that with high probability guarantees local convergence of Riemannian gradient descent for tensor train completion. We derive a new bound for the orthogonal projections onto the tangent spaces based on the harmonic mean of the unfoldings' singular values and introduce a notion of core coherence for tensor trains. We also extend the results to tensor train completion with auxiliary subspace information and obtain the corresponding local convergence guarantees.

Tensor train completion: local recovery guarantees via Riemannian optimization

TL;DR

The paper addresses TT completion from limited observations by formulating it as a Riemannian optimization problem on the TT manifold and derives local convergence guarantees for RGD under RIP-type conditions. A curvature bound based on the harmonic mean of the TT unfoldings' smallest singular values is established, along with a core-coherence measure to quantify TT incoherence. The authors prove that a tangent-space RIP holds with high probability for random samples and derive explicit sample-size bounds for TT recovery and TT completion, including extensions to auxiliary subspace information that reduce the required samples. These results quantify when TT-based methods can reliably recover or complete high-dimensional tensors from sparse observations and provide guidance for sample complexity in large-scale settings.

Abstract

In this work, we estimate the number of randomly selected elements of a tensor that with high probability guarantees local convergence of Riemannian gradient descent for tensor train completion. We derive a new bound for the orthogonal projections onto the tangent spaces based on the harmonic mean of the unfoldings' singular values and introduce a notion of core coherence for tensor trains. We also extend the results to tensor train completion with auxiliary subspace information and obtain the corresponding local convergence guarantees.

Paper Structure

This paper contains 19 sections, 21 theorems, 208 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Our contributions
  4. Curvature bound
  5. Local convergence of Riemannian gradient descent
  6. Tensor recovery
  7. Tensor completion
  8. Recovery guarantees
  9. Tensor train completion with auxiliary subspace information
  10. Proofs of main results
  11. Curvature bound
  12. Local convergence of Riemannian gradient descent: tensor train recovery
  13. Local convergence of Riemannian gradient descent: tensor train completion
  14. Recovery guarantees
  15. Discussion
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let the matrix $A$ have incoherent column and row spaces intro:eq:incoherent and assume that the index set $\Omega$ is chosen uniformly at random with Then the RIP intro:eq:rip holds with high probability.

Figures (2)

  • Figure 1: Numerically computed median of $\prod_{j = 1}^k \chi^2(5)$.
  • Figure 2: Phase plot of the Riemannian gradient descent for $n = 50$, $r = 3$, and varying number of dimensions $d$. The values between 0 and 1 are the frequencies of successful recovery for the given parameters. The orange (solid) and blue (dashed) curves correspond to $|\Omega| = d^2 r^2 n \log(n) / 10$ and $|\Omega| = d^{2.2} r^2 n \log(n) / 10$, respectively.

Theorems & Definitions (38)

  • Theorem 1.1: CandesRechtExact2009a, Theorem 4.1 and RechtSimpler2011, Theorem 6
  • Theorem 1.2: DingChenLeave2020, Theorem 2
  • Theorem 1.3: WeiEtAlGuarantees2016, Theorem 2.2
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Lemma 3.6
  • ...and 28 more