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Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion

HanQin Cai, Jian-Feng Cai, Juntao You

TL;DR

The paper addresses robust low-rank Hankel matrix completion from partial, Hankel-structured observations corrupted by sparse outliers. It introduces Hankel Structured Gradient Descent (HSGD), a non-convex, factorized algorithm that enforces Hankel structure and low rank while iteratively estimating outliers, achieving linear convergence under mild sampling and incoherence assumptions. Theoretical results establish local linear convergence, initialization guarantees, and favorable sample and outlier-tolerance bounds, with improved convergence speed in well-conditioned cases. Empirical results on synthetic data and large-scale NMR signals demonstrate that HSGD is faster and more robust than state-of-the-art methods, making it practical for large-scale, structured matrix completion tasks.

Abstract

We study the robust matrix completion problem for the low-rank Hankel matrix, which detects the sparse corruptions caused by extreme outliers while we try to recover the original Hankel matrix from the partial observation. In this paper, we explore the convenient Hankel structure and propose a novel non-convex algorithm, coined Hankel Structured Gradient Descent (HSGD), for large-scale robust Hankel matrix completion problems. HSGD is highly computing- and sample-efficient compared to the state-of-the-arts. The recovery guarantee with a linear convergence rate has been established for HSGD under some mild assumptions. The empirical advantages of HSGD are verified on both synthetic datasets and real-world nuclear magnetic resonance signals.

Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion

TL;DR

The paper addresses robust low-rank Hankel matrix completion from partial, Hankel-structured observations corrupted by sparse outliers. It introduces Hankel Structured Gradient Descent (HSGD), a non-convex, factorized algorithm that enforces Hankel structure and low rank while iteratively estimating outliers, achieving linear convergence under mild sampling and incoherence assumptions. Theoretical results establish local linear convergence, initialization guarantees, and favorable sample and outlier-tolerance bounds, with improved convergence speed in well-conditioned cases. Empirical results on synthetic data and large-scale NMR signals demonstrate that HSGD is faster and more robust than state-of-the-art methods, making it practical for large-scale, structured matrix completion tasks.

Abstract

We study the robust matrix completion problem for the low-rank Hankel matrix, which detects the sparse corruptions caused by extreme outliers while we try to recover the original Hankel matrix from the partial observation. In this paper, we explore the convenient Hankel structure and propose a novel non-convex algorithm, coined Hankel Structured Gradient Descent (HSGD), for large-scale robust Hankel matrix completion problems. HSGD is highly computing- and sample-efficient compared to the state-of-the-arts. The recovery guarantee with a linear convergence rate has been established for HSGD under some mild assumptions. The empirical advantages of HSGD are verified on both synthetic datasets and real-world nuclear magnetic resonance signals.
Paper Structure (14 sections, 13 theorems, 113 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 13 theorems, 113 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Assumptions and notation
  3. Related work and main contributions
  4. Proposed method
  5. Recovery guarantee
  6. Numerical experiments
  7. Synthetic examples
  8. Nuclear magnetic resonance spectroscopy
  9. Conclusion remarks
  10. Proofs of theoretical results
  11. Technical lemmas
  12. Proof of Theorem \ref{['thm:Initialization']} (guaranteed initialization)
  13. Proof of Theorem \ref{['thm:convergence']} (local convergence)
  14. Supporting lemmas

Key Result

Theorem 1

Suppose amp:Bernoulliamp:incoherenceamp:sparsity hold with $p\gtrsim \mathcal{O} ((c_s^2\mu^2 r^2\log n)/n)$ and $\alpha\lesssim \mathcal{O} (1/( c_s\mu r\kappa^2))$. Choose the parameters $\lambda=1/16$, $\gamma_k\in [1 + 1/b_0,2]$ with some fixed $b_0\geq 1$, and $\eta=\tilde{c}/\sigma_1^{\natural

Figures (5)

  • Figure 1: Empirical phase transition for HSGD, PartialSAP, and RobustEMaC: Rank vs. rate of outliers. $50$ entries are sampled in all testing problems.
  • Figure 2: Empirical phase transition for HSGD, PartialSAP, and RobustEMaC: Number of samples vs. rate of outliers. All testing problems have rank $10$.
  • Figure 3: Empirical phase transition for HSGD, PartialSAP, and RobustEMaC: Number of samples vs. rank. $10\%$ of samples are corrupted by outliers in all testing problems.
  • Figure 4: Experimental results for speed tests between HSGD and PartialSAP. Left: Dimension vs. runtime. Middle: Dimension vs. runtime with error bar for HSGD only. Right: Relative error vs. runtime.
  • Figure 5: Power spectrum of the noisy original NMR signal, the observed signal ($p=30\%$ and $\alpha=30\%$), and HSGD recovered signal (upside down). Note that the observed signal in picture is rescaled by $1/p$, which is a common method to offset the energy loss due to partial observation.

Theorems & Definitions (29)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 19 more