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Efficient Federated Low Rank Matrix Recovery via Alternating GD and Minimization: A Simple Proof

Namrata Vaswani

TL;DR

This work presents a streamlined proof of the sample complexity guarantee for solving the LRCS problem via Alternating Gradient Descent and Minimization (AltGDmin). By switching to the 2-norm subspace distance ${\mathrm{SD}}_2$ and employing a direct gradient-based analysis, the authors obtain an improved per-iteration and initialization sample complexity, with ${m q}\ge C\kappa^4\mu^2(n+q) r(\kappa^4 r+\log(1/\varepsilon))$ and a contraction in ${\mathrm{SD}}_2({\bm U},{\bm U}^*)$ at each GDmin step. Initialization is handled through truncated spectral methods and Wedin’s $\sin\Theta$ theorem, ensuring a good starting subspace with ${\mathrm{SD}}_2({\bm U}_0,{\bm U}^*)\le \delta_0$ under reasonable sampling. The results yield faster convergence and reduced sample requirements in federated LR matrix recovery, with potential impact on dynamic MRI and multi-task learning scenarios where column-wise measurements are distributed. All mathematical notation is carefully bounded within $...$ to support precise reproducibility and SEO relevance.

Abstract

This note provides a significantly simpler and shorter proof of our sample complexity guarantee for solving the low rank column-wise sensing problem using the Alternating Gradient Descent (GD) and Minimization (AltGDmin) algorithm. AltGDmin was developed and analyzed for solving this problem in our recent work. We also provide an improved guarantee.

Efficient Federated Low Rank Matrix Recovery via Alternating GD and Minimization: A Simple Proof

TL;DR

This work presents a streamlined proof of the sample complexity guarantee for solving the LRCS problem via Alternating Gradient Descent and Minimization (AltGDmin). By switching to the 2-norm subspace distance and employing a direct gradient-based analysis, the authors obtain an improved per-iteration and initialization sample complexity, with and a contraction in at each GDmin step. Initialization is handled through truncated spectral methods and Wedin’s theorem, ensuring a good starting subspace with under reasonable sampling. The results yield faster convergence and reduced sample requirements in federated LR matrix recovery, with potential impact on dynamic MRI and multi-task learning scenarios where column-wise measurements are distributed. All mathematical notation is carefully bounded within to support precise reproducibility and SEO relevance.

Abstract

This note provides a significantly simpler and shorter proof of our sample complexity guarantee for solving the low rank column-wise sensing problem using the Alternating Gradient Descent (GD) and Minimization (AltGDmin) algorithm. AltGDmin was developed and analyzed for solving this problem in our recent work. We also provide an improved guarantee.
Paper Structure (17 sections, 9 theorems, 46 equations, 1 algorithm)

This paper contains 17 sections, 9 theorems, 46 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement, notation, and algorithm
  3. Problem statement and assumption and notation
  4. Notation
  5. Review of AltGDmin algorithm lrpr_gdmin
  6. New Guarantee
  7. Discussion
  8. New proof: GDmin iterations
  9. Definitions and preliminaries
  10. Minimization step
  11. New bounds on the expected gradient and deviation from it
  12. GD step
  13. New proof: Initialization
  14. Results taken from lrpr_gdmin
  15. Obtaining the SD error bound for initialization
  16. ...and 2 more sections

Key Result

Theorem 3.1

Assume that Assumption right_incoh holds. Set $\eta = 0.4 / m {\sigma_{\max}^\star}^2$ and $T = C \kappa^2 \log( 1 /\epsilon)$. If and $m \ge C \max(\log n, \log q, r) \log (1/\epsilon)$, then, with probability (w.p.) at least $1 - n^{-10}$, The time complexity is $mqnr \cdot T = mqnr \cdot \kappa^2 \log( 1 /\epsilon)$. The communication complexity is $nr\cdot T = nr \cdot \kappa^2 \log( 1 /\e

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Lemma 4.1: lrpr_gdmin
  • proof
  • Theorem 4.2
  • Lemma 4.3
  • Theorem 4.4
  • Lemma 5.1: lrpr_gdmin
  • Lemma 5.4: lrpr_gdmin
  • Corollary 5.5
  • ...and 1 more