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Near-optimal Rank Adaptive Inference of High Dimensional Matrices

Frédéric Zheng, Yassir Jedra, Alexandre Proutiere

TL;DR

This work addresses high-dimensional matrix estimation from linear measurements by developing rank-adaptive inference that learns the effective rank from data. It introduces instance-specific lower bounds that depend on the target matrix spectrum and covariate design, and proposes a Thresholded Least Squares Estimator (T-LSE) that combines a Least Squares estimator with universal singular value thresholding to achieve near-optimal performance. Theoretical results provide finite-sample, rate-optimal bounds in multivariate regression and linear dynamical system identification, along with a refined analysis of matrix denoising via singular value thresholding. Empirical results corroborate the theory, showing that adaptivity yields substantial gains when the singular spectrum decays or alignment between design and target is imperfect, with potential impact on multivariate regression and dynamic system identification tasks.

Abstract

We address the problem of estimating a high-dimensional matrix from linear measurements, with a focus on designing optimal rank-adaptive algorithms. These algorithms infer the matrix by estimating its singular values and the corresponding singular vectors up to an effective rank, adaptively determined based on the data. We establish instance-specific lower bounds for the sample complexity of such algorithms, uncovering fundamental trade-offs in selecting the effective rank: balancing the precision of estimating a subset of singular values against the approximation cost incurred for the remaining ones. Our analysis identifies how the optimal effective rank depends on the matrix being estimated, the sample size, and the noise level. We propose an algorithm that combines a Least-Squares estimator with a universal singular value thresholding procedure. We provide finite-sample error bounds for this algorithm and demonstrate that its performance nearly matches the derived fundamental limits. Our results rely on an enhanced analysis of matrix denoising methods based on singular value thresholding. We validate our findings with applications to multivariate regression and linear dynamical system identification.

Near-optimal Rank Adaptive Inference of High Dimensional Matrices

TL;DR

This work addresses high-dimensional matrix estimation from linear measurements by developing rank-adaptive inference that learns the effective rank from data. It introduces instance-specific lower bounds that depend on the target matrix spectrum and covariate design, and proposes a Thresholded Least Squares Estimator (T-LSE) that combines a Least Squares estimator with universal singular value thresholding to achieve near-optimal performance. Theoretical results provide finite-sample, rate-optimal bounds in multivariate regression and linear dynamical system identification, along with a refined analysis of matrix denoising via singular value thresholding. Empirical results corroborate the theory, showing that adaptivity yields substantial gains when the singular spectrum decays or alignment between design and target is imperfect, with potential impact on multivariate regression and dynamic system identification tasks.

Abstract

We address the problem of estimating a high-dimensional matrix from linear measurements, with a focus on designing optimal rank-adaptive algorithms. These algorithms infer the matrix by estimating its singular values and the corresponding singular vectors up to an effective rank, adaptively determined based on the data. We establish instance-specific lower bounds for the sample complexity of such algorithms, uncovering fundamental trade-offs in selecting the effective rank: balancing the precision of estimating a subset of singular values against the approximation cost incurred for the remaining ones. Our analysis identifies how the optimal effective rank depends on the matrix being estimated, the sample size, and the noise level. We propose an algorithm that combines a Least-Squares estimator with a universal singular value thresholding procedure. We provide finite-sample error bounds for this algorithm and demonstrate that its performance nearly matches the derived fundamental limits. Our results rely on an enhanced analysis of matrix denoising methods based on singular value thresholding. We validate our findings with applications to multivariate regression and linear dynamical system identification.

Paper Structure

This paper contains 46 sections, 25 theorems, 122 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation.
  3. Preliminaries
  4. Related Work
  5. Instance-specific Sample Complexity Lower Bounds
  6. Lower bounds for rank-constrained algorithms
  7. Lower bounds for rank-adaptive algorithms
  8. Matrix Denoising via Singular Value Thresholding
  9. The Thresholded Least Squares Estimator
  10. Multivariate regression
  11. Linear System Identification
  12. Conclusion
  13. Proofs of results presented in Section \ref{['sec:lb']}
  14. Tools for the change-of-measure argument
  15. Computation of the expected log-likelihood ratio.
  16. ...and 31 more sections

Key Result

Lemma 4.1

For any $k\leq d / 2$, there exists a finite $\sqrt{k}$-packing ${\cal P}_k^d$ of $\mathrm{St}_k^{d}(\mathbb{R})$ whose cardinality is larger than $2^{kd}$. By $\sqrt{k}$-packing, we mean that for any $Q, R \in {\cal P}_{k}^d$, $Q\neq R$ implies $\Vert Q-R\Vert_{\textup{F}}\ge \sqrt{k}/C$ where $C\

Figures (4)

  • Figure 1: Frobenius error (left) and effective rank (right) vs noise level $b$, $(r=10)$.
  • Figure 2: Comparison of Lemma \ref{['lem:trade_off_k']} with the upper bound from chatterjee2015matrix as a function of the threshold $\tau$.
  • Figure 3: Multivariate regression: Frobenius error (left) and effective rank (right) vs noise level $b$, $(r=45)$.
  • Figure 4: System identification: Frobenius error (left) and effective rank (right) vs noise level $b$, $(r=10)$.

Theorems & Definitions (50)

  • Lemma 4.1
  • Definition 4.2: $(\varepsilon,\delta, r)$-stability
  • Theorem 4.3
  • Definition 4.4: $(\varepsilon,\delta)$-stability
  • Theorem 4.5
  • Theorem 5.1
  • Lemma 5.2
  • Lemma 6.1
  • Theorem 6.2
  • Theorem 6.3
  • ...and 40 more