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Uniform Recovery Guarantees for Quantized Corrupted Sensing Using Structured or Generative Priors

Junren Chen, Zhaoqiang Liu, Meng Ding, Michael K. Ng

TL;DR

The paper addresses uniform recovery in quantized corrupted sensing where measurements are contaminated by unknown corruption and then quantized via a dithered uniform quantizer. It develops a unified framework based on quantized product embedding (QPE) to enable uniform recovery guarantees for both structured priors (via constrained or unconstrained Lasso) and generative priors, using a single realization of a sub-Gaussian sensing matrix and dithering. The results show an $\mathcal{O}(m^{-1/2})$ decay in estimation error for both constrained Lasso with structured priors and for generative priors, with a detailed discussion of the cost of uniformity versus nonuniform results; unconstrained Lasso exhibits amplified dependence on the structured parameters. Experimental evaluations on synthetic and real data (e.g., MNIST, CelebA) validate the uniform recovery behavior under fixed sensing ensembles and illustrate the practical impact of quantization level $\delta$ and noise $E$ on recovery performance. Overall, the work provides a near-unified, QPE-based approach to uniform recovery in quantized corrupted sensing with both structured and generative priors, and offers refined insights into the trade-offs between uniformity, measurement count, and prior complexity.

Abstract

This paper studies quantized corrupted sensing where the measurements are contaminated by unknown corruption and then quantized by a dithered uniform quantizer. We establish uniform guarantees for Lasso that ensure the accurate recovery of all signals and corruptions using a single draw of the sub-Gaussian sensing matrix and uniform dither. For signal and corruption with structured priors (e.g., sparsity, low-rankness), our uniform error rate for constrained Lasso typically coincides with the non-uniform one [Sun, Cui and Liu, 2022] up to logarithmic factors. By contrast, our uniform error rate for unconstrained Lasso exhibits worse dependence on the structured parameters due to regularization parameters larger than the ones for non-uniform recovery. For signal and corruption living in the ranges of some Lipschitz continuous generative models (referred to as generative priors), we achieve uniform recovery via constrained Lasso with a measurement number proportional to the latent dimensions of the generative models. Our treatments to the two kinds of priors are (nearly) unified and share the common key ingredients of (global) quantized product embedding (QPE) property, which states that the dithered uniform quantization (universally) preserves inner product. As a by-product, our QPE result refines the one in [Xu and Jacques, 2020] under sub-Gaussian random matrix, and in this specific instance we are able to sharpen the uniform error decaying rate (for the projected-back projection estimator with signals in some convex symmetric set) presented therein from $O(m^{-1/16})$ to $O(m^{-1/8})$.

Uniform Recovery Guarantees for Quantized Corrupted Sensing Using Structured or Generative Priors

TL;DR

The paper addresses uniform recovery in quantized corrupted sensing where measurements are contaminated by unknown corruption and then quantized via a dithered uniform quantizer. It develops a unified framework based on quantized product embedding (QPE) to enable uniform recovery guarantees for both structured priors (via constrained or unconstrained Lasso) and generative priors, using a single realization of a sub-Gaussian sensing matrix and dithering. The results show an decay in estimation error for both constrained Lasso with structured priors and for generative priors, with a detailed discussion of the cost of uniformity versus nonuniform results; unconstrained Lasso exhibits amplified dependence on the structured parameters. Experimental evaluations on synthetic and real data (e.g., MNIST, CelebA) validate the uniform recovery behavior under fixed sensing ensembles and illustrate the practical impact of quantization level and noise on recovery performance. Overall, the work provides a near-unified, QPE-based approach to uniform recovery in quantized corrupted sensing with both structured and generative priors, and offers refined insights into the trade-offs between uniformity, measurement count, and prior complexity.

Abstract

This paper studies quantized corrupted sensing where the measurements are contaminated by unknown corruption and then quantized by a dithered uniform quantizer. We establish uniform guarantees for Lasso that ensure the accurate recovery of all signals and corruptions using a single draw of the sub-Gaussian sensing matrix and uniform dither. For signal and corruption with structured priors (e.g., sparsity, low-rankness), our uniform error rate for constrained Lasso typically coincides with the non-uniform one [Sun, Cui and Liu, 2022] up to logarithmic factors. By contrast, our uniform error rate for unconstrained Lasso exhibits worse dependence on the structured parameters due to regularization parameters larger than the ones for non-uniform recovery. For signal and corruption living in the ranges of some Lipschitz continuous generative models (referred to as generative priors), we achieve uniform recovery via constrained Lasso with a measurement number proportional to the latent dimensions of the generative models. Our treatments to the two kinds of priors are (nearly) unified and share the common key ingredients of (global) quantized product embedding (QPE) property, which states that the dithered uniform quantization (universally) preserves inner product. As a by-product, our QPE result refines the one in [Xu and Jacques, 2020] under sub-Gaussian random matrix, and in this specific instance we are able to sharpen the uniform error decaying rate (for the projected-back projection estimator with signals in some convex symmetric set) presented therein from to .
Paper Structure (37 sections, 23 theorems, 227 equations, 11 figures, 1 table)

This paper contains 37 sections, 23 theorems, 227 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Technical Contributions and Technically Related Works
  4. Paper Outline
  5. Preliminaries
  6. Sub-Gaussian Random Variable
  7. Covering Number and Kolmogorov Entropy
  8. Gaussian Width and Gaussian Complexity
  9. Dithered Uniform Quantization
  10. Main Results
  11. Structured Priors with Constrained Lasso
  12. Structured Priors with Unconstrained Lasso
  13. Generative Priors
  14. Experimental Results
  15. Structured Priors
  16. ...and 22 more sections

Key Result

Theorem 3.2

\newlabelthm10 Under assump1--assump3, we define the constraint sets Suppose that the positive scalars $(\zeta,\rho_1,\rho_2)$ and the sample size $m$ satisfy for small enough $(c_1,c_2,c_3)$. If for large enough $C_4$ it holds that then with probability exceeding on a single draw of $(\bm{\Phi},\bm{\epsilon},\bm{\tau})$, the following uniform error bound holds true for all $(\bm{x^\star},\bm

Figures (11)

  • Figure 1: From Non-Uniformity to Uniformity: (Left) Sparse Recovery via Constrained Lasso (\ref{['coro1']}); (Right) Sparse Recovery via Unconstrained Lasso (\ref{['coro3']})
  • Figure 2: Log-log error curves for the constrained Lasso under Gaussian or Bernoulli measurements.
  • Figure 3: Log-log error curves for the unconstrained Lasso under Gaussian or Bernoulli measurements.
  • Figure 4: Reconstructed images for digits 8 and 1 of MNIST under Gaussian measurements with $\sigma =2$ and $\delta = 10$.
  • Figure 5: Reconstructed images for digits 8 and 1 of MNIST under Bernoulli measurements with $\sigma =10$ and $\delta = 20$.
  • ...and 6 more figures

Theorems & Definitions (57)

  • Definition 3.1: Structured Set
  • Theorem 3.2: Uniform Recovery via Constrained Lasso
  • Proof 1
  • Remark 3.3: The Cost of Uniformity: Constrained Lasso with Structured Priors
  • Remark 3.4: The Role of Quantization Resolution $\delta$
  • Corollary 3.5: Sparse Signal and Sparse Corruption
  • Remark 3.6: Elimination of Logarithmic Factors
  • Corollary 3.7: Low-Rank Signal and Sparse Corruption
  • Remark 3.8
  • Remark 3.9: Related Works and the Novelty of Our Results
  • ...and 47 more