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One-bit Compressed Sensing using Generative Models

Swatantra Kafle, Geethu Joseph, Pramod K. Varshney

TL;DR

This work tackles one-bit compressed sensing by introducing a deep generative-model prior to constrain the search for sparse signals. The reconstruction is performed by optimizing over the generator's latent space, yielding a nonconvex but data-efficient procedure with theoretical guarantees. The authors derive a measurement bound showing that a modest number of Gaussian measurements suffices, extend the analysis to noisy measurements, and validate the approach on MNIST, Fashion-MNIST, and Omniglot, demonstrating improved recovery of both amplitude and direction over traditional algorithms. The results indicate practical benefits for low-resource sensing and signal domains with rich structure, while also outlining limitations related to generator capacity and domain shifts.

Abstract

This paper addresses the classical problem of one-bit compressed sensing using a deep learning-based reconstruction algorithm that leverages a trained generative model to enhance the signal reconstruction performance. The generator, a pre-trained neural network, learns to map from a low-dimensional latent space to a higher-dimensional set of sparse vectors. This generator is then used to reconstruct sparse vectors from their one-bit measurements by searching over its range. The presented algorithm provides an excellent reconstruction performance because the generative model can learn additional structural information about the signal beyond sparsity. Furthermore, we provide theoretical guarantees on the reconstruction accuracy and sample complexity of the algorithm. Through numerical experiments using three publicly available image datasets, MNIST, Fashion-MNIST, and Omniglot, we demonstrate the superior performance of the algorithm compared to other existing algorithms and show that our algorithm can recover both the amplitude and the direction of the signal from one-bit measurements.

One-bit Compressed Sensing using Generative Models

TL;DR

This work tackles one-bit compressed sensing by introducing a deep generative-model prior to constrain the search for sparse signals. The reconstruction is performed by optimizing over the generator's latent space, yielding a nonconvex but data-efficient procedure with theoretical guarantees. The authors derive a measurement bound showing that a modest number of Gaussian measurements suffices, extend the analysis to noisy measurements, and validate the approach on MNIST, Fashion-MNIST, and Omniglot, demonstrating improved recovery of both amplitude and direction over traditional algorithms. The results indicate practical benefits for low-resource sensing and signal domains with rich structure, while also outlining limitations related to generator capacity and domain shifts.

Abstract

This paper addresses the classical problem of one-bit compressed sensing using a deep learning-based reconstruction algorithm that leverages a trained generative model to enhance the signal reconstruction performance. The generator, a pre-trained neural network, learns to map from a low-dimensional latent space to a higher-dimensional set of sparse vectors. This generator is then used to reconstruct sparse vectors from their one-bit measurements by searching over its range. The presented algorithm provides an excellent reconstruction performance because the generative model can learn additional structural information about the signal beyond sparsity. Furthermore, we provide theoretical guarantees on the reconstruction accuracy and sample complexity of the algorithm. Through numerical experiments using three publicly available image datasets, MNIST, Fashion-MNIST, and Omniglot, we demonstrate the superior performance of the algorithm compared to other existing algorithms and show that our algorithm can recover both the amplitude and the direction of the signal from one-bit measurements.

Paper Structure

This paper contains 27 sections, 9 theorems, 54 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Our Contributions
  4. System Model and the Generative Model-Based Algorithm
  5. Theoretical Analysis: Measurement Bound
  6. Discussion
  7. Extension to Noisy Measurement Model
  8. Comparison With the Results in liu2020sample
  9. Simulation Results
  10. Noiseless Setting
  11. MSE Performance
  12. NMSE Performance
  13. Noisy Setting
  14. Additive Noise in Measurements
  15. Sign Flips in Measurements
  16. ...and 12 more sections

Key Result

Theorem 1

Suppose that Assumptions 1-3 hold, the ground truth $\boldsymbol{x}^*$ is such that it satisfies $\left\Vert\boldsymbol{x}^*\right\Vert = 1$, and the measurement vector $\boldsymbol{y}$ follows the model given by eq:model. Suppose $\tilde{\boldsymbol{z}}$ minimizes the cost function in eq:cost to wi with probability at least $1-4\exp\left(-c\epsilon^2 m\right)$, the following holds,

Figures (6)

  • Figure 1: Reconstruction performance of our algorithm compared with YPplan2012robust, BIHTbinarystableEmbedd, and GenModel_pgdliu2020sample as a function of number of measurements $m$ in the noiseless setting.
  • Figure 2: Reconstruction performance of our algorithm, GenModel_pgd, BIHT, and YP as a function of number of measurements $m$ when $v_n = 0.1$ and $\alpha = 0.85$.
  • Figure 3: Reconstruction performance of our algorithm, GenModel_pgd, BIHT, and YP as a function of sign-flip probability when $m=784$ and $v_n = 0.1$.
  • Figure 4: Reconstructions performance of our algorithm, GenModel_pgd, BIHT, and YP as a function of measurement matrix uncertainty, $v_\Delta$, when $m = 1500$, $\alpha=1$, and, $v_n = 0$.
  • Figure 5: The first row shows the original images, the second, third and fourth rows are the reconstruction images using BIHT, YP and our algorithm, respectively when $m = 784$ in the noiseless setting.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Proposition 1: liu2020sample
  • Lemma 1: plan2012robust
  • Definition 1: Gaussian mean width plan2012robust
  • Lemma 2: plan2012robust
  • ...and 3 more