Compressed Sensing using Generative Models (1703.03208v1)

Abstract: The goal of compressed sensing is to estimate a vector from an underdetermined system of noisy linear measurements, by making use of prior knowledge on the structure of vectors in the relevant domain. For almost all results in this literature, the structure is represented by sparsity in a well-chosen basis. We show how to achieve guarantees similar to standard compressed sensing but without employing sparsity at all. Instead, we suppose that vectors lie near the range of a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. Our main theorem is that, if $G$ is $L$-Lipschitz, then roughly $O(k \log L)$ random Gaussian measurements suffice for an $\ell_2/\ell_2$ recovery guarantee. We demonstrate our results using generative models from published variational autoencoder and generative adversarial networks. Our method can use $5$-$10$x fewer measurements than Lasso for the same accuracy.

• The paper demonstrates that using generative models for compressed sensing can achieve robust signal recovery with significantly fewer measurements, supported by the novel S-REC theoretical framework.
• The proposed algorithm optimizes latent representations in VAEs and GANs, with experiments showing 5-10 times fewer measurements needed for comparable reconstruction accuracy on datasets like MNIST and celebA.
• The method offers practical benefits for imaging applications, reducing costs and acquisition time while enabling high-resolution reconstructions in resource-constrained environments.

The Theoretical and Empirical Advancements in Compressed Sensing Using Generative Models

"Compressed Sensing using Generative Models" presents a novel approach to the problem of compressed sensing by leveraging advanced generative models, specifically Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs), to estimate a vector from noisy, underdetermined linear measurements without relying on traditional sparsity constraints. This paper provides both theoretical and empirical evidence that these generative models significantly reduce the number of measurements required for accurate recovery compared to existing methods like Lasso, which depend on sparsity.

Main Contributions

1. Theoretical Foundations:
   • The core theoretical contribution is a generalization of the Restricted Eigenvalue Condition (REC) called the Set-Restricted Eigenvalue Condition (S-REC). This generalization allows robustness guarantees for recovery via generative models that are not necessarily sparse but instead lie near the range of a generative function $G: R^k \to R^n$.
   • The paper establishes that for an $L$-Lipschitz generative model $G$, approximately $O(k \log L)$ random Gaussian measurements are sufficient for an $\ell_2/\ell_2$ recovery guarantee. This is a significant theoretical advancement since it opens the possibility of using a wide range of neural network-based generative models.
   • Two main theorems are presented which provide probabilistic bounds on measurement errors and reconstruction errors, showing that generative models can achieve comparable or superior recovery guarantees with fewer measurements than traditional sparsity-based methods.
2. Empirical Validation and Algorithm:
   • The authors present an efficient algorithm to recover an unknown vector by optimizing the representation $z \in R^k$ such that the corresponding image $G(z)$ has minimal measurement error.
   • Empirical results using published VAE and GAN models demonstrate that the algorithm can use 5-10 times fewer measurements than Lasso for the same reconstruction accuracy. The models were tested on MNIST and celebA datasets, showing effective reconstructions with significantly fewer measurements.
   • The paper also explores the noise tolerance of the proposed method, showing robustness to noise compared to traditional methods.
3. Practical Implications:
   • The approach outlined has broad implications for fields reliant on compressed sensing, such as medical imaging (MRI, CT scans) and other domains where acquiring measurements is costly or limited by physical constraints. Lowering the number of measurements not only reduces cost and time but can also potentially improve the feasibility of high-resolution imaging in resource-constrained environments.
   • Importantly, the proposed method is flexible to the advancements in generative models. As the generative models improve, particularly in their ability to capture the distribution of high-dimensional data (e.g., images), the efficacy of the compressed sensing algorithm will also enhance.

Future Directions

The findings in this paper pave the way for several future research directions:

1. Scaling Generative Models:
   • Improving the flexibility and power of generative models could directly enhance the recovery performance for compressed signals. Future work could explore deeper and more complex architectures or incorporate emerging generative models beyond VAEs and GANs.
2. Application to Other Domains:
   • Extending the methodology to other forms of data beyond images, such as audio, video, and possibly 3D spatial data, to explore compressed sensing where traditional sparsity may not be as effective.
3. Integrating Sparsity and Generative Models:
   • Investigating hybrid models that combine the strengths of sparsity and generative approaches. For example, exploring how dictionary learning or learned sparsity constraints within generative models can further reduce measurement requirements.
4. Optimization Techniques:
   • Enhancing optimization algorithms for faster and more effective convergence in high-dimensional generative model spaces. This could include more advanced gradient-based methods or integration with reinforcement learning techniques for adaptive optimization.
5. Robustness and Generalization:
   • Analyzing the robustness of the method across different types of noise and measurement distortions. Additionally, understanding the generalization capacity of the generative models when trained on one dataset type but applied to another.

In conclusion, "Compressed Sensing using Generative Models" exemplifies a significant theoretical and practical advance in the field of compressed sensing. By moving beyond sparsity towards leveraging the power of generative models, the authors have set the stage for more efficient and effective reconstruction algorithms applicable to a wide range of real-world problems.