Learning to Invert: Signal Recovery via Deep Convolutional Networks

Abstract: The promise of compressive sensing (CS) has been offset by two significant challenges. First, real-world data is not exactly sparse in a fixed basis. Second, current high-performance recovery algorithms are slow to converge, which limits CS to either non-real-time applications or scenarios where massive back-end computing is available. In this paper, we attack both of these challenges head-on by developing a new signal recovery framework we call {\em DeepInverse} that learns the inverse transformation from measurement vectors to signals using a {\em deep convolutional network}. When trained on a set of representative images, the network learns both a representation for the signals (addressing challenge one) and an inverse map approximating a greedy or convex recovery algorithm (addressing challenge two). Our experiments indicate that the DeepInverse network closely approximates the solution produced by state-of-the-art CS recovery algorithms yet is hundreds of times faster in run time. The tradeoff for the ultrafast run time is a computationally intensive, off-line training procedure typical to deep networks. However, the training needs to be completed only once, which makes the approach attractive for a host of sparse recovery problems.

• The paper presents DeepInverse, a deep convolutional network that learns to invert undersampled measurements for rapid signal recovery.
• It significantly reduces computational time by leveraging offline training, outperforming traditional compressive sensing methods by several orders of magnitude.
• Experimental results demonstrate that DeepInverse achieves comparable accuracy to state-of-the-art methods, enabling practical real-time applications in fields like medical imaging and remote sensing.

Learning to Invert: Signal Recovery via Deep Convolutional Networks

This paper presents an innovative approach to tackling the persistent challenges in compressive sensing (CS) through deep learning, specifically introducing the DeepInverse framework. The primary objectives are to address two notable limitations in compressive sensing: the non-exact sparsity of real-world data in a fixed basis and the sluggish convergence of high-performance recovery algorithms, which typically precludes their use in real-time applications without significant computational resources.

Problem Formulation and Background

The research focuses on solving the ill-posed inverse problem of recovering signals from undersampled linear measurements, formalized as recovering $\mathbf{x} \in \mathbb{R}^N$ from $$\mathbf{y} = \boldsymbol \Phi \mathbf{x} \in \mathbb{R}^M$$ with $M \ll N$. This underdetermined system requires exploiting the signal's inherent structure, typically thought of within the compressive sensing framework as sparse representation in some basis.

DeepInverse Framework

DeepInverse employs a deep convolutional network (DCN) to learn the inverse mapping from measurement vectors to original signals. In contrast to traditional methods reliant on sparsity-regularized convex optimization or greedy algorithms, this method leverages a trained network capable of rapid inference. The network is trained on representative images, learning both a suitable signal representation and an inverse mapping that approximates existing recovery algorithms. The experiments demonstrate that despite the computationally demanding offline training phase, DeepInverse achieves notable speed-ups during signal recovery, being hundreds of times faster than current state-of-the-art methods while delivering comparable accuracy.

Experimental Results and Comparisons

Extensive empirical evaluations highlight DeepInverse's efficacy. The framework compares favorably against existing algorithms such as D-AMP, total variation minimization, and P-AMP, particularly in settings requiring low latency. Notably, DeepInverse achieves a significant reduction in computational time, approximately by several orders of magnitude, making it suitable for real-time applications that necessitate immediate response.

Practical Implications and Future Work

The implications of this research are significant for practical applications where speed is paramount and computational resources may be limited. Instances include medical imaging and remote sensing, where rapid recovery processes could facilitate enhanced real-time analytics. Future work could explore extending the network's capacity, potentially through deeper architectures that could capitalize on progressively abstract representations of the data, thereby improving recovery performance even further.

Moreover, DeepInverse is adaptable beyond CS problems, where measurement matrices are not just arbitrary but structurally constrained, offering a unified approach without subdividing signals. This generalizes its applicability compared to prior works that required block-specific measurements.

Conclusion

The DeepInverse framework represents a substantial advance in the application of deep learning to signal recovery, addressing longstanding challenges in compressive sensing by delivering high-speed and efficient recovery without compromising accuracy. The work's potential expansions promise continued improvements in both algorithmic performance and practical applicability in various computational contexts.