Accurate Prediction of Phase Transitions in Compressed Sensing via a Connection to Minimax Denoising (1111.1041v2)

Abstract: Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula that characterizes the allowed undersampling of generalized sparse objects. The formula applies to Approximate Message Passing (AMP) algorithms for compressed sensing, which are here generalized to employ denoising operators besides the traditional scalar soft thresholding denoiser. This paper gives several examples including scalar denoisers not derived from convex penalization -- the firm shrinkage nonlinearity and the minimax nonlinearity -- and also nonscalar denoisers -- block thresholding, monotone regression, and total variation minimization. Let the variables eps = k/N and delta = n/N denote the generalized sparsity and undersampling fractions for sampling the k-generalized-sparse N-vector x_0 according to y=Ax_0. Here A is an n\times N measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve delta = delta(eps) separating successful from unsuccessful reconstruction of x_0 by AMP is given by: delta = M(eps| Denoiser), where M(eps| Denoiser) denotes the per-coordinate minimax mean squared error (MSE) of the specified, optimally-tuned denoiser in the directly observed problem y = x + z. In short, the phase transition of a noiseless undersampling problem is identical to the minimax MSE in a denoising problem.

• The paper derives a formula connecting phase transition boundaries in compressed sensing with the minimax MSE of optimally tuned denoisers.
• The paper extends AMP algorithms to incorporate diverse denoisers such as block thresholding and total variation minimization, outperforming traditional approaches.
• The paper demonstrates that nonconvex and minimax shrinkage methods yield superior signal recovery compared to conventional soft thresholding techniques.

Accurate Prediction of Phase Transitions in Compressed Sensing via Minimax Denoising

The paper presents a comprehensive analysis of the phase transitions in compressed sensing, linking them to a minimax denoising framework. Compressed sensing is a technique that allows for the accurate recovery of sparse signals from undersampled measurements. The precise limits to such undersampling are crucial for both theoretical foundations and practical implementations. This work provides a formula predicting these limits, focusing on Approximate Message Passing (AMP) algorithms, which are generalized to incorporate a variety of denoisers beyond traditional scalar soft thresholding.

Key Contributions

1. Phase Transition Formula: The authors derive a formula describing the phase transition boundary as a function of undersampling and sparsity levels. The phase transition is directly related to the minimax mean squared error (MSE) in a denoising problem. The notion that the phase transition curve $$separating successful reconstruction from failure is equivalent to$$, where $$ denotes the minimax MSE achieved by the optimally-tuned denoiser, is central to this paper.
2. Generalization of AMP Algorithms: AMP algorithms are extended to use diverse denoisers, including non-scalar types like block thresholding, monotone regression, and total variation minimization. The paper examines these denoisers in depth, showing several examples where nonconvex penalization can outperform traditional convex approaches, such as LASSO and positively-constrained LASSO.
3. Minimax Shrinkage: By exploring various denoisers, including firm shrinkage and minimax shrinkage, the paper demonstrates that AMP algorithms using these denoisers outperform those employing traditional denoisers. The minimax nonlinearity across arbitrary functions yields a superior MSE compared to soft and firm thresholding.
4. Block Thresholding and Structured Sparsity: For block-sparse vectors, AMP algorithms with block-separable denoisers exhibit phase transitions closely aligned with those predicted by the minimax MSE framework. Interestingly, as the block size increases, the minimax MSE tends to the ideal limit, highlighting the effectiveness of James-Stein shrinkage over block soft thresholding.
5. Versatility in Structural Sparsity: The applicability of the AMP framework extends beyond simple sparsity to structured sparsity patterns, such as monotone and total variation-sparse vectors. Calculations of the minimax MSE for these alternative structures elucidate the phase transition dynamics and provide a robust theoretical underpinning for empirical observations.

Implications and Speculation on Future Developments

The implications of this research are vast, suggesting that accurate predictions of phase transitions can be made for a wide array of sparsity and undersampling scenarios, provided the minimax framework is adhered to. This unified approach to characterizing the limits of undersampling offers insights into the design of more efficient algorithms and improved signal reconstruction techniques.

Furthermore, the development of nonconvex denoisers and their integration into AMP algorithms opens new avenues for exploring sparse signal recovery. Future work might include extending the state evolution tools used herein, rigorously establishing the robustness of the minimax framework across more problem domains and denoising methods.

This theoretical advancement also poses potential enhancements to practical applications in machine learning, signal processing, and other areas relying on sparse and structured data modeling. The continuing evolution of AMP algorithms, driven by findings like these, could inform an advanced generation of compressed sensing methodologies better aligned with contemporary computational challenges.

Overall, the paper provides a rigorous mathematical underpinning for understanding phase transitions in compressed sensing, drawing valuable connections between high-dimensional statistical learning and practical algorithmic performance.