Optimal estimation with arbitrary error metrics in compressed sensing

2013 Information Theory and Applications Workshop (ITA)

2013

Abstract-We consider the problem of reconstructing a signal from noisy measurements in linear mixing systems. The reconstruction performance is usually quantified by standard error metrics such as squared error, whereas we consider any additive error metric. Under the assumption that relaxed belief propagation (BP) can compute the posterior in the large system limit, we propose a simple, fast, and highly general algorithm that reconstructs the signal by minimizing the user-defined error metric. For two example metrics, we provide performance analysis and convincing numerical results. Finally, our algorithm can be adjusted to minimize the ∞ error, which is not additive. Interestingly, ∞ minimization only requires to apply a Wiener filter to the output of relaxed BP.

“…Because the replica method has not been rigorously justified, our results are given as claims. More detailed discussions of the claims can be found in Tan et al [2].…”

Section: B Theoretical Resultsmentioning

confidence: 99%

“…W911NF-04-D-0003. Portions appeared at the IEEE Statistical Signal Processing workshop (SSP), Aug. 2012 [1], and additional manuscripts [2,3].…”

Section: A Motivationmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

“…Equation (9) minimizes a single-variable function. We have shown in detail [1,2] how to perform this minimization in two example error metrics, and summarize the results in Section III-C.…”

Section: A Algorithmmentioning

confidence: 99%

See 3 more Smart Citations

Signal reconstruction in linear mixing systems with different error metrics

Tan

2013 Information Theory and Applications Workshop (ITA)

2013

Performance regions in compressed sensing from noisy measurements

Zhu

2013 47th Annual Conference on Information Sciences and Systems (CISS)

2013

Self Cite

Abstract-In this paper, compressed sensing with noisy measurements is addressed. The theoretically optimal reconstruction error is studied by evaluating Tanaka's equation. The main contribution is to show that in several regions, which have different measurement rates and noise levels, the reconstruction error behaves differently. This paper also evaluates the performance of the belief propagation (BP) signal reconstruction method in the regions discovered. When the measurement rate and the noise level lie in a certain region, BP is suboptimal with respect to Tanaka's equation, and it may be possible to develop reconstruction algorithms with lower error in that region.

Signal Estimation With Additive Error Metrics in Compressed Sensing

Tan

Carmon²,

IEEE Trans. Inform. Theory

2014

Self Cite

Compressed sensing typically deals with the estimation of a system input from its noise-corrupted linear measurements, where the number of measurements is smaller than the number of input components. The performance of the estimation process is usually quantified by some standard error metric such as squared error or support set error. In this correspondence, we consider a noisy compressed sensing problem with any additive error metric. Under the assumption that the relaxed belief propagation method matches Tanaka's fixed point equation, we propose a general algorithm that estimates the original signal by minimizing the additive error metric defined by the user. The algorithm is a pointwise estimation process, and thus simple and fast. We verify that our algorithm is asymptotically optimal, and we describe a general method to compute the fundamental information-theoretic performance limit for any additive error metric. We provide several example metrics, and give the theoretical performance limits for these cases. Experimental results show that our algorithm outperforms methods such as relaxed belief propagation (relaxed BP) and compressive sampling matching pursuit (CoSaMP), and reaches the suggested theoretical limits for our example metrics.