Finite Sample Analysis of Approximate Message Passing Algorithms

Abstract

Approximate message passing (AMP) refers to a class of efficient algorithms for statistical estimation in high-dimensional problems such as compressed sensing and low-rank matrix estimation. This paper analyzes the performance of AMP in the regime where the problem dimension is large but finite. For concreteness, we consider the setting of high-dimensional regression, where the goal is to estimate a high-dimensional vector ${\beta }_0$ from a noisy measurement $y=A{\beta }_0+w$. AMP is a low-complexity, scalable algorithm for this problem. Under suitable assumptions on the measurement matrix $A$, AMP has the attractive feature that its performance can be accurately characterized in the large system limit by a simple scalar iteration called state evolution. Previous proofs of the validity of state evolution have all been asymptotic convergence results. In this paper, we derive a concentration inequality for AMP with i.i.d. Gaussian measurement matrices with finite size $n\times N$. The result shows that the probability of deviation from the state evolution prediction falls exponentially in $n$. This provides theoretical support for empirical findings that have demonstrated excellent agreement of AMP performance with state evolution predictions for moderately large dimensions. The concentration inequality also indicates that the number of AMP iterations $t$ can grow no faster than order $\frac{\log n}{\log \log n}$ for the performance to be close to the state evolution predictions with high probability. The analysis can be extended to obtain similar non-asymptotic results for AMP in other settings such as low-rank matrix estimation.

To appear in IEEE Transactions on Information Theory

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• Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices
• Capacity-achieving Sparse Superposition Codes via Approximate Message Passing Decoding
• Approximate message-passing decoder and capacity-achieving sparse superposition codes
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• An Approximate Message Passing Framework for Side Information
• The Error Probability of Sparse Superposition Codes with Approximate Message Passing Decoding
• Efficient Massive Machine Type Communication (mMTC) via AMP
• Optimal Number of Measurements in a Linear System with Quadratically Decreasing SNR
• Linear Operator Approximate Message Passing (OpAMP)