[Paper Review] Sequential adaptive compressed sampling via Huffman codes

Akram Aldroubi, Haichao Wang+1 more|ArXiv.org|Oct 27, 2008

Sparse and Compressive Sensing Techniques23 references10 citations

TL;DR

This paper proposes a deterministic, adaptive compressed sensing method using Huffman coding principles to sequentially sample sparse signals via binary vectors. By leveraging signal probability distributions, it achieves an expected total cost (sampling and reconstruction) of at most $ s\log n + 2s $, ensuring exact recovery in noiseless cases and stable performance under noise.

ABSTRACT

There are two main approaches in compressed sensing: the geometric approach and the combinatorial approach. In this paper we introduce an information theoretic approach and use results from the theory of Huffman codes to construct a sequence of binary sampling vectors to determine a sparse signal. Unlike other approaches, our approach is adaptive in the sense that each sampling vector depends on the previous sample. The number of measurements we need for a k-sparse vector in n-dimensional space is no more than O(k log n) and the reconstruction is O(k).

Motivation & Objective

• To develop a deterministic, adaptive sampling scheme for sparse signals that minimizes the total cost of sampling and reconstruction.
• To leverage probability distributions of signal components to reduce the expected number of measurements required.
• To ensure exact recovery in noiseless conditions and stable reconstruction under noisy measurements.
• To provide a computationally efficient alternative to standard compressed sensing methods relying on RIP matrices or $\ell_1$-minimization.
• To demonstrate that the method remains robust even when the assumed signal distribution is inaccurate.

Proposed method

• Constructs binary sampling vectors adaptively using Huffman coding principles, where each vector is determined based on previous measurement outcomes.
• Uses a Huffman tree structure to encode the positions of nonzero components in an $s$-sparse signal, assigning shorter codes to higher-probability positions.
• Employs a sequential sampling strategy: each sampling vector is chosen based on prior measurements to focus on likely locations of nonzero entries.
• Performs reconstruction by traversing the Huffman tree using measurement results, identifying nonzero components with minimal queries.
• Treats the signal as a random vector with known or assumed probability distribution to minimize expected sampling cost.
• Uses a deterministic construction of sampling vectors that does not require random matrix ensembles or RIP conditions.

Experimental results

Research questions

• RQ1 Can an adaptive, deterministic compressed sensing method be designed to minimize the expected total cost of sampling and reconstruction for sparse signals?
• RQ2 How does the use of Huffman coding principles reduce the number of required measurements compared to non-adaptive methods?
• RQ3 What is the theoretical upper bound on the expected total cost (measurements + reconstruction) for recovering an $s$-sparse signal in $\mathbb{R}^n$?
• RQ4 How does the method perform under noisy measurements, and is it stable to noise corruption?
• RQ5 Does the method remain effective even when the assumed signal distribution is incorrect or suboptimal?

Key findings

• The expected total cost (sampling and reconstruction) for recovering an $s$-sparse vector in $\mathbb{R}^n$ is bounded by $ s\log n + 2s $, regardless of the true signal distribution.
• In the noiseless case, the method achieves exact recovery of the sparse signal using the derived adaptive sampling and reconstruction procedure.
• For $n = 32768$, the mean number of sampling vectors needed for $s=13$ was 111.88, close to the theoretical bound of $13\log(32768) = 195$.
• The CPU time for reconstruction remained under 0.15 seconds even for $s=150$ and $n=1024$, demonstrating high computational efficiency.
• Under noisy conditions, the relative $\ell^2$ error increased linearly with sparsity $s$, with a slope of approximately 0.6% per unit $s$.
• The relative error also increased linearly with noise level $N$, indicating predictable and stable degradation under increasing noise.