[Paper Review] Stronger L2/L2 Compressed Sensing; Without Iterating

Vasileios Nakos, Zhao Song|arXiv (Cornell University)|Mar 7, 2019

Sparse and Compressive Sensing Techniques79 references4 citations

TL;DR

This paper presents the first sublinear-time ℓ₂/ℓ₂ compressed sensing scheme that achieves optimal measurement complexity without iteration, using a novel non-iterative framework based on coordinate heaviness and random partitioning. It improves decoding time and reduces column sparsity compared to prior work, resolving two open questions from Gilbert et al. (STOC 2010).

ABSTRACT

We consider the extensively studied problem of $\ell_2/\ell_2$ compressed sensing. The main contribution of our work is an improvement over [Gilbert, Li, Porat and Strauss, STOC 2010] with faster decoding time and significantly smaller column sparsity, answering two open questions of the aforementioned work. Previous work on sublinear-time compressed sensing employed an iterative procedure, recovering the heavy coordinates in phases. We completely depart from that framework, and give the first sublinear-time $\ell_2/\ell_2$ scheme which achieves the optimal number of measurements without iterating; this new approach is the key step to our progress. Towards that, we satisfy the $\ell_2/\ell_2$ guarantee by exploiting the heaviness of coordinates in a way that was not exploited in previous work. Via our techniques we obtain improved results for various sparse recovery tasks, and indicate possible further applications to problems in the field, to which the aforementioned iterative procedure creates significant obstructions.

Motivation & Objective

• To eliminate iterative recovery procedures in sublinear-time ℓ₂/ℓ₂ compressed sensing, which have been a bottleneck in efficiency and scalability.
• To reduce column sparsity of the sensing matrix while maintaining optimal measurement complexity, addressing a key open problem in the field.
• To achieve faster decoding time and improved algorithmic efficiency for sparse recovery tasks without sacrificing accuracy.
• To develop a new theoretical framework based on coordinate heaviness and random partitioning that enables non-iterative recovery.
• To provide a foundation for future applications in streaming, network monitoring, and data mining where low update and recovery times are critical.

Proposed method

• Proposes a non-iterative recovery framework that directly identifies heavy coordinates using random partitioning and thresholding, avoiding phase-by-phase recovery.
• Employs a block-based random partitioning of coordinates into groups of size s, where each block is sampled via a random walk process to detect heavy components.
• Uses a set query data structure (Lemma 1.8) to efficiently recover approximate values of selected coordinates with high probability.
• Applies interval forest sparse recovery (Algorithm 1) with precision ε/10 to identify a candidate support set L of size O(k/ε).
• Implements a pruning procedure (Algorithm 2) to refine the support set L into a smaller set S of size O(k), ensuring ℓ₂ approximation error is within (1+ε/2) of the optimal.
• Combines probabilistic analysis using random walk concentration (Lemma C.1) to guarantee that the sum of heavy blocks exceeds a threshold with high probability.

Experimental results

Research questions

• RQ1 Can ℓ₂/ℓ₂ compressed sensing be achieved in sublinear time without iterative recovery phases?
• RQ2 What is the minimal column sparsity achievable in a sensing matrix for optimal ℓ₂/ℓ₂ recovery with sublinear decoding time?
• RQ3 Can a non-iterative framework based on coordinate heaviness and random partitioning outperform iterative schemes in both measurement complexity and decoding speed?
• RQ4 What is the trade-off between measurement count, decoding time, and column sparsity in ℓ₂/ℓ₂ recovery under the for-each model?
• RQ5 Can this framework be extended to other sparse recovery guarantees or applications in data streaming and network monitoring?

Key findings

• The proposed algorithm achieves optimal ℓ₂/ℓ₂ recovery with O(k log(n/k)) measurements, matching known lower bounds.
• Decoding time is sublinear in n, significantly faster than iterative schemes that require multiple rounds of recovery.
• Column sparsity of the sensing matrix is reduced to O(log n), improving upon the previous state of the art and resolving an open question.
• The algorithm succeeds with constant probability ≥ 3/4, and the overall failure probability is bounded by 1/4 via union bound over subroutines.
• The framework enables improved performance in various sparse recovery tasks, including heavy hitter detection and network monitoring, where low update and query time are essential.
• The non-iterative design avoids the scalability and latency limitations inherent in iterative recovery, opening new directions for practical deployment.