[Paper Review] Improving CUR Matrix Decomposition and the Nystr\"{o}m Approximation via Adaptive Sampling

Shusen Wang, Zhihua Zhang|arXiv (Cornell University)|Mar 18, 2013

Sparse and Compressive Sensing Techniques14 references15 citations

TL;DR

This paper proposes adaptive sampling algorithms for CUR matrix decomposition and the Nyström method that achieve improved relative-error bounds without assumptions on the data matrix. By establishing a general theoretical bound for adaptive sampling, the authors develop efficient, low-complexity algorithms that avoid storing the full matrix in RAM while maintaining high approximation accuracy.

ABSTRACT

The CUR matrix decomposition and Nyström method are two important low-rank matrix approximation techniques. The Nyström method approximates a positive semidefinite matrix in terms of a small number of its columns, while CUR approximates an arbitrary data matrix by a small number of its columns and rows. Thus, the CUR decomposition can be regarded as an extension of the Nyström method. In this paper we establish a more general bound for the adaptive column/row sampling algorithm, based on which we propose improved CUR and Nyström algorithms with expected relative-error bounds. The proposed CUR and Nyström algorithms also hold low time complexity and can avoid maintaining the whole data matrix in RAM. In addition, we give theoretical analysis for the lower bounds of the conventional Nyström method and the ensemble Nyström method. In our work we make no assumption on the data matrix, and our main theoretical results established in this paper are novel and encouraging.

Motivation & Objective

• To improve the theoretical guarantees of CUR matrix decomposition and the Nyström method through adaptive sampling.
• To develop algorithms with expected relative-error bounds that do not require assumptions on the data matrix.
• To reduce time complexity and avoid storing the entire data matrix in main memory.
• To analyze lower bounds for conventional and ensemble Nyström methods.
• To establish novel theoretical results that are both general and encouraging for low-rank approximation.

Proposed method

• Derives a general bound for adaptive column/row sampling in low-rank approximation, forming the foundation for improved algorithms.
• Proposes new CUR and Nyström algorithms based on this bound, ensuring expected relative-error performance.
• Employs adaptive sampling strategies that prioritize informative columns and rows to enhance approximation quality.
• Designs algorithms with low time complexity suitable for large-scale data processing.
• Avoids full matrix storage by processing data in a streaming or incremental fashion.
• Provides theoretical analysis of lower bounds for conventional and ensemble Nyström methods.

Experimental results

Research questions

• RQ1 How can adaptive sampling improve the relative-error bounds of CUR and Nyström approximations?
• RQ2 What is the theoretical performance limit of adaptive sampling in low-rank matrix approximation?
• RQ3 Can the proposed algorithms achieve high accuracy without assuming any structure on the data matrix?
• RQ4 How do the proposed methods compare to existing Nyström and CUR approaches in terms of error and efficiency?
• RQ5 What are the theoretical lower bounds for conventional and ensemble Nyström methods?

Key findings

• The proposed adaptive sampling algorithm achieves improved expected relative-error bounds for both CUR and Nyström methods.
• The theoretical framework applies generally without assumptions on the data matrix, making it broadly applicable.
• The algorithms maintain low time complexity and avoid storing the entire data matrix in RAM.
• The paper establishes novel theoretical lower bounds for conventional and ensemble Nyström methods.
• The main theoretical results are described as both novel and encouraging for future research.
• The approach extends the Nyström method to arbitrary matrices via the CUR framework, enhancing interpretability and applicability.