[Paper Review] A Simple Approach to Optimal CUR Decomposition

Haishan Ye, Yujun Li+1 more|arXiv (Cornell University)|Nov 5, 2015

Sparse and Compressive Sensing Techniques15 references3 citations

TL;DR

This paper presents a novel, efficient approach to optimal CUR decomposition and near-optimal column reconstruction using only leverage score sampling, eliminating the need for BSS or adaptive sampling. It achieves the first O(nnz(A)) optimal CUR algorithm and extends to the Nyström method with O(nnz(A)) or Õ(n²) runtime, significantly improving efficiency over prior methods.

ABSTRACT

Prior optimal CUR decomposition and near optimal column reconstruction methods have been established by combining BSS sampling and adaptive sampling. In this paper, we propose a new approach to the optimal CUR decomposition and near optimal column reconstruction by just using leverage score sampling. In our approach, both the BSS sampling and adaptive sampling are not needed. Moreover, our approach is the first $O(\mathrm{nnz}(\A))$ optimal CUR algorithm where $\A$ is a data matrix in question. We also extend our approach to the Nystr{ö}m method, obtaining a fast algorithm which runs $ ilde{O}(n^{2})$ or $O(\mathrm{ nz}(\A))$

Motivation & Objective

• To develop a simpler and more efficient approach to optimal CUR decomposition without relying on complex sampling strategies.
• To eliminate the need for BSS sampling and adaptive sampling in CUR decomposition, simplifying the algorithmic pipeline.
• To achieve an O(nnz(A)) time complexity for optimal CUR decomposition, improving upon prior methods.
• To extend the proposed method to the Nyström method, enabling fast and efficient low-rank approximation.

Proposed method

• The method uses only leverage score sampling to select columns and rows for the CUR decomposition, replacing the need for BSS and adaptive sampling.
• Leverage scores are computed to identify the most informative columns and rows, ensuring high-quality low-rank approximation.
• The algorithm constructs the CUR decomposition directly from the sampled columns and rows, maintaining optimality in approximation error.
• The approach is extended to the Nyström method by applying the same leverage score sampling strategy to kernel matrices.
• Theoretical analysis shows that the method achieves optimal approximation error with high probability using only O(nnz(A)) operations.
• The method is designed to be both simple and efficient, avoiding iterative or adaptive sampling steps.

Experimental results

Research questions

• RQ1 Can optimal CUR decomposition be achieved using only leverage score sampling, without BSS or adaptive sampling?
• RQ2 What is the computational complexity of a CUR decomposition algorithm based solely on leverage score sampling?
• RQ3 Can the leverage score-based approach be extended to the Nyström method for kernel matrix approximation?
• RQ4 Does the proposed method achieve optimal approximation error with minimal computational cost?

Key findings

• The proposed method achieves optimal CUR decomposition using only leverage score sampling, eliminating the need for BSS and adaptive sampling.
• The algorithm runs in O(nnz(A)) time, making it the first O(nnz(A)) optimal CUR algorithm.
• The method provides near-optimal column reconstruction by leveraging the same sampling strategy.
• The approach is successfully extended to the Nyström method, achieving O(nnz(A)) or Õ(n²) runtime.