Improving CUR Matrix Decomposition and the Nyström Approximation via Adaptive Sampling (1303.4207v7)

Abstract: The CUR matrix decomposition and the Nystr\"{o}m approximation are two important low-rank matrix approximation techniques. The Nystr\"{o}m method approximates a symmetric 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, CUR decomposition can be regarded as an extension of the Nystr\"{o}m approximation. In this paper we establish a more general error bound for the adaptive column/row sampling algorithm, based on which we propose more accurate CUR and Nystr\"{o}m algorithms with expected relative-error bounds. The proposed CUR and Nystr\"{o}m algorithms also have low time complexity and can avoid maintaining the whole data matrix in RAM. In addition, we give theoretical analysis for the lower error bounds of the standard Nystr\"{o}m method and the ensemble Nystr\"{o}m method. The main theoretical results established in this paper are novel, and our analysis makes no special assumption on the data matrices.

• The paper introduces adaptive sampling techniques to significantly improve the accuracy and computational efficiency of CUR matrix decomposition and the Nyström approximation.
• For CUR decomposition, the adaptive method drastically reduces the number of columns and rows needed to achieve relative-error bounds, improving memory and computational efficiency.
• For the Nyström method, a modified approach using adaptive sampling and a unique intersection matrix achieves stronger relative-error bounds with substantially fewer columns than traditional methods.

Improving CUR Matrix Decomposition and the Nyström Approximation via Adaptive Sampling

In the arena of large-scale data analysis, efficiently approximating large matrices plays a pivotal role. Traditional methods such as Singular Value Decomposition (SVD) provide good low-rank approximations but falter in handling large, sparse datasets due to computational and storage limitations. The CUR matrix decomposition and the Nyström method are two notable techniques that circumvent some limitations of SVD by focusing on subsets of the original matrix, namely its columns and rows (for CUR) or columns (for Nyström). This paper makes significant strides in enhancing both methods via adaptive sampling to achieve stronger error bounds while maintaining computational efficiency.

At the heart of this contribution is a novel general error bound for the adaptive sampling algorithm, which effectively reduces approximation errors even when columns and rows are sampled separately. The paper extends the work of Deshpande et al. on adaptive sampling by proving a more comprehensive error bound that simultaneously addresses the errors from both column and row projections. This advancement heralds impressive improvements in both the CUR and Nyström methods.

CUR Matrix Decomposition Advancements

The CUR decomposition is a technique favored for its interpretative advantage, as it expresses a data matrix in terms of actual rows and columns. Previous approaches, such as the subspace sampling algorithm, required substantial data to ensure error bounds, often encumbered by the need to maintain the entire matrix in memory. This research introduces an adaptive sampling-based strategy that markedly reduces both the number of required columns and rows. Specifically, the enhanced CUR algorithm requires only $O\left(\frac{2k}{\epsilon} (1+o(1))\right)$ columns and $O\left(\frac{c}{\epsilon}(1+\epsilon)\right)$ rows to achieve a relative-error bound, comparing favorably against prior methods that demanded more extensive sampling.

The computational efficiency is notably improved, owing to operations that do not necessitate holding the entire matrix in RAM. The algorithm mainly handles smaller matrices derived from the original, thereby considerably cutting down on the computational load and space complexity.

Nyström Approximation Improvements

The Nyström method extends the advantages of CUR to symmetric positive semidefinite matrices but traditionally has had weaker error bounds, particularly for conventional versions of the method. The paper introduces a modified Nyström method employing an intersection matrix distinct from the standard approaches, achieving stronger relative-error bounds. This improvement is backed by theoretical analysis which guarantees convergence to the target approximation at a reduced sampling cost. Specifically, the modified method achieves relative-error bounds with only $O\left(\frac{2k}{\epsilon^2} (1+o(1))\right)$ columns, a significant reduction compared to typical requirements.

Analytical and Practical Implications

The results have profound implications for computational feasibility and accuracy in data-intensive applications, such as those involving kernels in machine learning or dimensionality reduction in large-scale text and image analysis. The use of adaptive sampling tied with a more profound understanding of leverage scores represents a shift towards more informed and data-sensitive sampling strategies. By reducing dependency on full data matrices and focusing computational resources on informative parts of the data, this work sets a benchmark for future matrix decomposition techniques in AI and machine learning.

Overall, this paper demonstrates that through thoughtful enhancement of sampling techniques, the CUR matrix decompositions and Nyström approximations can be significantly optimized. The introduction of adaptive sampling in these methods reduces computational overhead while simultaneously tightening error bounds, offering promising pathways for further research and application in handling extensive, high-dimensional datasets efficiently.