Literature survey on low rank approximation of matrices (1606.06511v1)

Abstract: Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic algorithms for low rank approximation. But these techniques are very expensive $(O(n^{3})$ operations are required for $n\times n$ matrices). There are several randomized algorithms available in the literature which are not so expensive as the classical techniques (but the complexity is not linear in n). So, it is very expensive to construct the low rank approximation of a matrix if the dimension of the matrix is very large. There are alternative techniques like Cross/Skeleton approximation which gives the low-rank approximation with linear complexity in n . In this article we review low rank approximation techniques briefly and give extensive references of many techniques.

Overview of Low Rank Approximation Techniques

The paper by N. Kishore Kumar and J. Schneider presents a comprehensive literature survey on low rank matrix approximation, exploring both deterministic and randomized algorithms. Low rank approximation plays a crucial role in numerical linear algebra, offering computational efficiency and data compression advantages across a variety of applications, including image processing, data mining, and machine learning.

Classical Techniques

The exposition begins with classical deterministic techniques for low rank approximation, such as Singular Value Decomposition (SVD), pivoted QR decomposition, and rank revealing QR factorization (RRQR). SVD is notably recognized for providing optimal low rank approximations and minimizing error in terms of matrix norms. However, despite their efficacy, these methods are computationally demanding, typically requiring O(n³) operations for an n x n matrix, rendering them impractical for large-scale datasets.

Randomized Algorithms

The authors transition to discussing randomized algorithms as alternatives that alleviate computational burden. Randomized approaches, including subsampling and random projection methods, enable sublinear complexity, offering fast and scalable solutions for large matrices. These techniques provide high-accuracy approximations with minimal failure probability, making them suitable for real-world applications where data access is limited to a few passes.

Cross/Skeleton Approximation

A key contribution of the survey is the emphasis on decomposition techniques such as Cross/Skeleton and Pseudoskeleton approximations. These techniques leverage matrix sparsity by selecting specific rows and columns to form approximate matrices with complexity linear in matrix dimensions. The accuracy of these methods hinges on selecting submatrices with maximal volume, although practical implementations often necessitate heuristic strategies due to NP-hardness issues associated with finding optimal submatrices.

Implications and Future Directions

The exploration of these low rank approximation algorithms has numerous implications, both theoretically and practically. On a theoretical level, improving algorithms to handle large matrices efficiently continues to be an area of active research, with potential impacts on fields reliant on computational linear algebra. Practically, leveraging the efficiency of these approaches could transform applications in machine learning, signal processing, and scientific computing.

The paper prompts future research to consider hybrid approaches combining deterministic and randomized elements to potentially achieve optimal trade-offs between accuracy, computational cost, and robustness. Continued refinement in choosing sample matrices and optimizing composition procedures could further enhance algorithmic performance, driving advancements in AI and other computational areas reliant on matrix approximations.

Conclusion

Overall, this survey encapsulates the landscape of low rank approximation techniques, highlights the strengths and limitations of current approaches, and offers insight into future developments that could shape the domain of numerical linear algebra further. By synthesizing existing literature, the authors contribute a valuable reference for researchers focused on advancing efficient and accurate matrix approximation methods.