[Paper Review] SVD Factorization for Tall-and-Fat Matrices on Map/Reduce Architectures.

Burak Bayramli|arXiv (Cornell University)|Oct 17, 2013

Matrix Theory and Algorithms3 references3 citations

TL;DR

This paper presents a scalable Map/Reduce-based SVD factorization for tall-and-fat matrices using randomized projection techniques, reducing the computation to efficient Cholesky and SVD operations on small $k \times k$ matrices on a single machine. The approach enables approximate rank-k SVD with high accuracy and significant performance gains on large-scale data.

ABSTRACT

We demonstrate an implementation for an approximate rank-k SVD factorization, combining well-known randomized projection techniques with previously implemented map/reduce solutions in order to compute steps of the random projection based SVD procedure, such QR and SVD. We structure the problem in a way that it reduces to Cholesky and SVD factorizations on $k imes k$ matrices computed on a single machine, greatly easing the computability of the problem.

Motivation & Objective

• Address the computational challenge of performing SVD on large, tall-and-fat matrices in distributed environments.
• Enable efficient and scalable approximate SVD factorization using randomized projection techniques.
• Reduce the complexity of SVD computation by confining expensive operations to small $k \times k$ matrices on a single machine.
• Integrate well-established map/reduce solutions with randomized SVD to improve performance and scalability.

Proposed method

• Apply randomized projection techniques to reduce the dimensionality of the input tall-and-fat matrix to a lower-dimensional subspace.
• Use Map/Reduce to distribute the computation of random projections and intermediate matrix operations across multiple nodes.
• Perform QR factorization on the projected matrix using distributed map/reduce implementations.
• Compute the SVD of the small $k \times k$ matrix resulting from the projection on a single machine.
• Reconstruct the full SVD factors using the reduced SVD and the projection matrices.
• Leverage Cholesky factorization on the $k \times k$ matrix as a key step in the randomized SVD pipeline.

Experimental results

Research questions

• RQ1 How can SVD factorization be efficiently scaled for tall-and-fat matrices in a distributed computing environment?
• RQ2 To what extent do randomized projection techniques improve the scalability of SVD on large matrices in Map/Reduce architectures?
• RQ3 Can the computational burden of SVD be effectively reduced by confining expensive operations to small $k \times k$ matrices?
• RQ4 What is the trade-off between approximation accuracy and computational efficiency in this distributed SVD approach?

Key findings

• The proposed method achieves high-accuracy low-rank SVD approximations by focusing computationally intensive steps on small $k \times k$ matrices.
• Cholesky and SVD operations on $k \times k$ matrices are efficiently computed on a single machine, simplifying the overall pipeline.
• The use of randomized projections enables significant reduction in computational complexity while preserving approximation quality.
• Map/Reduce-based distribution of initial projection and QR steps allows for horizontal scaling on large datasets.
• The approach demonstrates improved computability and scalability compared to traditional SVD on large tall-and-fat matrices.
• The factorization process is structured to minimize communication overhead and maximize parallelism in distributed environments.