Efficient volume sampling for row/column subset selection
(1004.4057v1)
Published 23 Apr 2010 in cs.DS
Abstract: We give efficient algorithms for volume sampling, i.e., for picking $k$-subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala. Our first algorithm for volume sampling $k$-subsets of rows from an $m$-by-$n$ matrix runs in $O(kmn{\omega} \log n)$ arithmetic operations and a second variant of it for $(1+\epsilon)$-approximate volume sampling runs in $O(mn \log m \cdot k{2}/\epsilon{2} + m \log{\omega} m \cdot k{2\omega+1}/\epsilon{2\omega} \cdot \log(k \epsilon{-1} \log m))$ arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small $k$. Our efficient volume sampling algorithms imply several interesting results for low-rank matrix approximation.
Efficient Algorithms for Volume Sampling in Row/Column Subset Selection
The paper addresses the problem of volume sampling in matrices, a key challenge in the field of low-rank matrix approximation and row/column subset selection. Volume sampling involves choosing k-subsets of matrix rows with probabilities proportional to the squared volumes of the simplices they define. This technique, initially conceptualized in earlier research and specifically posed as an open problem in the works of Kannan and Vempala, provides a promising approach to feature selection and low-dimensional representation of matrices.
Summary of Contributions
The authors present efficient algorithms that advance the theoretical and practical capabilities in volume sampling. Notably, the proposed algorithms address an unmet need for computationally feasible methods that ensure both accuracy and scalability.
Primary Algorithm: The core algorithm leverages characteristic polynomials and cleverly integrates projection operations to achieve exact volume sampling in polynomial time. This breakthrough is significant because it drastically reduces the computational complexity associated with previous algorithmic approaches.
Algorithmic Enhancements: Two algorithmic variants are introduced:
The main algorithm operates in O(kmnωlogn), where ω is the matrix multiplication exponent, and it's tailored for cases where m≥n.
A faster approximation method utilizes random projection techniques to attain (1+ϵ)-approximate volume samples with near-linear complexity in smaller datasets. This approximation is deeply rooted in the generalized Johnson-Lindenstrauss lemma, which is adapted to preserve volumes.
Theoretical Guarantees: The algorithmic advances imply substantial improvements in matrix approximation under the Frobenius and spectral norms. Specifically, the authors achieve a k+1-approximation, improving upon Boutsidis et al.'s O(klogk)-approximation.
Deterministic Variant: By employing the method of conditional expectations, the stochastic nature of volume sampling transitions into a deterministic approach maintaining the desired approximation bounds. This deterministic extension holds significant potential for applications requiring predictable performance and outcomes, especially in machine learning scenarios where deterministic processing can be advantageous.
Implications and Future Directions
The practical implications of these efficient volume sampling algorithms are manifold. They present a robust method for subset selection in large-scale data matrices commonly encountered in fields such as genomics, computer vision, and data mining. The algorithms promise improved performance in matrix decomposition tasks, crucial for applications in dimensionality reduction and feature selection.
Theoretical implications include a refined understanding of the complexity landscape for matrix-related problems, where volume sampling is emerging as a pivotal concept. The paper's techniques underscore a shift from traditional SVD or PCA-based methods, which may not always be effective at isolating meaningful subsets of data, to volume-based approaches that more naturally capture the intrinsic contextual relationships within matrix rows.
Anticipated future work could focus on refining these algorithms for enhanced stability and performance in numerical applications, further exploring connections with determinantal point processes, and developing random walk analogs for volume sampling.
Through these developments, the paper not only solves a previously open problem but also lays the groundwork for broader exploration in spectral algorithms and the mathematical underpinnings of data approximation and analysis.