Gauss quadrature for matrix inverse forms with applications
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Jegelka_Gauss quadrature.pdf
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374.41 KB
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Author(s)
Li, Chengtao
Sra, Suvrit
Jegelka, Stefanie Sabrina
Date Issued
June 2016
Journal
International Conference on Machine Learning
Publisher
Proceedings of Machine Learning Research
Citation
Li, Chengtao, Suvrit Sra, and Stefanie Jegelka. "Gauss quadrature for matrix inverse forms with applications." International Conference on Machine Learning, 20-22 June 2016, New York, New York, PMLR, 2016.
Version
Original manuscript
Abstract
We present a framework for accelerating a spectrum of machine learning algorithms that require computation of bilinear inverse forms u[superscript T] A[superscript −1]u, where A is a positive definite matrix and u a given
vector. Our framework is built on Gauss-type quadrature and easily scales to large, sparse matrices. Further, it allows retrospective computation of lower and upper bounds on u[superscript T] > A[superscript −1]u, which in
turn accelerates several algorithms. We prove that these bounds tighten iteratively and converge at a linear (geometric) rate. To our knowledge, ours is the first work to demonstrate these key properties of Gauss-type quadrature, which is a classical and deeply studied topic. We illustrate empirical consequences of our results by using quadrature to accelerate machine learning tasks involving determinantal point processes and submodular optimization, and observe tremendous speedups in several
instances.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
http://proceedings.mlr.press/v48
