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The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.
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Straggler Robust Distributed Matrix Inverse Approximation
A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications [1]. This has both numerical stability and complexity issues, as well as high expected time to compute. We address the latter issue, by proposing an algorithm which uses a black-box least squares optimization solver as a subroutine, to give an estimate of the inverse (and pseudoinverse) of real nonsingular matrices; by estimating its columns. This also gives it the flexibility to be performed in a distributed manner, thus the estimate can be obtained a lot faster, and can be made robust to stragglers. Furthermore, we assume a centralized network with no message passing between the computing nodes, and do not require a matrix factorization; e.g. LU, SVD or QR decomposition beforehand.
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- Award ID(s):
- 1838179
- PAR ID:
- 10206907
- Date Published:
- Journal Name:
- IEEE Statistical Signal Processing Workshop
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
