A Probabilistic Subspace Bound with Application to Active Subspaces
| Autor: | John T. Holodnak, Ilse C. F. Ipsen, Ralph C. Smith |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Monte Carlo method
Numerical Analysis (math.NA) 010103 numerical & computational mathematics Positive-definite matrix 01 natural sciences Linear subspace 15A18 15A23 15A60 15B10 35J25 60G60 65N30 65C06 65C30 65F15 65D05 010101 applied mathematics Approximation error FOS: Mathematics Applied mathematics Mathematics - Numerical Analysis 0101 mathematics Intrinsic dimension Random variable Analysis Eigendecomposition of a matrix Subspace topology Mathematics |
| Zdroj: | SIAM Journal on Matrix Analysis and Applications. 39:1208-1220 |
| ISSN: | 1095-7162 0895-4798 |
| DOI: | 10.1137/17m1141503 |
| Popis: | Given a real symmetric positive semi-definite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative error in S due to randomization. The bounds do not depend on the matrix dimensions but only on the numerical rank (intrinsic dimension) of E. Our approach resembles the low-rank approximation of kernel matrices from random features, but our accuracy measures are more stringent. In the context of parameter selection based on active subspaces, where S is computed via Monte Carlo sampling, we present a bound on the number of samples so that with high probability the angle between the dominant subspaces of E and S is less than a user-specified tolerance. This is a substantial improvement over existing work, as it is a non-asymptotic and fully explicit bound on the sampling amount n, and it allows the user to tune the success probability. It also suggests that Monte Carlo sampling can be efficient in the presence of many parameters, as long as the underlying function f is sufficiently smooth. |
| Databáze: | OpenAIRE |
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