A Probabilistic Subspace Bound with Application to Active Subspaces

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Given a real symmetric positive semidefinite 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 nonasymptotic 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.