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A Probabilistic Analysis of Linear Operator Testing

Abstract We test for β-conformance of an implementation linear operator A to a specification linear operator S where the operator domain and range are separable Hilbert spaces and the domain space F is equipped with a Gaussian measure μ. Given an error bound e>0 and a tolerance parameter β∈(0, 1), we want to determine either that there is an element f in a ball Bq of radius q in domain F such that ‖Sf−Af‖>e or that Aβ-conforms to S on a set of measure at least 1−β in the ball Bq; i.e., μq(f: ‖Sf−Af‖⩽e)⩾1−β where μq is the truncated Gaussian measure to Bq. We present a deterministic algorithm that solves this problem and uses almost a minimal number of tests where each test is an evaluation of operators S and A at an element of F. We prove that optimal tests are conducted on the eigenvectors of the covariance operator of μ. They are universal; they are independent of the operators under consideration and other problem parameters. We show that finite testing is conclusive in this probabilistic setting. In contrast, finite testing is inconclusive in the worst and average case settings; see [5, 7]. We discuss the upper and lower bounds on the minimal number of tests. For q=∞ we derive the exact bounds on the minimal number of tests, which depend on β very weakly. On the other hand, for a finite q, the bounds on the minimal number of tests depend on β more significantly. We explain our approach by an example with the Wiener measure.