University of Colorado, Boulder
Tuesday, June 16, 4:00 PM EDT (2:00 PM MDT)
A video of this talk is available to NIST staff in the Math channel on NISTube, which is accessible from the NIST internal home page.
Abstract: The curse of dimensionality presents a fundamental challenge for approximating, integrating, or optimizing functions with many parameters in applications throughout science and engineering. In this talk, we trace one origin of the curse of dimensionality to the number of epsilon-balls required to cover the domain—a quantity that grows exponentially as the dimension increases. To mitigate this difficulty, we introduce a function-dependent metric on the input domain using the Lipschitz matrix: a generalization of the scalar Lipschitz constant. This metric reduces the number of epsilon-balls required to cover the domain and consequently decreases computational cost. One important case is when the Lipschitz matrix is low rank: this indicates the function is a ridge function and that complexity scales with the rank of the Lipschitz matrix, not the dimension of the domain. The Lipschitz matrix also has several other applications. For example, the Lipschitz matrix provides a notion of uncertainty complementing uncertainty based on Gaussian processes. Additionally the Lipschitz matrix can be used for design of experiments, providing more efficient space filling designs and generalizing Latin hypercube sampling to non-orthogonal directions.
Bio: Jeffrey Hokanson is a Postdoctoral Fellow at the University of Colorado Boulder working with Paul Constantine. He received his Ph.D. at Rice University under the supervision of Mark Embree. His research interests include model reduction, rational approximation, and parameter space dimension reduction.