Recorded 26 September 2024.Gregory Beylkin of the University of Colorado Boulder presents "On algorithms in high dimensions" at IPAM's Analyzing High-dimensional Traces of Intelligent Behavior Workshop.
Abstract: The talk will review two representations of multivariate functions and associated algorithms that avoid the "curse of dimensionality" in high dimensions. First, we will consider separated representations of functions
and operators and briefly discuss their applications to multivariate regression and ML. In the statistics literature representations of such form appear under the names “parallel factorization” or “canonical decomposition” and has been used primarily to analyze data on a grid (typically in dimension d=3) and do not construct the underlying function. We view separated representations as a nonlinear method to track a function in a high-dimensional space while using a small number of parameters. Note that a traditional approach to characterizing a wide class of low-complexity functions using smoothness and decay of derivatives does not appear to work in high dimensions.
Second, we will consider multivariate mixtures (a more general class of functions than separated representations), the corresponding reduction algorithm and several applications in numerics as well as in data science.
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