Interpolation for Physical Big Data
Time and place
3 PM on Wednesday, February 18th, 2015; SH 377
Matthew Hirn (École normale supérieure, Paris)
Abstract
The notion of Big Data is becoming increasingly ubiquitous in academia, industry, as well as general discourse. While such data sets take numerous forms and span myriad fields, common amongst them is that they are massive and high dimensional. The analysis and extraction of new information from these data sets can often be posed as an interpolation problem mathematically, in which the function to be interpolated corresponds to an important property of the data. However, interpolation algorithms must not only scale efficiently with the size of the data set, but also the dimension. Precise interpolations may require a number of examples that is exponential in the dimension, and are thus intractable. In order to circumvent this curse of dimensionality, functional regularity as well as data regularity can be exploited, depending on the particular problem. In this talk we will touch upon three areas of research, each of which exploits different regularity priors: manifold learning, minimal smooth interpolations, and sparse regressions over families of functions. The applications we will have in mind are derived from physics. Physical functionals are usually computed as solutions of variational problems or from solutions of partial differential equations, which may require huge computations for complex systems. Quantum chemistry calculations of molecular ground state energies is such an example. Learning strategies do not simulate the physical system but estimate solutions by interpolating values provided by a training set of known examples. A specific point of interest in this talk will be the sparse regression of quantum molecular energies in smaller approximation spaces that take advantage of the physical regularity of the system. We introduce deep multiscale learning architectures, which compute such approximations, with iterated wavelet transforms. Numerical applications are shown for molecular energies in quantum chemistry, in relation with Density Functional Theory.
