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AARMS Scientific Machine Learning Seminar: Simone Brugiapaglia (Concordia University)
March 8, 2022 @ 11:00 am - 12:00 pm
The curse of dimensionality and the blessings of sparsity and Monte Carlo sampling: From polynomial approximation to deep learning in high dimensions
In data science and scientific computing, the approximation of high-dimensional functions from pointwise samples is a ubiquitous task, which is made intrinsically difficult by the so-called curse of dimensionality. In this talk, we will illustrate how to alleviate the curse thanks to the “blessings” of sparsity and Monte Carlo sampling.
First, we will consider the case of sparse polynomial approximation via compressed sensing. Focusing on the case where the target function is smooth, but possibly highly anisotropic, we will show how to obtain sample complexity bounds only mildly affected by the curse of dimensionality, near-optimal accuracy guarantees, stability to unknown errors corrupting the data, and rigorous convergence rates of algebraic and exponential type.
Then, we will illustrate how the mathematical toolkit of sparse polynomial approximation can be employed to obtain a “practical existence theorem” for deep learning in the context of high-dimensional Hilbert-valued function approximation. This result shows not only the existence of neural networks with desirable approximation properties, but also how to compute them via a suitable training procedure in order to achieve best-in-class performance guarantees.
We will conclude by discussing ongoing and future research directions.