Section: New Results
A probabilistic approach of high-dimensional least-squares approximations
Participant : Erwan Faou.
The main goal of this work is to derive and analyze new schemes for the numerical approximation of least-squares problems set on high dimensional spaces. This work  ,originates from the Statistical Analysis of Distributed Multipoles (SADM) algorithm introduced by Chipot et al. in 1998 for the derivation of atomic multipoles from the quantum mechanical electrostatic potential mapped on a grid of points surrounding a molecule of interest. The main idea is to draw subsystems of the original large least-square problem and compute the average of the corresponding distribution of solutions as an approximation of the original solution. Moreover, this method not only provides a numerical approximation of the solution, but a global statistical distribution reflecting the accuracy of the physical model used.
Strikingly, it turns out that this kind of approach can be extended to many situations arising in computational mathematics and physics. The principle of the SADM algorithm is in fact very general, and can be adapted to derive efficient algorithms that are robust with the dimension of the underlying space of approximation. This provides new numerical methods that are of practical interest for high dimensional least-squares problems where traditional methods are impossible to implement.
The goal of this paper is twofold:
Give a general mathematical framework, and analyze the consistency, convergence and cost of these new algorithms in an abstract setting and in specific situations where calculations can be made explicit (Wishart or subgaussian distribution). The main outcome is that the subsystems drawn from the original system have to be chosen rectangular and not square (as initially proposed in the SADM method) to obtain convergent and efficient algorithms.
Apply these results to revisit and improve the SADM method. This is mainly done in Section 5 by considering the three-point charge model of water.