Recursive estimation of the conditional geometric median in Hilbert spaces

TL;DR

This paper introduces a recursive stochastic gradient algorithm for robust online estimation of the conditional geometric median in Hilbert spaces, with proven convergence properties and practical applications in high-dimensional data analysis.

Contribution

It presents a novel recursive estimator based on stochastic gradient methods for the conditional geometric median in Hilbert spaces, with theoretical convergence guarantees and real-world high-dimensional data applications.

Findings

Proves almost sure convergence and L2 rates of the estimator.

Establishes asymptotic normality with optimal convergence rate.

Demonstrates effectiveness in high-dimensional television audience data.

Abstract

A recursive estimator of the conditional geometric median in Hilbert spaces is studied. It is based on a stochastic gradient algorithm whose aim is to minimize a weighted L1 criterion and is consequently well adapted for robust online estimation. The weights are controlled by a kernel function and an associated bandwidth. Almost sure convergence and L2 rates of convergence are proved under general conditions on the conditional distribution as well as the sequence of descent steps of the algorithm and the sequence of bandwidths. Asymptotic normality is also proved for the averaged version of the algorithm with an optimal rate of convergence. A simulation study confirms the interest of this new and fast algorithm when the sample sizes are large. Finally, the ability of these recursive algorithms to deal with very high-dimensional data is illustrated on the robust estimation of television audience profiles conditional on the total time spent watching television over a period of 24 hours.