Statistics/Probability Seminar
Speaker: Sabyasachi Chatterjee, University of Illinois Urbana-Champaign
Host: Parthanil Roy
Title: A New Min–Max Representation for Univariate Total Variation Denoising
Time, day and date: 4:00:00 PM – 5:00:00 PM, Tuesday, August 11
Venue: Ramanujan Hall
Abstract: Total variation denoising (TVD) is a classical and widely used method for estimating signals with piecewise constant structure. Although the estimator is defined through a simple convex optimization problem and has been studied extensively, no explicit pointwise characterization of the fitted values was previously known.
In this talk, I will present a new min–max/max–min representation for TVD that gives an explicit pointwise description of the fitted values in terms of local averages over intervals. This representation reveals a hidden multiscale structure underlying the estimator, provides new insight into its bias and variance, and leads to sharp pointwise risk bounds that adapt to the local smoothness of the underlying signal.
I will then show that this pointwise perspective extends naturally to quantile regression. In this setting, the fitted values admit an exact characterization as an interval determined by local order statistics, reflecting the non-strict convexity of the quantile loss. This characterization yields new structural insights into quantile TVD, including automatically non-crossing quantile curves without the need for additional constraints.
The talk is based on the following two papers:
(1) https://arxiv.org/pdf/2410.03041
(2) https://arxiv.org/pdf/2605.01237