Denoising Flows on Trees

TL;DR

This paper investigates the estimation of flows on trees, analyzing the least squares estimator's performance and establishing minimax bounds, with results depending on the tree's diameter and path length.

Contribution

It provides a theoretical analysis of flow estimation on trees, extending isotonic regression to structured tree data and characterizing estimator risk in different regimes.

Findings

Risk bounds depend on tree diameter and path length.

Least squares estimator performs optimally in both regimes.

Comparison with isotonic regression risk bounds enhances understanding.

Abstract

We study the estimation of flows on trees, a structured generalization of isotonic regression. A tree flow is defined recursively as a positive flow value into a node that is partitioned into an outgoing flow to the children nodes, with some amount of the flow possibly leaking outside. We study the behavior of the least squares estimator for flows, and the associated minimax lower bounds. We characterize the risk of the least squares estimator in two regimes. In the first regime the diameter of the tree grows at most logarithmically with the number of nodes. In the second regime, the tree contains long paths. The results are compared with known risk bounds for isotonic regression.