Method of Moments for Estimation of Noisy Curves

TL;DR

This paper introduces a method for recovering high-dimensional piecewise linear curves from noisy data using third moment tensors, establishing sample complexity bounds and demonstrating practical recovery techniques.

Contribution

It presents a novel approach leveraging third moments for curve recovery, with theoretical analysis and practical algorithms validated by experiments.

Findings

Sample complexity scales with at least 6 for accurate recovery.

Recovery from third moment tensor is locally well-posed.

Numerical experiments confirm the effectiveness of the proposed methods.

Abstract

In this paper, we study the problem of recovering a ground truth high dimensional piecewise linear curve $C^*(t):[0, 1]\to\mathbb{R}^d$ from a high noise Gaussian point cloud with covariance $\sigma^2I$ centered around the curve. We establish that the sample complexity of recovering $C^*$ from data scales with order at least $\sigma^6$. We then show that recovery of a piecewise linear curve from the third moment is locally well-posed, and hence $O(\sigma^6)$ samples is also sufficient for recovery. We propose methods to recover a curve from data based on a fitting to the third moment tensor with a careful initialization strategy and conduct some numerical experiments verifying the ability of our methods to recover curves. All code for our numerical experiments is publicly available on GitHub.