Method of Moments for Estimation of Noisy Curves

TL;DR

This work tackles the problem of recovering a high-dimensional piecewise-linear curve from a highly noisy Gaussian sample cloud. It proves a fundamental $O(σ^6)$ lower bound on the sample complexity and shows that recovery is locally well-posed when relying on the third moment, enabling algorithms that fit a curve via a third-moment tensor. The proposed two-phase algorithm combines unbiased moment estimation, tensor power method-based subspace recovery, reordering of predicted subspaces, and a final relaxed-moment optimization, with numerical validation demonstrating strong performance in high-noise regimes. The findings offer a principled, moment-based approach to curve recovery in high dimensions and provide a foundation for extending to more general curve classes and robustness analyses.

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 $σ^2I$ centered around the curve. We establish that the sample complexity of recovering $C^*$ from data scales with order at least $σ^6$. We then show that recovery of a piecewise linear curve from the third moment is locally well-posed, and hence $O(σ^6)$ 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.

Figures (9)

• Figure 1: An illustration of the noisy curve recovery problem.
• Figure 2: Results of a cloud tracing algorithm (detailed in §\ref{['sec:cloud-tracing']} of the supplementary material) on a cloud coming from a piecewise linear curve with low noise (left) and high noise (right). In both cases, a ground truth curve in blue is shown with an accompanying point cloud. When the noise is low (i.e., the scale of the noise is smaller than the scale of the geometric features of the curve), the structure of the underlying curve is still visible in the point cloud, and the curve can be successfully traced through (in red). When the noise is large, the local geometric structure of the curve is destroyed and the tracing approach fails.
• Figure 3: An illustration of a toy ideal case for Algorithm \ref{['alg:chain-building']}.
• Figure 4: Four uncurated experimental results for curves with $M=16$ segments, ambient dimension $d = 24$, noise level $\sigma^2=1/4$ and moments estimated from point cloud with $N = 10^8$ points. We show the ground truth curve and a subsample of 1000 points from the point cloud in blue. The output of Algorithm \ref{['alg:full-alg']} is given by the dashed orange line. The output of the baseline algorithm with matching to all three moments is shown faintly in red. All curves are projected down to the first two dimensions of $\mathbb{R}^{24}$
• Figure 5: Four uncurated experimental results for curves with $M=32$ segments, ambient dimension $d = 48$, and access to ground truth moments. We show the ground truth curve in blue. The output of Algorithm \ref{['alg:full-alg']} is given by the dashed orange line. The output of the baseline algorithm with matching to all three moments is shown faintly in red. All curves are projected down to the first two dimensions of $\mathbb{R}^{48}$

Theorems & Definitions (41)

• Proposition 2.1: Moments of Piecewise Linear Curves
• proof
• Definition 2.2: Relaxed Curve Moments
• Lemma 2.3: Jacobians of Relaxed Moments
• Proposition 3.2
• Corollary 3.3
• Proposition 3.4
• Theorem 3.5
• proof
• Theorem 4.1