[Paper Review] Manifold Reconstruction and Denoising from Scattered Data in High Dimension via a Generalization of $L_1$-Median

Shira Faigenbaum-Golovin, David I. W. Levin|arXiv (Cornell University)|Dec 23, 2020

Advanced Vision and Imaging38 references4 citations

TL;DR

This paper proposes Manifold Locally Optimal Projection (MLOP), a non-convex optimization method that reconstructs and denoises low-dimensional manifolds from high-dimensional scattered data by generalizing the L1-median to higher dimensions. It achieves O(h²) approximation accuracy, converges linearly near local minima, and avoids dimensionality reduction while handling noise, outliers, and non-uniform sampling effectively.

ABSTRACT

In this paper, we present a method for denoising and reconstruction of low-dimensional manifold in high-dimensional space. We suggest a multidimensional extension of the Locally Optimal Projection algorithm which was introduced by Lipman et al. in 2007 for surface reconstruction in 3D. The method bypasses the curse of dimensionality and avoids the need for carrying out dimensional reduction. It is based on a non-convex optimization problem, which leverages a generalization of the outlier robust L1-median to higher dimensions while generating noise-free quasi-uniformly distributed points reconstructing the unknown low-dimensional manifold. We develop a new algorithm and prove that it converges to a local stationary solution with a bounded linear rate of convergence in case the starting point is close enough to the local minimum. In addition, we show that its approximation order is $O(h^2)$, where $h$ is the representative distance between the given points. We demonstrate the effectiveness of our approach by considering different manifold topologies with various amounts of noise, including a case of a manifold of different co-dimensions at different locations.

Motivation & Objective

• To address the challenge of reconstructing and denoising low-dimensional manifolds from high-dimensional, noisy, and non-uniformly sampled data.
• To overcome the limitations of classical methods that rely on normal estimation or assume smoothness and noise-free data.
• To develop a method that avoids dimensionality reduction and is robust to outliers and varying sampling density.
• To provide a noise-free, quasi-uniformly distributed reconstruction of the underlying manifold in high-dimensional space.
• To establish theoretical convergence and approximation error bounds for the proposed algorithm.

Proposed method

• The method extends the Locally Optimal Projection (LOP) algorithm to high dimensions using a multidimensional generalization of the L1-median for robustness to outliers.
• It solves a non-convex optimization problem that minimizes a cost function combining data fidelity and a regularization term promoting quasi-uniform point distribution.
• The algorithm iteratively updates point positions to minimize the objective, leveraging local projections and L1-median-based weighting in high-dimensional space.
• Convergence is proven to a local stationary solution with a bounded linear rate when initialized near a local minimum.
• The method does not require surface parameterization, normal estimation, or prior knowledge of the manifold's intrinsic dimension.
• The complexity scales linearly with ambient dimension n and is independent of the intrinsic dimension d.

Experimental results

Research questions

• RQ1 Can a non-convex optimization framework based on generalized L1-median effectively reconstruct low-dimensional manifolds from high-dimensional noisy data?
• RQ2 Does the proposed method achieve better approximation accuracy than classical methods in high-dimensional, noisy settings?
• RQ3 Can the method converge linearly to a local minimum without requiring dimensionality reduction or normal estimation?
• RQ4 How does the method perform on manifolds with varying co-dimensions and non-uniform sampling?
• RQ5 What is the theoretical approximation order of the method in terms of representative point spacing h?

Key findings

• The MLOP algorithm converges to a local stationary solution with a bounded linear rate of convergence when the initial point is sufficiently close to a local minimum.
• The method achieves an approximation order of O(h²), where h is the representative distance between input points.
• The reconstruction accuracy is superior to that of noise-free data when using local PCA, due to the quasi-uniform sampling induced by the algorithm.
• For noise levels of 0.1 and 0.2, the median PCA error increased only slightly, from 7.9 to 8.2, demonstrating robustness to noise.
• The method effectively handles complex topologies, including manifolds with varying co-dimensions across different regions.
• The algorithm’s complexity is linear in the ambient dimension n and does not depend on the intrinsic dimension d, enabling scalability to high dimensions.