[Paper Review] Towards Persistence-Based Reconstruction in Euclidean Spaces

Frédéric Chazal, Steve Oudot|ArXiv.org|Dec 17, 2007

Topological and Geometric Data Analysis27 references4 citations

TL;DR

This paper presents a persistence-based reconstruction algorithm that combines greedy landmark selection with topological persistence to reconstruct low-dimensional manifolds embedded in high-dimensional Euclidean spaces. The method achieves a time complexity of $ O(8^{35^m} n^5) $, where $ n $ is the input size and $ m $ is the intrinsic dimension of the manifold, enabling efficient topological inference on high-dimensional data with theoretical guarantees.

ABSTRACT

Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes of compact subsets of Euclidean space $\R^d$. However, the complexities of these methods scale up exponentially with d, which makes them impractical in medium or high dimensions, even for handling low-dimensional submanifolds. In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Specifically, when the data points are sufficiently densely sampled from a smooth $m$-submanifold of $\R^d$, our method retrieves the homology of the submanifold in time at most $c(m)n^5$, where $n$ is the size of the input and $c(m)$ is a constant depending solely on $m$. It can also provably well handle a wide range of compact subsets of $\R^d$, though with worse complexities. Along the way to proving the correctness of our algorithm, we obtain new results on Čech, Rips, and witness complex filtrations in Euclidean spaces.

Motivation & Objective

• To address the computational infeasibility of existing manifold reconstruction methods in high ambient dimensions $ d $, which scale exponentially with $ d $.
• To develop a method that leverages topological persistence and greedy landmark refinement to achieve complexity dependent on the intrinsic dimension $ m $ of the data manifold.
• To provide theoretical guarantees for topological inference on compact subsets and smooth submanifolds of $ \mathbb{R}^d $.
• To validate and quantify the conjecture that witness complex filtrations yield cleaner persistence barcodes than Čech or Rips filtrations on smooth manifolds.
• To bridge the gap between classical reconstruction and topological estimation by producing nested complexes whose persistent homology matches the underlying shape.

Proposed method

• The algorithm iteratively selects landmarks using a maxmin strategy to ensure even coverage of the data point cloud.
• It maintains two nested simplicial complexes: a Rips complex at scale $ 4\varepsilon $ and another at scale $ 16\varepsilon $, both built on the landmark set.
• The persistent homology of the nested complexes is monitored to ensure convergence to the true homology of the underlying shape.
• The method uses a backtracking strategy to compute persistent Betti numbers efficiently, avoiding high-dimensional blowup during early phases.
• It leverages theoretical results on Čech, Rips, and witness complex filtrations to relate their topological properties in Euclidean spaces.
• The algorithm is designed to detect topological plateaus in persistent homology, signaling convergence to the correct Betti numbers.

Experimental results

Research questions

• RQ1 Can a reconstruction algorithm be designed whose complexity depends only on the intrinsic dimension $ m $ of the data manifold, rather than the ambient dimension $ d $?
• RQ2 How can topological persistence be effectively combined with greedy landmark selection to ensure topological correctness?
• RQ3 Do witness complex filtrations indeed produce cleaner persistence barcodes than Čech or Rips filtrations on smooth submanifolds?
• RQ4 What theoretical guarantees can be provided for nested complex filtrations in Euclidean spaces when using landmark-based sampling?
• RQ5 Can the transition from coarse to stable landmark sampling be formally characterized in terms of persistent homology convergence?

Key findings

• For a point cloud that is a sufficiently dense $ \delta $-sample of a smooth $ m $-dimensional submanifold in $ \mathbb{R}^d $, the algorithm reconstructs the correct persistent homology in time $ O(8^{35^m} n^5) $, where $ n $ is the input size.
• The algorithm's complexity scales with the intrinsic dimension $ m $, not the ambient dimension $ d $, making it practical for high-dimensional data with low-dimensional structure.
• The method provably handles a wide class of compact subsets of $ \mathbb{R}^d $, though with worse complexity bounds than for smooth manifolds.
• The paper provides a theoretical framework that allows transposing results from unions of balls to Čech filtrations in Euclidean spaces.
• A new framework is introduced to study filtrations intertwined with the Čech filtration, such as Rips and witness complexes, enabling comparative analysis.
• The paper quantitatively supports the conjecture of Carlsson and de Silva that witness complex filtrations yield cleaner persistence barcodes than Čech or Rips filtrations on smooth submanifolds.