Recovering the homology of immersed manifolds

TL;DR

The paper tackles recovering the homology of an abstract manifold from samples of its immersion in Euclidean space. It introduces a measure-theoretic lift of the tangent bundle into a lift space and uses DTM-filtrations to read the homotopy type of the original manifold from a lifted measure, without requiring knowledge of the ambient or intrinsic dimension. A key contribution is the normal reach concept, which quantifies self-intersections and facilitates probabilistic bounds; tangent spaces are estimated via local covariance matrices under Wasserstein stability, yielding consistency and robustness. The framework yields a practical, stable pipeline where persistent homology barcodes from the lifted measure reveal the homology of the original manifold, with data-driven guidance and theoretical guarantees. Numerical illustrations on synthetic data accompany the theory, and code/Notebooks are provided for reproducibility and SEO-friendly indexing.

Abstract

Given a sample of an abstract manifold immersed in some Euclidean space, we describe a way to recover the singular homology of the original manifold. It consists in estimating its tangent bundle -- seen as subset of another Euclidean space -- in a measure theoretic point of view, and in applying measure-based filtrations for persistent homology. The construction we propose is consistent and stable, and does not involve the knowledge of the dimension of the manifold. In order to obtain quantitative results, we introduce the normal reach, which is a notion of reach suitable for an immersed manifold.

Figures (36)

• Figure 1: Left: The abstract manifold $\mathcal{M}_0$, a circle. Middle: The immersion $\mathcal{M} \subset \mathbb{R}^2$, known as the lemniscate of Bernoulli. Right: The observation $X$.
• Figure 2: Two views of the submanifold $\check \mathcal{M} \subset \mathbb R^2 \times \mathrm{M}(\mathbb R^2) \simeq \mathbb R^6$, projected in a 3-dimensional subspace via Principal Component Analysis (PCA). Observe that it does not self-intersect. The initial set $\mathcal{M}$ is represented in Figure \ref{['Paper2:fig:lemniscate']}.
• Figure 3: Left: Persistence barcode of the 1-homology of the Čech filtration of $\mathcal{M}$ in the ambient space $\mathbb R^2$. One reads the 1-homology of the lemniscate. Right: Persistence barcode of the 1-homology of the Čech filtration of $\check \mathcal{M}$ in the lift space $\mathbb R^2 \times \mathrm{M}(\mathbb R^2)$. At the beginning of the barcode, one reads the 1-homology of a circle. Parameter $\gamma = 2$
• Figure 4: Left: The set $\mathrm{supp}( \check{\mu}_0 ) = \check \mathcal{M}$, where $\mu$ is the uniform measure on $\mathcal{M}$ (see Figure \ref{['Paper2:fig:lemniscate']}). Right: The set $\mathrm{supp}( \check \nu )$, where $\nu$ is the empirical measure on $X$. Parameters $\gamma = 2$ and $r=0{.}1$.
• Figure 5: Persistence barcodes of the 0-homology (left) and 1-homology (right) of the DTM-filtration of the lifted measure $\check \nu$. Observe that the homology of the circle is salient on these barcodes (one large red bar and one large green bar). Parameters $\gamma = 2$, $r=0{.}1$ and $m=0{.}01$.

Theorems & Definitions (90)

• Theorem 2.1: federer1959curvature
• Theorem 2.2: aamari:hal-01521955
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 2.6
• Theorem 2.7: Chazal_Geometricinference
• Theorem 2.8: anai2020dtm
• Corollary 2.9
• Definition 3.1