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Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction

Mattéo Clémot, Julie Digne, Julien Tierny

TL;DR

TopoAE++ advances topology-aware dimensionality reduction by extending TopoAE to preserve $PH^{1}$ cycles through a cascade distortion loss that enforces isometric filling of 1-cycles. It introduces a fast planar $PH$ computation algorithm tailored to 2D embeddings, enabling practical optimization, and demonstrates a strong balance between topological accuracy and visual fidelity on synthetic and real datasets. Theoretical results clarify when $PH^{0}$ preservation guarantees hold and motivate the new $PH^{1}$-aware formulation, while empirical results show competitive $PD^{1}$ distances and improved cycle visualization compared to baselines. The work provides an open-source C++ implementation and outlines directions toward higher-dimensional latent spaces and homology levels, broadening the applicability of cycle-aware DR in complex data landscapes.

Abstract

This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation. First, we provide a novel theoretical analysis of its associated loss and show that a zero loss indeed induces identical persistence pairs (in high and low dimensions) for the $0$-dimensional persistent homology (PH$^0$) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PH$^d$ for $d\ge 1$. Based on this observation, we introduce a novel generalization of TopoAE to $1$-dimensional persistent homology (PH$^1$), called TopoAE++, for the accurate generation of cycle-aware planar embeddings, addressing the above failure case. This generalization is based on the notion of cascade distortion, a new penalty term favoring an isometric embedding of the $2$-chains filling persistent $1$-cycles, hence resulting in more faithful geometrical reconstructions of the $1$-cycles in the plane. We further introduce a novel, fast algorithm for the exact computation of PH for Rips filtrations in the plane, yielding improved runtimes over previously documented topology-aware methods. Our method also achieves a better balance between the topological accuracy, as measured by the Wasserstein distance, and the visual preservation of the cycles in low dimensions. Our C++ implementation is available at https://github.com/MClemot/TopologicalAutoencodersPlusPlus.

Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction

TL;DR

TopoAE++ advances topology-aware dimensionality reduction by extending TopoAE to preserve cycles through a cascade distortion loss that enforces isometric filling of 1-cycles. It introduces a fast planar computation algorithm tailored to 2D embeddings, enabling practical optimization, and demonstrates a strong balance between topological accuracy and visual fidelity on synthetic and real datasets. Theoretical results clarify when preservation guarantees hold and motivate the new -aware formulation, while empirical results show competitive distances and improved cycle visualization compared to baselines. The work provides an open-source C++ implementation and outlines directions toward higher-dimensional latent spaces and homology levels, broadening the applicability of cycle-aware DR in complex data landscapes.

Abstract

This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation. First, we provide a novel theoretical analysis of its associated loss and show that a zero loss indeed induces identical persistence pairs (in high and low dimensions) for the -dimensional persistent homology (PH) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PH for . Based on this observation, we introduce a novel generalization of TopoAE to -dimensional persistent homology (PH), called TopoAE++, for the accurate generation of cycle-aware planar embeddings, addressing the above failure case. This generalization is based on the notion of cascade distortion, a new penalty term favoring an isometric embedding of the -chains filling persistent -cycles, hence resulting in more faithful geometrical reconstructions of the -cycles in the plane. We further introduce a novel, fast algorithm for the exact computation of PH for Rips filtrations in the plane, yielding improved runtimes over previously documented topology-aware methods. Our method also achieves a better balance between the topological accuracy, as measured by the Wasserstein distance, and the visual preservation of the cycles in low dimensions. Our C++ implementation is available at https://github.com/MClemot/TopologicalAutoencodersPlusPlus.

Paper Structure

This paper contains 51 sections, 6 theorems, 17 equations, 23 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. General purpose dimensionality reduction
  4. Globally topology-aware dimensionality reduction
  5. Background
  6. Geometric structures
  7. Persistent homology
  8. Persistence homology computation
  9. Geometric interpretation of persistent homology
  10. Topological Autoencoders
  11. Topological Autoencoder Analysis
  12. Relation to the Wasserstein distance
  13. Counter-example for higher dimensional PH
  14. Cascade Distortion
  15. Formulation
  16. ...and 36 more sections

Key Result

Lemma 1

For any point clouds $X$ and $Z$ of equal size, we have the following inequality: Besides, under general position hypothesis (unique pairwise distances), if $\mathcal{L}_{\mathrm{TAE}}^{0}(X,Z)=0$, then $\mathcal{D}_{\mathop{\mathrm{Rips}}\nolimits}^{0}(X)=\mathcal{D}_{\mathop{\mathrm{Rips}}\nolimits}^{0}(Z)$ and the $\text{PH}^{0}$ pairs are the same, i.e. $\mathsf{MST}(X)=\math

Figures (23)

  • Figure 1: Comparison of DR methods on three synthetic 3D point clouds (see \ref{['sec:test_data']} for a description), along with the metric distortion $\mathop{\mathrm{\mathfrak{D}}}\nolimits(X,Z)$ between the input $X$ and its planar embedding $Z$, the Wasserstein distance $\mathop{\mathrm{PD\mathcal{W}^1}}\nolimits(X,Z)= \mathcal{W}_2\bigl(\mathcal{D}_{\mathop{\mathrm{Rips}}\nolimits}^{1}(X),\mathcal{D}_{\mathop{\mathrm{Rips}}\nolimits}^{1}(Z)\bigr)$ between their respective 1-dimensional persistence diagrams, and the running time $t$ in seconds. The best value for an indicator is written in bold, the second best is underlined. In the second line, a generator of the most persistent $\text{PH}^{1}$ pair in the high-dimensional input is projected in the 2D embedding in transparency and slightly smoothed for visualization purposes (to distinguish it from the point cloud). The generator color map depicts its arc-length parameterization. In the third line, the input -- that is sampled around the edges of a tetrahedron in 3D -- features 3 significantly persistent $\text{PH}^{1}$ pairs, and a generator for each is represented both in the input and in the 2D embeddings with a specific color. Quantitatively, our approach (TopoAE++) generates planar embeddings with competitive Wasserstein distances. Qualitatively, it produces planar embeddings yielding less crossings of the projected high-dimensional persistent generators (colored curves), thereby producing visualizations that depict more faithfully the topological handles present in high dimensions.
  • Figure 2: Left: geometric graphs for a point cloud $X$ in the plane. Its $\mathsf{MST}$ is depicted in red. The edges from the $\mathsf{RNG}$ which are not in the $\mathsf{MST}$ are shown in continuous yellow, while those which are in the $\mathsf{UG}$ but not in the $\mathsf{RNG}$ are shown in dashed yellow. The remaining Delaunay edges are shown in gray. Three lenses are shown: the purple one is devoid of points of $X$, therefore the associated yellow edge belongs to the $\mathsf{RNG}$; the blue one -- associated with the dashed yellow edge -- is devoid of points of the link of that edge but contains another point of $X$ (the blue one), therefore this edge is in the $\mathsf{UG}$ but not in the $\mathsf{RNG}$; the green one contains the green point and the associated gray edge is not in $\mathsf{RNG}$ nor in $\mathsf{UG}$. Right: within the highlighted bottom $\mathsf{RNG}$-polygon, a $\mathsf{MML}$ triangulation has replaced the Delaunay triangulation, with its longest edge highlighted in black. Note that for the two other polygons, the Delaunay triangulation is already a $\mathsf{MML}$ triangulation within these polygons.
  • Figure 3: Rips complexes of the same planar point cloud $X$, for increasing diameter thresholds $r$ (only the 2-skeletons, i.e., the vertices, edges and triangles, of the simplicial complexes are shown). This threshold increase induces a sequence of nested simplicial complexes, whose topology varies along the process. From left to right, the number of connected components is successively $n$, then $2$, then $1$, while the number of handles is successively $0$, $2$, $1$, $2$, $1$ and $0$.
  • Figure 4: Two point clouds (red and blue) in the plane represented with their $\mathsf{RNG}$, along with their respective 1-dimensional persistence diagrams (right). The edges in $\mathsf{RNG}\setminus\mathsf{MST}$ are highlighted in bold and yellow. The number of non-diagonal points in each diagram is exactly the number of $\mathsf{RNG}$-polygons in the associated point cloud (3 for the red one, 2 for the blue one), and also the number of $\mathsf{RNG}\setminus\mathsf{MST}$ edges. The optimal assignment inducing the Wasserstein distance $\mathcal{W}_2$ between them is shown in black.
  • Figure 5: Some steps of two executions of the EliminateBoundaries\ref{['algo:eliminateBoundaries']} procedure run on the 2-simplices $\sigma_0$ (top line) and $\sigma_1$ (bottom line), corresponding to the creation of two $\text{PH}^{1}$ pairs in the previous planar point cloud (with its smallest cycle removed for clarity). Unpaired edges -- which are in $\mathsf{RNG}(X)\setminus\mathsf{MST}(X)$ -- are depicted in yellow. At each step, $\mathop{\mathrm{\mathrm{cascade}}}\nolimits[\sigma]$ is shown in red, and its boundary $\partial(\mathop{\mathrm{\mathrm{cascade}}}\nolimits[\sigma])$ is highlighted in bold, with its longest edge, i.e., $\tau=\text{Youngest}\bigl(\partial(\mathop{\mathrm{\mathrm{cascade}}}\nolimits[\sigma])\bigr)$, dotted. When it exists, $\mathop{\mathrm{\mathrm{cascade}}}\nolimits\bigl[\mathop{\mathrm{\mathrm{partner}}}\nolimits[\tau]\bigr]$ is shown is blue. A persistent pair $(\tau, \sigma)$ is created when $\mathop{\mathrm{\mathrm{partner}}}\nolimits[\tau]=\varnothing$, i.e., when $\tau$ coincides with a yellow unpaired edge (rightmost images). In both executions, $\mathop{\mathrm{Rips}}\nolimits_{\delta(\sigma)}^1$ is shown in light gray.
  • ...and 18 more figures

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof : Proof outline
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 2 more