Convolutional Persistence Transforms

TL;DR

This work introduces convolutional persistence, a topology-based feature pipeline that pre-processes data on simplicial cubical complexes with local filters before computing persistent homology. By framing convolutions as local motifs, CPT connects to the Persistent Homology Transform to establish generic injectivity, while delivering stability guarantees and flexible, data-driven vectorizations. Empirically, CPT improves classification performance across multiple image- and PDE-based datasets, even with random filters and simple vectorizations like total persistence. The approach offers computational benefits through downsampling and has potential for broader adoption in topological data analysis and related learning tasks.

Abstract

In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution filter as a local motif, the persistence diagram of the resulting convolution describes the way the motif is distributed across the simplicial complex. This pipeline, which we call convolutional persistence, extends the capacity of topology to observe patterns in such data. Moreover, we prove that (generically speaking) for any two labeled complexes one can find some filter for which they produce different persistence diagrams, so that the collection of all possible convolutional persistence diagrams is an injective invariant. This is proven by showing convolutional persistence to be a special case of another topological invariant, the Persistent Homology Transform. Other advantages of convolutional persistence are improved stability, greater flexibility for data-dependent vectorizations, and reduced computational complexity for certain data types. Additionally, we have a suite of experiments showing that convolutions greatly improve the predictive power of persistence on a host of classification tasks, even if one uses random filters and vectorizes the resulting diagrams by recording only their total persistences.

Figures (17)

• Figure 1: The original image is shown on the left. The image has interesting topological structure but the sublevel-set persistence is trivial. On the right, we show the result of convolving with a $2 \times 2$-filter with a value of $1$ in each pixel. The sublevel-set persistence of the convolved image is more informative; at height zero, there are three connected components, corresponding to three regions in the image where the convolution is zero. These regions then merge at an immediate value, due to those parts of the image where the local patches are more similar to the filter.
• Figure 2: Top-left: A simplicial complex with filtration values attached to vertices, edges, and squares. Top-right through bottom-left: sublevel-sets associated with different threshold values. Bottom-right: barcodes in dimensions zero and one. The zero-dimensional homology $H_0$ barcode contains four bars, since at $\alpha=0$ there are four connected components. Two bars die at $\alpha=1$, since at that threshold value there are only two connected components. Finally, $\alpha=2$ sees the merger of these connected component, so another bar dies at $\alpha=2$ and the last persists to infinity. In one-dimensional homology $H_1$, three bars are born at $\alpha = 2$, when three loops appear in the sublevel-set, and one of these bars dies at $\alpha=3$, when that loop is killed off by the introduction of a square.
• Figure 3: A $2 \times 2$ image can be turned into a complex in one of two ways. Left: A complex with four top-dimensional cubes, with function values, indicated using color, extended to vertices and edges via the upper-* rule. Right: A complex with four vertices, with function values extended to the edges and interior square via the lower-* rule.
• Figure 4: Two persistence diagram $D_1$ and $D_2$, one in black and the other in blue. An optimal matching between these diagrams is shown, containing both diagonal and non-diagonal pairings. The resulting $p$-Wasserstein distance is then $W_{p}(D_1,D_2) = \sqrt[p]{d_1^p + d_2^p + d_3^p + d_4^p + d_5^p}$.
• Figure 5: Two functions on the same grid with the same persistence, using either sublevel- or superlevel-set filtrations. Color scheme: {black: 0, grey: 0.5, white: 1}. All persistence diagrams consist of the single point $(0,1)$ in dimension $0$.

Theorems & Definitions (48)

• Theorem 1: cohen2007stability
• Corollary 1
• proof
• Definition 1: skraba2020wasserstein, Definition 4.1
• Theorem 2: skraba2020wasserstein, Theorem 4.8
• Theorem 3: skraba2020wasserstein, Theorem 5.1
• Definition 2
• Proposition 4
• proof
• Definition 3