Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH

TL;DR

Non-Isotropic Persistent Homology (NIPH) addresses the limitation that standard persistent homology depends on a single metric and may miss geometric structure. It introduces a pipeline that varies the distance function along directions with scaling factors, computes persistence diagrams, matches death distributions via optimal transport, and extracts orientation $\varphi$, orientational variance $V$, and scaling $s$ by optimizing peak alignment. The method is demonstrated on synthetic data and road networks, showing accurate orientation recovery and meaningful scaling/variance estimates, outperforming PCA in orientation detection. This work provides a general framework to encode geometric information through metric variations, with potential applications in shape analysis and network data.

Abstract

Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration. Most filtrations used for persistent homology depend (implicitly) on a chosen metric, which is typically agnostically chosen as the standard Euclidean metric on $\mathbb{R}^n$. Recent work has tried to uncover the 'true' metric on the point cloud using distance-to-measure functions, in order to obtain more meaningful persistent homology results. Here we propose an alternative look at this problem: we posit that information on the point cloud is lost when restricting persistent homology to a single (correct) distance function. Instead, we show how by varying the distance function on the underlying space and analysing the corresponding shifts in the persistence diagrams, we can extract additional topological and geometrical information. Finally, we numerically show that non-isotropic persistent homology can extract information on orientation, orientational variance, and scaling of randomly generated point clouds with good accuracy and conduct some experiments on real-world data.

Figures (17)

• Figure 1: Schematic of Non-Isotropic Persistent Homology (NIPH). Step 1. Produce different versions of the input point cloud by applying directional scaling in direction $\mathbf{u}$ and scaling-factor $s$ according to elements in input set $D$. Step 2. Compute PH of desired degree (the diagram displays 0th persistent homology) for input point cloud and each of the scaled point clouds. Step 3. Compute optimal transport between weighted death time distributions of input point cloud and each of the scaled point clouds. Step 4. Compute multiplicative factor of shift for each death time. Extract maxima and use optimisation problem to compute preferred orientations, scaling-factor, and orientational variance of point cloud, as seen in the illustrations on the right.
• Figure 2: Data features not captured by standard PH.Left: Persistent homology will not distinguish circles and ellipses by their death time $\varepsilon$. Right: Persistent homology cannot distinguish orientations and orientational variances of the data set by death times $\varepsilon$.
• Figure 3: Illustrative Experiments. Point cloud with ellipses of various radii in orientation $\varphi = 0^\circ$, $15^\circ$ and $30^\circ$, scaling factor $s=2$ and orientational variance $V=0$. Top: Part of point cloud. Middle: Death density diagram of $1$-dimensional persistent homology. Bottom: Multiplicative shift diagram used to extract information on orientation, orientational variance, and scaling. In the left diagram, the blue curve represents scaling parallel to the directional scaling of the ellipses. Thus there is no change in the death times and the peak of the curve is at $\sim 1.0$. The orange curve represents scaling in a direction almost orthogonal to the scaling of the data points, hence the death times are multiplied by the factor of the scaling. This is represented by the peak almost reaching $2.0$. The red and green curve represent scaling roughly at a direction of $45^\circ$ to the original scaling in the point cloud. When we change the orientation of the point cloud the orange peak will move to the left and the the red peak will move to the right. The mult. death shift diagrams are easy to interpret and concise.
• Figure 4: Quantitative performance of NIPH. We have run NIPH on a point cloud sampled from 200 oriented rectangles with different orientational variances and $s=2$. We show the root of the mean squared error of the predictions of NIPH depending on the orientational variance of $X$.
• Figure 5: Predictions of NIPH in high orientational variance setting.$x$-axis: True $\varphi$. Blue:$\varphi$. Green:$\sqrt{V}$. (True value: $0.5$). Orange: Scaling $s$. (True value: $1.5$).

Theorems & Definitions (15)

• Definition 2.1: Simplicial Complex
• Definition 2.2: Vietoris--Rips Complex
• Definition 2.3: Optimal transport
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4: Multiplicative Death Shift
• Definition 3.5: Expected Peak
• Definition 3.6: Outlier metric, cf. Anai2020
• Theorem 3.7: Theoretical guarantees