Related papers: Intrinsic Topological Transforms via the Distance …
The Persistent Homology Transform (PHT) was introduced in the field of Topological Data Analysis about 10 years ago, and has since been proven to be a very powerful descriptor of Euclidean shapes. The PHT consists of scanning a shape from…
We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with…
In this paper we consider two topological transforms that are popular in applied topology: the Persistent Homology Transform (PHT) and the Euler Characteristic Transform (ECT). Both of these transforms are of interest for their mathematical…
The space of embedded submanifolds plays an important role in applications such as computational anatomy and shape analysis. We can define two different classes on Riemannian metrics on this space: so-called outer metrics are metrics that…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
This overview article makes the case for how topological concepts can enrich research in machine learning. Using the Euler Characteristic Transform (ECT), a geometrical-topological invariant, as a running example, I present different use…
Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information,…
We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties…
The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape…
We present an isometry and parametrisation invariant of embeddings of $S^1$ into Euclidean space. We do so by representing the distance between pairs of points on the embedded circle as a function on a M\"obius band, the two-point finite…
Topological transforms have been very useful in statistical analysis of shapes or surfaces without restrictions that the shapes are diffeomorphic and requiring the estimation of correspondence maps. In this paper we introduce two…
The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel…
Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point x to S). Given a…
The Transformer architecture has achieved tremendous success in natural language processing, computer vision, and scientific computing through its self-attention mechanism. However, its core components-positional encoding and attention…
Here we shall consider the topology and dynamics associated to a wide class of matchbox manifolds, including a large selection of tiling spaces and all minimal matchbox manifolds of dimension one. For such spaces we introduce topological…
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to…
Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence barcodes and their…
Topological data analysis is an emerging mathematical concept for characterizing shapes in multi-scale data. In this field, persistence diagrams are widely used as a descriptor of the input data, and can distinguish robust and noisy…
Topological phases are generally characterized by topological invariants denoted by integer numbers. However, different topological systems often require different topological invariants to measure, such as geometric phases, topological…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…