The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations.

A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose a framework that optimizes the choice of wavelet for a dataset of graphs, such that their associated persistence diagrams capture features of the graphs that are best suited to a given data science problem. Since the spectral wavelet signature of a graph is derived from its Laplacian, our framework encodes geometric properties of graphs in their associated persistence diagrams and can be applied to graphs without a priori node attributes. We apply our framework to graph classification problems and obtain performances competitive with other persistence-based architectures. To provide the underlying theoretical foundations, we extend the differentiability result for ordinary persistent homology to extended persistent homology.

In this report, various deduplication methods are described in order to assist Vet-AI with the removal of redundant clinical codes from their database system. Their system currently operates whereby clinicians enter codes for diagnoses, leaving open the possibility that multiple clinicians assign the same code to disparate diagnoses. It is also possible that two new entries in the database may be the same diagnosis, with synonymous terminology used. By formulating this as a graph problem, we sought to reduce redundancies by identifying the most probable duplicated codes. A probabilistic model was used, where the probability that two codes are duplicates is a function of a suitable similarity measure (e.g. the Hamming distance). A heuristic method for graph edge pruning is also outlined, based on the application of principles of logical consistency.

A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose a framework that optimises the choice of wavelet for a dataset of graphs, such that their associated persistence diagrams capture features of the graphs that are best suited to a given data science problem. Since the spectral wavelet signature of a graph is derived from its Laplacian, our framework encodes geometric properties of graphs in their associated persistence diagrams and can be applied to graphs without a priori vertex features. We demonstrate how our framework can be coupled with different persistence diagram vectorisation methods for various supervised and unsupervised learning problems, such as graph classification and finding persistence maximising filtrations, respectively. To provide the underlying theoretical foundations, we extend the differentiability result for ordinary persistent homology to extended persistent homology.Recently, persistent homology [ZC05; EH08] has been applied as a feature map that explicitly represents topological and geometric features of a graph as a set of persistence diagrams (a.k.a. barcodes). In the context of our discussion, the persistent homology of a graph G = (V, E) depends on a vertex function f : V → R. In the case where a vertex function is not given with the data, several schemes have been proposed in the literature to assign vertex functions to graphs in a consistent way. For example, vertex functions can be constructed using local geometric descriptions of vertex neighbourhoods, such as discrete curvature [ZW19] and heat kernel signatures [Car+20b].