topology

Topology

Topological analysis approaches, with a recent focus on integrating deep learning with topology.

Topologically Densified Distributions

We study regularization in the context of small sample-size learning with over-parameterized neural networks. Specifically, we shift focus from architectural properties, such as norms on the network weights, to properties of the internal …

Connectivity-Optimized Representation Learning via Persistent Homology

We study the problem of learning representations with controllable connectivity properties. This is beneficial in situations when the imposed structure can be leveraged upstream. In particular, we control the connectivity of an autoencoder’s latent …

Graph Filtration Learning

We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this end, we …

Learning Representations of Persistence Barcodes

We consider the problem of supervised learning with summary representations of topological features in data. In particular, we focus on persistent homology, the prevalent tool used in topological data analysis. As the summary representations, …

Constructing Shape Spaces from a Topological Perspective

We consider the task of constructing (metric) shape space(s) from a topological perspective. In particular, we present a generic construction scheme and demonstrate how to apply this scheme when shape is interpreted as the differences that remain …

Deep Learning with Topological Signatures

Inferring topological and geometrical information from data can offer an alternative perspective in machine learning problems. Methods from topological data analysis, eg, persistent homology, enable us to obtain such information, typically in the …

Statistical Topological Data Analysis - A Kernel Perspective

We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several …

Topological Descriptors of Histology Images

The purpose of this study is to investigate architectural characteristics of cell arrangements in breast cancer histology images. We propose the use of topological data analysis to summarize the geometric information inherent in tumor cell …

On the detection of simple points in higher dimensions using cubical homology

Simple point detection is an important task for several problems in discrete geometry, such as topology preserving thinning in image processing to compute discrete skeletons. In this paper, the approach to simple point detection is based on …