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Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features,…
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines…
Data exploration and visualization systems are of great importance in the Big Data era, in which the volume and heterogeneity of available information make it difficult for humans to manually explore and analyse data. Most traditional…
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…
Persistent Homology is a powerful tool in Topological Data Analysis (TDA) to capture topological properties of data succinctly at different spatial resolutions. For graphical data, shape, and structure of the neighborhood of individual data…
This paper proposes FREEtree, a tree-based method for high dimensional longitudinal data with correlated features. Popular machine learning approaches, like Random Forests, commonly used for variable selection do not perform well when there…
A central problem in data-driven scientific inquiry is how to interpret structure in noisy, high-dimensional data. Topological data analysis (TDA) provides a solution via persistent homology, which encodes features of interest as…
When solving ill-posed inverse problems, one often desires to explore the space of potential solutions rather than be presented with a single plausible reconstruction. Valuable insights into these feasible solutions and their associated…
Treemaps have been widely applied to the visualization of hierarchical data. A treemap takes a weighted tree and visualizes its leaves in a nested planar geometric shape, with sub-regions partitioned such that each sub-region has an area…
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…
We use methods from computational algebraic topology to study functional brain networks, in which nodes represent brain regions and weighted edges encode the similarity of fMRI time series from each region. With these tools, which allow one…
Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to…
Long lived topological features are distinguished from short lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and…
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…
Embedding learning transforms discrete data entities into continuous numerical representations, encoding features/properties of the entities. Despite the outstanding performance reported from different embedding learning algorithms, few…
Existing research highlights the crucial role of topological priors in image segmentation, particularly in preserving essential structures such as connectivity and genus. Accurately capturing these topological features often requires…
We use methods from topological data analysis to study the topological features of certain distributions of string vacua. Topological data analysis is a multi-scale approach used to analyze the topological features of a dataset by…
Nervous systems are characterized by neurons displaying a diversity of morphological shapes. Traditionally, different shapes have been qualitatively described based on visual inspection and quantitatively described based on morphometric…
Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional…
Interacting, self-propelled particles such as epithelial cells can dynamically self-organize into complex multicellular patterns, which are challenging to classify without a priori information. Classically, different phases and phase…