Related papers: Topology-Preserving Dimensionality Reduction via I…
In a regression setting we propose algorithms that reduce the dimensionality of the features while simultaneously maximizing a statistical measure of dependence known as distance correlation between the low-dimensional features and a…
This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation.…
Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way…
Dimensionality reduction is crucial both for visualization and preprocessing high dimensional data for machine learning. We introduce a novel method based on a hierarchy built on 1-nearest neighbor graphs in the original space which is used…
This work concerns the theoretical foundations of persistence-based topological data analysis. We develop theory of topological inference in the multidimensional persistence setting, and directly at the (topological) level of filtrations…
The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used…
Dimensionality reduction is often used as an initial step in data exploration, either as preprocessing for classification or regression or for visualization. Most dimensionality reduction techniques to date are unsupervised; they do not…
An important new trend in additive manufacturing is the use of optimization to automatically design industrial objects, such as beams, rudders or wings. Topology optimization, as it is often called, computes the best configuration of…
The Vietoris-Rips filtration for an $n$-point metric space is a sequence of large simplicial complexes adding a topological structure to the otherwise disconnected space. The persistent homology is a key tool in topological data analysis…
Dimension reduction algorithms are a crucial part of many data science pipelines, including data exploration, feature creation and selection, and denoising. Despite their wide utilization, many non-linear dimension reduction algorithms are…
Data visualization is the process by which data of any size or dimensionality is processed to produce an understandable set of data in a lower dimensionality, allowing it to be manipulated and understood more easily by people. The goal of…
Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise while distorting global topology. We show that standard…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
This article studies the robust version of persistent homology based on trimming methodology to capture the geometric feature through support of the data in presence of outliers. Precisely speaking, the proposed methodology works when the…
Persistence-based topological optimization deforms a point cloud $X \subset \mathbb{R}^d$ by minimizing objectives of the form $L(X) = \ell(\mathrm{Dgm}(X))$, where $\mathrm{Dgm}(X)$ is a persistence diagram. In practice, optimization is…
Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data…
Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the…
Text documents are complex high dimensional objects. To effectively visualize such data it is important to reduce its dimensionality and visualize the low dimensional embedding as a 2-D or 3-D scatter plot. In this paper we explore…
Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…
Visualizing very large matrices involves many formidable problems. Various popular solutions to these problems involve sampling, clustering, projection, or feature selection to reduce the size and complexity of the original task. An…