44 citations · 109 across the 21 of their papers we have counts for
24 papers
An Extended Topological Model For High-Contrast Optical Flow
Brad Turow, Jose A. Perea
In this paper, we identify low-dimensional models for dense core subsets in the space of high-contrast optical flow patches sampled from the Sintel dataset. In particul…
Estimation of Persistence Diagrams via the Three Gap Theorem
Luis Suarez Salas, Jose A. Perea
The time delay (or Sliding Window) embedding is a technique from dynamical systems to reconstruct attractors from time series data. Recently, descriptors from Topological Data Anal…
Discrete Approximate Circle Bundles
Brad Turow, Jose A. Perea
In this paper, we introduce discrete approximate circle bundles, a class of objects designed to serve as the data science analog of circle bundles from algebraic topology. We show…
-Equivariant Dimensionality Reduction on Stiefel Manifolds
Andrew Lee, Harlin Lee, Jose A. Perea +2
Many real-world datasets live on high-dimensional Stiefel and Grassmannian manifolds, and respectively, and benefit from projection onto l…
Topological Data Analysis of Electroencephalogram Signals for Pediatric Obstructive Sleep Apnea
Shashank Manjunath, Jose A. Perea, Aarti Sathyanarayana
Topological data analysis (TDA) is an emerging technique for biological signal processing. TDA leverages the invariant topological features of signals in a metric space for robust…
Learning on Persistence Diagrams as Radon Measures
Alex Elchesen, Iryna Hartsock, Jose A. Perea +1
Persistence diagrams are common descriptors of the topological structure of data appearing in various classification and regression tasks. They can be generalized to Radon measures…
Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction
Luis Scoccola, Hitesh Gakhar, Johnathan Bush +4
The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a -dimensional hole in the…
FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles
Luis Scoccola, Jose A. Perea
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 topolo…
Move Schedules: Fast persistence computations in coarse dynamic settings
Matthew Piekenbrock, Jose A. Perea
Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with simplices. Its output is a particular decomposition of th…
Approximate and discrete Euclidean vector bundles
Luis Scoccola, Jose A. Perea
We introduce -approximate versions of the notion of Euclidean vector bundle for , which recover the classical notion of Euclidean vector bundle whe…
Sliding Window Persistence of Quasiperiodic Functions
Hitesh Gakhar, Jose A. Perea
A function is called quasiperiodic if its fundamental frequencies are linearly independent over the rationals. With appropriate parameters, the sliding window point clouds of such…
Adaptive template systems: Data-driven feature selection for learning with persistence diagrams
Luis Polanco, Jose A. Perea
Feature extraction from persistence diagrams, as a tool to enrich machine learning techniques, has received increasing attention in recent years. In this paper we explore an adapti…
Künneth Formulae in Persistent Homology
Hitesh Gakhar, Jose A. Perea
The classical Künneth formula in algebraic topology describes the homology of a product space in terms of that of its factors. In this paper, we prove Künneth-type theorems for the…
Coordinatizing Data With Lens Spaces and Persistent Cohomology
Luis Polanco, Jose A. Perea
We introduce here a framework to construct coordinates in \emph{finite} Lens spaces for data with nontrivial 1-dimensional persistent cohomology, . Said coo…
Approximating Continuous Functions on Persistence Diagrams Using Template Functions
Jose A. Perea, Elizabeth Munch, Firas A. Khasawneh
The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The…
Geometric Data Analysis Across Scales via Laplacian Eigenvector Cascading
Joshua L. Mike, Jose A. Perea
We develop here an algorithmic framework for constructing consistent multiscale Laplacian eigenfunctions (vectors) on data. Consequently, we address the unsupervised machine learni…
Topological Time Series Analysis
Jose A. Perea
Time series are ubiquitous in our data rich world. In what follows I will describe how ideas from dynamical systems and topological data analysis can be combined to gain insights f…
A Brief History of Persistence
Jose A. Perea
Persistent homology is currently one of the more widely known tools from computational topology and topological data analysis. We present in this note a brief survey on the evoluti…
Sparse Circular Coordinates via Principal -Bundles
Jose A. Perea
We present in this paper an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, we derive, from a…
Twisty Takens: A Geometric Characterization of Good Observations on Dense Trajectories
Boyan Xu, Christopher J. Tralie, Alice Antia +2
In nonlinear time series analysis and dynamical systems theory, Takens' embedding theorem states that the sliding window embedding of a generic observation along trajectories in a…
Chatter Classification in Turning Using Machine Learning and Topological Data Analysis
Firas A. Khasawneh, Elizabeth Munch, Jose A. Perea
Chatter identification and detection in machining processes has been an active area of research in the past two decades. Part of the challenge in studying chatter is that machining…
(Quasi)Periodicity Quantification in Video Data, Using Topology
Christopher J. Tralie, Jose A. Perea
This work introduces a novel framework for quantifying the presence and strength of recurrent dynamics in video data. Specifically, we provide continuous measures of periodicity (p…
Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations
Jose A. Perea
We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how t…
Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis
Jose Perea, John Harer
We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding wind…