• Publications
  • Influence
Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis
TLDR
It is shown that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension.
SW1PerS: Sliding windows and 1-persistence scoring; discovering periodicity in gene expression time series data
TLDR
In biological systems with low noise, i.e. where periodic signals with interesting shapes are more likely to occur, SW1PerS can be used as a powerful tool in exploratory analyses, and is shown to be the most shape-agnostic of the evaluated methods.
(Quasi)Periodicity Quantification in Video Data, Using Topology
TLDR
This work provides continuous measures of periodicity (perfect repetition) and quasiperiodicity (superposition of periodic modes with non-commensurate periods), in a way which does not require segmentation, training, object tracking or 1-dimensional surrogate signals.
Persistent homology of toroidal sliding window embeddings
  • Jose A. Perea
  • Mathematics
    IEEE International Conference on Acoustics…
  • 20 March 2016
TLDR
This paper proves theorems which guide the choice of window size and embedding dimension, and describes the associated persistent homology of sliding window embeddings of quasi-periodic signals.
Approximating Continuous Functions on Persistence Diagrams Using Template Functions
TLDR
This paper describes a mathematical framework for featurizing the persistence diagram space using template functions, and discusses two example realizations of these functions: tent functions and Chybeyshev interpolating polynomials.
A Klein-Bottle-Based Dictionary for Texture Representation
TLDR
It is shown that most of the patches from a given image can be projected onto K yielding a finite sample of S⊂K, whose underlying probability density function can be represented in terms of Fourier-like coefficients, which in turn can be estimated from S.
SW 1 PerS : Sliding Windows and 1-Persistence Scoring ; Discovering Periodicity in Gene Expression Time Series Data
TLDR
A novel method, SW1PerS, for quantifying periodicity in time series data is presented, performed directly, without presupposing a particular shape or pattern, by evaluating the circularity of a high-dimensional representation of the signal.
Geometric Data Analysis Across Scales via Laplacian Eigenvector Cascading.
TLDR
An algorithmic framework for constructing consistent multiscale Laplacian eigenfunctions (vectors) on data is developed, and it is shown via examples that cascading accelerates the computation of graph Laplacan eigenvectors, and more importantly, that one obtains consistent bases of the associated eigenspaces across scales.
A Brief History of Persistence
TLDR
A brief survey on the evolution of persistent homology is presented, starting from the subject's computational inception more than 20 years ago, to the more modern categorical and representation-theoretic point of view.
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