Persistent Cohomology and Circular Coordinates

Abstract

Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.

A Topological Low-Pass Filter for Quasi-Periodic Signals

“…Signals that have a hidden state space are identifiable using the topology of delay embeddings [7], a concept that can be traced to a paper by Takens [8]. Many papers have discussed ways to find the hidden state of a dynamical system; recovering the phase space from measurements [9], [10], [11], [12], [13].…”

Topological analysis and visualization of cyclical behavior in memory reference traces

“…Our goal in the current work is to find and visualize higher-order structures in reference traces that may extend through time, forming cycles that may be executed multiple times. de Silva et al [10] and Wang et al [28] present approaches to finding topological features, such as circles and branches, in general point sets.…”

“…With the technical tools described in [10,28], we now give an overview of our algorithm. Here we assume basic knowledge in topology and homology.…”

“…This relates cohomology with circular coordinates. It implies that if X has non-trivial 1-dimensional cohomology class α ∈ H 1 (X; Z), we can construct a continuous function θ : X → S 1 from α (see [10] for a formal proof).…”

“…2. Use the concept of persistent cohomology [10,28] to detect a significant cohomology class in K, and convert such a class into a circle-valued function θ : X → S 1 . 3.…”