High-Dimensional Geometry of Sliding Window Embeddings of Periodic Videos

We explore the high dimensional geometry of sliding windows of periodic videos. Under a reasonable model for periodic videos, we show that the sliding window is necessary to disambiguate all states within a period, and we show that a video embedding with a sliding window of an appropriate dimension lies on a topological loop along a hypertorus. This hypertorus has an independent ellipse for each harmonic of the motion. Natural motions with sharp transitions from foreground to background have many harmonics and are hence in higher dimensions, so linear subspace projections such as PCA do not accurately summarize the geometry of these videos. Noting this, we invoke tools from topological data analysis and cohomology to parameterize motions in high dimensions with circular coordinates after the embeddings. We show applications to videos in which there is obvious periodic motion and to videos in which the motion is hidden

Topological feature selection for time series data

We use tools from applied topology for feature selection on vector-valued time series data. We employ persistent homology and sliding window embeddings to quantify the coordinated dynamics of time series. We describe an algorithm for gradient descent to assign scores, or weights, to the variables of the time series based on their contribution to the dynamics as quantified by persistent homology; the result is a convex combination of a subset of the variables. In this setting, we prove persistence vineyards are piecewise linear and we give a simple formula for the derivatives of the vines. We demonstrate our method of topological feature selection with synthetic data and C. elegans neuronal data.Comment: 15 page

Topological data analysis of C. elegans locomotion and behavior

We apply topological data analysis to the behavior of C. elegans, a widely-studied model organism in biology. In particular, we use topology to produce a quantitative summary of complex behavior which may be applied to high-throughput data. Our methods allow us to distinguish and classify videos from various environmental conditions and we analyze the trade-off between accuracy and interpretability. Furthermore, we present a novel technique for visualizing the outputs of our analysis in terms of the input. Specifically, we use representative cycles of persistent homology to produce synthetic videos of stereotypical behaviors.Comment: 27 pages, 15 figures/table