Recent advances in time series classification have largely focused on methods that either employ deep learning or utilize other machine learning models for feature extraction. Though successful, their power often comes at the requirement of computati
onal complexity. In this paper, we introduce GeoStat representations for time series. GeoStat representations are based off of a generalization of recent methods for trajectory classification, and summarize the information of a time series in terms of comprehensive statistics of (possibly windowed) distributions of easy to compute differential geometric quantities, requiring no dynamic time warping. The features used are intuitive and require minimal parameter tuning. We perform an exhaustive evaluation of GeoStat on a number of real datasets, showing that simple KNN and SVM classifiers trained on these representations exhibit surprising performance relative to modern single model methods requiring significant computational power, achieving state of the art results in many cases. In particular, we show that this methodology achieves good performance on a challenging dataset involving the classification of fishing vessels, where our methods achieve good performance relative to the state of the art despite only having access to approximately two percent of the dataset used in training and evaluating this state of the art.
Many algorithms for surface registration risk producing significant errors if surfaces are significantly nonisometric. Manifold learning has been shown to be effective at improving registration quality, using information from an entire collection of
surfaces to correct issues present in pairwise registrations. These methods, however, are not robust to changes in the collection of surfaces, or do not produce accurate registrations at a resolution high enough for subsequent downstream analysis. We propose a novel algorithm for efficiently registering such collections given initial correspondences with varying degrees of accuracy. By combining the initial information with recent developments in manifold learning, we employ a simple metric condition to construct a measure on the space of correspondences between any pair of shapes in our collection, which we then use to distill soft correspondences. We demonstrate that this measure can improve correspondence accuracy between feature points compared to currently employed, less robust methods on a diverse dataset of surfaces from evolutionary biology. We then show how our methods can be used, in combination with recent sampling and interpolation methods, to compute accurate and consistent homeomorphisms between surfaces.
In the case of some fractals, sampling with average values on cells is more natural than sampling on points. In this paper we investigate this method of sampling on $SG$ and $SG_{3}$. In the former, we show that the cell graph approximations have the
spectral decimation property and prove an analog of the Shannon sampling theorem.. We also investigate the numerical properties of these sampling functions and make conjectures which allow us to look at sampling on infinite blowups of $SG$. In the case of $SG_{3}$, we show that the cell graphs have the spectral decimation property, but show that it is not useful for proving an analogous sampling theorem.
