🍩 Database of Original & NonTheoretical Uses of Topology
(found 4 matches in 0.00304s)


Generalized Penalty for Circular Coordinate Representation (2020)
Hengrui Luo, Alice Patania, Jisu Kim, Mikael VejdemoJohanssonAbstract
Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for highdimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account sparsity in highdimensional applications. We use a generalized penalty function instead of an \$L_\2\\$ penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analysis to support our claim that circular coordinates with generalized penalty will accommodate the sparsity in highdimensional datasets under different sampling schemes while preserving the topological structures. 
Decoding of Neural Data Using Cohomological Feature Extraction (2019)
Erik Rybakken, Nils Baas, Benjamin DunnAbstract
We introduce a novel datadriven approach to discover and decode features in the neural code coming from large population neural recordings with minimal assumptions, using cohomological feature extraction. We apply our approach to neural recordings of mice moving freely in a box, where we find a circular feature. We then observe that the decoded value corresponds well to the head direction of the mouse. Thus, we capture head direction cells and decode the head direction from the neural population activity without having to process the mouse's behavior. Interestingly, the decoded values convey more information about the neural activity than the tracked head direction does, with differences that have some spatial organization. Finally, we note that the residual population activity, after the head direction has been accounted for, retains some lowdimensional structure that is correlated with the speed of the mouse. 
Geometric Anomaly Detection in Data (2020)
Bernadette J. Stolz, Jared Tanner, Heather A. Harrington, Vidit NandaAbstract
The quest for lowdimensional models which approximate highdimensional data is pervasive across the physical, natural, and social sciences. The dominant paradigm underlying most standard modeling techniques assumes that the data are concentrated near a single unknown manifold of relatively small intrinsic dimension. Here, we present a systematic framework for detecting interfaces and related anomalies in data which may fail to satisfy the manifold hypothesis. By computing the local topology of small regions around each data point, we are able to partition a given dataset into disjoint classes, each of which can be individually approximated by a single manifold. Since these manifolds may have different intrinsic dimensions, local topology discovers singular regions in data even when none of the points have been sampled precisely from the singularities. We showcase this method by identifying the intersection of two surfaces in the 24dimensional space of cyclooctane conformations and by locating all of the selfintersections of a Henneberg minimal surface immersed in 3dimensional space. Due to the local nature of the topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors.