🍩 Database of Original & NonTheoretical Uses of Topology
(found 8 matches in 0.006405s)


Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations (2018)
Jose A. PereaAbstract
We present a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction. 
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. 
Crystallographic Interacting Topological Phases and Equvariant Cohomology: To Assume or Not to Assume (2020)
Daniel Sheinbaum, Omar Antolín CamarenaAbstract
For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for nondegenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not assume shortrange entanglement nor the existence of quasiparticles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the \$P\$equivariant cohomology of a \$d\$dimensional torus gives rise to different interacting phases. We compare our results to bosonic symmorphic crystallographic SPT phases and to noninteracting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases. 
Positive Alexander Duality for Pursuit and Evasion (2017)
Robert Ghrist, Sanjeevi KrishnanAbstract
Considered is a class of pursuitevasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the complement of a coverage region in a Euclidean space over time. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in the general case. The principal tools are (1) a refinement of the Čech cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the BorelMoore homology of the coverage gaps with a positive cone encoding time orientation, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable using techniques of computational polyhedral geometry and linear programming. 
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. 
Branching and Circular Features in High Dimensional Data (2011)
B. Wang, B. Summa, V. Pascucci, M. VejdemoJohanssonAbstract
Large observations and simulations in scientific research give rise to highdimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of highdimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto lowdimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original highdimensional data. Our solution is to utilize topological techniques to recover important structures in highdimensional data that contains nontrivial topology. Specifically, we are interested in highdimensional branching structures. We construct local circlevalued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Additionally, we study the effects of global circular structures on visualizations. Our results reveal neverbeforeseen structures on realworld data sets from a variety of applications.