Related papers: Multivariate Topology Simplification
Subspace clustering aims to group data points into multiple clusters of which each corresponds to one subspace. Most existing subspace clustering approaches assume that input data lie on linear subspaces. In practice, however, this…
We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on…
This paper introduces topological data analysis. Starting from notions of a metric space and some elementary graph theory, we take example sets of data and demonstrate some of their topological properties. We discuss simplicial complexes…
We use the techniques of birational algebraic geometry and some combinatorial arguments related to weighted trees to study the structure of resolutions of compactifications of hypothetical counterexamples to the two-dimensional Jacobian…
Entity alignment has always had significant uses within a multitude of diverse scientific fields. In particular, the concept of matching entities across networks has grown in significance in the world of social science as communicative…
We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the…
In a world abundant with diverse data arising from complex acquisition techniques, there is a growing need for new data analysis methods. In this paper we focus on high-dimensional data that are organized into several hierarchical datasets.…
We introduce a bivariate version of topological complexity, $\mathrm{TC}(f,g)$, associated with two continuous maps $f\colon X\to Z$ and $g\colon Y\to Z$. This invariant measures the minimal number of continuous motion planning rules…
Many computations in robotics can be dramatically accelerated if the robot configuration space is described as a collection of simple sets. For example, recently developed motion planners rely on a convex decomposition of the free space to…
As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for…
Multivariate time-dependent data, where multiple features are observed over time for a set of individuals, are increasingly widespread in many application domains. To model these data we need to account for relations among both time…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
Clustering aims to form groups of similar data points in an unsupervised regime. Yet, clustering complex datasets containing critically intertwined shapes poses significant challenges. The prevailing clustering algorithms widely depend on…
A motion of a mechanism is a curve in its configuration space (c-space). Singularities of the c-space are kinematic singularities of the mechanism. Any mobility analysis of a particular mechanism amounts to investigating the c-space…
The construction of the topologically protected code space of Kitaev's model for fault-tolerant quantum computation is extended from complex semisimple to arbitrary finite-dimensional Hopf algebras admitting pairs in involution. One input…
Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between…
The systems without symmetries, e.g. the spatial and chiral symmetries, are generally thought to be improper for topological study and no conventional integral topological invariant can be well defined. In this work, with multi-band…
The mapper construction is a powerful tool from topological data analysis that is designed for the analysis and visualization of multivariate data. In this paper, we investigate a method for stitching a pair of univariate mappers together…
In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we…
Dimensionality reduction techniques are powerful tools for data preprocessing and visualization which typically come with few guarantees concerning the topological correctness of an embedding. The interleaving distance between the…