Related papers: The distilled Vietoris Rips filtration for persist…
We introduce a new flavor of functor cocalculus, called \emph{poset cocalculus}, as a tool for studying approximations in topological data analysis. Given a functor from a distributive lattice to a model category, poset cocalculus produces…
Dataset distillation is the technique of synthesizing smaller condensed datasets from large original datasets while retaining necessary information to persist the effect. In this paper, we approach the dataset distillation problem from a…
Particle probability hypothesis density filtering has become a promising means for multi-target tracking due to its capability of handling an unknown and time-varying number of targets in non-linear non-Gaussian system. However, its…
In this paper, we introduce DistDD, a novel approach within the federated learning framework that reduces the need for repetitive communication by distilling data directly on clients' devices. Unlike traditional federated learning that…
The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from…
As a pivotal technique for improving the defense of deep models, adversarial robustness transfer via distillation has demonstrated remarkable success in conventional image classification tasks. However, this paradigm encounters critical…
The persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors…
We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence…
We introduce persistence matching diagrams induced by set mappings of metric spaces, based on 0-persistent homology of Vietoris-Rips filtrations. Also, we present a geometric definition of the persistence matching diagram that is more…
In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show…
We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-$k$ in persistent homology, for a either the \cech or the Vietoris--Rips filtration built…
In recent years, persistent homology (PH) has been successfully applied to real-world data in many different settings. Despite significant computational advances, PH algorithms do not yet scale to large datasets preventing interesting…
The Mayer-Vietoris theorem is known for its wide applications, especially in determining homology. In fact, this theorem provides us with a long exact sequence, where the underlying homology groups fit in. However, this theorem does not…
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the…
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…
We present a fast algorithm for computing discrete cubical homology of graphs over finite fields with an appropriate characteristic. This algorithm improves on several computational steps compared to constructions in the existing…
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at…
In topological data analysis (TDA), a longstanding challenge is to recognize underlying geometric structures in noisy data. One motivating examples is the shape of a point cloud in Euclidean space given by image. Carlsson et al. proposed a…
Topological data analysis leverages topological features to analyze datasets, with applications in diverse fields like medical sciences and biology. A key tool of this theory is the persistence diagram, which encodes topological information…
Geometric graphs form an important family of hidden structures behind data. In this paper, we develop an efficient and robust algorithm to infer a graph skeleton of a high-dimensional point cloud dataset (PCD). Previously, there has been…