Related papers: Decorated Merge Trees for Persistent Topology
Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric…
Pairwise comparison of graphs is key to many applications in Machine learning ranging from clustering, kernel-based classification/regression and more recently supervised graph prediction. Distances between graphs usually rely on…
Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics…
Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
Multiparameter persistence modules come up naturally in topological data analysis and topological robotics. Given a metric graph $(X,\delta)$, the second configuration space of $(X,\delta)$ with proximity parameters (for example, the…
This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…
Decision tree (DT) attracts persistent research attention due to its impressive empirical performance and interpretability in numerous applications. However, the growth of traditional yet widely-used univariate decision trees (UDTs) is…
We propose a network for semantic mapping called the Dense Dilated Convolutions Merging Network (DDCM-Net) to provide a deep learning approach that can recognize multi-scale and complex shaped objects with similar color and textures, such…
Distance measures play an important role in shape classification and data analysis problems. Topological distances based on Reeb graphs and persistence diagrams have been employed to obtain effective algorithms in shape matching and scalar…
As a modern ensemble technique, Deep Forest (DF) employs a cascading structure to construct deep models, providing stronger representational power compared to traditional decision forests. However, its greedy multi-layer learning procedure…
Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…
We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…
In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this…
We define and study several new interleaving distances for persistent cohomology which take into account the algebraic structures of the cohomology of a space, for instance the cup product or the action of the Steenrod algebra. In…
We introduce Joint Probability Trees (JPT), a novel approach that makes learning of and reasoning about joint probability distributions tractable for practical applications. JPTs support both symbolic and subsymbolic variables in a single…
The tree is an essential data structure in many applications. In a distributed application, such as a distributed file system, the tree is replicated.To improve performance and availability, different clients should be able to update their…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
We study rotation-robust learning for image inputs using Convolutional Model Trees (CMTs) [1], whose split and leaf coefficients can be structured on the image grid and transformed geometrically at deployment time. In a controlled MNIST…
This paper introduces an efficient algorithm for persistence diagram computation, given an input piecewise linear scalar field $f$ defined on a $d$-dimensional simplicial complex $K$, with $d \leq 3$. Our work revisits the seminal algorithm…