Related papers: Functional Data Representation with Merge Trees

In recent years, manifold methods have moved into focus as tools for dimension reduction. Assuming that the high-dimensional data actually lie on or close to a low-dimensional nonlinear manifold, these methods have shown convincing results…

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this…

Functional MRI is a neuroimaging technique that analyzes the functional activity of the brain by measuring blood-oxygen-level-dependent signals throughout the brain. The derived functional features can be used for investigating brain…

Semi-structured networks (SSNs) merge the structures familiar from additive models with deep neural networks, allowing the modeling of interpretable partial feature effects while capturing higher-order non-linearities at the same time. A…

The increasing complexity of data requires methods and models that can effectively handle intricate structures, as simplifying them would result in loss of information. While several analytical tools have been developed to work with complex…

The paper introduces two new aggregation functions to encode structural knowledge from tree-structured data. They leverage the Canonical and Tensor-Train decompositions to yield expressive context aggregation while limiting the number of…

For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute…

Context: Edge graphs are graphs whose edges are labelled with identifiers, and nodes can have multiple edges between them. They are used to model a wide range of systems, including networks with distances or degrees of connection and…

Tree-based methods are powerful nonparametric techniques in statistics and machine learning. However, their effectiveness, particularly in finite-sample settings, is not fully understood. Recent applications have revealed their surprising…

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize…

In this work we answer an open question asked by Johnson--Scoville. We show that each merge tree is represented by a discrete Morse function on a path. Furthermore, we present explicit constructions for two different but related kinds of…

Recent work showed that compiling functional programs to use dense, serialized memory representations for recursive algebraic datatypes can yield significant constant-factor speedups for sequential programs. But serializing data in a…

Statistical depth is the act of gauging how representative a point is compared to a reference probability measure. The depth allows introducing rankings and orderings to data living in multivariate, or function spaces. Though widely applied…

In machine learning it is common to interpret each data point as a vector in Euclidean space. However the data may actually be functional i.e.\ each data point is a function of some variable such as time and the function is discretely…

Succinct data structures give space-efficient representations of large amounts of data without sacrificing performance. They rely one cleverly designed data representations and algorithms. We present here the formalization in Coq/SSReflect…

Archetype and archetypoid analysis can be extended to functional data. Each function is represented as a mixture of actual observations (functional archetypoids) or functional archetypes, which are a mixture of observations in the data set.…

Connected operators are filtering tools that act by merging elementary regions of an image. A popular strategy is based on tree-based image representations: for example, one can compute an attribute on each node of the tree and keep only…

We investigate the pertinence of methods from algebraic topology for text data analysis. These methods enable the development of mathematically-principled isometric-invariant mappings from a set of vectors to a document embedding, which is…

We consider the analysis of high dimensional data given in the form of a matrix with columns consisting of observations and rows consisting of features. Often the data is such that the observations do not reside on a regular grid, and the…

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…