Related papers: Local biplots for multi-dimensional scaling, with …
Multidimensional scaling visualizes dissimilarities among objects and reduces data dimensionality. While many methods address symmetric proximity data, asymmetric and especially three-way proximity data (capturing relationships across…
Most biological data are multidimensional, posing a major challenge to human comprehension and computational analysis. Principal component analysis is the most popular approach to rendering two- or three-dimensional representations of the…
Jacobi sets are an important tool to study the relationship between functions. Defined as the set of all points where the function's gradients are linearly dependent, Jacobi sets extend the notion of critical point to multifields. In…
Dimension reduction and visualization is a staple of data analytics. Methods such as Principal Component Analysis (PCA) and Multidimensional Scaling (MDS) provide low dimensional (LD) projections of high dimensional (HD) data while…
We propose a visualization method to understand the effect of multidimensional projection on local subspaces, using implicit function differentiation. Here, we understand the local subspace as the multidimensional local neighborhood of data…
The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the…
Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multi-field data, topological analysis requires simultaneous advances both mathematically…
We adapt concepts, methodology, and theory originally developed in the areas of multidimensional scaling and dimensionality reduction for multivariate data to the functional setting. We focus on classical scaling and Isomap -- prototypical…
One develops a fast computational methodology for principal component analysis on manifolds. Instead of estimating intrinsic principal components on an object space with a Riemannian structure, one embeds the object space in a numerical…
A trivializing map is a field transformation whose Jacobian determinant exactly cancels the interaction terms in the action, providing a representation of the theory in terms of a deterministic transformation of a distribution from which…
Often the relation between the variables constituting a multivariate data space might be characterized by one or more of the terms: ``nonlinear'', ``branched'', ``disconnected'', ``bended'', ``curved'', ``heterogeneous'', or, more general,…
Large-scale perturbations in the microbiome constitution are strongly correlated, whether as a driver or a consequence, with the health and functioning of human physiology. However, understanding the difference in the microbiome profiles of…
Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way…
Principal component analysis is a widely-used method for the dimensionality reduction of a given data set in a high-dimensional Euclidean space. Here we define and analyze two analogues of principal component analysis in the setting of…
We present network embedding algorithms that capture information about a node from the local distribution over node attributes around it, as observed over random walks following an approach similar to Skip-gram. Observations from…
Multidimensional data distributions can have complex topologies and variable local dimensions. To approximate complex data, we propose a new type of low-dimensional ``principal object'': a principal cubic complex. This complex is a…
High dimensional and heterogeneous count data are collected in various applied fields. In this paper, we look closely at high-resolution sequencing data on the microbiome, which have enabled researchers to study the genomes of entire…
Multivariate distributions are fundamental to modeling. Discrete copulas can be used to construct diverse multivariate joint distributions over random variables from estimated univariate marginals. The space of discrete copulas admits a…
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…
Classical Biplot Methods allow for the simultaneous representation of individuals (rows) and variables (columns) of a data matrix. For Binary data, Logistic biplots have been recently developed.When data are nominal, linear or even binary…