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Additive principal components (APCs for short) are a nonlinear generalization of linear principal components. We focus on smallest APCs to describe additive nonlinear constraints that are approximately satisfied by the data. Thus APCs fit…
Constellations are partial algebras that are one-sided generalisations of categories. It has previously been shown that the category of inductive constellations is isomorphic to the category of left restriction semigroups. Here we consider…
Data analysis often requires methods that are invariant with respect to specific transformations, such as rotations in case of images or shifts in case of images and time series. While principal component analysis (PCA) is a widely-used…
This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in field of the group…
In several application domains, high-dimensional observations are collected and then analysed in search for naturally occurring data clusters which might provide further insights about the nature of the problem. In this paper we describe a…
The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the…
We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…
Nowadays, the number of emerging embedded systems rapidly grows in many application domains, due to recent advances in artificial intelligence and internet of things. The main inherent specification of these application-specific systems is…
We present an extension of sparse PCA, or sparse dictionary learning, where the sparsity patterns of all dictionary elements are structured and constrained to belong to a prespecified set of shapes. This \emph{structured sparse PCA} is…
Principal Component Analysis (PCA) finds the best linear representation of data, and is an indispensable tool in many learning and inference tasks. Classically, principal components of a dataset are interpreted as the directions that…
Data integration is the problem of combining multiple data groups (studies, cohorts) and/or multiple data views (variables, features). This task is becoming increasingly important in many disciplines due to the prevalence of large and…
Covariances from categorical variables are defined using a regular simplex expression for categories. The method follows the variance definition by Gini, and it gives the covariance as a solution of simultaneous equations. The calculated…
We present an unsupervised learning analysis of correlation hierarchies in the quarter-filled simple and extended Hubbard models by applying principal component analysis (PCA) to exact-diagonalization (ED) data on 3x4 and 4x4 cylindrical…
Principal Component Analysis (PCA) is a commonly used tool for dimension reduction in analyzing high dimensional data; Multilinear Principal Component Analysis (MPCA) has the potential to serve the similar function for analyzing tensor…
The idea of representation has been used in various fields of study from data analysis to political science. In this paper, we define representativeness and describe a method to isolate data points that can represent the entire data set.…
The quantum or quantum field theory concept of a complex wave function is useful for understanding the information transport in classical statistical generalized Ising models. We relate complex conjugation to the discrete transformations…
Principal Component Analysis (PCA) is the workhorse tool for dimensionality reduction in this era of big data. While often overlooked, the purpose of PCA is not only to reduce data dimensionality, but also to yield features that are…
Formal Concept Analysis (FCA) is a mathematical framework for knowledge representation and discovery. It performs a hierarchical clustering over a set of objects described by attributes, resulting in conceptual structures in which objects…
We generalise the notion of separable equivalence, originally presented by Linckelmann (2011), to an equivalence relation on additive categories. We use this generalisation to show that from an initial equivalence between two algebras we…
Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise. The maximum likelihood solution for the model is an eigenvalue problem on the…