Related papers: Angles between subspaces and their tangents
Integrated gradients is prevalent within machine learning to address the black-box problem of neural networks. The explanations given by integrated gradients depend on a choice of base-point. The choice of base-point is not a priori obvious…
We introduce a novel framework for clustering a collection of tall matrices based on their column spaces, a problem we term Subspace Clustering of Subspaces (SCoS). Unlike traditional subspace clustering methods that assume vectorized data,…
A problem that is frequently encountered in a variety of mathematical contexts, is to find the common invariant subspaces of a single, or set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea…
Principal Component Analysis (PCA) is a highly useful topic within an introductory Linear Algebra course, especially since it can be used to incorporate a number of applied projects. This method represents an essential application and…
Let $\mathcal{S}$ be a finite set of integer points in $\mathbb{R}^d$, which we assume has many symmetries, and let $P\in\mathbb{R}^d$ be a fixed point. We calculate the distances from $P$ to the points in $\mathcal{S}$ and compare the…
In quantum mechanics, the connection between the operator algebraic realization and the logical models of measurement of state observables has long been an open question. In the approach that is presented here, we introduce a new…
Posets are discrete mathematical structures which are ubiquitous in a broad range of data analysis and machine learning applications. Research connecting posets to the data science domain has been ongoing for many years. In this paper, a…
We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants…
We introduce the Primary Gasing Triangle, a right triangle with a hypotenuse of 1 unit, to define the primary trigonometric functions: sine and cosine. This triangle serves as the foundational element in a new approach to learning…
In this work we provide detailed estimates of maximal principal angles between subspaces and we analyze their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in…
In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with…
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems,…
Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers-Ramanujan…
Principal Component Analysis (PCA) is a workhorse of modern data science. While PCA assumes the data conforms to Euclidean geometry, for specific data types, such as hierarchical and cyclic data structures, other spaces are more…
We derive the Laws of Cosines and Sines in the super hyperbolic plane using Minkowski supergeometry and find the identical formulae to the classical case, but remarkably involving different expressions for cosines and sines of angles which…
This paper addresses the numerical computation of critical angles between two convex cones in finite-dimensional Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary…
Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell…
A $d$-dimensional simplex in Euclidean space is called orthocentric if all of its altitudes intersect at a single point, referred to as the orthocenter. We explicitly compute the internal and external angles at all faces of an orthocentric…
Distributions, i.e., subsets of tangent bundles formed by piecing together subspaces of tangent spaces, are commonly encountered in the theory and application of differential geometry. Indeed, the theory of distributions is a fundamental…