Related papers: Right-left asymmetry of the eigenvector method: A …
It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In…
The alignment of shapes has been a crucial step in statistical shape analysis, for example, in calculating mean shape, detecting locational differences between two shape populations, and classification. Procrustes alignment is the most…
Methods for the design of physical parameterization schemes that possess certain invariance properties are discussed. These methods are based on different techniques of group classification and provide means to determine expressions for…
Reward design in reinforcement learning and optimal control is challenging. Preference-based alignment addresses this by enabling agents to learn rewards from ranked trajectory pairs provided by humans. However, existing methods often…
Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve…
We study analytically the Chalker-Mehlig mean diagonal overlap $\mathcal{O}(z)$ between left and right eigenvectors associated with a complex eigenvalue $z$ of $N\times N$ matrices in the real Ginibre ensemble (GinOE). We first derive a…
For research to go in the right direction, it is essential to be able to compare and quantify performance of different algorithms focused on the same problem. Choosing a suitable evaluation metric requires deep understanding of the pursued…
Ranking or assessing centrality in multivariate and non-Euclidean data is difficult because there is no canonical order and many depth notions become computationally fragile in high-dimensional or structured settings. We introduce a…
The problem to compute the vertices of a polytope given by affine inequalities is called vertex enumeration. The inverse problem, which is equivalent by polarity, is called the convex hull problem. We introduce `approximate vertex…
This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum…
As machine learning informs increasingly consequential decisions, different metrics have been proposed for measuring algorithmic bias or unfairness. Two popular fairness measures are calibration and equality of false positive rate. Each…
For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue. In…
In this paper, we describe a new algorithm that approximates the extreme eigenvalue/eigenvector pairs of a symmetric matrix. The proposed algorithm can be viewed as an extension of the Jacobi eigenvalue method for symmetric matrices…
Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues…
Algorithmic fairness is receiving significant attention in the academic and broader literature due to the increasing use of predictive algorithms, including those based on artificial intelligence. One benefit of this trend is that algorithm…
Mixed linear regression involves the recovery of two (or more) unknown vectors from unlabeled linear measurements; that is, where each sample comes from exactly one of the vectors, but we do not know which one. It is a classic problem, and…
Interleaving is an online evaluation approach for information retrieval systems that compares the effectiveness of ranking functions in interpreting the users' implicit feedback. Previous work such as Hofmann et al (2011) has evaluated the…
For the generalized eigenvalue problem, a quotient function is devised for estimating eigenvalues in terms of an approximate eigenvector. This gives rise to an infinite family of quotients, all entirely arguable to be used in estimation.…
Diagonalizing a matrix $A$, that is finding two matrices $P$ and $D$ such that $A = PDP^{-1}$ with $D$ being a diagonal matrix needs two steps: first find the eigenvalues and then find the corresponding eigenvectors. We show that we do not…
Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for…