Related papers: Computationally efficient orthogonalization for pa…
This paper proposes a Newton-type method to solve numerically the eigenproblem of several diagonalizable matrices, which pairwise commute. A classical result states that these matrices are simultaneously diagonalizable. From a suitable…
We show that the objective function of conventional k-means clustering can be expressed as the Frobenius norm of the difference of a data matrix and a low rank approximation of that data matrix. In short, we show that k-means clustering is…
Multi-view learning is frequently used in data science. The pairwise correlation maximization is a classical approach for exploring the consensus of multiple views. Since the pairwise correlation is inherent for two views, the extensions to…
To reduce human error and prejudice, many high-stakes decisions have been turned over to machine algorithms. However, recent research suggests that this does not remove discrimination, and can perpetuate harmful stereotypes. While…
Many tasks in data mining and related fields can be formalized as matching between objects in two heterogeneous domains, including collaborative filtering, link prediction, image tagging, and web search. Machine learning techniques,…
Given the noisy pairwise measurements among a set of unknown group elements, how to recover them efficiently and robustly? This problem, known as group synchronization, has drawn tremendous attention in the scientific community. In this…
Incomplete pairwise comparison matrices offer a natural way of expressing preferences in decision making processes. Although ordinal information is crucial, there is a bias in the literature: cardinal models dominate. Ordinal models usually…
In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization-free methods are preferred for large-scale problems for…
We give an effective method to compute the entropy for polynomials orthogonal on a segment of the real axis that uses as input data only the coefficients of the recurrence relation satisfied by these polynomials. This algorithm is based on…
The low-rank matrix completion problem can be succinctly stated as follows: given a subset of the entries of a matrix, find a low-rank matrix consistent with the observations. While several low-complexity algorithms for matrix completion…
Applications in machine learning and data mining require computing pairwise Lp distances in a data matrix A. For massive high-dimensional data, computing all pairwise distances of A can be infeasible. In fact, even storing A or all pairwise…
We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…
Meaningful comparison between sets of observations often necessitates alignment or registration between them, and the resulting optimization problems range in complexity from those admitting simple closed-form solutions to those requiring…
We describe an algorithm that, given any full-rank matrix A having fewer rows than columns, can rapidly compute the orthogonal projection of any vector onto the null space of A, as well as the orthogonal projection onto the row space of A,…
We present a new theoretical and numerical assessment methodology for a one-dimensional process chain with general applicability to management problems such as the optimization of decision chains or production chains. The process is thereby…
In this study, a new form of quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve…
Kiers (1991) considered the orthogonal rotation in PCAMIX, a principal component method for a mixture of qualitative and quantitative variables. PCAMIX includes the ordinary principal component analysis (PCA) and multiple correspondence…
With the increasing deployment of machine learning systems in practice, transparency and explainability have become serious issues. Contrastive explanations are considered to be useful and intuitive, in particular when it comes to…
Two matrices $A$ and $B$ are called unitary (resp. orthogonal) equivalent if $AU=VB$ for two unitary (resp. orthogonal) matrices $U$ and $V$. Using trace identities, criteria are given for simultaneous unitary, orthogonal or complex…
Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality,…