arXiv research

Study asymptotics of unitary matrix elements in quantum mechanics.

Paper maps Hamiltonians and line elements in manifolds.

New methods combine matrix elements and machine learning for more accurate LHC measurements.

Many tasks require finding groups of elements in a matrix of numbers, symbols or class likelihoods. One approach is to use efficient bi- or tri-linear factorization techniques including PCA, ICA, sparse matrix factorization and plaid analysis. These techniques are not appropriate when addition and multiplication of mat…

This paper improves matrix completion by leveraging element importance and non-uniform sampling.

A new method speeds up ALS for recommender systems by subsampling key elements.

In this paper we form relations for the determination of the elements of the Eötvös matrix of the Earth's normal gravity field. In addition a relation between the Gauss curvature of the normal equipotential surface and the Gauss curvature of the actual equipotential surface both passing through the point P is presented…

Origin-destination (OD) matrices are often used in urban planning, where a city is partitioned into regions and an element (i, j) in an OD matrix records the cost (e.g., travel time, fuel consumption, or travel speed) from region i to region j. In this paper, we partition a day into multiple intervals, e.g., 96 15-min …

A new method uses SPDEs to efficiently model random fields on complex domains.

For any matrix A in R^(m x n) of rank ρ, we present a probability distribution over the entries of A (the element-wise leverage scores of equation (2)) that reveals the most influential entries in the matrix. From a theoretical perspective, we prove that sampling at most s = O ((m + n) ρ^2 ln (m + n)) entries of the ma…

Matrix completion, i.e., the exact and provable recovery of a low-rank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraint---known as {\em incoherence}---on its row and column spaces. In these cases, the subset of elements is sam…

This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -- with or without self-intersections -- we associate an element of (the fractio…

In this paper we give a new basis, $Λ$, for the Homflypt skein module of the solid torus, $\mathcal{S}({\rm ST})$, which was predicted by Jozef Przytycki, using topological interpretation. The basis $Λ$ is different from the basis $Λ^{\prime}$, discovered independently by Hoste--Kidwell \cite{HK} and Turaev \cite{Tu} w…

We consider analysis of relational data (a matrix), in which the rows correspond to subjects (e.g., people) and the columns correspond to attributes. The elements of the matrix may be a mix of real and categorical. Each subject and attribute is characterized by a latent binary feature vector, and an inferred matrix map…

Efficiently factorizes coupled matrix tensor data for better accuracy and speed.

Predict missing movie ratings or graph embeddings with low rank matrices.

Polynomial sketch approximates functions of low-rank matrices efficiently.

BlockEcho method improves imputation of block-wise missing data.

Attention tokens are group elements, negated algebra norms.

We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML).

New finite element method for complex forms in any dimension.

Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that c…

Suppose, we are given a set of $n$ elements to be clustered into $k$ (unknown) clusters, and an oracle/expert labeler that can interactively answer pair-wise queries of the form, "do two elements $u$ and $v$ belong to the same cluster?". The goal is to recover the optimum clustering by asking the minimum number of quer…

The paper analyzes how Gaussian kernel parameters affect posterior covariance in Gaussian processes.

Proposes a method to identify elements in a skewness matrix for multivariate skew-elliptical distributions.

Modified Newton step for online learning reduces matrix size for large datasets.

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…

Quantum theory uses modular group representations to assign invariants to 3-manifolds.

Near-convex archetypal analysis improves interpretability and fitting error in NMF.

We study a data model in which the data matrix D can be expressed as D = L + S + C, where L is a low rank matrix, S an element-wise sparse matrix and C a matrix whose non-zero columns are outlying data points. To date, robust PCA algorithms have solely considered models with either S or C, but not both. As such, existi…

As a typical dimensionality reduction technique, random projection can be simply implemented with linear projection, while maintaining the pairwise distances of high-dimensional data with high probability. Considering this technique is mainly exploited for the task of classification, this paper is developed to study th…

This paper classifies reversible and strongly reversible elements in affine groups.

We know that any element A of the group SO(3) can be represented as A = A1 A2 A1', where A1, A1' are elements of SO1(2)={A is an element of SO(3) | Ae1=e1}, and SO2(2)={A is an element of SO(3) | Ae2=e2} . This fact is known as Euler's angle. When this situation, a matrix A is called the generator. In the present paper…

We consider the problem of Graphical lasso with an additional $\ell_{\infty}$ element-wise norm constraint on the precision matrix. This problem has applications in high-dimensional covariance decomposition such as in \citep{Janzamin-12}. We propose an ADMM algorithm to solve this problem. We also use a continuation st…

This paper addresses how well we can recover a data matrix when only given a few of its elements. We present a randomized algorithm that element-wise sparsifies the data, retaining only a few its elements. Our new algorithm independently samples the data using sampling probabilities that depend on both the squares ($\e…

We analyze the relationship between the covering of the Jacobi group and the squeezed states. We attach some nonclassical states to the Jacobi group. The matrix elements of the Jacobi group are presented.

We study how well one can recover sparse principal components of a data matrix using a sketch formed from a few of its elements. We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the original data matrix, then one can recover a near optimal solution to the optimiza…

Constructs CAT(0) actions for certain groups without unipotent elements.

Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is distorted by a (typically unknown) nonlinear transformation. This paper addresses the…

This paper considers probabilistic estimation of a low-rank matrix from non-linear element-wise measurements of its elements. We derive the corresponding approximate message passing (AMP) algorithm and its state evolution. Relying on non-rigorous but standard assumptions motivated by statistical physics, we characteriz…

Matrix formulas for super Teichmüller spaces generalize previous work and yield super λ-lengths.

This paper develops a Bayesian procedure for estimation and forecasting of the volatility of multivariate time series. The foundation of this work is the matrix-variate dynamic linear model, for the volatility of which we adopt a multiplicative stochastic evolution, using Wishart and singular multivariate beta distribu…

Given $n$ i.i.d. observations of a random vector $(X,Z)$, where $X$ is a high-dimensional vector and $Z$ is a low-dimensional index variable, we study the problem of estimating the conditional inverse covariance matrix $Ω(z) = (E[(X-E[X \mid Z])(X-E[X \mid Z])^T \mid Z=z])^{-1}$ under the assumption that the set of non…

This paper considers a restriction to non-negative matrix factorization in which at least one matrix factor is stochastic. That is, the elements of the matrix factors are non-negative and the columns of one matrix factor sum to 1. This restriction includes topic models, a popular method for analyzing unstructured data.…

Bayesian non-linear matrix completion tackles large, sparse data.

Analogous exponential map defined for Hopf algebras.

Linear dimensionality reduction techniques are powerful tools for image analysis as they allow the identification of important features in a data set. In particular, nonnegative matrix factorization (NMF) has become very popular as it is able to extract sparse, localized and easily interpretable features by imposing an…