arXiv research

Minimal spectral radii found for specific matrix types.

Groups of matrices with integer-like entries are studied.

Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding low-rank approximations to Boolean matrices. In this paper we give what is to the best …

The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N…

Paper proves Fujimoto's conjecture for even m ≥ 4.

Knots and 4-manifolds linked via matrix kinking.

We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a rando…

The article classifies 6D flat solvmanifolds by analyzing conjugacy classes of matrices.

This paper shows universality in spectrum behavior for random inner-product kernel matrices in polynomial regime.

Twisted Neumann--Zagier matrices for quantum invariants.

The abstract discusses resurgent functions in quantum knot invariants.

New method estimates sparse covariance matrices in logit mixtures.

Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…

We show that the Nielsen-Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum SU(n)-representations, for any fixed integer $n \geq 2$. In the Pseudo-Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) SU(2)-TQFT represen…

The problem of faithfulness of the (reduced) Burau representation for $n =4$ is known to be equivalent to the problem of whether certain two matrices $A$ and $B$ generate a free group of rank two. It is known that $A^3$ and $B^3$ generate a free group of rank two \cite{9}, \cite{10}, \cite{4}. We prove that they also g…

In 1981 Masur proved the existence of a dense geodesic in the moduli space for a Teichmüller space. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove …

Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances…

The paper develops algorithms for Boolean matrix factorization using IP and heuristics.

Affine manifolds are called integral if there is an atlas such that all transition maps are affine transformations with integer matrices of linear parts. In this paper we describe all complete integral affine structures on compact three-dimensional manifolds up to a finite-sheeted covering. Also a complete list of inte…

Study irreducible SU(2) representations for knots in 3D.

The paper studies algebraic integer relations and sequences converging to 4.

In compressed sensing problems, $\ell_1$ minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than $\e…

For two positive integers m and n, we let ${\mathcal P}_n$ be the open convex cone in ${\mathbb R}^{n(n+1)/2}$ consisting of positive definite n x n real symmetric matrices and let ${\mathbb R}^{(m,n)}$ be the set of all m x n real matrices. In this article, we investigate differential operators on the non-reductive ma…

We show that for any positive integer $n$, the maps $x \in \mathbb{C}^n \mapsto \{\left|\langle x, z_i \rangle \right|^2\}_{i=1}^{4n} \in \mathbb{R}^{4n}$, where $z_i$ are the columns of four $n\times n$ unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of emb…

This paper proposes exact and approximation algorithms for Sparse PCA, improving interpretability and scalability.

Let $R$ be an infinite commutative ring with identity and $n\geq 2$ be an integer. We prove that for each integer $i=0,1,\cdots ,n-2,$ the $L^{2}$-Betti number $b_{i}^{(2)}(G)=0,$ $\$when $G=\mathrm{GL}_{n}(R)$ the general linear group, $\mathrm{SL}_{n}(R)$ the special linear group, $$ the group generated by…

Defines cross product for m vectors in n-dimensional spaces.

The symplectic representation of mapping classes is not surjective for certain types of mapping classes.

I-BBS identifies latent sub-manifolds from distance matrices, robust to noise.

We present power low rank ensembles (PLRE), a flexible framework for n-gram language modeling where ensembles of low rank matrices and tensors are used to obtain smoothed probability estimates of words in context. Our method can be understood as a generalization of n-gram modeling to non-integer n, and includes standar…

Classifies cobounded hyperbolic actions of metabelian groups.

The Schatten quasi-norm can be used to bridge the gap between the nuclear norm and rank function, and is the tighter approximation to matrix rank. However, most existing Schatten quasi-norm minimization (SQNM) algorithms, as well as for nuclear norm minimization, are too slow or even impractical for large-scale problem…

Algorithm calculates Jones polynomial from Goeritz matrix.

New method improves portfolio selection by filtering noisy covariance matrices.

A flat complete causal Lorentzian manifold is called {\it strictly causal} if the past and the future of each its point are closed near this point. We consider strictly causal manifolds with unipotent holonomy groups and assign to a manifold of this type four nonnegative integers (a signature) and a parabola in the con…

A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riema…

Starting from considering deeper relationship between conjugacy classes and irreducible representations of a finite group $G$, we find some quite simple $R-$matrice defined by using finite groups. This construction produces many sets (or topological spaces) admitting braid group actions. We introduce conceptions "exten…

We analyze kernel matrices in polynomial high-dimensional settings and explain double descent in KRR.

The paper solves a maximum entropy sampling problem with efficient algorithms and performance guarantees.

Matrix factorization is a key tool in data analysis; its applications include recommender systems, correlation analysis, signal processing, among others. Binary matrices are a particular case which has received significant attention for over thirty years, especially within the field of data mining. Dictionary learning …

We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity. We give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality. The approa…

Study centers of quantum tori and skein algebras for even roots of unity.

This brief report (6 pages) was written in 1983 but never published. It concerns the hyperbolic 3-orbifolds obtained as quotients of hyperbolic 3-space by the group of invertible 2 by 2 matrices whose entries are integers in the imaginary quadratic extension of Q of discriminant D. For values D > -100 the topological t…

New q-deformed integers help compute Jones polynomials efficiently.

Generalized Steinberg module presentation for Gaussian and Eisenstein integers.

We present a new proof of Thurston's theorem that the unit ball of a seminorm on $\mathbf{R}^d$ taking integer values on $\mathbf{Z}^d$ is a polyhedra defined by finitely many inequalities with integer coefficients.

Surgery obstructions extended to integer homology spheres using Heegaard Floer homology.

IDF++ improves integer discrete flows for lossless compression.