Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
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48 results for “Matrix groups”
New neural networks for non-commutative data.
problem No existing neural networks suitable for non-commutative data.
method Developed compact matrix quantum group equivariant neural networks.
result Characterized weight matrices for easy compact matrix quantum groups.
We propose improved methods to identify stock groups using the correlation matrix of stock price changes. By filtering out the marketwide effect and the random noise, we construct the correlation matrix of stock groups in which nontrivial high correlations between stocks are found. Using the filtered correlation matrix…
A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscs.LG
New method constructs equivariant neural networks for arbitrary matrix groups.
problem Challenges in constructing equivariant neural networks for complex groups.
method Completely general algorithm for solving equivariant layers of matrix groups.
result Constructs multilayer perceptrons equivariant to multiple groups including O(1,3), O(5), Sp(n), and Rubik's cube group.
Matrix Lie Groups as 3-Dimensional Almost Paracontact Almost Paracomplex Riemannian Manifoldsmath.DG
Study connects Lie groups to specific Riemannian manifolds.
problem Understanding Lie groups through Riemannian manifold properties.
method Investigates Lie groups as 3D almost paracontact almost paracomplex Riemannian manifolds.
result Established correspondence between Lie algebra and matrix representation.
Low-dimensional representations of matrix groups and group actions on CAT(0) spaces and manifoldsmath.GT
We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.
Proposes Robust Matrix Factorization with Grouping Effect (GRMF) for better performance and robustness.
problem Improves matrix factorization by incorporating grouping effect for better performance and robustness.
method Integrates grouping effect into matrix factorization, using an efficient alternating minimization framework with DC programming and ADMM.
result Demonstrates improved performance and robustness compared to five benchmark algorithms on real-world data sets with outliers and noise.
The paper explores continuous inverse ambiguous functions on various Lie groups.
problem Existence of continuous inverse ambiguous functions on Lie groups.
method Investigation of continuous inverse ambiguous functions on specific Lie groups.
result Existence of continuous inverse ambiguous functions on various Lie groups.
We propose a privacy-enhanced matrix factorization recommender that exploits the fact that users can often be grouped together by interest. This allows a form of "hiding in the crowd" privacy. We introduce a novel matrix factorization approach suited to making recommendations in a shared group (or nym) setting and the …
New graph Hamiltonicity via cohomology of Artin groups.
problem Characterizing Hamiltonicity in graphs using cohomology.
method Defining a new graph from matrices and analyzing cohomology.
result New graph Hamiltonicity characterization via cohomology.
A Diagrammatic Approach to Improve Computational Efficiency in Group Equivariant Neural Networkscs.LG
New algorithm speeds up group equivariant neural networks computations.
problem Challenging computations in group equivariant neural networks.
method Diagrammatic framework based on category theory for matrix multiplication.
result Exponential improvement in time complexity for matrix multiplication.
This paper develops a new class of nonconvex regularizers for low-rank matrix recovery. Many regularizers are motivated as convex relaxations of the matrix rank function. Our new factor group-sparse regularizers are motivated as a relaxation of the number of nonzero columns in a factorization of the matrix. These nonco…
Equivalence proven between equivariant K-theory and K-homology for certain matrix group actions.
problem Establishing Poincaré duality for proper cocompact matrix group actions.
method Using equivariant K-theory and K-homology, with geometric models of Baum and Douglas.
result Poincaré duality holds between equivariant K-theory and K-homology for -spin manifolds with compact quotient.
Study asymptotics of unitary matrix elements in quantum mechanics.
problem Asymptotic behavior of unitary matrix elements in quantum mechanics.
method Uses Berezin-Toeplitz quantization and symplectic geometry.
result Recover asymptotics of Wigner's d-matrix elements for spin representations.
Structured Conformal Inference for Matrix Completion with Applications to Group Recommender Systemsstat.ME
Develops a method to estimate uncertainty for group-level recommendations in matrix completion.
problem Uncertainty estimation for group-level recommendations in matrix completion.
method Structured conformal inference method combining any matrix completion algorithm.
result Stronger group-level guarantees through structured calibration.
Matrix completion has a long-time history of usage as the core technique of recommender systems. In particular, 1-bit matrix completion, which considers the prediction as a ``Recommended'' or ``Not Recommended'' question, has proved its significance and validity in the field. However, while customers and products aggre…
GAME improves matrix completion by considering subgroup-specific latent structures.
problem Heterogeneous data with overlapping categories, smoothing away subgroup-specific variation.
method Group-Aware Matrix Estimation (GAME) with overlapping nuclear-norm penalties.
result GAME outperforms global low-rank estimators in structured missingness regimes.
We consider the problem of matrix completion with side information (\textit{inductive matrix completion}). In real-world applications many side-channel features are typically non-informative making feature selection an important part of the problem. We incorporate feature selection into inductive matrix completion by p…
The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some kno…
We obtain minimal dimension matrix representations for each indecomposable five-dimensional Lie algebra over and justify in each case that they are minimal. In each case a matrix Lie group is given whose matrix Lie algebra provides the required representation.
There are studied Lie groups considered as almost hypercomplex Hermitian-Norden manifolds, which are integrable and have the lowest dimension four. It is established a correspondence of the derived Lie algebras of types of invariant hypercomplex structures and the explicit matrix representation of their Lie groups. The…
Starting from considering deeper relationship between conjugacy classes and irreducible representations of a finite group , we find some quite simple matrice defined by using finite groups. This construction produces many sets (or topological spaces) admitting braid group actions. We introduce conceptions "exten…
Study Alexander matrices for link quandles and their relation to knot invariants.
problem Understanding Alexander matrices for link quandles and their applications to knot invariants.
method Investigate -twisted Alexander matrices and their connection to quandle cocycle invariants.
result Show that -twisted Alexander invariants of knot quandles are stronger than those of knot groups.
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…
Roots of a compact Lie groupmath.DG
This expository article introduces the topic of roots in a compact Lie group. Compared to the many other treatments of this standard topic, I intended for mine to be relatively elementary, example-driven, and free of unnecessary abstractions. Some familiarity with matrix groups and with maximal tori is assumed. This ar…
Study optimizes estimation of orthogonal and rotation matrices from noisy data.
problem Estimating orthogonal and rotation matrices from noisy data.
method Iterative polar decomposition algorithm initialized by spectral methods.
result Algorithm achieves optimal error rate of $(1+o(1)) rac{σ^2 d(d-1)}{2np}$.
Researchers develop a new quantum invariant using a matrix dilogarithm for 3-manifolds.
problem Developing quantum invariants for 3-manifolds.
method Using a matrix dilogarithm and quantum groups.
result The matrix dilogarithm can be considered as a 6-symbol.
Attention tokens are group elements, negated algebra norms.
problem Attention mechanism for matrix Lie groups
method Lie-Algebra Attention
result Score matches learned kernel on invariant poses, outperforms vector-token baseline
Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.
problem Computing isotropy subgroups of orthogonal similarity on symmetric matrices.
method Algorithmic procedure solving a Toeplitz matrix equation.
result Structure of isotropy subgroups described.
Quantum theory uses modular group representations to assign invariants to 3-manifolds.
problem Assigning invariants to 3-manifolds via modular group representations.
method Projective representations of the modular group derived from a noncommutative torus.
result Computed traces and determinants of matrices associated with modular group elements.
Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
problem Computing isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
method Algorithm for solving a matrix equation to compute isotropy subgroups.
result Computed isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
Linear and Quadratic Discriminant analysis (LDA/QDA) are common tools for classification problems. For these methods we assume observations are normally distributed within group. We estimate a mean and covariance matrix for each group and classify using Bayes theorem. With LDA, we estimate a single, pooled covariance m…
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.
Private algorithms approximate matrices with private data.
problem Approximate matrices with same spectrum using private data.
method Differential privacy algorithms for unitary orbit optimization.
result Upper and lower bounds on approximation error.
In this paper, we apply tools from the random matrix theory (RMT) to estimates of correlations across volatility of various assets in the S&P 500. The volatility inputs are estimated by modeling price fluctuations as GARCH(1,1) process. The corresponding correlation matrix is constructed. It is found that the distribut…
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…
In this paper, we study the effects of different prior and likelihood choices for Bayesian matrix factorisation, focusing on small datasets. These choices can greatly influence the predictive performance of the methods. We identify four groups of approaches: Gaussian-likelihood with real-valued priors, nonnegative prio…
Classifies matrices in the quaternionic hyperbolic unitary group.
problem Understanding the structure of matrices in the quaternionic hyperbolic unitary group.
method Used complex representation and characteristic polynomial to study matrices.
result Computed the characteristic polynomial and studied its sign.
We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N,g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Ka…
Algorithm counts intersections of normal curves efficiently.
problem Efficiently solving word problems in mapping class groups of punctured surfaces.
method Fast algorithm for counting intersections of normal curves on triangulated surfaces.
result Efficient solution of the word problem for mapping class groups of punctured surfaces.
A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
Let S be a compact Riemann surfaces of genus g >= 2 and G a conformal automoprhism group of order n acting on S. In this paper we give the definition of an adapted generating set and an adapted basis for the first homology group of such a compact Riemann surface. This generating set and basis reflect the action of G in…
We discuss the question of how to pick a matrix uniformly (in an appropriate sense) at random from groups big and small. We give algorithms in some cases, and indicate interesting problems in others.
We show that for k at least 3, given any matrix in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z^k is the given matrix.
Canonical Symplectic Representations for Prime Order Conjugacy Classes of the Mapping-class Groupmath.GT
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…
Flow Matching on Lie Groupsmath.DG
Generative model learns from simpler distributions on Lie groups.
problem Learning from complex Lie group data.
method Substituting exponential curves for line segments on Lie groups.
result Simple, intrinsic, and fast implementation for generative modelling.
Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.
problem Analyse Stokes phenomena in Poisson-Lie groups and quantum groups.
method Use Ug-valued Stokes phenomena to construct quantum group U_hg and relate it to Poisson-Lie group G*.
result Show that Ug-valued Stokes phenomena can be obtained as a semiclassical limit of the KZ associator.
We address the task of identifying densely connected subsets of multivariate Gaussian random variables within a graphical model framework. We propose two novel estimators based on the Ordered Weighted (OWL) norm: 1) The Graphical OWL (GOWL) is a penalized likelihood method that applies the OWL norm to the lowe…