arXiv research

New neural networks for non-commutative data.

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…

Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.

Equivalence proven between equivariant K-theory and K-homology for certain matrix group actions.

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 construct membrane homology groups $\h(M)$ associated with each compact connected oriented smooth manifold, and show that $\h(M)$ is matrix graded algebra.

In this paper we construct the Differential calculus on the Hopf Group Coalgebra introduced by Turaev [10]. We proved that the concepts introduced by S.L.Woronowicz in constructing Differential calculus on Hopf Compact Matrix Pseudogroups (Quantum Groups)[7] can be adapted to serve again in our construction.

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear $G$-manifolds (1996). Also we improve the Bredon--Floyd theorem from compact groups $G$ (1960).

The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.

Global analysis of Dixmier traces and Wodzicki residues on compact Lie groups.

This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study "Infinite groups as geometric objects", as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can f…

New paradigm for Neural ODEs stabilizes training and improves model performance.

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is t…

Proves a conjecture about matrix orders for pseudo-Anosov maps.

Researchers study injectivity of magnetic and thermostatic nonabelian ray transforms on compact surfaces.

In this article we study the Hofer geometry of a compact Lie group $K$ which acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized Hofer norms on the Lie algebra of $K$ are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global res…

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimi…

Attention tokens are group elements, negated algebra norms.

The stability and robustness of compact schemes for parabolic PDEs are analyzed.

Study Brownian motion on Grassmann manifold using matrix stochastic calculus.

Estimates Laplace eigenvalues and diameter for Lie group metrics.

The coadjoint orbits of compact Lie groups carry many Kähler structures, which include a Riemannian metric and a complex structure. We provide a fairly explicit formula for the Levi-Civita connection of the Riemannian metric, and we use the complex structure to give a fairly explicit construction of a canonical Dirac o…

A geometric algorithm is introduced for finding a symplectic basis of the first integral homology group of a compact Riemann surface, which is a $p$-cyclic covering of ${\mathbb C} P^1$ branched over 3 points. The algorithm yields a previously unknown symplectic basis of the hyperelliptic curve defined by the affine eq…

We show that integration over a $G$-manifold $M$ can be reduced to integration over a minimal section $Σ$ with respect to an induced weighted measure and integration over a homogeneous space $G/N$. We relate our formula to integration formulae for polar actions and calculate some weight functions. In case of a compact …

Extends pseudo-differential operators theory to compact Lie groups.

Formula for sectional curvatures on matrix groups.

Linear representations help embed manifolds into matrix spaces.

The paper studies how the singularity of character varieties changes when representations are extended.

Abstract: Determinants and formulas for operators on various spaces.

We consider height functions on symmetric spaces $M\cong G/K$ embedded in the associated matrix Lie group $G$. In particular we study the relationship between the critical sets of the height function on $G$ and its restriction to $M$. Also we prove that the gradient flow on $M$ can be integrated by means of a generaliz…

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…

Random matrix ensembles yield uniform distributions on manifolds.

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…

Efficiently optimizes orthogonal and Stiefel matrices on parallel units.

New method constructs equivariant neural networks for arbitrary matrix groups.

Study connects Lie groups to specific Riemannian manifolds.

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.

Unified treatment of eigenvalue processes using Riemannian geometry.

Tensor completion estimates missing components by exploiting the low-rank structure of multi-way data. The recently proposed methods based on tensor train (TT) and tensor ring (TR) show better performance in image recovery than classical ones. Compared with TT and TR, the projected entangled pair state (PEPS), which is…

The paper explores 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 …

Convexity proven for sums of angles of unitary paths.

New graph Hamiltonicity via cohomology of Artin groups.

New algorithm speeds up group equivariant neural networks computations.

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…

We develop the scattering theory of general conformally compact metrics. For low frequencies, the domain of the scattering matrix is shown to be frequency dependent. In particular, generalized eigenfunctions exhibit L^2 decay in directions where the asymptotic curvature is sufficiently negative. The scattering matrix i…

Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.