arXiv research

Group-invariant neural networks improve approximation accuracy for symmetric functions.

Paper describes invariants of slice regular functions' automorphism group.

Derives representations invariant under crystallographic groups for functions.

Study invariant minimizers in convex functions under amenable groups.

Machine learning uses invariant theory to restrict function classes.

The study bounds Dehn functions of coabelian subgroups using a second BNSR invariant.

In this study, a novel feature coding method that exploits invariance for transformations represented by a finite group of orthogonal matrices is proposed. We prove that the group-invariant feature vector contains sufficient discriminative information when learning a linear classifier using convex loss minimization. Ba…

The article studies Hamiltonian flows on surface group representations induced by invariant multi-functions.

In this paper, we discuss relations among several invariants of 3-manifolds including Meyer's function, the eta-invariant, the von Neumann rho-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.

A function group is a finitely generated Kleinian group with an invariant connected component of its region of discontinuity. An extended function group is a finitely generated extended Kleinian group that contains orientation reversing elements and keep invariant a connected components of its region of discontinuity. …

New filling functions for groups with coefficients show different asymptotic behavior.

The paper classifies invariant structures on complex almost Abelian groups.

The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2].…

Solves symplectic and conformal symplectic group actions equivalence problem.

We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by U. Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. We als…

The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial differential operators (derivatives) and a finite system of invariants, such that any …

Researchers find a way to bound the complexity of certain subgroup geometric invariants.

Study presents a method to induce a generalized neural network from joint group invariant functions.

Study expanding gradient Ricci solitons with Euclidean base.

Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of…

The paper proves that relative Dehn functions are invariant under quasi-isometry.

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by t…

Given an action of a Lie group on a smooth manifold, we discuss the induced action on the Hochschild cohomology of smooth functions, and notions of invariance on this space. Depending on whether one considers invariance of cochains or invariance of cohomology classes, two different spaces of invariants arise. We perfor…

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function $n^2\log n$ and other small Dehn functions. Some of these groups have undecidable conjugacy problem.

The study of Morse functions on 3-manifolds and their Reeb graphs.

In this paper we consider the problem of group invariant subspace clustering where the data is assumed to come from a union of group-invariant subspaces of a vector space, i.e. subspaces which are invariant with respect to action of a given group. Algebraically, such group-invariant subspaces are also referred to as su…

Energy functional for Legendrian knots in Heisenberg group, invariant under PU(2,1).

Study of Gödel Universe as Lie group with specific metric.

We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.

We analyze in this paper a random feature map based on a theory of invariance I-theory introduced recently. More specifically, a group invariant signal signature is obtained through cumulative distributions of group transformed random projections. Our analysis bridges invariant feature learning with kernel methods, as …

We give some new methods, based on Lipschitz extension theorems, for bounding filling invariants of subsets of nonpositively curved spaces. We apply our methods to find sharp bounds on higher-order Dehn functions of Sol_{2n+1}, horospheres in euclidean buildings, Hilbert modular groups, and certain S-arithmetic groups.

Invariants for singular knots and links using non-commutative cocycles.

In this paper, we shall be concerned with a relation between TQFTs and cut and paste invariants introduced by Karras, Kreck, Neumann and Ossa. Cut and paste invariants, or SK invariants, are functions on the set of smooth manifolds that are invariant under the cutting and pasting operation. Central to the work in this …

Universal MLPs with a single hidden layer can learn any function.

We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an invariant of homology spheres. We apply this study to give a new purely algebrai…

The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.

We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal invariant of a one-parameter group is known then a complete set of functionally independ…

Study on homological Dehn functions of groups of type $FP_2$.

Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.

Study on braids' invariant growth and counting functions.

Study shows how to reduce data needed for learning under geometric constraints.

We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with ar…

In this paper, we develop a theory about the relationship between $G$-invariant/equivariant functions and deep neural networks for finite group $G$. Especially, for a given $G$-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip $G$-actions and each affine t…

Analytic realization of Thom-Smale complex for G-manifolds.

Morse inequalities for noncompact manifolds with group action.

In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(2|1). The usual quantum group invariant of links associated to (generic) representations of sl(2|1) is trivial. However, we modify this construction and define a nontrivial link …

Quantum invariants from $U_h\mathfrak{sl}(2|1)$ are q-holonomic.

Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has been devoted to encoding invariance under symmetry transformations into neural n…