arXiv research

Novel representer theorem for metric and preference learning in RKHSs.

Survey of recent Kleinian representation convergence results.

We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is at the core of kernel methods in machine learning as it makes…

Study k-positive surface group representations and their degenerations.

New examples of embeddings defy Anosov representation limits.

Geometrically, Kostant's Convexity Theorem is extended to submetries with a fat section.

Study local structure of knot group representations into SL(n,C).

New representations defined for groups and graphs, with applications to stable representations.

Entropy rigidity theorem for cusped Hitchin representations.

We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is the core of kernel methods in machine learning as it makes th…

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group $G$ on a smooth or analytic manifold $M$ with a rigid $\mathrm{A}$-structure $σ$. It generalizes Gromov's centralizer and representation theorems to the case where $R(G)$ is split solvable and $G/R(G…

We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{ö}nflies theorem. As an application, we show that one can read off oriented matroids from arrangement…

We generalize a theorem of Burde and de Rham characterizing the zeros of the Alexander polynomial. Given a representation of a knot group $π$, we define an extension of $π$, the Crowell group. For any GL(n,C) representation of $π$, the zeros of the associated twisted Alexander polynomial correspond to representations o…

The necessary and sufficient conditions for existence of a generalized representer theorem are presented for learning Hilbert space-valued functions. Representer theorems involving explicit basis functions and Reproducing Kernels are a common occurrence in various machine learning algorithms like generalized least squa…

Revisits the connection between neural networks and the Kolmogorov-Arnold theorem.

We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an aut…

The standard approach for dealing with the ill-posedness of the training problem in machine learning and/or the reconstruction of a signal from a limited number of measurements is regularization. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augme…

The article discusses extensions of Harish-Chandra's admissibility theorem.

SL(n) covariant valuations on Orlicz spaces are represented and characterized.

New proof of Alesker's Irreducibility Theorem using localization techniques.

PARIS reduces imbalanced regression datasets by pruning uninformative samples.

Link groups can only have certain SU(2) representations.

Proves a generalization of a multiplicity one theorem for specific groups.

Paper proves almost all stabilizer subgroups of Thompson's group satisfy Alexander's theorem.

This paper presents the first use of graph neural networks (GNNs) for higher-order proof search and demonstrates that GNNs can improve upon state-of-the-art results in this domain. Interactive, higher-order theorem provers allow for the formalization of most mathematical theories and have been shown to pose a significa…

Developed algebraic theory of bonded braids, proving Markov theorem.

Represents neural networks as solutions to inverse problems in Banach spaces.

We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical t…

In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(Γ) of a compact Lie group $Γ$ to the complex K-theory of the classifying space $BΓ$. For infi…

In the paper "Pappus's theorem and the modular group", R. Schwartz constructed a 2-dimensional family of faithful representations $ρ_Θ$ of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ into the group $\mathscr{G}$ of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subg…

Geometric models for representations up to homotopy using simplicial vector bundles.

The paper proves approximation and interpolation theorems for maxfaces with singularities.

the main theorem gives a sufficient condition for a n elements of SL(2,R) to generate a free group.The idea behind it is to use a nonorientable version of the Dehn-Wolpert-Goldman twist and to sew it with the original representation of a free group to get representation of the closed surfase group and then to apply Gol…

Paper develops efficient estimator for Hawkes processes using representer theorem.

Describes representations of modular group into SL(3,R)/SO(3).

In this paper, we proved a rigidity theorem of the Hodge metric for concave horizontal slices and a local rigidity theorem for the monodromy representation.

In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…

We prove the existence of optimal strategies for agents with cumulative prospect theory preferences who trade in a continuous-time illiquid market, transcending known results which pertained only to risk-averse utility maximizers. The arguments exploit an extension of Skorohod's representation theorem for tight sequenc…

Factorizes discrete representations of finitely generated groups into PSL(2, R).

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A_infty functor from the representations up to homotopy of a Lie algebroid to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several exampl…

Paper uses index theorem to relate symplectic bundle signature to surface group representation in real symplectic group.

The paper classifies fiber structures of discontinuity domains for Anosov representations.

We prove universality theorems ("Murphy's Laws") for representation schemes of fundamental groups of closed 3-dimensional manifolds. We show that germs of SL(2,C)-representation schemes of such groups are essentially the same as germs of schemes of over rational numbers.

The article extends Thurston's Grafting Theorem to signed spaces and defines a framed monodromy map.

We give a new proof of Markov's classical theorem relating any two closed braid representations of the same knot or link. The proof is based upon ideas in a forthcoming paper by the authors, "Stabilization in the braid groups". The new proof of the classical Markov theorem is used by Nancy Wrinkle in her forthcoming ma…

DM improves self-supervised transfer learning by matching target distributions.

Let $Γ$ be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of $Γ$ in SO(n,1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the repr…

Study of instantons and knot representations in 3-manifolds.