arXiv research

Generative model learns to autoencode and generate sets of images.

Minimal generating sets of Reidemeister moves identified and classified.

Minimal sets of moves for isotopic knots and trivalent graphs identified.

The paper analyzes set-to-set matching with neural networks, focusing on theoretical generalization.

A new method for generating sets and graphs without requiring exchangeability.

The study classifies and characterizes totally symmetric sets in the general linear group.

Consider a general machine learning setting where the output is a set of labels or sequences. This output set is unordered and its size varies with the input. Whereas multi-label classification methods seem a natural first resort, they are not readily applicable to set-valued outputs because of the growth rate of the o…

Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $…

The paper defines generalized s-manifolds and explores their polars and antipodal sets.

SCHA-VAE generates novel data from limited examples using hierarchical context aggregation.

We present a study of generalization for data-dependent hypothesis sets. We give a general learning guarantee for data-dependent hypothesis sets based on a notion of transductive Rademacher complexity. Our main result is a generalization bound for data-dependent hypothesis sets expressed in terms of a notion of hypothe…

Cutting and pasting image segments feels intuitive: the choice of source templates gives artists flexibility in recombining existing source material. Formally, this process takes an image set as input and outputs a collage of the set elements. Such selection from sets of source templates does not fit easily in classica…

Paper studies generic dynamics of MCFs with spherical singularities.

Proves a minimal generating set for a specific group of mapping classes.

The paper finds minimal generating sets and abelianizes the quasitoric braid group.

Study of CB generating sets for infinite-type surfaces.

Paper solves the minimal generating set problem for singular Reidemeister moves.

PAC-Bayesian theory applied to data-dependent hypothesis sets yields uniform generalization bounds.

Every convex set in a generic Riemannian manifold has peculiar properties.

We determine a set of generators for the Brunnian braids on a general surface $M$ for $M\not=S^2$ or $\RP^2$. For the case $M=S^2$ or $\RP^2$, a set of generators for the Brunnian braids on $M$ is given by our generating set together with the homotopy groups of a 2-sphere.

New theory uses probability sets for data variability, improving machine learning.

A new definition for vector fields extends the Jacobi set concept.

This paper finds minimal sets of generators for mapping class groups of specific surfaces.

We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-cas…

Abstract: Proves generic torus diffeomorphisms act parabolically and non-properly on fine curve graph and have generalized rotation sets.

We introduce the new notion of Bianchi-convex sets, a generalization of convex sets of algebraic curvature tensors inspired by the second Bianchi identity. It turns out that Hamilton's maximum principle for the Ricci flow can be generalized for Bianchi-convex sets.

We study groups generated by three half-turns in the Lobachevsky $3$-space and their quotient orbifolds. These generalized triangle groups are closely related to the arbitrary 2-generator Kleinian groups. Our main result is a classification of the singular sets of the generalized triangle orbifolds. We also present a m…

Generative model learns conditional distributions on collective variable levels.

We improve adversarial robustness calibration analysis for broader hypothesis sets.

We show a general theorem of existence of temporal foliations in a general causal set, under mild constraints. Then we study automorphisms of infinite causal sets (which satisfy further requirements) and show that they fall under one of two types: 1) Automorphims that induce automorphisms of spacelike hypersurfaces in …

Study spectral settings of generalized Laplacians on homogeneous spaces.

We consider the fundamental group $π$ of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of $π$ has finite diameter with respect to the word metric given by this set. This is in contrast with a result of Danny Calegari that…

We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

Confidence measures for the generalization error are crucial when small training samples are used to construct classifiers. A common approach is to estimate the generalization error by resampling and then assume the resampled estimator follows a known distribution to form a confidence set [Kohavi 1995, Martin 1996,Yang…

Minimal generating sets found for Kim-Manturov groups.

Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver (1999). This provides constructive tools for exploring algebras of differential inva…

In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 e…

The study confirms most Cantor sets are in general position for all projections.

Generative adversarial training can be generally understood as minimizing certain moment matching loss defined by a set of discriminator functions, typically neural networks. The discriminator set should be large enough to be able to uniquely identify the true distribution (discriminative), and also be small enough to …

This paper reviews the functional aspects of statistical learning theory. The main point under consideration is the nature of the hypothesis set when no prior information is available but data. Within this framework we first discuss about the hypothesis set: it is a vectorial space, it is a set of pointwise defined fun…

Study on non-classical generating sets in Fuchsian Schottky groups.

For $n$ at least 7 and $n$ equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on $n$ strands. These generating sets are of the smallest possible cardinality. For $n$ equal to 4 or 6, we give generating sets of size three. We also prove that the commutator subgroup of the braid …

Proves convexity of level sets of general inverse σ_k equations.

A-GPS learns to generate Pareto sets efficiently with user preferences.

Deep Sets improve reinforcement learning agent's object-centered navigation and generalization.

Domain generalization is the problem of assigning labels to an unlabeled data set, given several similar data sets for which labels have been provided. Despite considerable interest in this problem over the last decade, there has been no theoretical analysis in the setting of multi-class classification. In this work, w…

We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. Th…

Generic level sets in mean curvature flow are BV solutions.