arXiv research

Generative model for morphisms in free categories learns from wiring diagrams.

New approach uses neural networks to learn program structure and parameters.

It is well known that the opposite F^{op} of the category F of finitely generated free groups is a Lawvere theory for groups, and also that F is a free symmetric monoidal category on a commutative Hopf monoid, or, in other words, a PROP for commutative Hopf algebras. In this paper, we give a direct, combinatorial proof…

Recent work on single-view 3D reconstruction shows impressive results, but has been restricted to a few fixed categories where extensive training data is available. The problem of generalizing these models to new classes with limited training data is largely open. To address this problem, we present a new model archite…

Study multiplicity-free covering of graded manifolds, proving equivalence of categories.

We introduce the notion of a positive opetope and positive opetopic cardinals as certain finite combinatorial structures. The positive opetopic cardinals to positive-to-one polygraphs are like simple graphs to free omega-categories over omega-graphs, c.f. [MZ]. In particular, they allow us to give an explicit combinato…

Study homotopy sheaves on categories and their presheaves, proving descent properties.

Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the categ…

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having an affine structure on a manifold is equivalent to having a flat torsion-free c…

Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.

Study proposes learning optimal priors from data for better Bayesian inference.

The paper discusses strictification and non-strictification of monoidal categories.

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a…

This paper extends link invariants using functors on nanophrases.

Study proves topological complexity and LS-category inequalities for specific groups and manifolds.

We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We also obtain some general results on the relations between the fundamenta…

We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient…

The main contribution of this article is a new prior distribution over directed acyclic graphs, which gives larger weight to sparse graphs. This distribution is intended for structured Bayesian networks, where the structure is given by an ordered block model. That is, the nodes of the graph are objects which fall into …

Study probabilistic category and complexity bounds, comparing with classical invariants.

We define a category $v\mathcal{T}$ of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to $v\mathcal{T}$ which induces an equivalence of categories. On the other hand, we show that $v\mathcal{T}$ is universal among ribbon c…

CITE algorithm provides anytime-valid certification of model outputs.

New PROP structure defined for automorphism group cohomology.

Constructs 2-representations and 2-vector bundles for Lie 2-groups.

Learning the network structure underlying data is an important problem in machine learning. This paper introduces a novel prior to study the inference of scale-free networks, which are widely used to model social and biological networks. The prior not only favors a desirable global node degree distribution, but also ta…

Study on polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.

Bayesian model for discrete data with conditional transformations.

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor…

We consider the stochastic multi-armed bandit problem with a prior distribution on the reward distributions. We are interested in studying prior-free and prior-dependent regret bounds, very much in the same spirit as the usual distribution-free and distribution-dependent bounds for the non-Bayesian stochastic bandit. B…

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C, W, z, z^*) consist…

Bayesian semi-supervised learning for multi-class classification.

The paper defines a new metric space invariant and computes it for various manifolds.

Study automorphism groups' action on Jacobi diagrams, leading to new decompositions.

Blade uses diffusion priors to accurately and calibratedly infer complex systems.

Introduces injective category number for continuous maps, linking classical and contemporary research.

Paper proposes redundancy-free features for zero-shot object recognition.

Model dynamic customer sensitivities across categories.

The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.

We introduce the category of generalized Courant algebroids and show that it admits a free object on any anchored vector bundle. The free Courant algebroid is built from two components: the generalized Courant algebroid associated to a symmetric Leibniz algebroid and the free symmetric Leibniz algebroid on an anchored …

Unit-free approach to Jacobi geometry and Hamiltonian mechanics.

Kontsevich and Soibelman introduced a notion of orientation data on Calabi-Yau category. It can be viewed as a consistent choice of spin structure on moduli space of objects in the given category. The orientation data plays an important role in Donaldson-Thomas theory. Let X be a projective, simply connected and torsio…

We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a "fusion" product with respect to triples of paths. The equivalence is established by explicit fun…

The paper connects sectional category and parametrized Borsuk-Ulam property for fibrations.

Compared to machines, humans are extremely good at classifying images into categories, especially when they possess prior knowledge of the categories at hand. If this prior information is not available, supervision in the form of teaching images is required. To learn categories more quickly, people should see important…

New algebraic presentations for 3D cobordisms and 4D 2-handlebodies.

Racks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and their relation to the theory of permutations. In particular, we compute the centers of the categories and describe power operations…

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a f…

We introduce a model-based reconstruction framework with deep learned (DL) and smoothness regularization on manifolds (STORM) priors to recover free breathing and ungated (FBU) cardiac MRI from highly undersampled measurements. The DL priors enable us to exploit the local correlations, while the STORM prior enables us …

The paper studies posets from decompositions in symmetric monoidal categories.