arXiv research

The paper aims to mathematically define and learn abstractions from data.

Generative models use latent abstractions to create images.

A distinctive property of human and animal intelligence is the ability to form abstractions by neglecting irrelevant information which allows to separate structure from noise. From an information theoretic point of view abstractions are desirable because they allow for very efficient information processing. In artifici…

Tabular Q-Learning with learned state abstractions solves continuous control tasks.

Learning transferable knowledge across similar but different settings is a fundamental component of generalized intelligence. In this paper, we approach the transfer learning challenge from a causal theory perspective. Our agent is endowed with two basic yet general theories for transfer learning: (i) a task shares a c…

The paper develops a framework for abstracting causal models using category theory.

Abstract cone operators prove scalar curvature comparisons on singular manifolds.

Math theory explains how neural networks learn abstract representations.

moment maps arise as a generalization of genuine moment maps on symplectic manifolds when the symplectic structure is discarded, but the relation between the mapping and the action is kept. Particular examples of abstract moment maps had been used in Hamiltonian mechanics for some time, but the abstract notion originat…

A site-specific Gordian distance between two spatial embeddings of an abstract graph is the minimal number of crossing changes from one to another where each crossing change is performed between two previously specified abstract edges of the graph. It is infinite in some cases. We determine the site-specific Gordian di…

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André-Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous man…

Abstract machinery finds obstructions to uniform positive scalar curvature.

Developed a theory of ultradifferentiable sheafs with applications.

According to a mainstream position in contemporary cognitive science and philosophy, the use of abstract compositional concepts is both a necessary and a sufficient condition for the presence of genuine thought. In this article, we show how the ability to develop and utilise abstract conceptual structures can be achiev…

COTA learns abstraction maps from data without complete SCM knowledge.

Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the loc…

Abstract perspective on quadratic programming for optimal portfolio allocation.

We show that the theory of stable complex $G$-cobordisms, for a torus $G$, is embedded into the theory of stable complex $G$-cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex $G$-cobordism theory does not…

Abstract Lie algebroids generalize Lie algebroids to abstract categories.

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…

The classical duality theory of Kantorovich and Kellerer for the classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice $\cal{X}$ with a order unit. The primal problem is given as the supremum o…

We construct a map from knots to (abstract) 2-knots which can be extended to higher dimensions; this map is the natural "knot" counterpart for "braid" theory of groups $G_{n}^{k}$.

Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.

We rephrase some well-known results in Donaldson-Thomas theory in terms of (formal families of) Frobenius type and CV-structures on a vector bundle in the sense of Hertling. We study these structures in an abstract setting, and prove a convergence result which is relevant to the case of triangulated categories. An appl…

This article constructs the moduli stack of torsionfree $G$-jet-structures in homotopy type theory with one monadic modality. This yields a construction of this moduli stack for any $\infty$-topos equipped with any stable factorization systems. In the intended applications of this theory, the factorization systems are …

Generative models characterized through learning theory.

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de …

Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.

This short note provides an overview of some theorems and conjectures obtained by the author and his collaborators. It is an extended abstract for the Oberwolfach workshop "New Trends in Teichmüller Theory and Mapping Class Groups", 2 September - 8 September 2018.

Motivated by the relationship between symplectic fibrations and classical Yang-Mills theories, we study the closeness of a $n$-form (n=2,3) defined on the total space of a fibration as a simple model for an abstract field theory. We introduce 2-plectic fibrations and interpret geometrically the corresponding equations …

Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an …

We provide an interpretation of the APS index theorem of Piazza-Schick and Zeidler in terms of coarse homotopy theory. On the one hand we propose a motivic version of the boundary value problem, the index theorem, and the associated secondary invariants. On the other hand, we discuss in detail how the abstract version …

Study delta invariant of curves on rational surfaces using topological methods.

This work concludes a series of four papers on the foundational theory of orbifolds and stacks. We apply the abstract theory, developed in its predecessors, to orbifolds derived from manifolds. Specifically, we show how the very concrete topological base spaces associated to such orbifolds can be described and manipula…

Abstracts index for ML4H workshop at NeurIPS 2019.

Abstraction is a fundamental part when learning behavioral models of systems. Usually the process of abstraction is manually defined by domain experts. This paper presents a method to perform automatic abstraction for network protocols. In particular a weakly supervised clustering algorithm is used to build an abstract…

Constructs moduli stacks of quiver bundles and applies to Higgs bundles.

We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications, and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings.

Geometric duality connects graph isomorphism and knot equivalence.

Abstract: Almost Kähler Hodge numbers vary with metric choices.

We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A moti…

Unified framework for causal models at different levels of abstraction.

In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable (vector) density is known to generate the so--called {\em conserved Noether currents}. It turns out that along any section of the relevant gauge--natural bundle this density is the divergence of a skew--symmetric (tenso…

Abstract MDPs enable strategic exploration and fast reward transfer in complex environments.

This is a survey of motivations, constructions and applications of higher prequantum geometry. In section 1 we highlight the open problem of prequantizing local field theory in a local and gauge invariant way, and we survey how a solution to this problem exists in higher differential geometry. In section 2 we survey ex…

It is well-known that there are a number of relations between theoretical finance theory and information theory. Some of these relations are exact and some are approximate. In this paper we will explore some of these relations and determine under which conditions the relations are exact. It turns out that portfolio the…

Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.

We construct infinitely many non-diffeomorphic examples of $5$-dimensional contact manifolds which are tight, admit no strong fillings, and do not have Giroux torsion. We obtain obstruction results for symplectic cobordisms, for which we give a proof not relying on the polyfold abstract perturbation scheme for SFT. The…