arXiv research

The paper explores simple and relatively simple transformation groups and their universal coverings.

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension $n$, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on t…

Simple bounds for covariance and Gram matrices across various settings.

Finite index subgroups of relatively hyperbolic groups have equal index.

The recent crisis and the following flight to simplicity put most derivative businesses around the world under considerable pressure. We argue that the traditional modeling techniques must be extended to include product design. We propose a quantitative framework for creating products which meet the challenge of being …

In this paper, we present our general results about traversing flows on manifolds with boundary in the context of the flows on surfaces with boundary. We take advantage of the relative simplicity of $2D$-worlds to explain and popularize our approach to the Morse theory on smooth manifolds with boundary, in which the bo…

The paper develops formulas for hyperbolic simplices based on edge lengths.

Shellable tilings on simplicial complexes help understand their structure.

Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.

The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.

Neural networks favor simple features over complex ones, even when complex features are available.

Algorithms are increasingly used to aid, or in some cases supplant, human decision-making, particularly for decisions that hinge on predictions. As a result, two additional features in addition to prediction quality have generated interest: (i) to facilitate human interaction and understanding with these algorithms, we…

Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.

OSIRIS reduces variance in off-policy evaluation by omitting irrelevant states.

Study PL bordism theories with quantitative bounds on filling simplices.

Framework reduces simplicity bias in NNs, improving OOD generalization and robustness.

New framework shows $C^*$-simplicity for groups without certain subalgebras.

In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

Research reveals simplicity bias in random logistic map, impacting data analysis and forecasting.

Unimodular classification of symmetric matrix map-germs.

The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.

Similar simplices can be inscribed in most smoothly embedded spheres.

We propose the segmented iHMM (siHMM), a hierarchical infinite hidden Markov model (iHMM) that supports a simple, efficient inference scheme. The siHMM is well suited to segmentation problems, where the goal is to identify points at which a time series transitions from one relatively stable regime to a new regime. Conv…

We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter …

Simplicial sets deformation retract onto transverse simplices.

We generalize the very well known boundary operator of the ordinary singular homology theory, defined in many books about algebraic topology. We describe a variant of this ordinary simplicial boundary operator where the usual boundary (n-1)-simplices of each n-simplex are replaced by combinations of internal (n-1)- sim…

Normal 4-pseudomanifolds with one or two singular vertices are derived from specific operations.

The paper explores how simplicity leads to better out-of-distribution generalization in models.

The study reveals simplicity bias in neural networks leading to better compositional mappings.

Study on simplicity of Lie skew braces, proving new results for compact cases.

Triangulations of R^n have at least tensor rank of determinant simplices.

Two-layer networks favor simple features, especially in complex datasets.

Adam avoids simplicity bias in neural networks, leading to better generalization.

We give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology.

Neural nets learn simple distributions first, then more complex ones.

The paper proves eigenvalues are simple for specific operators on bundles.

Study analyzes risks and opportunities in blockchain currency markets.

Ancient formula connects volume forms and infinitesimal square volumes in manifolds.

Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristi…

The study improves representation learning bounds using data-dependent Gaussian mixtures.

Maps discrete manifolds to partitions to define new manifolds.

Compressed imitation learning uses simplicity priors for efficient expert behavior copying.

AB-testing is a very popular technique in web companies since it makes it possible to accurately predict the impact of a modification with the simplicity of a random split across users. One of the critical aspects of an AB-test is its duration and it is important to reliably compute confidence intervals associated with…

Study the boundary operator property on simplicial complexes, proving essential properties for Hodge theory.

In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible surfaces which are not tori. Next, such incompressible surfaces are related to …

Proves simplicity of Lyapunov exponents for specific Anosov flows.

This paper proposes a new method for efficient data compression using Bayesian neural networks.