The paper develops formulas for hyperbolic simplices based on edge lengths.
problem Understanding the geometry of hyperbolic simplices using only edge lengths.
method Develops geometric formulas for hyperbolic simplices based on edge lengths.
result Distance and projection formulas in hyperbolic simplices.
Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.
problem Understanding the structure of trees in Outer space.
method Associate simplices to R-trees and estimate their dimensions. result Estimates the dimensions of maximal simplices for both rational and irrational trees.
The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.
problem Conditions for Riemannian connections and semi-simplicity of Lie algebras.
method Using almost product structures and spray, the paper provides necessary and sufficient conditions for these properties.
result Equivalence of semi-simplicity of Lie algebras to derived ideal coincidence, interiority of derivations, and adjoint representation semi-simplicity.
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.
Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.
problem Understanding geodesic simplices in pseudo-hyperbolic space.
method Cohomological interpretation and necessary/sufficient condition formulation.
result Every ideal geodesic polytope in (2,2) pseudo-hyperbolic space has finite volume. Study PL bordism theories with quantitative bounds on filling simplices.
problem Understanding PL bordism theories with geometric constraints.
method Quantitative analysis of PL manifolds and exotic theories.
result Bounding the number of simplices in fillings of cycles.
Framework reduces simplicity bias in NNs, improving OOD generalization and robustness.
problem Simplicity bias in deep learning models leads to biased predictions and poor OOD generalization.
method Proposes a framework that regularizes conditional mutual information to encourage use of diverse features.
result Demonstrates effectiveness in various settings, enhancing OOD generalization and robustness.
New framework shows C∗-simplicity for groups without certain subalgebras.
problem Characterizing C∗-simplicity of groups. method Introducing confined subalgebras and Uniformly Recurrent States.
result A countable discrete group is C∗-simple if it has no non-trivial amenable confined subalgebras. In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.
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…
Research reveals simplicity bias in random logistic map, impacting data analysis and forecasting.
problem Simplicity bias in dynamical systems and its impact on data analysis and prediction.
method Examined the logistic map and random logistic map, focusing on simplicity bias and noise effects.
result Simplicity bias is observable in the random logistic map, persisting even with small noise levels.
The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.
problem Generalizing the Apollonius theorem for m-simplices.
method Direct generalization of the theorem to m-simplices in n-dimensional space.
result Applications in geometry and optimization, including minimal surface enclosures, simplex thickness, and root-finding methods.
Similar simplices can be inscribed in most smoothly embedded spheres.
problem Inscribing families of similar simplices in spheres.
method Diffeomorphic mapping and techniques from previous work on inscribing triangles.
result A dense family of spheres allows inscribing similar simplices of every pose.
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.
problem Deformation retraction of simplicial sets.
method Showed deformation retraction of singular simplicial set onto transverse simplices.
result Singular simplicial set deformation retracts 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…
The paper explores how simplicity leads to better out-of-distribution generalization in models.
problem Understanding the theoretical principles behind out-of-distribution (OOD) generalization in modern models.
method Examining diffusion models in image generation to analyze compositional generalization abilities and develop a theoretical framework for simplicity-based OOD generalization.
result The true, generalizable model corresponds to the simplest among consistent models, and this simplicity can be quantified and used to establish sample complexity guarantees.
The study reveals simplicity bias in neural networks leading to better compositional mappings.
problem Understanding when and how to encourage neural networks to learn compositional mappings.
method Examined compositional mappings through coding length and gradient descent dynamics.
result Neural networks tend to learn the simplest bijections, explaining their good generalization.
Study on simplicity of Lie skew braces, proving new results for compact cases.
problem Simplicity of Lie skew braces, focusing on compact connected cases.
method Reviewing correspondence, investigating ideals and rigidity, proving main result for compact Lie skew braces.
result Compact connected simple Lie skew braces are either trivial or have simple underlying Lie groups.
Triangulations of R^n have at least tensor rank of determinant simplices.
problem Understanding the minimum number of simplices in periodic triangulations of R^n.
method Proving lower bounds on the number of simplices in periodic triangulations of R^n.
result Lower bounds on the number of simplices in periodic triangulations of R^n.
Two-layer networks favor simple features, especially in complex datasets.
problem Simplicity bias in neural networks over-reliing on simple features.
method Characterization of two-layer neural networks with small weights and gradient flow.
result Features learned in middle training stages are more useful for out-of-distribution transfer.
Adam avoids simplicity bias in neural networks, leading to better generalization.
problem Simplicity bias in neural networks trained with SGD.
method Comparison of Adam and GD on binary classification tasks with Gaussian data.
result Adam leads to richer and more diverse features, improving 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.
problem Understanding how neural networks generalize from simple to complex functions.
method Stochastic gradient descent training, synthetic data, CIFAR10, ImageNet pre-training.
result Neural networks initially use lower-order statistics, then higher-order ones.
The paper proves eigenvalues are simple for specific operators on bundles.
problem Eigenvalue simplicity for connection Laplacian and G-simplicity on bundles. method Analyzes connections on vector bundles and principal bundles, proving eigenvalue simplicity for a residual set of connections.
result Eigenvalues of the connection Laplacian and Laplace-Beltrami operator are simple for specified conditions.
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.
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…
Maps discrete manifolds to partitions to define new manifolds.
problem Creating manifolds from discrete structures.
method Mapping discrete d-manifolds onto (k+1)-partite complexes to define new manifolds.
result Defines a (d-k)-manifold from simplices in G mapped to P.
Compressed imitation learning uses simplicity priors for efficient expert behavior copying.
problem Efficiently learn expert behaviors with minimal data.
method Utilizes policy simplicity as a prior for sample-efficient imitation learning.
result Significantly higher scores achieved with limited expert demonstrations.
Proves simplicity of Lyapunov exponents for specific Anosov flows.
problem Proving all Lyapunov exponents have multiplicity 1 for certain Anosov flows.
method Perturbative results for flows, modification of eigenvalues, Markov partition, and simplicity criterion.
result In a C1-open and Ck-dense set of Anosov flows, all Lyapunov exponents have multiplicity 1. Julia accelerates machine learning in various fields with balance of efficiency and simplicity.
problem Efficiency and simplicity in machine learning algorithms.
method Developed and applied Julia language in machine learning.
result Julia balances efficiency and simplicity for machine learning.
New oriented matroids from simplex triangulations.
problem Constructing oriented matroids from triangulations.
method Using polyhedral matching fields and matroids over hyperfields.
result Generalized construction of matroids over hyperfields.
We construct invariants of four-dimensional piecewise-linear manifolds, represented as simplicial complexes, with respect to rebuildings that transform a cluster of three 4-simplices having a common two-dimensional face in a different cluster of the same type and having the same boundary. Our construction is based on t…
We propose a new embedding method which is particularly well-suited for settings where the sample size greatly exceeds the ambient dimension. Our technique consists of partitioning the space into simplices and then embedding the data points into features corresponding to the simplices' barycentric coordinates. We then …
Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
New datasets reveal neural networks can rely on simple features, leading to poor generalization.
problem Neural networks' reliance on simple features can lead to poor generalization and robustness.
method Designing datasets with varying levels of simplicity and incorporating non-robustness.
result Neural networks can exclusively rely on the simplest feature, leading to poor performance on complex data.
Geometric approach clusters intersecting manifolds with high probability.
problem Clustering intersecting d-dimensional manifolds.
method Compute locality graph on d-simplices using dihedral angles, then compute LAPD to separate manifold components.
result The method separates manifold components with high probability under random sampling.
Characterizes when differential forms have weak exterior derivatives based on limiting behavior of integration over simplices.
problem Characterizing differential forms with weak exterior derivatives.
method Uses integration over simplices to characterize the limiting behavior of differential forms.
result Proves a direct analogue of the Bourgain-Brezis-Mironescu characterization for differential forms.
Study simplicial volume via foliated simplices and duality.
problem Calculate simplicial volume using foliated simplices and duality.
method Defined real singular foliated homology, constructed foliated fundamental class, and established isometric isomorphism with measurable bounded cohomology.
result Norm of foliated fundamental class equals simplicial volume of M. Beside simplices, n-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter n-cubes are not classified. We show that there is no hyperbolic Coxeter n-cube for n≥ 6, and provide a full classification for n≤5. Our methods, which are essentially of combin…
We study the sample complexity of learning a high-dimensional simplex from a set of points uniformly sampled from its interior. Learning of simplices is a long studied problem in computer science and has applications in computational biology and remote sensing, mostly under the name of `spectral unmixing'. We theoretic…
Spectral regularization simplifies sequence models by focusing on grammatical simplicity.
problem Sequence modeling challenges in learning tasks.
method Introduces spectral regularization based on Hankel matrices and trace norm, addressing bi-infinite matrices with an unbiased estimator.
result Demonstrates spectral regularization's potential benefits on Tomita grammars.
Finite simplicial complexes dominate certain manifolds with a bounded number of simplices.
problem Understanding the finite domination of manifolds by simplicial complexes.
method Proving that a manifold can be dominated by the n-skeleton of a finite simplicial complex with a bounded number of simplices. result The total number of simplices in the n-skeleton is bounded above by a constant depending only on n and the embolic volume of the manifold. Researchers study the normalizing constant of a continuous categorical distribution.
problem Understanding the normalizing constant of the continuous categorical distribution.
method Characterize numerical behavior and present theoretical and methodological advances.
result The normalizing constant can be written in closed form using elementary functions.
When Daan Krammer and Stephen Bigelow independently proved that braid groups are linear, they used the Lawrence-Krammer-Bigelow representation for generic values of its variables q and t. The t variable is closely connected to the traditional Garside structure of the braid group and plays a major role in Krammer's alge…
Proves cup product homomorphism for bounded cohomology on negatively curved manifolds.
problem Understanding cup product behavior in bounded cohomology.
method Analyzes map Ψ* associating closed forms to bounded cohomology classes via integration.
result Proves Ψ* preserves cup product in sufficiently high degrees.
Properly embedded simplices in a convex divisible domain Ω⊂RPd behave somewhat like flats in Riemannian manifolds, so we call them flats. We show that the set of codimension-1 flats has image which is a finite collection of disjoint virtual (d−1)-tori in the compact quotient manifold. I…
In this paper we construct and study a new 15-vertex triangulation X of the complex projective plane $\CP^2$. The automorphism group of X is isomorphic to S4×S3. We prove that the triangulation X is the minimal by the number of vertices triangulation of $\CP^2$ admitting a chess colouring of four-dimens…