arXiv research

Proof confirms perfect representation in deep learning models.

New theory explains how self-supervised learning converges, advancing AI research.

New mixed-platonic 3-manifolds from different polyhedra types.

We show that if a cusped hyperbolic manifold is Platonic, i.e., can be decomposed into isometric Platonic solids, it can also be decomposed into geodesic ideal tetrahedra.

We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedra…

Skeleta of Platonic solids are factored into spheres.

Constant Mean Curvature n-noids with Platonic Symmetries

The paper explains emergent phenomena in deep learning using entropic forces.

We borrow a classical construction from the study of rational billiards in dynamical systems known as the "unfolding construction" and show that it can be used to study the automorphism group of a Platonic surface. More precisely, the monodromy group, or deck group in this case, associated to the cover of a regular pol…

We interpret Coxeter's truncated braid groups in terms of Platonic solids.

We construct new embedded self-shrinkers of genus 3, 5, 7, 11 and 19 using variational methods. Our self-shrinkers resemble doublings of the Platonic solids and were discovered numerically by D. Chopp in 1994.

A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of…

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representati…

The problem of classifying, upto isometry (or similarity), the orientable spherical, Euclidean and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. In the spherical and hyperbolic cases, this allows us to complete the classification begun by…

Study of translation covers of platonic solids reveals monodromy group structures.

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology…

We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of p…

Minimal surfaces span periodic curves in 3D space.

New periodic polyhedra found in curved spaces.

A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is …

Develops a stochastic approach to financial market delays.

Extends linear representation hypothesis to categorical and hierarchical concepts in LLMs.

We study the strong predictable representation property in filtrations initially enlarged with a random variable L. We prove that the strong predictable representation property can always be transferred to the enlarged filtration as long as the classical density hypothesis of Jacod (1985) holds. This generalizes the ex…

Hypothesis identifies useful transformations for self-supervised learning.

In this paper, we will construct an example of a closed Riemann surface $X$ that can be realized as a quotient of a triply periodic polyhedral surface $Π\subset \mathbb{R}^3$ where the Weierstrass points of $X$ coincide with the vertices of $Π.$ First we construct $Π$ by attaching Platonic solids in a periodic manner a…

STRAND: A single representation for hypothesis testing and vectorisation of persistence diagrams

A framework for hypothesis testing on attributed graphs using sampling.

Cyclification of orbifolds explained in cohesive higher topos theory.

If all but two vertices of a triangulated sphere have degrees divisible by $k$, then the exceptional vertices are not adjacent. This theorem is proved for $k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can asso…

Enhances RL agents with predictive internal representations.

The paper defines a hypothesis space for deep learning using DNNs.

MSA compares neural representations' intrinsic geometry for better understanding.

Method measures weight similarity in neural networks using normalization and statistical inference.

Differential privacy is a statistical concept that can be explained through hypothesis testing.

IMA addresses non-identifiability in nonlinear ICA by assuming orthogonal Jacobian columns.

Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than $2-$Sphere. We classify some semi equivelar maps on surface of Euler characteristic -1 and show that none of these are vertex transitive. We establish existence of 12-covered triangula…

Deep learning's anomalous generalization explained by standard frameworks.

The paper explores linear representations in language models using counterfactuals.

Probably the most important problem in machine learning is the preliminary biasing of a learner's hypothesis space so that it is small enough to ensure good generalisation from reasonable training sets, yet large enough that it contains a good solution to the problem being learnt. In this paper a mechanism for {\em aut…

The paper refutes the manifold hypothesis for image data and proposes the union of manifolds hypothesis.

This work lists and describes the main recent strategies for building fixed-length, dense and distributed representations for words, based on the distributional hypothesis. These representations are now commonly called word embeddings and, in addition to encoding surprisingly good syntactic and semantic information, ha…

Let $Σ_g$ be a compact, connected, orientable surface of genus $g \geq 2$. We ask for a parametrization of the discrete, faithful, totally loxodromic representations in the deformation space ${\rm Hom}(π_1(Σ_g), {\rm SU}(3,1))/{\rm SU}(3,1)$. We show that such a representation, under some hypothesis, can be determined …

A classical result of H. S. M. Coxeter asserts that a certain quotient $B(m,n)$ of the braid group $B(m)$ on $m$ strands is finite if and only if $(m,n)$ corresponds to the type of one of the five Platonic solids. If ${\bf k}$ is a knot or virtual knot, one can study similar quotients $G({\bf k}, n)$ for the correspond…

The Familiarity Hypothesis explains deep open set methods' success in detecting novel objects.

New method reduces bias in NLI models using ensemble adversarial training.

When analyzing empirical data, we often find that global linear models overestimate the number of parameters required. In such cases, we may ask whether the data lies on or near a manifold or a set of manifolds (a so-called multi-manifold) of lower dimension than the ambient space. This question can be phrased as a (mu…

Proposes iCaRL for nonlinear OOD generalization in causal settings.

Unified theory for representation learning using learnable functions.