arXiv research

Grid homology invariant proved for lens space links.

New method constructs moduli spaces of Lagrangian surfaces in CP^2 from grid diagrams.

It will be shown that according to theorems of K. Menger, every neuron grid if identified with a curve is able to preserve the adopted qualitative structure of a data space. Furthermore, if this identification is made, the neuron grid structure can always be mapped to a subset of a universal neuron grid which is constr…

The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.

We define combinatorial invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing [OST08] and prove that they are equivalent to the invariants defined in [BVVV13] and [LOSS09]. We use these combinatorial invariants to characterize index one grid diagrams for knots …

In this paper we study topological surfaces as gridded surfaces in the 2-dimensional scaffolding of cubic honeycombs in Euclidean and hyperbolic spaces.

Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian representatives of K in (S^3, ξ_\st), where ξ_\st is the standard, tight contact structure.…

The paper defines grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology.

Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.

Researchers find a Steenrod square for link Floer homology.

This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.

The paper studies how grid cell patterns emerge in neural networks.

We present a sparse grid high-order alternating direction implicit (ADI) scheme for option pricing in stochastic volatility models. The scheme is second-order in time and fourth-order in space. Numerical experiments confirm the computational efficiency gains achieved by the sparse grid combination technique.

Study on Matérn covariance approximations on grids, finding issues with high-frequency aliasing.

Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.

M-CaStLe discovers causal structures in multivariate space-time data.

A new method uses SVM classification to efficiently compute confidence sets.

Paper tackles RL for power grid topology optimization.

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Moti…

Decades of research on the neural code underlying spatial navigation have revealed a diverse set of neural response properties. The Entorhinal Cortex (EC) of the mammalian brain contains a rich set of spatial correlates, including grid cells which encode space using tessellating patterns. However, the mechanisms and fu…

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics which are isometric along asymptotic lines and …

We present a combinatorial proof for the existence of the sign refined grid homology in lens spaces, and a self contained proof that $\partial_{\mathbb{Z}}^2 = 0$. We also present a Sage program that computes $\widehat{\mathrm{GH}} (L(p,q),K;\mathbb{Z})$, and provide empirical evidence supporting the absence of torsion…

For low-dimensional data sets with a large amount of data points, standard kernel methods are usually not feasible for regression anymore. Besides simple linear models or involved heuristic deep learning models, grid-based discretizations of larger (kernel) model classes lead to algorithms, which naturally scale linear…

Computes elastic grids that approximate 3D surfaces without physical simulations.

New method solves elliptic equations on manifolds without grids.

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

Minimal grid diagrams found for 13-crossing prime knots.

This paper improves MADDPG's performance in discrete grid-world scenarios.

Context: One of the black arts of data mining is learning the magic parameters which control the learners. In software analytics, at least for defect prediction, several methods, like grid search and differential evolution (DE), have been proposed to learn these parameters, which has been proved to be able to improve t…

Proposes a new convolutional neural network for non-grid data.

A simpler edge-based discretization method without dual volumes.

We introduce a model based off-the-grid image reconstruction algorithm using deep learned priors. The main difference of the proposed scheme with current deep learning strategies is the learning of non-linear annihilation relations in Fourier space. We rely on a model based framework, which allows us to use a significa…

Half grid diagrams prove every link can be represented by a special type of grid diagram.

Grid homology confirms the Upsilon invariant in knot theory.

GridPyM handles grid diagrams for knot theory.

Grid homology theory for spatial graphs extends skein sequence.

Improved tracking of tangled point sources using Riemannian metrics.

Extends knot invariant to filtered grid complexes.

New method finds grid diagrams for many fibered knots.

Grid homology properties for MOY graphs studied.

New trading strategy beats traditional grid in crypto markets.

We suggest a novel method of clustering and exploratory analysis of temporal event sequences data (also known as categorical time series) based on three-dimensional data grid models. A data set of temporal event sequences can be represented as a data set of three-dimensional points, each point is defined by three varia…

Develops equivariant grid homology for strongly invertible knots.

New method uses tensor networks to price multi-asset options efficiently.

Computes homology of an obstruction chain complex in grid homology.

Grid homology shows knot unknotting lower bound.

SKI accelerates GP inference with sparse grids to handle higher dimensions.

The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.