Counting tripods on a flat torus using lattice point counting.
problem Counting finite BPS webs in flat torus geometry.
method Lattice point counting techniques in .
result Asymptotic counting result for tripods on the torus.
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48 results for “Lattice Problems”
Proves a lattice version of the Atiyah-Singer index theorem.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a -theoretic formula for an index-type invariant.
result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
Backward stochastic difference equations on lattices with application to market equilibrium analysismath.PR
Study BSΔE on lattices for asset price analysis.
problem Optimal investment and market equilibrium analysis in asset price models.
method Backward stochastic difference equations on lattices.
result Applications to optimal investment and market equilibrium analysis.
L-CNNs learn gauge invariant quantities on lattices.
problem Learning gauge invariant quantities on lattices.
method Novel convolutional layer preserving gauge equivariance and forming Wilson loops.
result L-CNNs can approximate any gauge covariant function on the lattice.
L-CNNs preserve gauge symmetry in lattice simulations.
problem Breaking gauge symmetry in neural network models.
method Lattice gauge equivariant convolutional neural networks (L-CNNs).
result L-CNNs represent gauge invariant functions on the lattice.
SLR tackles sparse linear regression problems, showing hardness for efficient algorithms.
problem Sparse linear regression with noisy data and k-sparse solutions.
method Reduction from lattice problems to SLR instances, showing hardness.
result Hardness of SLR instances, even for isotropic Gaussian design matrices.
The paper proposes a method to sample quantum field configurations using neural operators and flows.
problem Sampling lattice field configurations from Boltzmann distributions in quantum field theories.
method Approximating a time-dependent neural operator to map between free and target theories, discretizing to a normalizing flow, and training to diffeomorphism.
result The method can generalize to larger lattice sizes when pre-trained on smaller ones, improving efficiency.
Study on tilings of the plane with two types of tiles of varying areas.
problem Classifying tilings with minimal interface length.
method Analysis of isoperimetric configurations for different lattice types and tile areas.
result Three distinct tilings configurations found based on tile area ratio.
Tropical Geometry and Piecewise-Linear Approximation of Curves and Surfaces on Weighted Latticescs.LG
Tropical geometry and weighted lattices improve curve and surface fitting.
problem Fitting max- tropical curves and surfaces to data.
method Max- algebra, weighted lattices, morphological adjunctions.
result Optimal piecewise-linear regression for max- curves and surfaces.
Continuous LWEcs.CC
Introduces a continuous version of LWE problem.
problem Hardness of learning mixtures of Gaussians.
method Polynomial-time quantum reduction from CLWE to lattice problems.
result CLWE shares hardness with LWE.
L-CNNs preserve gauge symmetry in neural networks.
problem Applying machine learning to lattice gauge theory while preserving gauge symmetry.
method L-CNNs use gauge equivariance to construct a gauge equivariant convolutional layer and bilinear layer.
result L-CNNs achieve higher accuracy in non-linear regression tasks compared to non-equivariant CNNs.
Novel method for learning Gaussian graphical models from paired data.
problem Learning Gaussian graphical models for dependent groups.
method Introducing twin order to explore the search space more efficiently.
result The twin order makes the model space a distributive lattice, leading to more efficient model exploration.
New property identifies arithmetic lattices from nonuniform lattices.
problem Characterizing arithmetic lattices among nonuniform lattices.
method Introduced Bounded Clustering (B-C) property.
result B-C property uniquely identifies arithmetic lattices.
The paper defines and studies discrete p-density and compression-radius profiles of lattice knots.
problem Understanding geometric properties of lattice knots.
method Develops a framework for discrete p-density and compression-radius profiles of lattice knots, studying them on length-filtered sets and finite move-graph exploration.
result Density and compression-radius values are not monotone, illustrating distinct optimization problems.
Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.
problem Determining the stick number and edge length of knots in a hexagonal lattice.
method Introducing a linear transformation between lattices to prove strict inequalities and classifying knots.
result Only trefoil and figure-eight knots are 11-stick knots in the hexagonal lattice.
We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology o…
We present an intriguing question about lattice points in triangles where Pick's formula is "almost correct". The question has its origin in knot theory, but its statement is purely combinatorial. After more than 30 years the topological question was recently solved, but the lattice point problem is still open.
Arithmetic groupsmath.GT
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
We give a simple example showing that a knot or link diagram that lies in the lattice is not necessarily the projection of a lattice stick knot or link in the lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the lat…
GPU-accelerated particle methods outperform neural samplers in LFT benchmarks.
problem High-dimensional multimodal sampling problems in lattice field theory.
method GPU-accelerated particle Monte Carlo methods (Sequential Monte Carlo and nested sampling).
result These methods match or outperform neural samplers in sample quality and wall-clock time.
We find coordinates, the metric tensor, the inverse metric tensor and the Laplace-Beltrami operator for the orbit space of Hamiltonian SU(2) gauge theory on a finite, rectangular lattice. This is done using a complete axial gauge fixing. The Gribov problem can be completely solved, with no remaining gauge ambiguities.
We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature . We assume in addition that the Lie groups under consideration are simply-connected, indecomposable and solvable. Then their centre is one- or two-dimensional. In both cases, a parametrisation of the set of such L…
New rigidity theorem for product of lattices.
problem Understanding quasi-isometry of product lattices.
method Demonstrated rigidity for product of non-uniform rank one lattice and nilpotent lattice.
result Any quasi-isometric group is an extension of a non-uniform rank one lattice by a nilpotent lattice.
We explore hybrid subgroups of certain non-arithmetic lattices in . We show that all of Mostow's lattices are virtually hybrids; moreover, we show that some of these non-arithmetic lattices are hybrids of two non-commensurable arithmetic lattices in .
This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…
The paper finds incommensurable lattices in complex models of Baumslag-Solitar groups.
problem Locally finite 2-complexes and their automorphism groups contain incommensurable lattices.
method Constructing lattices in combinatorial models of Baumslag-Solitar groups and analyzing their properties.
result The constructed lattices are incommensurable and have specific properties like isomorphic Cayley graphs.
Comparison of Lattice-Free and Lattice-Based Sequence Discriminative Training Criteria for LVCSReess.AS
Sequence discriminative training criteria have long been a standard tool in automatic speech recognition for improving the performance of acoustic models over their maximum likelihood / cross entropy trained counterparts. While previously a lattice approximation of the search space has been necessary to reduce computat…
Vertex distortion measures how far lattice knots deviate from straight lines.
problem Measuring how much lattice knots deviate from straight paths.
method Analogous to smooth knots, study vertex distortion in lattice knots.
result Vertex distortion is 1 only for the unknot and can be arbitrarily high.
Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…
In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n,1) it …
We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the ra…
The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.
problem Residual finiteness of lattices in and existence of smooth projective surfaces.
method Proved residual finiteness of certain lattices and constructed surfaces using central extensions.
result First examples of residually finite lattices in and construction of surfaces with specific fundamental groups.
New method learns dependencies in high-dimensional data without graph assumptions.
problem Learning dependencies in nonparametric and high-dimensional settings.
method Neighbourhood lattice decomposition for nonparametric CI learning.
result Compact, non-graphical representation of CI exists in any graphical model.
We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with…
A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…
Deriving option prices from operational-time Markov lattices
problem Option pricing
method Operational-time Markov lattice
result Derives option-pricing equations from an operational-time Markov lattice
In this article we describe cell decompositions of the moduli space of Riemann surfaces and their relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational convex polytopes which come equipped with natural integer points and a volume form. …
The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…
New method finds lattice polygons that can be dissected into triangles with integer areas.
problem Finding lattice polygons that can be dissected into triangles with integer areas.
method A new version of Sperner's Lemma.
result Simple and complete description of lattice polygons that can be dissected into triangles with integer areas.
We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenki…
L-CNNs approximate gauge actions, revealing fixed points with no lattice artifacts.
problem Approximating gauge actions with lattice artifacts.
method Lattice gauge-equivariant convolutional neural networks (L-CNNs).
result L-CNNs provide fixed point actions with no lattice artifacts.
The paper refines transformations of lattice diagrams and introduces dotted diagrams.
problem Investigating transformations and deformations of lattice diagrams and their associated dotted diagrams.
method Introducing dotted diagrams and investigating deformations of these diagrams, relating them to transformations of lattice diagrams.
result Refined results on the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams.
Classifies knots by lattice size, finding unknot ratios and crossing numbers.
problem Understanding the distribution of knots within different lattice sizes.
method Introduced a new knot classification by lattice size, analyzed ratios of unknots and knots with more than 10 crossings, and compared with theoretical estimates.
result Ratio of unknots decreases exponentially with lattice size, and computational results match theoretical estimates.
Let be a simply connected, solvable Lie group and a lattice in . The deformation space is the orbit space associated to the action of $\Aut(G)$ on the space of all lattice embeddings of into . Our main result generalises the classical rigidity theorems of Mal'tsev…
Proves effective slope gaps for lattice surfaces.
problem Proving effective slope gaps for lattice surfaces.
method Proves effective slope gap distribution for square torus and general lattice surfaces.
result Effective slope gap distribution result for lattice surfaces.
Optimal smooth subspaces approximate large data sets efficiently.
problem Approximating large data sets with invariant subspaces.
method Smooth functions under lattice translations or crystallographic groups, with optimal selection of Paley-Wiener space.
result Optimal lattice selection enhances approximation efficiency.
We study the arithmeticity of the Couwenberg-Heckman-Looijenga lattices in PU(n,1), and show that they contain a non-arithmetic lattice in PU(3,1) which is not commensurable to the non-arithmetic Deligne-Mostow lattice in PU(3,1).
Hardness proof for agnostically learning halfspaces from worst-case lattice problems.
problem Agnostically learning halfspaces in the presence of noise.
method Reduction to worst-case lattice problems (GapSVP, SIVP).
result No efficient algorithm can achieve misclassification error better than 1/2 - γ under given hardness assumptions.