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…
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48 results for “distributional lattices”
Distributional lattices on Riemannian symmetric spaces are studied, leading to new insights on random walks.
problem Understanding distributional lattices on Riemannian symmetric spaces.
method Introduced distributional lattices, used amenability equivalence, and developed graph speed for Poisson-Voronoi tessellations.
result Simple random walk on distributional lattices in nonamenable spaces has positive embedded speed.
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.
New MCMC method samples from lattice distributions efficiently.
problem Sampling from probability distributions on lattice structures.
method Metropolis-Hastings algorithm with a pull-back measure.
result The method is uniformly ergodic under certain conditions.
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.
The study examines Teichmüller distances in lattices and punctured tori.
problem Distribution of Teichmüller distances in moduli spaces.
method Uniform distribution of lattices, calculation of Teichmüller distances.
result Identifies distribution of distances in Teichmüller space.
New method for geodesics of multivariate normals, derived from a Toda lattice.
problem Computing geodesics of multivariate normal distributions.
method Using block Cholesky decomposition and a natural Riemannian submersion, a new Toda lattice type Lax pair is derived.
result A new Toda lattice type Lax pair derived from geodesics and block Cholesky decomposition.
New method uses neural maps to efficiently sample lattice QCD distributions.
problem Challenges in sampling Boltzmann distributions of lattice field theories.
method Sparse triangular transport maps exploiting conditional independence structure of lattice graphs.
result Sparse triangular maps achieve efficient sampling with linear time complexity in lattice size.
Generalizes Kauffman's clock theorem to surfaces.
problem Proving a lattice structure on graph states in various surfaces.
method Using matchings and graph orientations, extending Propp's results.
result Two generalizations of Kauffman's theorem for more surfaces.
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…
A new method called MCLMC avoids dissipation in sampling from canonical distributions.
problem Sampling from canonical distributions without dissipation.
method Microcanonical Langevin Monte Carlo (MCLMC) as a dissipation-free system of SDE.
result MCLMC converges faster than HMC for lattice φ^4 models.
Finitary estimates for the distribution of lattice orbits in homogeneous spaces I: Riemannian metricmath.DS
Effective estimates for lattice orbits in homogeneous spaces.
problem Distribution of lattice orbits in homogeneous spaces.
method Refined techniques on equidistribution of regions under flows.
result Effective convergence of orbit distribution to a limiting density.
Researchers compute gap distributions for saddle connection directions on specific translation surfaces.
problem Computing gap distributions for saddle connection directions on translation surfaces.
method Translation to dynamical question of return times to a transversal under the horocycle flow.
result Gap distributions have support at 0 and quadratic tail decay.
Paper solves NGCA for discrete distributions using LLL method.
problem Learning hidden non-Gaussian components in discrete distributions.
method Utilizes LLL lattice basis reduction method.
result Sample and computationally efficient algorithm for NGCA in discrete distributions.
Machine learning classifies topological phases in leaky photonic lattices.
problem Classifying topological phases in leaky photonic lattices using limited data.
method A fully connected neural network trained on bulk intensity measurements.
result Accurate determination of topological properties from intensity distributions.
We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of…
Constructs continuous analogues for binomial and Catalan numbers.
problem No specific problem stated; focuses on mathematical construction.
method Uses techniques from convex polytopes, lattice paths, and directed manifolds.
result Develops a continuous analogue for the binomial distribution.
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.
Improved sampling for gauge theory with SNFs.
problem Sampling from target probability distributions in gauge theory.
method Stochastic Normalizing Flows (SNFs) for lattice gauge theory.
result Promising scaling properties of SNFs with degrees of freedom.
Paper improves distributed mean estimation and variance reduction without relying on input norm.
problem Distributed mean estimation and variance reduction with large input norms.
method Quantization and lattice theory connection for improved error bounds.
result Output error bounds depend only on input distance, not norm.
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.
We explore a simple lattice field model intended to describe statistical properties of high frequency financial markets. The model is relevant in the cross-disciplinary area of econophysics. Its signature feature is the emergence of a self-organized critical state. This implies scale invariance of the model, without tu…
Random neural networks mapped to statistical physics models.
problem Design questions about neural networks.
method Mapped random neural networks to lattice models in statistical physics.
result Large scale behavior of random neural networks approximated by effective field theory.
Random knotting in very long off-lattice self-avoiding polygonscond-mat.stat-mech
Study on knotting in very long polymer chains, finding Poisson distribution for prime knot types.
problem Understanding knotting in very long polymer chains.
method Generated and analyzed polygons of size using tree data structure and pivot algorithm. Used new knot diagram simplification and invariant-free classification.
result Number of prime summands of knot type in a random -gon is well described by a Poisson distribution.
Hard to estimate -accurate scores without strong assumptions.
problem Estimating the score of unknown data distributions accurately.
method Reduction to generating samples and leveraging lattice-based cryptography hardness.
result Score estimation is computationally hard even with polynomial sample complexity.
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.
This paper studies posets associated with link diagrams and their algebraic properties.
problem Understanding the algebraic structure of posets derived from link diagrams.
method Associaed posets with link diagrams, proved distributivity, and described join irreducibles.
result Posets of Kauffman states are distributive lattices and isomorphic to coefficient quiver posets.
The paper explores triangulations on spheres and tori, extending clock theorems.
problem Investigating triangulations of spheres and tori with colored triangles.
method Analyzing matchings between white and black triangles, focusing on their lattices and state transitions.
result Clock theorems extend to spheres but not to tori, with different lattice structures.
New framework for detecting complex interactions in multivariate data.
problem Insufficient pairwise measures fail to capture multivariate data complexities.
method Lattice theory and operator functions to derive higher-order information-theoretic measures.
result Streitberg Information fully characterizes all interactions among variables.
Investigates stochastic networks on disordered lattices, converging to Brownian web in 2D.
problem Stochastic networks on disordered lattices.
method Directed spanning forests on randomly perturbed lattices.
result DSF converges to Brownian web in 2D under diffusive scaling.
Hybrid subgroups found in non-arithmetic PU(2,1) lattices.
problem Exploring hybrid subgroups in non-arithmetic PU(2,1) lattices.
method Exploring hybrid subgroups of certain non-arithmetic lattices in PU(2,1). Showing that Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
result Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
Data compression speeds up machine learning loss calculations.
problem Computational demand in calculating mean squared error for large datasets.
method Use rank-1 lattices to compress data, assigning weights based on original data and responses.
result Our QMC data compression algorithms can lead to arbitrary high convergence rates for smooth functions.
New non-arithmetic lattice found in PU(3,1)
problem Arithmeticity of Couwenberg-Heckman-Looijenga lattices
method Study of arithmeticity and non-arithmetic lattices in PU(n,1)
result Found a non-arithmetic lattice in PU(3,1) not commensurable to Deligne-Mostow lattice
We construct a Poincaré section for the horocycle flow on the modular surface , and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…
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.
New lattice stick knot condition identified.
problem Determining if 2D lattice knots project to 3D lattice sticks.
method Provided a necessary and sufficient condition.
result Identified a condition for 2D lattice knots to project to 3D lattice sticks.
Simplified proof for lattice link projections.
problem Necessary and sufficient condition for lattice link projections.
method Shorter and simpler proof of existing result.
result Simplified proof for lattice link projections.
Develops a new sampling method for gauge theories.
problem Sampling from gauge theories.
method Gauge-equivariant flows for variables.
result Constructs a class of flows respecting matrix conjugation symmetry.
Comparison of Lattice-Free and Lattice-Based Sequence Discriminative Training Criteria for LVCSReess.AS
This work compares lattice-free and lattice-based training criteria for LVCSR.
problem Improving acoustic model performance in speech recognition.
method Direct comparison of lattice-free and lattice-based sequence discriminative training criteria using GPU.
result Lattice-free MMI performance is comparable to lattice-based criteria, while lattice-based sMBR remains superior.
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 continue the study of the distribution of closed geodesics on nilmanifolds constructed from a simply connected 2-step nilpotent Lie group with a left invariant metric and a lattice. We consider a Lie group with an associated 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact sem…
New lattices from 4-manifolds show manifold properties.
problem Identifying distinct 4-manifolds from their lattices.
method Examined smooth, simply-connected 4-manifolds bounded by fixed homology 3-spheres.
result Found two distinct sets of Niemeier lattices.
Arithmetic groupsmath.GT
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
Upper bound for lattice stick number of spatial graphs.
problem Finding the minimum number of sticks in a cubic lattice to represent spatial graphs.
method Defined lattice stick number for spatial graphs and presented an upper bound in terms of crossing number.
result An upper bound for the lattice stick number of spatial graphs.
This article classifies reflective hyperbolic lattices of rank 4.
problem Classifying reflective hyperbolic lattices of a specific rank.
method Using automorphism groups and reflections to classify lattices.
result Complete classification of 1.2-reflective maximal anisotropic lattices of rank 4.
New thin subgroups found in special linear groups via bending techniques.
problem Finding thin subgroups of lattices in special linear groups.
method Techniques from convex projective geometry.
result Infinitely many non-commensurable lattices with thin subgroups.
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.
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.