New method estimates spatial weights matrix for lattice data, improving prediction accuracy.
problem Estimating spatial dependence structure for regular lattice data.
method Adaptive lasso with cross-sectional resampling to estimate sparse spatial weights matrix.
result Improves prediction accuracy of nitrogen dioxide concentrations.
Researchers calculate complexity of billiard paths in regular polygons.
problem Calculating the complexity of billiard paths in regular polygons.
method Counting saddle connections on lattice surfaces, focusing on combinatorial length.
result They answered a question about billiard language complexity in regular polygons.
We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…
A new surgery formula for knot lattice homology.
problem Developing a new surgery formula for knot lattice homology.
method Provided an iterable version of the surgery formula using doubly-filtered spaces and involutive data.
result Computed knot lattice spaces for specific knots and three-manifolds.
The Phi- relationship also known as Phi-factor appears in a number of lattice structures, mostly considering the lines within several separate circles or polygons. The paper considers a regular hexagonal tessellation as a lattice with the highest specific mechanical stiffness.
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…
Study Poisson boundaries of building lattices and generalize rigidity results.
problem Understanding Poisson boundaries of building lattices and their rigidity properties.
method Proved Poisson boundaries and used them to generalize rigidity results.
result Generalized rigidity results for morphisms and cocycles from lattices in buildings to groups with negative curvature.
Sharp bounds for spanning tree entropy in planar lattices.
problem Estimating spanning tree entropy in planar lattice graphs.
method Using hyperbolic geometry and polyhedra volumes.
result Proved bounds are easy to compute and provide excellent estimates.
In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differen…
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.
We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross polytopes. In the 2-dimensional case, the lattice is regular and it incorporates dyna…
We study cocompact lattices with dense projections in a product G1×G2 of locally compact groups and show, under the assumption that each Gi is a closed subgroup of the automorphism group Aut(Ti) of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only…
We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Briesk…
Deep Learning methods, specifically convolutional neural networks (CNNs), have seen a lot of success in the domain of image-based data, where the data offers a clearly structured topology in the regular lattice of pixels. This 4-neighbourhood topological simplicity makes the application of convolutional masks straightf…
Study flat metrics from right prisms, finding non-lattice surfaces with translation coverings.
problem Analyzing flat metrics from right regular prisms.
method Viewing prisms as n-differentials and analyzing unfoldings, proving translation coverings to hyperelliptic surfaces.
result Non-lattice surfaces admit translation coverings to hyperelliptic surfaces, allowing explicit computation of orbit closures and counting problems.
Investigates methods to regularize quantile regression for accurate predictions.
problem Improving accuracy and fairness in quantile regression predictions.
method Various regularization techniques including expected pinball loss, monotonicity constraints, and rate constraints.
result Deep lattice networks can maintain non-crossing quantiles and improve calibration and fairness.
Regular subgroups of SL3(R) are identified and ruled out.
problem Identifying and characterizing regular subgroups of SL3(R).
method Using Kapovich–Leeb–Porti and Guichard–Wienhard divergent subgroups criteria, and Oh's results.
result Regular subgroups of SL3(R) are precisely lattices in minimal horospherical subgroups.
Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.
problem Proving Doyle conjecture for hexagonal lattice circle packings.
method Using Liouville theorem of discrete harmonic functions based on logarithmic radii ratio observation.
result Proves rigidity of Doyle spirals in hexagonal lattice circle packings with bounded radii ratios.
We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the h…
This paper solves PDEs for embedding discrete lattices into smooth manifolds.
problem Embedding discrete lattices into smooth manifolds while preserving geometric and topological properties.
method Rigorous mathematical framework and analysis of partial differential equations (PDEs).
result Existence and regularity of solutions to PDEs under initial boundary conditions.
Paper recovers lattice signal partitions efficiently.
problem Estimating lattice partition from noisy data.
method Uses dyadic CART for computationally-efficient partition recovery.
result Consistently estimates partition with optimal error rate.
The paper proves actions of lattices in higher rank groups have cost one.
problem Fixed price question for higher rank semisimple Lie groups.
method Low intensity Poisson point processes and geometry of Voronoi tessellations.
result Proves all probability measure preserving actions of lattices in higher rank groups have cost one.
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…
Improved speech recognition for voice assistants by analyzing speech data.
problem Reducing false triggers in speech-enabled assistants.
method Post-processing LVCSR hypothesis lattice with a Bidirectional Lattice Recurrent Neural Network (LatticeRNN).
result LatticeRNN significantly improves detection accuracy over traditional methods.
New proof for symmetric spaces with rectangular lattices.
problem Characterizing symmetric spaces with rectangular unit lattices.
method Explicit construction of isometric embeddings and analysis of root systems.
result Symmetric spaces with rectangular unit lattices are symmetric R-spaces.
Let (W,S) be a Coxeter system with Davis complex Σ. The polyhedral automorphism group G of Σ is a locally compact group under the compact-open topology. If G is a discrete group (as characterised by Haglund--Paulin), then the set Vu(G) of uniform lattices in G is discrete. Whether the converse i…
We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n greater or equal to 8. These families give rise to expl…
LatticeNet segments 3D point clouds faster and more efficiently.
problem Challenges in applying CNNs to 3D point cloud data.
method Embeds point cloud geometry into a permutohedral lattice for fast convolutions.
result Achieves state-of-the-art performance in 3D segmentation.
Study finds the spectrum of a cubic Dirac operator on specific oscillator group manifolds.
problem Determining the spectrum of a cubic Dirac operator on oscillator group manifolds.
method Explicit decomposition of the regular representation and calculation of eigenspaces.
result Explicit eigenspaces and spectrum of the cubic Dirac operator determined.
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.
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. The study examines knot probabilities in confined lattice polygons.
problem Determining the relative knotting probabilities in confined lattice knots.
method Used Monte Carlo algorithms to enumerate conformations of lattice knots in a confined volume.
result Relative knotting probabilities are small, with the model dominated by unknots.
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 risk measures for incomplete markets without lattice structures.
problem Risk measures on incomplete markets without lattice structures.
method Study of risk measures without lattice structures, focusing on tractable dual representations and solid superspaces.
result Existence of a tractable dual representation equivalent to a Fatou-like property, and extension theorems under certain conditions.
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 d variables. New subgroup found in Lie groups with unusual properties.
problem Finding discrete subgroups with specific properties in Lie groups.
method Constructing a specific subgroup of a higher rank Lie group.
result Found a new subgroup that is dense, discrete, non-lattice, and non-tempered.
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 provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the thickened torus. A corollary is that the spanning tree entropy for each regular pla…
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 study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
Research on graph representation learning has received a lot of attention in recent years since many data in real-world applications come in form of graphs. High-dimensional graph data are often in irregular form, which makes them more difficult to analyze than image/video/audio data defined on regular lattices. Variou…
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.
The paper analyzes the spectra of compact quotients of the oscillator group.
problem Computing spectra of compact solvmanifolds.
method Classification of lattices, decomposition of representations, explicit computation of spectra.
result Explicit computation of the spectrum of the wave operator on compact locally-symmetric Lorentzian manifolds.
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…
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.
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 Z2 lattice is not necessarily the projection of a lattice stick knot or link in the Z3 lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the Z2 lat…
This paper classifies commensurability of Deligne-Mostow lattices.
problem Classifying commensurability among Deligne-Mostow lattices.
method Geometric and algebraic approaches to commensurability relations.
result 104 Deligne-Mostow lattices form 38 commensurability classes.