Random extrapolation speeds up coordinate descent for sparse and dense data.
problem Efficiently solving primal-dual coordinate descent for sparse and dense data.
method Adapts to sparsity and uses large step sizes for dense data, proving linear convergence under metric subregularity.
result Linear convergence under metric subregularity and optimal sublinear convergence rates in general convex-concave problems.
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
Maximal representations in symplectic lattices proven for most cases.
problem Understanding maximal representations in symplectic lattices.
method Analyzing mapping class group orbits and continuous deformations of maximal diagonal representations.
result Proof of maximal representations in most lattices of Sp(2n,R).
In each manifold M modeled on a finite or infinite dimensional cube [0,1]n we construct a closed nowhere dense subset S⊂M (called a spongy set) which is a universal nowhere dense set in M in the sense that for each nowhere dense subset A⊂M there is a homeomorphism h:M→M such that $h(A)\sub…
Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.
problem Characterizing dense points in the Bers boundary of Teichmüller space for infinite-type surfaces.
method Quasiconformal deformations and analytic conditions.
result Maximal cusps cannot approach certain points on the Bers boundary.
The paper bounds crossing numbers of dense graphs on surfaces.
problem Estimating the minimum number of edge crossings for dense graphs on surfaces.
method Proved lower and upper bounds on crossing numbers, providing explicit families of surfaces.
result Upper and lower bounds on crossing numbers match up to constant factors.
Polynomial-time test for detecting dense subgraphs in heterogeneous networks.
problem Detecting a planted community in heterogeneous networks.
method Proposes a polynomial-time test with a standard normal distribution null limiting distribution.
result The test is efficient and performs well in both simulations and real data.
Generic geodesic nets are dense in high-dimensional manifolds.
problem Density of non-closed geodesic nets in high-dimensional manifolds.
method Proving density for a generic metric on a manifold.
result Stationary geodesic nets that are not closed geodesics form a dense set.
We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing…
Detects dense subhypergraphs in random hypergraphs using low-degree polynomials.
problem Detecting a planted dense subhypergraph in a random hypergraph model.
method Degree-n^o(1) polynomials of adjacency tensor entries.
result Thresholds for detection in different density regimes.
Algorithm finds real line mapping from points under ordinal constraints.
problem Finding a mapping from points to real line under ordinal constraints.
method Approximation algorithm for dense case in O(n7)+(1/ε)O(1/ε1/8)n time. result Computes a solution satisfying (1−O(ε1/8))-fraction of all constraints. New models explain residual and dilated dense neural networks using sparse coding.
problem Lack of theoretical understanding of residual and dilated dense neural networks.
method Proposed Res-CSC and MSD-CSC models, derived mathematical relationships, implemented ISTA.
result Mathematical understanding of residual and dilated dense neural networks.
Generic Hitchin representations generate dense subgroups.
problem Understanding dense subgroups in SL_n(R) representations.
method Using a theorem by Rapinchuk, Benyash-Krivetz, and Chernousov.
result Generic Hitchin representations are strongly dense.
New lattices in higher dimensions have dense surface subgroups.
problem Finding dense subgroups in higher-dimensional arithmetic lattices.
method Exhibited nonuniform arithmetic lattices in SO(n,1).
result Contain Zariski-dense surface subgroups.
Study on horospheres in higher rank homogeneous spaces, proving density properties.
problem Density of horospheres in higher rank homogeneous spaces.
method Analyzing maximal horospherical subgroups and their minimal subsets in the context of Furstenberg boundary.
result Equivalence of horospherical limit points and density properties in higher rank homogeneous spaces.
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…
Sparse sampling method for tensor factorization and completion of high rank tensors.
problem Completion of high rank tensors with missing data in recommendation systems.
method Sparse measurements and message-passing algorithms in a high-dimensional limit.
result Theoretical insights and performance analysis of tensor factorization in dense limit.
New method estimates precision matrices without models, achieving dense, consistent, and model-free properties.
problem Lack of methods that are dense, consistent, and model-free for precision matrix estimation.
method General class of estimators that unify dense, consistent, and model-free properties within a nonasymptotic framework.
result Ridgeless regression exhibits the double descent phenomenon, establishing a precision matrix analogue to linear regression's double descent.
A model of associative memory is studied, which stores and reliably retrieves many more patterns than the number of neurons in the network. We propose a simple duality between this dense associative memory and neural networks commonly used in deep learning. On the associative memory side of this duality, a family of mo…
Generative models can still learn from contaminated data, but with limitations.
problem How much contamination can generative models tolerate?
method Characterized robustness under contaminated enumerations, proving generation is achievable for all countable collections if contamination fraction converges to zero.
result Generation under contamination is achievable for all countable collections if contamination fraction converges to zero, but dense generation is strictly less robust.
In this article, we study complete pseudo-Riemannian manifolds whose cone admits a parallel symmetric 2-tensorfield. The situation splits in three cases: nilpotent, decomposable or complex Riemannian. In the complex Riemannian and decomposable cases we provide a classification. In the nilpotent case, we are able to des…
We show that discrete synaptic weights can be efficiently used for learning in large scale neural systems, and lead to unanticipated computational performance. We focus on the representative case of learning random patterns with binary synapses in single layer networks. The standard statistical analysis shows that this…
Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of…
Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.
problem Characterizing dense subgroups of algebraic groups.
method Bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups.
result No C1-smooth slim limit set for higher rank semisimple algebraic groups. We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for approximating the partition function are Markov Chain Monte Carlo and Variational Method…
The study finds conditions for certain groups to be dense in a specific mathematical space.
problem Conditions for linear reflection groups to be dense in a projective space.
method Analyzes necessary and sufficient conditions for Zariski-density, applies to Coxeter groups and surface subgroups.
result Establishes conditions for Zariski-dense subgroups in SLn(Z) for various n. The Greenberg-Shalom hypothesis connects subgroup properties to lattice structures in Lie groups.
problem Understanding subgroup properties in Lie groups and their implications.
method Analyzing infinite discrete subgroups of semisimple Lie groups and their commensurators.
result An infinite discrete subgroup of a semisimple Lie group with a dense commensurator is a lattice in a product of some factors.
New representations of hyperbolic 3-manifold groups into larger groups.
problem Finding representations of hyperbolic 3-manifold groups into larger matrix groups.
method Holonomy representations from projective deformations of hyperbolic structures.
result First examples of strongly dense representations into SL(4,R) and SU(3,1). In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, …
Detects dense subhypergraphs in heterogeneous random hypergraphs.
problem Testing for the existence of a dense subhypergraph in heterogeneous random hypergraphs.
method Established detection boundaries and constructed asymptotically powerful and adaptive tests.
result Developed tests for distinguishing between null and alternative hypotheses.
Dense neural networks can't approximate all functions.
problem Approximation capabilities of dense neural networks.
method Model compression approach combining weak regularity lemma and graph neural networks.
result Existence of Lipschitz continuous functions not approximable by dense neural networks.
The key idea of current deep learning methods for dense prediction is to apply a model on a regular patch centered on each pixel to make pixel-wise predictions. These methods are limited in the sense that the patches are determined by network architecture instead of learned from data. In this work, we propose the dense…
Odd-dimensional SL(n,Q) contains dense surface subgroups.
problem Finding dense subgroups in SL(n,Q) for odd n.
method Constructing a continuous path of representations.
result Existence of dense surface subgroups in SL(n,Q) for odd n.
Classifies manifolds with dense conjugacy classes in their mapping class groups.
problem Classifying manifolds based on conjugacy classes in their mapping class groups.
method Analyzing connected orientable 2-manifolds and their mapping class groups.
result Mapping class groups of certain manifolds have dense conjugacy classes.
The paper finds dense subgroups in certain Lie groups.
problem Finding dense subgroups in Lie groups.
method Constructing dense surface subgroups in specific Lie groups.
result Uniform lattices contain infinitely many dense Hitchin representations.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
This paper shows stable mappings are never dense on non-compact manifolds.
problem Density of stable mappings on non-compact manifolds.
method Complementing Mather's theory, proving stability results for non-compact manifolds.
result The set of stable mappings is never dense on non-compact manifolds.
The complexity of horizontality in twistor spaces on tori is infinite.
problem Complexity of horizontality in twistor spaces on tori.
method Analyzing the complexity of horizontality in the twistor space associated with an oriented vector bundle over a torus.
result The complexity of horizontality in the twistor space is expressed by a dense subset of S2 when it is infinite. We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian manifold.
Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.
problem Dynamics of isoperiodic leaves in rank 1 affine invariant suborbifolds.
method Defines foliation FM and establishes density criterion.
result Establishes criterion for density of isoperiodic leaves.
Sharp boundaries for detecting dense subhypergraphs established.
problem Detecting dense subhypergraphs in random hypergraphs.
method Established sharp detection boundaries for known and unknown edge probabilities.
result Sharp detectable regions differ significantly from graph counterparts.
Method generates dense fields from sparse measurements without needing spatial statistics or examples.
problem Generating dense physical fields from sparse measurements.
method Introduces a differentiable numerical simulator into neural network training.
result Superior results on fluid mechanics problems compared to statistical and neural network methods.
Oja's algorithm has been the cornerstone of streaming methods in Principal Component Analysis (PCA) since it was first proposed in 1982. However, Oja's algorithm does not have a standardized choice of learning rate (step size) that both performs well in practice and truly conforms to the online streaming setting. In th…
Deforms surface groups to be Zariski dense in SL(n,R)
problem Finding Zariski dense surface groups in SL(n,R)
method Deforming K-integral representations of surface groups result Generalizes Long and Thistlethwaite's method to SL(n,R)
Paper tackles dense subgraph discovery with noisy feedback.
problem Discover dense subgraphs in edge-weighted graphs with noisy feedback.
method Proposes polynomial-time and scalable algorithms for dense subgraph discovery.
result Polynomial-time algorithm obtains nearly-optimal solution with high probability.
Wave fronts on certain surfaces become dense.
problem Density of wave fronts on surfaces.
method Proof of density for specific surfaces.
result Wave fronts become dense on flat torus, square billiard, Klein bottle, and cube surface.
Example shows dense subgroup of SL5(Z) not finitely presented.
problem Finding dense subgroups of SL5(Z) that are not finitely presented.
method Discussing an example of a Zariski-dense finitely generated subgroup of SL5(Z).
result Example shows a subgroup that is dense but not finitely presented.
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.