Every discrete subset in a complex domain is in a complex curve.
problem Embedding discrete subsets in complex domains.
method Proving every closed discrete subset is in a complex curve with any topology.
result Closed discrete subsets are contained in complex curves with any topology.
Rigidity theorem for discrete metric spaces embedded in Riemannian surfaces.
problem Understanding the rigidity of discrete metric spaces embedded in Riemannian surfaces.
method Proving that certain discrete metric spaces are rigidly embedded in the Euclidean plane or other Riemannian surfaces.
result Riemannian embeddings of certain discrete metric spaces are rigid, meaning they cannot be deformed without changing distances.
PixelCNN models can achieve state-of-the-art results on CIFAR-10 with exact likelihood computation.
problem Dequantization gap in modeling discrete data like images.
method Introducing subset flows to allow exact computation of likelihoods for discrete data.
result PixelCNN models trained with exact likelihood computation achieve state-of-the-art results on CIFAR-10.
Score function estimators improve k-subset sampling efficiency.
problem Efficiently sampling k-subsets in machine learning tasks. method Revisit score function estimators, using discrete Fourier transform and control variates.
result Efficient and unbiased gradient estimates for k-subset sampling. Abstract result on correlations of pairs in exponentially growing discrete subsets.
problem Pair correlations in exponentially growing discrete subsets with weight functions.
method Proved abstract result on correlations of pairs of elements in an exponentially growing discrete subset with a weight function.
result Distribution function of unscaled differences is t↦2δe−∣t∣, and pair correlation exhibits Poissonian behavior under certain conditions. Unique metric found for discrete curvature on spherical cone-metrics.
problem Finding a unique metric with prescribed curvature on spherical cone-metrics.
method Discrete conformal approach to spherical cone-metrics.
result Existence of a unique metric realizing prescribed curvature in each conformal class.
The group of C1-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations nV of Thompson's group V arise…
New non-rigid discrete groups found in hyperbolic spaces.
problem Uniqueness of conformal or spherical CR structures on spheres.
method Nilpotent Sierpiński carpet and stretching to construct non-rigid groups.
result Discrete hyperbolic groups can have non-rigid deformations.
A closed discrete subset A⊂C is called tame if C∖A is quasiconformally equivalent to C∖Z. By giving several criteria for A to be tame, we shall show that Z+iZ is not tame.
Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Isometric actions of G on M are (by definition) points in the space R := Hom(G, Isom(M)) with the compact open topology. Sample theorems: 1. The…
Proves critical exponent for Θ−positive representations in discrete subgroups.
problem Determining the critical exponent for Θ−positive representations. method Analyzes discrete subgroups Γ⊂PSL(2,R) and their geometric properties. result Equality of critical exponent holds if and only if Γ is a lattice for geometrically finite Γ. Universal inequalities for Laplacian eigenvalues on discrete groups.
problem Proving inequalities for Laplacian eigenvalues on discrete groups.
method Analyzing Laplacian eigenvalues with Dirichlet boundary conditions on subsets of discrete groups.
result Yang-type universal inequalities for Cayley graphs of amenable groups and the d-regular tree.
The paper proves the existence of minimal surfaces avoiding specific points.
problem Proving the existence of minimal surfaces avoiding specific points.
method Interpolation theorem for conformal minimal immersions avoiding hyperplanes.
result Existence of complete conformal minimal immersions avoiding prescribed points.
This paper provides an explicit form for symmetric differentials and their corresponding holomorphic functions.
problem Understanding the correspondence between symmetric differentials and L2 holomorphic functions on quotient spaces. method Explicit description of the correspondence between symmetric differentials and weighted L2-holomorphic functions. result Derivation of several applications based on the explicit form of the correspondence.
Researchers derive Markov properties of discrete DPPs.
problem Lack of statistical properties exploration for discrete DPPs.
method Derive Markov properties using graphical models.
result Markov properties of discrete DPPs can be expressed.
Compact hyperbolic complex manifolds are rigid under deformation.
problem Studying the deformation behavior of compact hyperbolic complex manifolds.
method Analyzing smooth families of compact complex manifolds over the unit disk and compact Riemann surfaces.
result The H-locus is either at most a discrete subset or the whole domain, depending on the family structure. Log-concavity proven for multinomial likelihoods under specific constraints.
problem Log-concavity of multinomial likelihoods under interval censoring constraints.
method Proved log-concavity by showing M-convex subsets of the discrete simplex.
result Likelihood function is completely log-concave.
No exact G₂-structures on compact Lie group quotients.
problem Existence of exact G₂-structures on compact quotients of Lie groups.
method Analyzing compact quotients of seven-dimensional Lie groups by co-compact discrete subgroups.
result Compact quotients of seven-dimensional Lie groups by co-compact discrete subgroups do not admit exact G₂-structures induced by left-invariant ones.
Finite graphs with specific curvature have limited harmonic functions and ends.
problem Graphs with nonnegative curvature outside a finite subset.
method Introducing discrete Gromov-Hausdorff convergence to study bounded harmonic functions.
result The space of bounded harmonic functions is finite dimensional, and the number of non-parabolic ends is finite.
We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that t…
Proposes a neural framework to select subsets efficiently across different models.
problem Lack of generalizability in subset selection methods for unseen architectures.
method Introduces a trainable subset selection framework, SubSelNet, that uses attention-based neural gadgets and subset samplers.
result SubSelNet generalizes across architectures and outperforms existing methods.
The study describes quotient spaces for specific discrete subgroups acting on complex projective spaces.
problem Understanding quotient spaces for discrete subgroups acting on complex projective spaces.
method Topological description of quotient spaces for specific discrete subgroups.
result Maximum number of complex projective lines in general position contained in Kulkarni's limit set is 4.
Let Γ be a lattice in a connected semisimple Lie group G with trivial center and no compact factors. We introduce a volume invariant for representations of Γ into G, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this vo…
New DKPP family controls positive and negative dependence in random subsets.
problem Challenges in seamlessly bridging probabilistic models for positive and negative dependence.
method Introduced DKPP family and developed computational methods for probabilistic operations and inference.
result Controllability of positive and negative dependence demonstrated through numerical experiments.
Locally-verifiable conditions ensure exactness of spline discrete de Rham complex.
problem Ensuring cohomological equivalence of spline discrete complex to continuous de Rham complex.
method Theoretical analysis and locally-verifiable sufficient conditions for exactness.
result Locally-verifiable conditions guarantee exactness of hierarchical B-spline discrete de Rham complex.
New algorithm approximates maximum of certain distributions on subsets.
problem Finding maximum of distributions on subsets.
method Connection between sampling and optimization via exchange inequalities and local random walks.
result Simple nearly-optimal approximation algorithm for MAP inference.
New methods merge discrete gradient fields from patches to correct errors.
problem Correctly merging partially defined discrete gradient fields from patches.
method Developed general and lean merging procedures for specific covering patterns.
result Corrected errors in merging discrete gradient fields from patches.
Algorithm reduces support of discrete measures by integrating against functions.
problem Efficiently reducing the support of discrete measures when N≫n. method Geometric characterization and greedy geometric sampling.
result A new measure with n+1 atoms has the same mean as original measure. A scalable algorithm for computing Wasserstein barycenters of streaming data.
problem Aggregating data from different, possibly non-identically distributed sources.
method Parallel, semi-discrete algorithm for continuous input distributions.
result Robust, streaming Wasserstein barycenter estimate that tracks nonstationary distributions.
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group G⊂PU(n,1) acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…
A Lie algebra is called nonsoliton if it does not admit a soliton inner product. We demonstrate that the subset of nonsoliton Lie algebras in the moduli space of indecomposable n-dimensional N-graded nilpotent Lie algebras is discrete if and only if n <= 7.
The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of d-dimensional cubes in Rd is ⌊(3d+1)/2⌋.
Holomorphic curves exiting bounded symmetric domains are asymptotically totally geodesic.
problem Understanding the asymptotic behavior of holomorphic curves in bounded symmetric domains.
method Proof by contradiction and rescaling, using the Poincaré-Lelong equation.
result Holomorphic curves exiting a bounded symmetric domain are asymptotically totally geodesic.
RIPE predicts and explains continuous/discrete data with sparse rule sets.
problem Predicting and explaining continuous/discrete data.
method RIPE infers a model from a sample, extracting a sparse set of hyperrectangles (rules) to partition the feature space.
result RIPE efficiently predicts and explains data, superior to other algorithms.
Determinantal point processes (DPPs) are random point processes well-suited for modeling repulsion. In machine learning, the focus of DPP-based models has been on diverse subset selection from a discrete and finite base set. This discrete setting admits an efficient sampling algorithm based on the eigendecomposition of…
A new sampling method accelerates inference in discrete probabilistic models.
problem Slow convergence of Markov chain Monte Carlo algorithms in discrete probabilistic models.
method Proposes a mixture of product distributions using semigradient information to accelerate convergence.
result Combining the new sampler with existing ones improves inference in various models.
Exponential localization of eigensections for Bochner-Schrödinger operator.
problem Understanding spectral properties of Bochner-Schrödinger operator on high tensor powers of Hermitian line bundles.
method Approximation of operator by model Schrödinger operator with constant magnetic field, analysis of spectrum.
result Spectrum of Bochner-Schrödinger operator in gaps is discrete and eigensections decay exponentially.
Smooth groupoid algebras are H-unital, with implications for algebraic and homological properties.
problem Understanding the structure of convolution algebras on Lie groupoids.
method Analyzing smooth functions and invariant subsets to prove H-unitality.
result H-unitality of groupoid algebras and their quotients, leading to excision properties.
Differentiable Greedy Networks improve sentence selection for claim verification.
problem Optimal selection of sentences for claim verification in FEVER task.
method Proposes a trainable subset selection algorithm based on submodular optimization and unfolds a greedy algorithm into a computational graph.
result Differentiable Greedy Network (DGN) outperforms other methods in precision and recall.
AdaWISH improves efficiency of discrete integration queries.
problem Efficiently solving discrete integration in high-dimensional spaces.
method Adaptive quantile queries to reduce query count.
result AdaWISH achieves the same approximation guarantee with fewer queries.
Combination theorems for convex projective geometry subgroups.
problem Understanding discrete subgroups in convex projective geometry.
method General combination theorems for discrete subgroups preserving properly convex open subsets.
result Free products of convex cocompact subgroups are convex cocompact.
Hybrid RL method optimizes trading by balancing continuous and discrete actions.
problem Optimal execution in algorithmic trading with continuous-discrete action space.
method Combines continuous and discrete RL agents for better trading decisions.
result Significantly outperforms existing methods in trading efficiency and stability.
Paper optimizes summarization of multiple document groups for better distinction.
problem Comparative document summarization to select representative documents from multiple groups.
method Formulated new objective functions based on binary classification and maximum mean discrepancy, using gradient-based optimization.
result Gradient-based optimization outperforms other methods in automatic and crowd-sourced evaluations.
Recently developed techniques have made it possible to quickly learn accurate probability density functions from data in low-dimensional continuous space. In particular, mixtures of Gaussians can be fitted to data very quickly using an accelerated EM algorithm that employs multiresolution kd-trees (Moore, 1999). In thi…
The ball in complex 2-space can contain curves of any shape.
problem Embedding complex curves of arbitrary topology in the ball of C2. method Proving existence of curves with any given topological type.
result Complete embedded complex curves of any topological type exist in the ball of C2. AQL uses amortized inference to handle high-dimensional action spaces in Q-learning.
problem Difficulty in maximizing over large action spaces in Q-learning.
method Replace expensive maximization over all actions with a maximization over a small subset sampled from a learned proposal distribution.
result AQL outperforms existing methods on continuous control tasks with up to 21 dimensional actions.
We investigate the systematic mechanism for designing fast mixing Markov chain Monte Carlo algorithms to sample from discrete point processes under the Dobrushin uniqueness condition for Gibbs measures. Discrete point processes are defined as probability distributions μ(S)∝exp(βf(S)) over all subsets $S\in 2^…
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.