We give a method for searching for thin positions of a given link.
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Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S3, k) from a thin position of k. In this article, we will prove that if a thin position of k induces a thin decomposition of double branched cover of (S3,k) by Howards and…
Proposes efficient training method for deep thin networks.
problem Deploying deep learning models with accuracy and compactness.
method Three-stage method: widen, warm up, fine tune.
result Deep thin networks trained with method outperform standard deep networks.
KSD Thinning uses KSD to thin MCMC samples efficiently.
problem Efficiently representing posterior distributions in Bayesian inference.
method KSD Thinning: retains only samples exceeding a KSD threshold.
result Established convergence and complexity tradeoffs for KSD Thinning.
New method characterizes thin links via Conway spheres and tangle decompositions.
problem Characterize thin links without relying on specific knot invariants.
method Developed a relative version of thinness for tangles and used it to characterize thinness via tangle decompositions along Conway spheres.
result Characterized thin links via Conway spheres and tangle decompositions.
Low-Rank Thinningstat.ML
New method reduces summary points for datasets while maintaining quality.
problem Thinning datasets to reduce summary points while maintaining quality.
method Low-rank analysis of sub-Gaussian thinning.
result Guarantees high-quality compression for any distribution and kernel.
Thin Tree Positionmath.CO
We introduce a method for creating a special type of tree, called a tree position, from a weighted graph. Leaves of the tree correspond to vertices of the original graph, and the tree edges contain information which can be used to partition these vertices. By repeatedly applying reducing operations to the tree position…
The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.
problem Finding thin Hitchin representations in non-uniform lattices of Lie groups.
method Arithmetic methods to construct thin Hitchin representations.
result Infinitely many orbits of thin Hitchin representations in non-uniform lattices.
New thin position for graphs in 3-manifolds combines ideas from knot theory and 3-manifold topology.
problem Defining a new thin position for graphs in 3-manifolds.
method Combining thin position concepts from knot theory and 3-manifold topology.
result Defines new invariants of knots, links, and graphs in 3-manifolds.
The paper examines rigidity of thin domains under specific boundary conditions.
problem Linear geometric rigidity of shallow thin domains with zero Dirichlet boundary conditions.
method Analyzes two scaling regimes for ε in (h, √h] and (√h, 1), proving rigidity formulas.
result Rigidity does not depend on curvature in the small parameter regime ε ∈ (h, √h].
Signature uniquely identifies piecewise linear surfaces up to thin homotopy.
problem Characterizing piecewise linear surfaces up to equivalence.
method Crossed module of piecewise linear surfaces and signature homomorphism.
result Signature uniquely characterizes surfaces up to translation and thin homotopy.
Thin Position for Tanglesmath.GT
If a tangle, K, in the 3-ball has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.
Algorithm builds surfaces in 3-manifolds, simplifying their description.
problem Finding and describing incompressible surfaces in 3-manifolds.
method Algorithmic approach to build and simplify descriptions of surfaces.
result Incompressible surfaces can be described in terms of thin/thick levels.
New proof shows most thin knots satisfy Cabling Conjecture.
problem Cabling Conjecture for thin knots using Heegaard Floer homology.
method Heegaard Floer homology and immersed curves techniques.
result Almost all thin knots satisfy the Cabling Conjecture.
We produce embeddings of knots in thin position that admit compressible thin levels. We also find the bridge number of tangle sums where each tangle is high distance.
The paper finds many thin subgroups isomorphic to Gromov-Piatetski-Shapiro lattices.
problem Understanding thin subgroups in special linear groups.
method Constructing and embedding non-arithmetic hyperbolic manifolds into SL(n+1)(R).
result Non-arithmetic lattices in SO(n,1) can be embedded into SL(n+1)(R) as thin subgroups.
A new stochastic version of BPS improves sampling from big datasets.
problem Efficiently sampling Bayesian posteriors in large datasets.
method Stochastic Bouncy Particle Sampler using thinning method of Poisson process.
result The algorithm outperforms other samplers in efficiency and mixing.
The thick-thin decomposition and the bilipschitz classification of normal surface singularitiesmath.AG
We describe a natural decomposition of a normal complex surface singularity into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…
Abby Thompson proved that if a link is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin…
Introduces thin-thick decomposition for real isolated singularities.
problem Classifying real isolated singularities.
method Introduces thin-thick decomposition as a blow-spherical invariant.
result Generalizes thin-thick decomposition for complex surface singularities to real isolated singularities.
Regularized Stein thinning improves MCMC output approximations.
problem Pathologies in Stein thinning leading to poor approximations.
method Theoretical analysis and regularization to improve KSD.
result Regularized Stein thinning alleviates pathologies and improves efficiency.
The paper studies eigenvalues in negatively curved spaces, showing their relationship to the thick-thin decomposition.
problem Investigating the relationship between eigenvalues and the geometric structure of negatively curved spaces.
method Analyzing the eigenvalues of the Laplacian on the thick part of a manifold's thick-thin decomposition.
result Eigenvalues in the thick part of a negatively curved manifold are significantly smaller than those in the entire manifold.
Automated malaria diagnosis from field slides achieves accurate results.
problem Challenges in analyzing field-prepared thin blood film microscopy images.
method Fully automated framework using machine learning, including CNNs trained on diverse field samples.
result Results are close to sufficient for drug resistance monitoring and clinical use-cases.
Thin groups found in specific lattices.
problem Embedding right-angled Coxeter groups in arithmetic lattices.
method Using Agol's unpublished argument, embedding in indefinite orthogonal groups.
result Irreducible right-angled Coxeter groups embed as thin subgroups.
Data thinning splits observations into independent parts for convolution-closed distributions.
problem Validation of unsupervised learning results in settings with limited data.
method Data thinning, splitting observations into independent parts following the same distribution.
result Data thinning provides an attractive alternative to cross-validation in settings with limited sample splitting.
A new method uses Transformers for efficient prediction of marked point processes.
problem Efficiently predicting the next event in a sequence given its history.
method Modeling conditional inter-event times with a mixture of log-normals and marks with a Transformer architecture.
result The method achieves state-of-the-art performance and is faster during inference.
New method allows generating independent data matrices from summary statistics.
problem Generating independent data matrices from summary statistics like mean and covariance.
method Thinning a Wishart random matrix based on sample mean and covariance.
result It is possible to generate two independent data matrices from summary statistics.
Kernel Thinningstat.ML
Kernel thinning compresses distributions more effectively than i.i.d. sampling or standard thinning.
problem Efficiently compressing distributions for better sampling and integration accuracy.
method Introduces kernel thinning, a procedure that compresses an n-point approximation of a distribution into a sqrt(n)-point approximation with comparable integration error.
result Kernel thinning achieves a maximum discrepancy in integration error of O_d(n^(-1/2) sqrt(log n)) in probability for compactly supported distributions and O_d(n^(-1/2) (log n)^(d+1/2) sqrt(log log n)) for sub-exponential distributions.
We exhibit the first examples of links which are homologically thin but not quasi-alternating. To show that they are not quasi-alternating, we argue that none of their branched double-covers bounds a negative definite 4-manifold with non-torsion H_1. Using this method, we also complete the determination of the quasi-al…
Cable knots are not thinmath.GT
Study shows -cables of non-trivial knots are not thin.
problem Proving -cables of non-trivial knots are not thin.
method Bordered Floer theory of Lipshitz-Ozsváth-Thurston and Zemke's theorem.
result Proves -cables of non-trivial knots are not thin.
Generalizes data thinning for various distributions.
problem Thinning random variables without losing information.
method Relaxes summation requirement to sufficiency.
result Generalizes thinning to more distributions.
Wu has shown that if a link or a knot in in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that is prime, then the thin sphere of lowest width also does not have any vertical c…
Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
problem Understanding the structure of arithmetic locally symmetric spaces.
method Analyzing thin parts and deducing asymptotic results on Betti numbers.
result Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
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.
Enhances supervised learning speed with KT algorithm.
problem Speed up supervised learning tasks with minimal loss.
method Generalizes kernel thinning to supervised learning, combining NW and KRR with KT.
result KT-based estimators offer superior computational and statistical efficiency.
We show that every thin position for a connected sum of small knots is obtained in an obvious way: place each summand in thin position so that no two summands intersect the same level surface, then connect the lowest minimum of each summand to the highest maximum of the adjacent summand below.
New infinite family of knots without special surgeries found.
problem Finding knots without specific surgeries.
method Identifying an infinite family of knots.
result First infinite family of strongly invertible L-space knots without Khovanov thin surgery.
Enhanced neural networks detect thin boundaries between different types of anomalies.
problem Detecting thin boundaries between different types of anomalies in outlier detection.
method Defined new types of anomalies, enhanced Multi-Layer Perceptron Neural Network with Genetic Algorithm.
result Reduced error in anomaly detection compared to baseline methods.
Let L be a link in the 3-sphere that is in thin position but not in bridge position and let P be a thin level sphere. We generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks that P can have, including identifying thin spheres with unique compressing disks. We also give con…
Compactifies group representations into thin triangle spaces.
problem Compactify group representations into geometric spaces.
method Geometric reformulation and extension of Culler-Morgan-Shalen theory.
result Ideal points are actions on real trees.
Survey of mathematical developments in gauge theory using thin homotopy.
problem Formalizing gauge fields as group homomorphisms on smooth connections.
method Introducing group structures on spaces of loops via thin homotopy equivalence.
result Clarified difference between thin and retrace equivalence for loops.
Paper proposes efficient online data thinning for expert analysis.
problem Large-scale streaming data exceeds human analysis capacity.
method Online anomaly detection using dynamic low-rank Gaussian mixture models.
result Proposed method reduces data to unique elements for timely analysis.
Study thin hyperbolic reflection groups and their properties.
problem Characterize and enumerate thin hyperbolic reflection groups.
method Analyze Zariski dense subgroups of hyperbolic isometries, apply Vinberg algorithm.
result All thin hyperbolic reflection groups are enumerable.
Defines thin position for 4-manifolds and trisections, finding key relations and diagrams.
problem Defining and analyzing thin position for 4-manifolds and their trisections.
method Introduced width of handle decompositions, defined thin position, and studied trisections.
result Described genus 2g+2 and g+2 trisection diagrams for sphere bundles.
We generalize the definition of thin position of Scharlemann and Thompson for compact orientable 3-manifolds with torus boundary components and introduce -sloped generalized Heegaard splittings. We examine its relationship to generalized Heegaard splittings of manifolds resulting from Dehn filling. We compare alpha-…
We show that, in the Teichmüller metric, "thin-framed triangles are thin"---that is, under suitable hypotheses, the variation of geodesics obeys a hyperbolic-like inequality. This theorem has applications to the study of random walks on Teichmüller space. In particular, an application is worked out for the action of th…
We unify the notions of thin position for knots and for 3-manifolds and survey recent work concerning these notions.
This paper shows how thinning Markov chains can improve statistical efficiency under certain conditions.
problem Improving statistical efficiency of Markov chain samplers by subsampling.
method Analyzes the cost of thinning and the autocorrelation decay of Markov chains to determine when thinning improves efficiency.
result Thinning Markov chain output can improve statistical efficiency if the autocorrelation decay is slow and the cost of computing a sample is high.