Extends model geometries for surface bundles over graphs using tight trees.
problem Constructing model geometries for surface bundles over graphs.
method Generalizing tight geodesics to tight trees and using them to construct model geometries.
result Uniformly Gromov-hyperbolic geometric model spaces equipped with geometric G−actions. Paper extends tree bijection for hyperbolic surfaces without requiring cusps.
problem Computing volumes and distances on hyperbolic surfaces without cusps.
method Extend tree bijection to half-tight cylinders, using Busemann function.
result Tree bijection can now be applied to surfaces without cusps.
The paper tightens bounds on distances between Reeb graphs.
problem Certifying quasi-universality of distances between Reeb graphs.
method Establishes tight bi-Lipschitz bounds for various distances.
result Proves strict universality of the functional contortion distance for contour trees and coincides with interleaving distance for merge trees.
Paper studies non-tight reconstruction threshold in a 4-state model with different in/out block mutations.
problem Non-tight reconstruction threshold in a 4-state symmetric model with different in-block and out-block mutations.
method Inspired by the q1+q2 stochastic block model, rigorously analyzes conditions for non-tightness of the reconstruction threshold. result Rigorously gives conditions for the non-tightness of the reconstruction threshold in a 4-state symmetric model.
New algorithm speeds up robustness verification for tree-based models.
problem Formal robustness verification of tree-based models, especially ensembles.
method Reformulated as max-clique problem on a multi-partite graph with bounded boxicity; developed efficient multi-level verification algorithm.
result Tight lower bounds on robustness of decision tree ensembles, hundreds of times faster than previous approach.
Structured prediction is used in areas such as computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions …
Researchers prove it's impossible to partially recover graph alignments in certain conditions.
problem Recovering vertex correspondence between two random graphs with correlated edges.
method Used the probabilistic method to build automorphisms between tree components of a subcritical Erdös-Rényi graph.
result Proved an impossibility result for partial recovery in the sparse regime with constant average degree and correlation.
Optimal sparse recovery with decision stumps achieves strong feature selection guarantees.
problem Sparse recovery of active features from high-dimensional data.
method Analysis of single-depth decision trees (decision stumps) for feature selection in linear regression.
result Tight sample performance guarantees for O(slogp), improving upon previous bounds. Many important optimization problems, such as the minimum spanning tree and minimum-cost flow, can be solved optimally by a greedy method. In this work, we study a learning variant of these problems, where the model of the problem is unknown and has to be learned by interacting repeatedly with the environment in the ba…
New method debiases feature importance in Random Forests.
problem MDI feature importance measure incorrectly assigns high importance to noisy features.
method Derive a new analytical expression for MDI and propose MDI-oob debiased feature importance measure.
result MDI-oob achieves state-of-the-art performance in feature selection from Random Forests.
Estimates piecewise polynomials and bounded variation functions using optimal decision trees.
problem Estimating piecewise smooth functions in general dimensions.
method Dyadic CART and Optimal Regression Tree (ORT) estimators for piecewise polynomials and bounded variation functions.
result Oracle inequalities and risk bounds for ORT estimators, demonstrating adaptivity and optimality.
The paper provides robustness guarantees for classifiers under Gaussian noise and discrete adversaries.
problem Ensuring robustness of classifiers against adversarial attacks.
method Explores robustness guarantees for ensembles of classifiers under Gaussian noise and discrete adversaries, tightening the guarantees with specific assumptions.
result The paper offers robustness guarantees and associated algorithms for discrete adversaries, demonstrating their effectiveness on image and molecule datasets.
Hierarchical Federated Learning bounds generalize using Wasserstein distance.
problem Bounding generalization error in Federated Learning with hierarchical sampling.
method Introduced a hierarchical sampling framework and derived generalization bounds using Wasserstein distance.
result Recover and strictly imply existing CMI bounds for bounded losses.
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…
New geodesics found that are not tight but still have useful properties.
problem Understanding geodesics in curve complexes and Teichmüller spaces.
method Introducing and studying weak tight geodesics with canonical constructions.
result Found examples of weak tight geodesics with gaps between them.
We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…
Study tight contact structures on figure-eight knot surgeries.
problem Classify tight contact structures on surgeries of figure-eight knot.
method Analyzes surgeries on figure-eight knot, determining tightness, symplectic fillability, and universality.
result First classification of tight contact structures on surgeries of figure-eight knot.
The study identifies conditions for algorithms to have tight generalization bounds.
problem Understanding which algorithms have tight generalization bounds.
method Analyzing conditions that preclude tight generalization bounds and identifying stable algorithms.
result Stable algorithms have tight generalization bounds, while unstable ones do not.
Surgery on knots always admits a tight contact structure.
problem Understanding tight contact structures on knots after surgery.
method Smooth (-r)-surgery on knots, using Heegaard Floer contact invariant.
result Tight contact structures detected for all knots after surgery.
Tight maps was introduced along tight homomorphisms by Burger, Iozzi and Wienhard with aims towards maximal representations. In this paper we classify tight maps into classical Hermitian symmetric spaces and give a partial result for the exceptional spaces.
Efficiently learns polytrees with known skeleton in polynomial time and sample complexity.
problem Learning polytrees with known skeleton structure.
method Proposes an efficient algorithm for learning d-polytrees in polynomial time and sample complexity when the skeleton is known. result Establishes finite-sample guarantees for efficient learning of d-polytrees. 3-manifold triangulations are Golod and tight, proven through a topological characterization.
problem Understanding Golodness and tightness in 3-manifold triangulations.
method Topological characterization of a polyhedral product for a tight-neighborly manifold triangulation.
result Golodness and tightness are equivalent for 3-manifold triangulations.
Simply connected spaces of tight frames identified.
problem Understanding the connectivity of spaces of tight frames.
method Viewing tight frames as elements of Stiefel manifolds and identifying simply connected spaces.
result Spaces of tight frames, including finite unit-norm tight frames, are simply connected.
The study finds many tight contact structures on hyperbolic 3-spheres.
problem Finding tight contact structures on hyperbolic 3-spheres.
method Constructing hyperbolic homology 3-spheres and analyzing their tight contact structures.
result Produces hyperbolic homology 3-spheres with multiple distinct tight contact structures.
Study tight contact structures on specific 3-manifolds.
problem Counting and constructing tight contact structures on plumbed 3-manifolds.
method Algorithm to construct stein diagrams for tight structures without Giroux torsion.
result Explicit algorithm to construct tight contact structures.
Classifies tight contact structures on surgeries of the Whitehead link.
problem Classifying tight contact structures on surgeries of the Whitehead link.
method Analyzes various surgeries on the Whitehead link to classify tight contact structures.
result Determines tight contact structures, Stein fillability, and virtually overtwisted properties.
5D contact structures are universally tight via Bourgeois construction.
problem Understanding tightness of 5D contact structures.
method Explicit construction of contact structures on VimesT2. result All constructed contact structures are universally tight.
Study finds tight contact structures on many hyperbolic 3-manifolds.
problem Existence of tight contact structures on hyperbolic 3-manifolds.
method Dehn surgeries on hyperbolic surface bundles.
result Existence of infinitely many hyperbolic 3-manifolds with tight contact structures.
In \cite{confol} Y. Eliashberg and W. Thurston gave a definition of tight confoliations. We give an example of a tight confoliation ξ on T3 violating the Thurston-Bennequin inequalities. This answers a question from \cite{confol} negatively. Although the tightness of a confoliation does not imply the Thurston-Benn…
In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces. We also prove that any lens space admits only finitely many tight contact structures.
Ozsvath-Szabo invariant classifies tight contact structures on L-spaces.
problem Classifying tight contact structures on L-spaces.
method Ozsvath-Szabo contact invariant.
result Complete classification invariant for tight contact structures.
Classifies real tight contact structures on lens spaces and solid tori.
problem Classifying real tight contact structures on specific 3-manifolds.
method Equivariant contact isotopy, real open book decompositions, and isolated real algebraic surface singularities.
result Unique real tight structures on S3 and RP3, at most one on L(p,±1), and bounds on the count. New proof of Giroux Correspondence for tight contact 3-manifolds.
problem Proving the Giroux Correspondence for tight contact 3-manifolds.
method Introducing tight Heegaard splittings, using refinement process, and translating moves between splittings to moves between open books.
result Proves the tight Giroux Correspondence for contact 3-manifolds.
The study finds algebraically overtwisted tight 3-manifolds via contact surgeries.
problem Finding algebraically overtwisted tight 3-manifolds.
method Executing Avdek's algorithm to perform contact surgeries.
result Contact surgeries on standard contact 3-sphere yield algebraically overtwisted and tight 3-manifolds.
Tight contact foliations can be made less restrictive.
problem Homotopy invariance of tightness in contact foliations.
method Turbulisation procedure to make tight foliations less restrictive.
result Tightness is not a homotopy invariant property.
We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yie…
We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group G acting on a G--space X, we prove that if G contains a strongly contracting eleme…
We determine all tight Lagrangian surfaces in S2×S2. In particular, globally tight Lagrangian surfaces in S2×S2 are nothing but real forms.
Study non-fibered links' relation to tight contact structures.
problem Understanding non-fibered links and their tight contact structures.
method Analyze non-fibered links with induced partial open books and contact structures.
result Strongly quasipositive non-fibered links induce tight contact structures, but the converse is not always true.
New 3D shapes found without certain special flows.
problem Finding 3D shapes without specific special flows.
method Rational surgeries on the figure eight knot.
result First infinite family of hyperbolic 3-manifolds without tight projectively Anosov flows.
The paper proves that the support norm of tight contact structures adds up.
problem Understanding the support norm of contact structures.
method Analyzing open books supporting contact structures.
result Additivity of the support norm for tight contact structures.
We prove gluing theorems for tight contact structures. In particular, we rederive (as special cases) gluing theorems due to Colin and Makar-Limanov, and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus 4 h…
We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.
Classifies tight contact structures on specific Seifert fibered manifolds.
problem Classifying tight contact structures on Seifert fibered manifolds.
method Constructed contact structures using Legendrian surgery and used convex surface theory for the upper bound.
result Found the lower and upper bounds for tight contact structures.
New evidence supports the Euler class one conjecture for tight contact structures.
problem Euler class one conjecture for taut foliations and tight contact structures.
method Analysis of tight contact structures and counterexamples to the conjecture.
result Counterexamples to the Euler class one conjecture for taut foliations are also Euler classes of tight contact structures.
The study finds tight contact structures without fillings in high dimensions.
problem Finding tight contact structures that cannot be filled by symplectic forms.
method Construction of specific contact structures on manifolds of various dimensions.
result Existence of tight contact structures without fillings in all dimensions n≥3 and for n=2 under certain conditions. This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is preserved under Legendrian surgery.
We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched cover…