Study loop ensembles on graphs, linking group theory and topology.
problem Understanding loop homotopy classes and homologies on graphs.
method Determined distributions of loop homotopy classes and homologies using the lower central series of the fundamental group.
result Distributions of loop homotopy classes and homologies defined by the lower central series of the fundamental group.
Study loop corrections in random feature models affecting training and test errors.
problem Analyzing loop corrections in random feature models to understand training and test errors.
method Statistical physics and effective field theory approach to study loop corrections.
result Derived loop corrections to training error, test error, and generalization gap.
Human analysts that use anomaly detection systems in practice want to retain the use of simple and explainable global anomaly detectors. In this paper, we propose a novel human-in-the-loop learning algorithm called GLAD (GLocalized Anomaly Detection) that supports global anomaly detectors. GLAD automatically learns the…
New algorithms reduce dynamic regret in online MDPs with changing losses.
problem Online MDPs with adversarial loss changes and known transitions.
method Dynamic regret measure, novel ensemble algorithms for three models.
result Provably optimal dynamic regret bounds for episodic SSP, improved bounds for predictable environments.
DriftMoE adapts to concept drifts in data streams efficiently.
problem Adapting to non-stationary data streams with concept drifts.
method Online Mixture-of-Experts (MoE) architecture with co-training framework.
result DriftMoE achieves competitive results with state-of-the-art stream learning adaptive ensembles.
DECAF optimizes molecular graphs for ensemble properties, improving drug design accuracy.
problem Designing molecules with ensemble properties rather than single conformations.
method DECAF uses Boltzmann-expected design with decoupled annealing flows to optimize molecular graphs.
result DECAF optimizes molecular graphs to shift ensemble properties towards targets, improving accuracy over single-conformer methods.
Study interbank lending and borrowing dynamics with heterogeneous mean field model.
problem Modeling systemic risk in a network of banks with varying capitalization.
method Developed a mean field type model with coupled diffusions to describe log-capitalization evolution.
result Existence of Nash equilibria in large-scale heterogeneous interbank networks.
Automated suggestions help train technicians diagnose incidents faster.
problem Manual and time-consuming incident diagnosis by train maintenance technicians.
method Developed and deployed a learning machine to suggest diagnostics to technicians.
result The model refines its accuracy through feedback from experts and uses feature engineering.
Hypothesis testing is an important cognitive process that supports human reasoning. In this paper, we introduce a computational hypothesis testing approach based on memory augmented neural networks. Our approach involves a hypothesis testing loop that reconsiders and progressively refines a previously formed hypothesis…
SRVs improve MSMs for Trp-cage miniprotein, revealing new folding states.
problem Constructing high-resolution MSMs for complex protein dynamics.
method Employing SRVs as feature set for MSM construction, leveraging slowest modes identified by SRVs.
result SRV-MSMs reveal new folding states and faster convergence.
Study smooth loops and loop bundles, relating to G2-structures.
problem Properties of smooth loops and their applications.
method Analyze smooth loops, introduce loop bundles, define torsion and curvature.
result Showed how loop bundles relate to G2-structures. We produce skew loops -- loops having no pair of parallel tangent lines -- homotopic to any loop in a flat torus or other quotient of R^n. The interesting case here is n=3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop in the unit sphere as tangent ind…
The paper proves T-duality and Hori formulae for winding loop spaces.
problem Realizing T-duality and Hori formulae for loop spaces.
method Proving T-duality and Hori formulae for winding q-loop spaces.
result T-duality and Hori formulae for winding q-loop spaces are proven.
Paper establishes loop space T-duality formulae and refines earlier work.
problem Establishing T-duality on loop spaces with background flux.
method Developed loop Hori map and twisted Bismut-Chern character.
result Loop Hori map induces quasi-isomorphism on loop spaces.
The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.
Rational loops played a central role in Uhlenbeck's construction of harmonic maps into U(n) (chiral model in physics), and they are generated by simple elements with one pole and one zero constructed from Hermitian projections. It has been believed for long time that nilpotent loops should be added to generate rational…
Kähler manifold loop space inherits Kähler structure and is complete.
problem Characterizing the geometric properties of loop spaces of Kähler manifolds.
method Proving the Kähler structure and completeness of the loop space.
result The loop space of a Kähler manifold is complete and inherits a Kähler structure.
New structure found in loops on quasi-surfaces.
problem Understanding operations on loops in quasi-surfaces.
method Defined a quasi-Lie bialgebra structure.
result Natural operations on loops form a quasi-Lie bialgebra.
Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most 9-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are is…
A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics. In particular, ellipsoids are the only closed surfaces without skew loops. We also prove results about skew loops on cylinders an…
Introduces string structures linking to loop spaces.
problem Understanding geometric string structures.
method Explains connections to loop spaces.
result String structures linked to loop spaces.
Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.
problem Understanding the homotopy properties of 6-manifolds over 4-manifolds.
method Analyzes the homotopy equivalence and rational homotopy of the total space of a sphere bundle over a 4-manifold.
result Looping a 6-manifold over a 4-manifold is homotopy equivalent to a product of loops on spheres.
New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.
problem Defining invariants for framed 3-manifolds with semi-simple Lie groups.
method Constructing graph complexes and cocycles, including self-loops, to define invariants.
result Higher-loop invariants can be defined by graph cocycles with or without self-loops.
Method calculates loop intersections on surfaces.
problem Computing intersections of loops on surfaces.
method Nielsen fixed point theory and Gröbner-Shirshov basis.
result Simple method to compute intersection numbers.
Geodesic loops on tetrahedra are studied in spherical and hyperbolic spaces.
problem Characterize geodesic loops on tetrahedra in different types of spaces.
method Analytical proofs for spherical and hyperbolic spaces.
result Existence and properties of geodesic loops on tetrahedra in various spaces.
New definition of twisted 1-loop invariant using Ptolemy coordinates.
problem Defining and proving properties of twisted 1-loop invariants.
method Alternative definition via Jacobian of Ptolemy coordinates.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial for hyperbolic once-punctured torus bundles.
Training-free looped transformers improve model performance without additional training.
problem Improving model performance without additional training or fine-tuning.
method A lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning.
result Our method improves model performance across various model families.
Study path spaces and their homology, extending loop products and coproducts.
problem Understanding the homology of path spaces in closed manifolds.
method Morse-Bott theory and homology operations.
result Complete computation of extended loop product and coproduct on spheres.
Study the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds.
problem Global topology of loops of contactomorphisms and Legendrians in non-orderable manifolds.
method Filtering loops by positivity and analyzing subspaces of the filtration.
result Homotopy groups of the space of loops are subgroups of the positive loops subspace.
This paper reformulates the p-adic Littlewood Conjecture using infinite loops.
problem The p-adic Littlewood Conjecture in number theory. method Introducing infinite loops mod n and linking them to the conjecture. result A real number α is a counterexample to the p-adic Littlewood Conjecture if and only if pkα is an infinite loop mod pm for all k. Complex surfaces show non-simply connected diffeomorphism groups with non-homotopic loops.
problem Complex surfaces with non-simply connected diffeomorphism groups and non-homotopic loops.
method Exhibited examples of complex surfaces.
result Diffeomorphism groups of complex surfaces are not simply-connected and contain non-homotopic loops.
New method evaluates AI stock prediction systems based on decision-making processes.
problem Lack of evaluation for AI systems' decision-making processes.
method Scores intermediate decision process using large language models and closed-loop reinforcement learning feedback.
result Composite behavioral score correlates with Sharpe ratio and reduces prediction error.
New basis and Schur-Weyl duality for loop Hecke algebra defined.
problem Define a new basis for the loop Hecke algebra.
method Use higher linear rewriting theory and combinatorics of Dyck paths.
result Yields a conjecture of Damiani-Martin-Rowell and provides a representation theoretic interpretation.
A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group. Transgressive loop group extensions are those that can be explored by finite-dimensional, higher-categorical geometry ov…
Study finds loops with specific curvature exist using Hardy's inequality.
problem Existence of closed planar loops with prescribed curvature.
method Variational approach, Hardy's inequality and associated functional space.
result Existence of loops with specific curvature proven.
We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial.
Study Brownian loops on hyperbolic surfaces, linking to Selberg zeta function.
problem Understanding Brownian loops on hyperbolic surfaces and their relation to Selberg zeta function.
method Computed mass of loops and related to Selberg zeta function for geometrically finite surfaces.
result Relate total loop mass to Selberg zeta function, providing probabilistic interpretations of determinants.
We introduce various versions of spin structures on free loop spaces of smooth manifolds, based on a classical notion due to Killingback, and additionally coupled to two relations between loops: thin homotopies and loop fusion. The central result of this article is an equivalence between these enhanced versions of spin…
Loop group method varies with base point choice.
problem Dependence of loop group method on base point choice.
method Analyzes how loop group method for harmonic maps varies with base point.
result Loop group method results depend on base point selection.
Study vortex loops as coadjoint orbits of diffeomorphisms.
problem Understanding vortex loops in terms of coadjoint orbits.
method Analyzing vortex loops as coadjoint orbits of area-preserving diffeomorphisms.
result Vortex loops are coadjoint orbits of the diffeomorphism group.
Sphere bundles over 4-manifolds are trivial after looping, except for two cases.
problem Understanding the triviality of sphere bundles over 4-manifolds.
method Analyzing the splitting of sphere bundles after looping.
result The loop spaces of total manifolds of sphere bundles are homotopy equivalent, except for two special cases.
New invariant measures loop iterations in algebraic structures.
problem Measuring the asymptotic behavior of loop iterations in algebraic structures.
method Introduced sign stability and cluster stretch factor to measure loops.
result Cluster algebraic entropies match cluster stretch factor.
This paper analyzes the impact of loops on bilevel optimization efficiency.
problem The impact of loops on the efficiency of bilevel optimization algorithms.
method Unified convergence analysis and computational complexity characterization for AID-BiO and ITD-BiO with and without loops.
result Loops in bilevel optimization can improve overall efficiency but increase per-step complexity.
The paper finds non-contractible loops of Legendrian tori from knot families.
problem Computing non-contractible loops of Legendrian tori from knot families.
method Using cord algebra of knots to compute Legendrian contact homology.
result Obtained an infinite family of non-contractible loops of Legendrian tori.
Characterizes visibility and geodesic loops in complex domains.
problem Visibility and geodesic loops in complex domains.
method Using quasi-geodesic frames to characterize visibility and geodesic loops.
result Characterizes visibility and existence of geodesic loops in Kobayashi complete hyperbolic and Gromov hyperbolic domains.
Framework learns robust control policies from expert demonstrations.
problem Adversarial robustness and closed-loop generalization in feedback control policies.
method Lipschitz-constrained loss minimization for certified robustness and generalization.
result Finite sample bound on policy learning error and robust closed-loop stability.
Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.
problem Existence and uniqueness of solutions to loop equations in generalized Frobenius manifolds.
method Proves existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
result Existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
Cartan calculus applied to string topology homology.
problem Understanding the structure of free loop spaces.
method Introduced Cartan calculus on loop homology, linked to string topology operations.
result Loop product and bracket behavior under Hodge decomposition.