The paper studies 1-equivariant harmonic map flow behavior from R² to S².
problem Examining the long-term dynamics of 1-equivariant harmonic map flow.
method Constructing and analyzing global solutions for the flow equation.
result Global solutions exhibit trichotomy in long-time asymptotic behavior.
New Kazdan-Warner problem for equivariant metrics on manifolds.
problem Equivariant scalar curvature functions on manifolds with group actions.
method Established equivariant analogue of Kazdan-Warner trichotomy.
result New class of totally G-positive pairs with positive constant scalar curvature.
The thesis shows how automorphisms of hyperbolic groups can be represented by train track maps.
problem Representing automorphisms of hyperbolic groups using train track maps.
method Using graphs of groups and Bestvina-Handel's irreducible train track maps, the thesis constructs relative train track maps.
result Outer automorphisms of finitely-generated word hyperbolic groups satisfy a dynamical trichotomy.
The article classifies curvature functions on compact manifolds with boundaries.
problem Prescribing scalar and mean curvature functions on compact manifolds with boundaries.
method Classification based on the sign of the first eigenvalue of the conformal Laplacian.
result A 'Trichotomy Theorem' for curvature functions is established.
The paper studies mean curvature flow with contact angles in high-dimensional cylinders.
problem Mean curvature flow with prescribed contact angles in a high-dimensional cylinder.
method Derives uniform-in-time gradient bounds and presents a trichotomy result for asymptotic behavior.
result The solution converges to a translating solution with positive speed when a specific condition is met.
This paper continues our study, initiated in [arXiv:1108.3370], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolic links, we show that the geometric type of these surfaces in the Thurston trichotomy is completely determined by a simple gr…
We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal p…
New methods improve neural directed link prediction across all sub-tasks.
problem Directed link prediction requires handling edge directionality and bidirectionality, not just edge existence.
method Proposes three strategies: Multi-Class Framework, Multi-Objective, and Scalarized approaches.
result Improved performance across all three sub-tasks of directed link prediction.
This paper explores conditions for positive scalar curvature on spin^c manifolds.
problem Conditions for the existence of metrics with positive scalar curvature on spin^c manifolds.
method Introduces a new scalar-valued function and uses it to study positivity conditions.
result Equivalence between positivity of the new scalar function and the old twisted scalar curvature.
New algorithm tackles multiclass transductive online learning with unbounded labels.
problem Characterizing optimal mistake bound for unbounded label spaces.
method Introducing new combinatorial dimensions (Level-constrained Littlestone and Branching dimensions) to characterize online learnability.
result Established trichotomy of possible minimax rates for unbounded label spaces: Θ(T), Θ(logT), or Θ(1). A theory for approximating complex concepts with simple decision trees.
problem Approximating complex concepts with simple decision trees.
method Introducing interpretable approximations, studying binary concept approximation by decision trees.
result A trichotomy of cases for approximating a binary concept by decision trees based on a simple class.
The paper introduces a new learning model that explains practical aspects of machine learning.
problem Understanding how quickly a concept class can be learned from examples in practical scenarios.
method Introducing a new learning model that considers fixed data sources and varying number of training examples.
result There are only three possible rates of universal learning: exponential, linear, or arbitrarily slow.
Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.
problem Understanding how edge subdivision impacts the Perron eigenvalue of tree Ricci matrices.
method Compressing branches into scalar feedback functions via Schur complement, reducing the spectral problem to a one-dimensional Chebyshev equation.
result Edge subdivision can decrease, preserve, or increase the Perron eigenvalue of tree Ricci matrices.
The paper solves scalar curvature problems under conformal deformation for Riemannian manifolds.
problem Solving scalar curvature problems under conformal deformation for Riemannian manifolds.
method Pointwise conformal deformation, Yamabe equation with Dirichlet boundary conditions.
result Positive, smooth solutions to the Yamabe equation with Dirichlet boundary conditions.
Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
Study apple tasting feedback in online binary classification, providing new insights into minimax expected mistakes.
problem Online binary classification with partial feedback (apple tasting).
method Combinatorial analysis, Littlestone dimension, Effective width.
result Established a trichotomy of minimax expected mistakes in the realizable setting.
Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly log-concave and log-convex densities, extensions to p-exponential …
Effective rank rigidity proved for cubulated groups with factor systems.
problem Rank rigidity in cubulated groups with factor systems.
method Exhibiting special pairs of hyperplanes and curtains for skewering.
result Effective form of rank rigidity proved for cubulated groups.
Study expands multiclass classification models with new rates and partial concept classes.
problem Multiclass classification with a bounded number of labels under various conditions.
method Extends traditional PAC model to distribution-dependent and data-dependent learning rates, characterizes optimal rates for universal and partial concept classes.
result Characterizes three types of learning rates (exponential, linear, arbitrarily slow) for fixed distributions and complexity measures for partial concept classes.
This paper studies universal rates of ERM for binary classification under agnostic learning.
problem The challenge of achieving universal rates of ERM for binary classification under agnostic learning.
method The paper explores the agnostic universal rates of ERM for binary classification, revealing three possible rates: e−n, o(n−1/2), or arbitrarily slow. result The paper provides a complete characterization of which concept classes fall into each of the three categories of agnostic universal rates.
This paper analyzes M-estimators under infinite-variance noise in high dimensions.
problem High-dimensional M-estimation with infinite-variance noise.
method Study of the Fenchel conjugate domain and its impact on risk.
result Exact risk of M-estimators under infinite-variance noise is derived.
A deterministic apple tasting learner is developed, confirming a conjecture and providing tight bounds for mistake bounds.
problem Determining the learnability of hypothesis classes in binary online classification with apple tasting feedback.
method Developed a deterministic apple tasting learner and proved tight bounds for mistake bounds.
result Deterministic apple tasting is feasible and provides tight bounds for mistake bounds.
Study on hemisphere threshold for Escobar functional on Riemannian manifolds, revealing mass and boundary invariant behaviors.
problem Analyzing the hemisphere threshold for the Escobar functional on compact Riemannian manifolds.
method Near-threshold landscape organization by boundary invariants, exact evaluation of weighted profile moments, Lyapunov-Schmidt correction, and blow-up analysis.
result At threshold, blow-ups concentrate at umbilic points with vanishing mass and gradient, leading to compactness and hemispherical rigidity.
The paper confirms isoperimetric conjectures on cubes and Gaussian slabs.
problem Isoperimetric inequalities on slabs and cubes.
method Analysis of weighted Riemannian manifolds and product spaces.
result The isoperimetric conjecture on the three-dimensional cube is confirmed for a new range of relative volumes.
Study on 2-valued dynamics on complex plane, showing some dynamics can't be group actions.
problem Whether 2-valued dynamics can be defined by the action of a 2-valued group.
method Construction of examples of dynamics that are or are not group actions.
result Some 2-valued dynamics on complex plane cannot be defined by the action of a 2-valued group.
The paper studies dynamic star-shaped risk measures and their representation.
problem Representing dynamic star-shaped risk measures and their properties.
method Representation theorems for dynamic monetary and star-shaped risk measures.
result Dynamic star-shaped risk measures can be represented as the lower envelope of a family of dynamic convex risk measures.
Study circles to understand dynamics and rigidity in homogeneous spaces.
problem Understanding dynamics and rigidity in infinite-volume homogeneous spaces.
method Addressing four questions about circle packings.
result Highlighting the interplay between dynamics, geometry, and rigidity.
A Riemmanian foliated dynamical system of 3-dimension (RFDS3) is a closed Riemannian 3-manifold with additional structures: foliation, dynamical system. In the context of arithmetic topology, it is a geometric/analytic analogue of an arithmetic scheme with a conjectural dynamical system suggested by C. De…
Paper connects dynamics of mechanical systems to Reeb dynamics.
problem Understanding dynamics in mechanical systems with Poisson structures.
method Using Jacobi bundle metrics and linear Poisson structures.
result Extends classical results on Reeb dynamics to mechanical systems.
Two heuristics solve dynamic multiple travelling salesmen problems.
problem Dynamic routing with unknown customers.
method Balanced dynamic closest vehicle heuristic and balanced dynamic assignment vehicle heuristic.
result Continuous approximation models for strategic dynamic routing.
In this paper we present a theoretical framework for studying coherent acceptability indices in a dynamic setup. We study dynamic coherent acceptability indices and dynamic coherent risk measures, and we establish a duality between them. We derive a representation theorem for dynamic coherent risk measures in terms of …
DOODL learns shared spectral dynamics across related dynamical systems.
problem Learning independent dynamical operators for each system limits discovery of shared structure.
method DOODL learns a dictionary of characteristic spectral dynamics on a manifold of related systems.
result DOODL achieves errors one to two orders of magnitude lower than independent operator estimation methods.
We propose a new class of mappings, called Dynamic Limit Growth Indices, that are designed to measure the long-run performance of a financial portfolio in discrete time setup. We study various important properties for this new class of measures, and in particular, we provide necessary and sufficient condition for a Dyn…
In this paper we present a theoretical framework for determining dynamic ask and bid prices of derivatives using the theory of dynamic coherent acceptability indices in discrete time. We prove a version of the First Fundamental Theorem of Asset Pricing using the dynamic coherent risk measures. We introduce the dynamic …
Dynamical-VAE learns causal dynamics from POMDPs using future information.
problem Learning accurate state representations from partial observations in POMDPs.
method Dynamical Variational Auto-Encoder (DVAE) with hindsight framework.
result DVAE uncovers causal graph more effectively than history-based methods.
Most real world phenomena such as sunlight distribution under a forest canopy, minerals concentration, stock valuation, exhibit nonstationary dynamics i.e. phenomenon variation changes depending on the locality. Nonstationary dynamics pose both theoretical and practical challenges to statistical machine learning algori…
Unified analysis of DLNs using DMFT reveals dynamics of loss convergence and generalization trade-offs.
problem Understanding the overall dynamics of diagonal linear networks (DLNs) in neural network training.
method Dynamical Mean-Field Theory (DMFT) applied to DLNs.
result Derives low-dimensional effective process capturing high-dimensional gradient flow dynamics.
dLDS models neural dynamics as sparse combinations of simpler components.
problem Understanding complex neural dynamics at a population level.
method Proposes a decomposed dynamical system model trained through dictionary learning.
result Model efficiently captures and demix diverse neural dynamics.
Reinforcement learning would enjoy better success on real-world problems if domain knowledge could be imparted to the algorithm by the modelers. Most problems have both hidden state and unknown dynamics. Partially observable Markov decision processes (POMDPs) allow for the modeling of both. Unfortunately, they do not p…
We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…
Framework for quantifying uncertainty in dynamic processes.
problem Quantifying uncertainty in dynamic stochastic processes.
method Define dynamic uncertainty sets and dynamic robust risk measures.
result Dynamic robust risk measures are time-consistent under specific uncertainty sets.
This survey clarifies dynamic network terminology and reviews GNN models for dynamic networks.
problem Ambiguity in dynamic network terminology and lack of GNN models for dynamic networks.
method Established consistent terminology and notation for dynamic networks, reviewed GNN models.
result Comprehensive survey of dynamic graph neural network models.
Framework infers Langevin dynamics from stochastic observations of latent systems.
problem Inferring non-stationary Langevin dynamics from indirect stochastic observations.
method Non-parametric framework explicitly modeling stochastic observation process and non-stationary latent dynamics.
result Correct inference of non-stationary dynamics requires accounting for non-equilibrium states and observation duration.
The paper introduces a dynamic MVP model using high-frequency financial data.
problem Capturing the dynamics of minimum variance portfolio weights in financial markets.
method Imposes autoregressive structure on MVP processes and uses CLIME and LASSO for estimation.
result Proposes DR-MVP model with established asymptotic properties.
The paper extends Vlasov kinetic theory to time-dependent dynamics using cosymplectic and cocontact manifolds.
problem Extending Vlasov kinetic theory to time-dependent dynamics.
method Introducing geometric kinetic theories within cosymplectic and cocontact manifolds.
result Alternative realizations of cosymplectic and cocontact kinetic theories linked via Poisson/momentum maps.
NDS learns dynamical models with prior knowledge, improving accuracy and efficiency.
problem Learning accurate dynamical models with limited data and varying dynamics.
method Neural Dynamical Systems (NDS) integrates prior knowledge in ODEs with neural networks to estimate parameters and predict states.
result NDS achieves higher accuracy and uses fewer samples compared to other methods.
New method learns population dynamics from snapshots, outperforming existing models.
problem Capturing periodic and other dynamical properties of population dynamics.
method Wasserstein Lagrangian Mechanics (WLM) for learning second-order dynamics from observed marginals.
result WLM outperforms existing methods across various dynamics, including vortex dynamics, embryonic development, and flocking.
Paper uses Chebyshev Tensors for accurate dynamic sensitivities and ISDA SIMM computation.
problem Computing dynamic sensitivities and initial margin for financial instruments.
method Uses Chebyshev Tensors in Monte Carlo simulations to compute dynamic sensitivities and ISDA SIMM.
result High accuracy and computational gains for FX swaps and Spread Options.