Paper characterizes optimal learning trajectories for high-dimensional nonlinear models.
problem Characterizing optimal learning trajectories in high-dimensional nonlinear models.
method Exploits maximum principle and dynamic programming for an optimal control problem of a gradient system.
result Constructs optimal learning trajectories leading to optimal model parameters.
The paper characterizes Legendre curves on trans-S-manifolds.
problem Characterizing Legendre trajectories on trans-S-manifolds.
method Obtained curvature characterizations and classified Legendre curves.
result Classified Legendre curves with linearly dependent Frenet frame fields.
Paper proposes a new descriptor for early trajectory characterization in matrix iterations.
problem Comparing early behavior of high-dimensional trajectories in nonlinear matrix iterations.
method Develops a two-channel fuzzy coordinate system using F-transform for compact representation.
result The descriptor achieves high R^2 values (mean = 0.6480) in approximating convergence lengths.
Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.
problem Characterizing metabelian distributions and geodesics in sub-Riemannian manifolds.
method Characterization of metabelian distributions in terms of principal bundle structures. Proof of geodesic properties for rank-2 distributions.
result For rank-2 metabelian distributions, geodesics are of class C 1 C^1 C 1 . This work characterizes how data augmentation shapes neural representations.
problem Understanding the impact of data augmentation on neural network representations.
method Embedding neural network hidden representations into a metric space invariant to transformations, analyzing shape-space trajectories.
result Increasing data augmentation strength leads to well-behaved trajectories in the embedded space, and different augmentation types steer representations in distinct directions.
Method detects trajectory outliers using Hodge Laplacian embeddings.
problem Detecting outliers in trajectory data on simplicial complexes.
method Flow-embeddings using Hodge 1-Laplacian of simplicial complexes.
result Classifies trajectories based on topological behavior.
Using movement primitive libraries is an effective means to enable robots to solve more complex tasks. In order to build these movement libraries, current algorithms require a prior segmentation of the demonstration trajectories. A promising approach is to model the trajectory as being generated by a set of Switching L…
TRAKNN detects rare atmospheric trajectories efficiently.
problem Detecting rare atmospheric anomalies over long periods.
method Unsupervised, recurrence-based kNN algorithm.
result Rare trajectories correspond to physical anomalies.
ELM combines machine learning and feature engineering for anomalous diffusion detection.
problem Quantitative characterization of anomalous diffusion from single trajectories.
method Extreme Learning Machine (ELM) combined with feature engineering.
result ELM achieves satisfactory performance in AnDi challenge tasks.
Deep learning models can infer individual trajectories from sparse data.
problem Learning individual dynamics from limited data points.
method Combining variational autoencoders (VAEs) with ordinary differential equations (ODEs) for dynamic modeling.
result Deep learning can recover individual trajectories from sparse data, but requires careful adaptation.
The paper defines symmetries in no-arbitrage markets.
problem Characterizing transformations preserving no-arbitrage.
method Geometric formalization in discrete time models.
result Local characterization of no-arbitrage symmetries.
Modeling continuous movement of entities in latent space for interaction timing.
problem Analyzing timing and frequency of instantaneous interactions.
method Latent position model with continuous trajectories.
result Individual trajectories estimated from interaction data.
Visual observations of dynamic phenomena, such as human actions, are often represented as sequences of smoothly-varying features . In cases where the feature spaces can be structured as Riemannian manifolds, the corresponding representations become trajectories on manifolds. Analysis of these trajectories is challengin…
PhysVarMix predicts diverse urban trajectories with physics constraints.
problem Predicting complex urban agent trajectories with multiple plausible scenarios.
method Physics-informed variational mixture model combining learning and physics constraints.
result Superior performance compared to existing methods on benchmark datasets.
We propose a new approach to the problem of neural network expressivity, which seeks to characterize how structural properties of a neural network family affect the functions it is able to compute. Our approach is based on an interrelated set of measures of expressivity, unified by the novel notion of trajectory length…
Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.
problem Long-term behavior of stochastic gradient descent in non-smooth settings.
method Functional central limit theorem applied to rescaled trajectory of SGD.
result Characterization of long-term fluctuations around the minimizer.
Trajectory-level supervision allows efficient offline reinforcement learning.
problem Offline reinforcement learning
method Developing a statistical theory for offline policy optimization from trajectory-level labels
result Proving a high-probability guarantee of order O ~ ( H 2 C s a ( π ⋆ ) / n ) \widetilde O(H^2\sqrt{C_{sa}(\pi^\star)/n}) O ( H 2 C s a ( π ⋆ ) / n ) The paper shows how data and algorithm interactions affect overparameterized linear regression generalization.
problem Understanding generalization in overparameterized linear regression.
method Introducing data-algorithm compatibility and performing data-dependent trajectory analysis with gradient descent.
result Early stopping iterates lead to better generalization than last-iterate analysis, with weaker restrictions.
Diffusion models' sampling paths lie in a low-dimensional subspace, resembling boomerangs.
problem Understanding the geometric structure of diffusion-based generative models.
method Characterization of deterministic sampling trajectories using low-dimensional subspace and kernel-estimated data modeling.
result Sampling trajectories in diffusion models are confined to a low-dimensional subspace and exhibit a boomerang shape.
Analyzes how quadratic differential trajectories change with variation, proving a wall-crossing formula.
problem Analyzing how the number of trajectories of quadratic differentials changes with variation.
method Proves an analytic wall-crossing formula using Fock-Goncharov coordinates and characterizes birational automorphisms.
result Characterizes certain birational automorphisms and computes Stokes automorphisms.
Proposes a model for generating survival trajectories and data.
problem Generating survival trajectories and data for new feature vectors.
method Autoencoder with specific structure, incorporating weighting scheme for robustness.
result Generates a prototype time-dependent trajectory for objects.
Study risk-controlling prediction sets for single trajectory data from dynamical systems.
problem Performance guarantees for risk-controlling prediction sets in single trajectory data from unknown stochastic dynamical systems.
method Used blocking and decoupling techniques to analyze performance guarantees under different data generating processes.
result Performance guarantees similar to iid setting when data is stationary and contractive, with graceful degradation otherwise.
We study geodesics along a noncompact Kerr-Newman instanton, where the asymptotic geometry is either de Sitter or anti-de Sitter. We use first integrals for the Hamilton-Jacobi equation to characterize trajectories both near and away from horizons. We study the interaction of geodesics with special features of the metr…
Paper analyzes EM algorithm's trajectory in 2MLR, revealing cycloid behavior.
problem Understanding the convergence and trajectory of EM algorithm in 2MLR.
method Explicit closed-form expressions for EM updates, recurrence relation derivation at population level.
result EM iterations lie on a cycloid trajectory, leading to theoretical estimate of convergence exponent.
Characterizes billiard and quasigeodesic flows in polyhedral convex bodies.
problem Characterizing billiard and quasigeodesic flows in polyhedral convex bodies.
method Alexandrov geometry methods.
result Optimal regularity result for convex bodies: billiard dynamics is continuous if boundary is of class C 2 , 1 \mathcal{C}^{2,1} C 2 , 1 . The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.
The paper examines the sampling dynamics of diffusion models using ODEs.
problem Understanding the sampling dynamics of diffusion models.
method Careful inspection of ODE-based sampling of SDEs, revealing structures and relationships.
result Established a theoretical relationship between optimal ODE-based sampling and mean-shift algorithm.
Develops anytime-valid stopping rules for SGD based on observed trajectory.
problem Stopping stochastic gradient descent (SGD) based on observed trajectory.
method Develops anytime-valid confidence sequences for stochastic gradient methods.
result Statistically valid, time-uniform stopping rules for SGD across convex and nonconvex settings.
New method detects OOD samples using neural network trajectories.
problem Lack of comprehensive layer exploration in OOD detection.
method Functional data perspective, analyzing sample trajectories through multi-layer classifier.
result Empirically validated as effective compared to state-of-the-art methods.
New analysis shows FM learns underlying dynamical structure, not just trajectory replay.
problem Understanding whether flow matching models learn transferable dynamical structure or merely replay trajectories.
method Derived velocity field implied by FM objective, characterized as a continuous-time dynamical system.
result FM models can be seen as parametric surrogates of nonparametric solutions, providing strong probabilistic forecasts.
Transformer-based models overfit financial time series data, leading to increased prediction variance.
problem Forecast collapse of transformer-based models under squared loss in financial time series.
method Theoretical analysis and numerical experiments on high-frequency EUR/USD exchange rate data.
result Increased model expressivity in Transformer-based models leads to spurious fluctuations without reducing bias, resulting in higher prediction variance.
The paper proposes a new method for clustering survival data using smoothed log-hazard trajectories.
problem Clustering survival data based on instantaneous risk dynamics.
method Functional Principal Component Analysis applied to B-spline smoothed log-hazard trajectories.
result The proposed method provides an interpretable representation of relative temporal risk dynamics.
The paper develops a new theory to understand deep learning optimization.
problem Understanding the dynamics of optimization in deep learning, especially in the edge of stability regime.
method Developed a central flow differential equation to describe the time-averaged trajectory of oscillatory optimizers.
result Central flows can predict long-term optimization trajectories with high numerical accuracy.
Bi-Mamba model predicts diffusion coefficients and exponents from short data.
problem Characterizing anomalous diffusion in complex systems.
method Bidirectional state-space deep learning architecture.
result Efficient inference of diffusion coefficient and exponent from short trajectories.
TD learning reduces prediction error in Markov chain problems.
problem Estimating value functions in Markov chains with temporal inconsistency.
method Temporal difference learning minimizes temporal inconsistency between successive estimates.
result TD learning can significantly reduce mean-squared error in value estimates.
Researchers improve visualization of neural network loss landscapes.
problem Understanding neural network generalization performance.
method Novel 'jump and retrain' procedure, non-linear dimensionality reduction (PHATE), computational homology.
result Improved visualization and quantification of neural network generalization performance.
The study characterizes infinite Riemann surfaces and their foliations using quadratic differentials.
problem Characterizing infinite Riemann surfaces and their foliations using quadratic differentials.
method Extending Hubbard-Masur theorem to infinite surfaces and analyzing Jenkins-Strebel differentials.
result Density of Jenkins-Strebel differentials and extension of Kerckhoff's formula for Teichmüller metric.
Study characterizes bladder motion using dynamic MRI and statistical analysis.
problem Limited volume coverage in dynamic MRI sequences hinders 3D shape reconstruction.
method 3D dense velocity measurements, LDDMM framework, statistical characterization, mean curvature changes, surface deformation analysis.
result Stable shape descriptor for characterizing bladder surface dynamics.
A 3 3 3 -dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part,…
We introduce a mixed-effects model to learn spatiotempo-ral patterns on a network by considering longitudinal measures distributed on a fixed graph. The data come from repeated observations of subjects at different time points which take the form of measurement maps distributed on a graph such as an image or a mesh. Th…
Characterizes optimal-speed quantum state evolution Hamiltonians.
problem Optimal-speed unitary time evolution of pure and quasi-pure quantum states.
method Construction of the manifold of pure states and isometry with flag manifold, characterization of equigeodesic vectors.
result Hamiltonians generating optimal-speed time evolution are fully characterized by equigeodesic vectors of the flag manifold.
The paper develops a neural network-based classifier for diffusion process drifts.
problem Classifying diffusion processes with distinct drift functions from discrete observations.
method Derives a Bayes rule and constructs a plug-in classifier using neural networks to estimate drifts.
result Establishes convergence rates for misclassification risk, highlighting benefits of diffusion structure.
Develops a numerical algorithm for stochastic impulse control using regression surrogates.
problem Optimal impulse control in stochastic processes.
method Generates statistical surrogates for continuation and intervention functions, recursively trained over simulated state trajectories.
result Demonstrates flexibility and extensibility of the numerical scheme through case studies.
Geometrically characterizes virtual nonlinear nonholonomic constraints using symplectic methods.
problem Characterizing virtual nonlinear nonholonomic constraints geometrically.
method Geometric characterization using symplectic structures and Chetaev equations.
result A unique control law exists to satisfy virtual constraints, and closed-loop dynamics are projections of uncontrolled dynamics.
Paper develops a hybrid DNN approach for RUL prediction with adaptive drift.
problem RUL estimation challenges in practice, especially online update and uncertainty quantification.
method Hybrid DNN approach with Wiener-based-degradation model and adaptive drift. LSTM-CNN for trajectory prediction and Bayesian inference for adaptive drift.
result Superior accuracy in RUL prediction demonstrated on turbofan engines data.
New method characterizes surface quadrilateral layouts as special immersions.
problem Characterize surface quadrilateral layouts mathematically.
method Characterizes quadrilateral layouts as special immersions of a cut representation of the surface into the Euclidean plane.
result Mathematically describes and generalizes integer grid maps.
A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and inve…
Exact simulation method for market impact estimation under various execution strategies.
problem Estimating market impact from observed price trajectories under different execution strategies.
method Conditional simulation of point processes under perturbed intensities.
result Exact, event-driven algorithm for reconstructing counterfactual paths.