Smooth Schrödinger Bridges improve trajectory inference by smoothing Gaussian processes.
problem Improving trajectory inference in applications like particle tracking.
method Generalizes Schrödinger Bridge problem to smooth Gaussian processes, solving the problem on phase space.
result The method outperforms existing methods on real datasets.
New algorithm infers smooth trajectories from unpaired snapshots.
problem Inferring smooth trajectories from unpaired snapshots.
method Lifts interpolation problem to phase space, regresses onto explicit acceleration field.
result Our algorithm is competitive or superior to existing methods on benchmark problems.
We consider billiard trajectories in a smooth convex body in Rd and estimate the number of distinct periodic trajectories that make exactly p reflections per period at the boundary of the body. In the case of prime p we obtain the lower bound (d−2)(p−1)+2, which is much better than the previous estimat…
Reinforcement learning optimizes robot trajectories for unknown dynamics.
problem Optimizing robot trajectories for systems with unknown dynamics.
method Curriculum learning with reinforcement learning to generate smooth trajectories.
result Reinforcement learning agent outperforms PID controllers in trajectory tracking.
We give lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. We apply a topological approach based on the calculation of cohomology of certain configuration spaces.
Optimal control trajectories have limited irregularities.
problem Regularity of time-optimal control trajectories in control-affine systems.
method Generic conditions on drift and controlled vector field are used to prove smoothness out of a countable set of times, up to K-th order iterated singularities.
result Control trajectories are smooth out of a countable set of times, with singularities limited to K-th order iterated.
FLUID uses flows to unify filtering and smoothing for complex systems.
problem Bayesian filtering and smoothing for high-dimensional nonlinear systems.
method FLUID encodes observation histories into a fixed summary statistic, using flows for filtering and smoothing.
result FLUID provides accurate approximations of filtering and smoothing distributions.
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 investigates how neural network weights evolve to monitor training progress.
problem Monitoring the training progress of neural networks in a cost-effective manner.
method Investigates the evolution of neural network weights in weight space.
result DNN models evolve on unique, smooth trajectories in weight space that can be used to track training progress.
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.
New algorithm speeds up RNN time series prediction by filtering noise.
problem Predicting smooth trajectories from noisy time series data.
method Analyzed RNN dynamics to propose an efficient noise filtering algorithm.
result Significant speedup in predictive process without accuracy loss.
Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (eg. direction, shape) or temporal (eg. speed) features of trajectories rather than trajectory's representation as a whole. This work is about empirically supported mathematical ideas behind splitting and merging…
In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in Rn. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiar…
Framework for continuous-time network data representation learning.
problem Learning reliable representations of dynamic network interactions.
method Three-stage process: intensity estimation, projection learning, evolving node representation construction.
result Trajectories satisfy structural and temporal coherence, providing robust inference.
Framework learns continuous dynamics from sparse trajectories.
problem Learning dynamics from sparsely sampled and high-dimensional trajectories.
method Interpolative Multi-Marginal Flow Matching (IMMFM) framework.
result IMMFM outperforms existing methods in forecasting and downstream tasks.
Develops a method to infer cell trajectories from RNA sequencing data.
problem Inferring cell trajectories from single cell RNA-sequencing data.
method Entropy-regularized optimal transport for global optimization.
result Proves and implements a method to recover ground truth trajectories from limited samples.
Dan Reznik discovered conserved quantities for ellipses using billiard maps.
problem Conservation laws in periodic billiard trajectories.
method Non-standard generating function for the billiard ball map.
result Proved identities valid for all smooth convex billiard tables.
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.
NESS improves neighbor embedding for smooth cell-state transitions in single-cell data.
problem Challenges in extracting smooth, low-dimensional representations from noisy single-cell data.
method Builds on PCS framework to develop NESS, a stable machine learning approach.
result NESS consistently yields useful biological insights across diverse single-cell datasets.
Lower bounds on periodic Finsler billiard trajectories in convex hypersurfaces.
problem Estimating the number of periodic Finsler billiard trajectories.
method Morse and Lusternik-Schnirelmann theories applied to extremal polygons inscribed in a smooth closed hypersurface.
result For prime r≥3, the number of r-periodic Finsler billiard trajectories is not less than (r−1)(d−2)+1. Billiard trajectories and geodesics are closely related geometrically.
problem Understanding the relationship between billiard trajectories and geodesics on surfaces.
method Establishing mutual approximation results for billiard trajectories and geodesic segments on surfaces.
result For Riemannian billiard tables, there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory.
Smooths GPS data with splines for noisy, irregularly sampled data.
problem Noisy, irregularly sampled GPS data with non-Gaussian noise.
method Smoothing splines with chosen spline order and tension parameter, allowing for non-Gaussian noise and outliers.
result Effective smoothing and interpolation of GPS data.
We give elementary constructions of manifold with corner structures and associative gluing maps on compactifications of spaces of infinite, half infinite, and finite Morse flow lines.
This work enables UAVs to autonomously form desired trajectories without needing a central plan.
problem Autonomous formation of complex trajectories in UAVs.
method Decentralized control system using geometric embeddings.
result Quadcopters self-organize into desired trajectories while maintaining separation.
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.
ARPs improve exploration and sample efficiency in continuous control tasks.
problem Limited exploration in continuous control tasks leading to low sample efficiency.
method Introduce autoregressive policies (ARPs) with temporally coherent standard normal distributions.
result ARPs enhance exploration and sample efficiency in both simulated and real-world domains.
This paper proposes a new algorithm for learning guidance rewards in RL.
problem Long-term temporal credit assignment in sparse or delayed reward environments.
method Surrogate RL objective with trajectory-space smoothing to learn guidance rewards.
result Guidance rewards can be learned without additional neural networks and have intuitive interpretation.
We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points A,B such that no finite set of points can block all billiard trajectories from A to B.
Bayesian inference models failure distributions in autonomous systems.
problem Estimating the distribution of failures in complex systems.
method Bayesian inference using system dynamics rollouts and gradient computation.
result Improves sample efficiency and parameter space coverage in autonomous systems.
This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric c…
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.
Study particle dynamics in non-differentiable fractal spaces.
problem Understanding motion in non-smooth, probabilistic geometries.
method Use fiber bundle theory to characterize multivalued geodesic trajectories.
result Developed a hybrid theory combining surface and stochastic process theories.
The paper is an informal report on joint work with Stefan Haller on Dynamics in relation with Topology and Spectral Geometry. By dynamics one means a smooth vector field on a closed smooth manifold; the elements of dynamics of concern are the rest points, instantons and closed trajectories. One discusses their counting…
KOMET identifies Koopman operators from model parameter trajectories to adapt to evolving data distributions.
problem Adaptation of parametric models to non-stationary environments.
method Data-driven framework using Koopman operator identification and Extended Dynamic Mode Decomposition (EDMD).
result KOMET achieves high autonomous-rollout accuracies of 0.981 to 1.000 over 100 time steps on various drifting datasets.
The study generalizes Santalo's formula and shows stability of trapping sets in Riemannian manifolds.
problem Stability of trapping sets in Riemannian manifolds under smooth perturbations of obstacles.
method Generalization of Santalo's formula applied to billiard trajectories in the exterior of obstacles.
result The measure of the set of trapped points depends continuously on perturbations of the obstacle.
vLGP recovers neural dynamics from spike trains, improving prediction and capturing complex patterns.
problem Recovering latent neural trajectories from noisy spike trains is challenging.
method vLGP combines generative model, history-dependent point process observation, and smoothness prior.
result vLGP achieves higher performance in predicting omitted spike trains and capturing neural dynamics.
This study models target trajectories using stochastic processes for efficient tracking.
problem Efficiently modeling and predicting target trajectories in continuous time.
method Decomposes trajectory modeling into deterministic and stochastic components using Gaussian or Student's-t processes. result Demonstrates superior performance in tracking maneuvering targets compared to existing methods.
3MSBM learns smooth trajectories from multiple snapshots.
problem Capturing long-range temporal dependencies in complex systems.
method Lifts dynamics to phase space, generalizes stochastic bridges to multi-marginal conditional problems, learns transport maps preserving intermediate marginals.
result Significantly improves convergence and scalability in capturing complex dynamics.
Bayesian Neural ODEs improve vessel trajectory prediction with better uncertainty estimates.
problem Challenges in predicting vessel trajectories from irregular AIS data.
method Adopted a Gaussian process (GP) kernel-based prior on the vector field evaluated at measurement points, combined with probabilistic multiple shooting for long trajectories.
result Improved accuracy and uncertainty quantification in vessel trajectory predictions.
We give lowed bounds on the number of periodic trajectories in strictly convex smooth billiards in Rm+1 for m≥3. For plane billiards (when m=1) such bounds were obtained by G. Birkhoff in the 1920's. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik - Schir…
We give topological lower bounds on the number of periodic and closed trajectories in strictly convex smooth billiards. We use variational reduction admitting a finite group of symmetries and apply topological approach based on equivariant Morse and Lusternik - Schnirelman theories. The paper continues results publishe…
Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
problem Accelerating convex optimization
method Hamiltonian dynamics
result Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
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.
Novel method for SDE calibration from sparse data using neural flows.
problem Calibrating SDEs from sparse, noisy observations.
method Characterization of posterior SDE using neural networks trained to solve a PDE with multiplicative updates.
result Significant improvement in scalability and accuracy compared to classical methods.
Generalizes smoothness conditions for optimization methods.
problem Optimization under non-uniform smoothness conditions.
method Develops a new analysis technique for bounding gradients.
result Obtains convergence rates for gradient descent and Nesterov's method.
When the Poincaré map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby trajectories in finite time. This result shows that the long-term behavior of a hybrid model…
CVF learns stable dynamical systems from trajectories.
problem Stable dynamical systems from sampled trajectories.
method Constructs vector fields with controlled contraction and curvature using convex optimization.
result CVF learns stable dynamical systems explicitly controlled by curvature and contraction.
A new method for generating samples without training, using smoothed score matching.
problem Generating samples efficiently and without training.
method Moment-matched score-smoothed overdamped Langevin dynamics (MM-SOLD).
result The method enables fast, robust, training-free sampling with competitive sample fidelity and diversity.