arXiv research

Smooth Schrödinger Bridges improve trajectory inference by smoothing Gaussian processes.

We consider billiard trajectories in a smooth convex body in $\mathbb R^d$ 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…

In this paper, we propose a reinforcement learning-based algorithm for trajectory optimization for constrained dynamical systems. This problem is motivated by the fact that for most robotic systems, the dynamics may not always be known. Generating smooth, dynamically feasible trajectories could be difficult for such sy…

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.

We consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set o…

FLUID uses flows to unify filtering and smoothing for complex systems.

The paper proposes a new method for clustering survival data using smoothed log-hazard trajectories.

The paper investigates how neural network weights evolve to monitor training progress.

Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.

New algorithm speeds up RNN time series prediction by filtering noise.

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 ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiar…

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…

Framework for continuous-time network data representation learning.

Framework learns continuous dynamics from sparse trajectories.

Develops a method to infer cell trajectories from RNA sequencing data.

Develops anytime-valid stopping rules for SGD based on observed trajectory.

NESS improves neighbor embedding for smooth cell-state transitions in single-cell data.

A comprehensive methodology is provided for smoothing noisy, irregularly sampled data with non-Gaussian noise using smoothing splines. We demonstrate how the spline order and tension parameter can be chosen a priori from physical reasoning. We also show how to allow for non-Gaussian noise and outliers which are typical…

Billiard trajectories and geodesics are closely related geometrically.

We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dimensional Finsler space with possibly irreversible Finsler metric. An example of such a system is a billiard in a sufficiently weak magnetic field. The $r$-periodic Fin…

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.

PhysVarMix predicts diverse urban trajectories with physics constraints.

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 $B$.

This paper proposes a new algorithm for learning guidance rewards in RL.

Bayesian inference models failure distributions 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.

Study particle dynamics in non-differentiable fractal spaces.

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.

When governed by underlying low-dimensional dynamics, the interdependence of simultaneously recorded population of neurons can be explained by a small number of shared factors, or a low-dimensional trajectory. Recovering these latent trajectories, particularly from single-trial population recordings, may help us unders…

This study models target trajectories using stochastic processes for efficient tracking.

Dan Reznik found, by computer experimentation, a number of conserved quantities associated with periodic billiard trajectories in ellipses. We prove some of his observations using a non-standard generating function for the billiard ball map. In this way, we also obtain some identities valid for all smooth convex billia…

Bayesian Neural ODEs improve vessel trajectory prediction with better uncertainty estimates.

3MSBM learns smooth trajectories from multiple snapshots.

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…

We give lowed bounds on the number of periodic trajectories in strictly convex smooth billiards in $\R^{m+1}$ for $m\ge 3$. 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…

Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.

The paper develops a new theory to understand deep learning optimization.

Novel method for SDE calibration from sparse data using neural flows.

Generalizes smoothness conditions for optimization methods.

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…

Reinforcement learning algorithms rely on exploration to discover new behaviors, which is typically achieved by following a stochastic policy. In continuous control tasks, policies with a Gaussian distribution have been widely adopted. Gaussian exploration however does not result in smooth trajectories that generally c…

We present a probabilistic language model for time-stamped text data which tracks the semantic evolution of individual words over time. The model represents words and contexts by latent trajectories in an embedding space. At each moment in time, the embedding vectors are inferred from a probabilistic version of word2ve…

A new method for generating samples without training, using smoothed score matching.

VLBM learns MDP transitions from limited data, improving OPE performance.

Let $T\subset \R^{m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partial T$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed intial point $A\in X$, a prescribed final point $B\in X$ and make a prescribed number $n$ of reflections at the bo…