The paper provides uniform length estimates for trajectories on flat cone surfaces.
problem Estimating the length of trajectories on flat cone surfaces.
method Using self-intersection numbers and constants depending only on the flat metric, the paper focuses on convex flat cone spheres with a positive curvature gap and a fixed number of singularities.
result Uniform two-sided estimates for trajectory lengths on convex flat cone spheres are obtained.
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 C1. Study on overlaps of singular vectors in Gaussian matrix submatrices.
problem Analyzing overlaps of singular vectors in submatrices of Gaussian matrices.
method Utilizes dynamics of singular vectors and specific resolvents for Brownian trajectories.
result Explicit forms for limiting rescaled mean squared overlaps in the bulk of spectra.
We show an example providing a significance in geometric control theory of the existence of the dependence locus of a system of vector fields in particular, the generic appearance of non-trivial singular trajectories embedded in the dependence locus.
Study of ants' movement rules on a 6D space, revealing distribution structures and singular trajectories.
problem Understanding the movement patterns of ants in a 6D space.
method Analyzing mechanical system rules to derive distribution structures and singular trajectories.
result Distributions and singular trajectories of ants' movement rules in a 6D space.
A new neural network captures and explains trajectory patterns.
problem Analyzing complex spatial trajectories in urban planning and neuroscience.
method Composite Signal Neural Networks (CompSNN) combining three interpretable ANN modules.
result CompSNN outperforms individual modules and visualizes useful signal parts.
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…
Bounds on saddle connections on flat spheres with conical singularities.
problem Counting saddle connections on flat spheres with conical singularities.
method Geometry of immersed disks and explicit upper bounds.
result Explicit upper bounds on the number and lengths of saddle connections.
Unified framework for multi-view diffusion geometries using intertwined diffusion trajectories.
problem Constructing multi-view diffusion geometries with flexible view interaction and fusion.
method Intertwined multi-view diffusion trajectories (MDTs) as a class of inhomogeneous diffusion processes.
result Established theoretical properties and derived diffusion distances and embeddings.
We introduce a dynamical-systems approach for the study of the Sard problem in sub-Riemannian Carnot groups. We show that singular curves can be obtained by concatenating trajectories of suitable dynamical systems. As an applications, we positively answer the Sard problem in some classes of Carnot groups.
New method for analyzing learning dynamics in singular models.
problem Challenges in analyzing learning of singular models with no one-to-one parameter space.
method Relative reparameterization technique to extract regular sub-models.
result Demonstrated differences in convergence behavior due to algorithmic and intrinsic aspects.
A generative model with a disentangled representation allows for independent control over different aspects of the output. Learning disentangled representations has been a recent topic of great interest, but it remains poorly understood. We show that even for GANs that do not possess disentangled representations, one c…
In this paper, we define a relative Morse complex for manifold with boundary using the handlebody decomposition of the manifold. We prove that the homology of the relative Morse complex is isomorphic to the relative singular homology. Furthermore, we construct A∞-category structure on the relative Morse complex…
Method estimates number of clusters in Block Markov Chain trajectories.
problem Challenges in choosing number of clusters for sequential data.
method Spectral embedding and density-based clustering.
result Asymptotically consistent method for estimating clusters.
In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of the transform R that arises from this model. As a consequence, we sh…
Study of caustics in Einstein-dust system, showing spacetime singularities and diverging curvature.
problem Understanding caustics and singularities in the Einstein-dust system.
method Established local existence result for spherically symmetric spacetimes containing caustics, constructed from solutions to a PDE problem.
result Obtained spherically symmetric spacetimes with diverging curvature and singular boundary.
A new geometric concept, the dead direction, bridges singular learning theory and information geometry.
problem The gap between singular learning theory and information geometry.
method Introducing the dead direction, a unit vector along degenerating Fisher metric, and showing its KL order can be recovered.
result The KL order of the dead direction can be recovered as the decay rate of the directional Fisher curvature, providing a handle on singular geometry.
This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler ℓ∞ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that…
We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapt…
We study the problem of discriminative sub-trajectory mining. Given two groups of trajectories, the goal of this problem is to extract moving patterns in the form of sub-trajectories which are more similar to sub-trajectories of one group and less similar to those of the other. We propose a new method called Statistica…
We provide a rigorous numerical computation method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system x′=f(x,y,ε),y′=εg(x,y,ε) with one-dimensional slow variable y. Our validation procedure is based on topological tools called isolatin…
Investors' strategic trading affects asset prices, modeled as a game.
problem Investors' trading rates influence asset prices in dynamic markets.
method Model as a non-zero sum singular stochastic differential game, establishing equivalence between best-response and auxiliary control problems.
result Unique Nash equilibrium is deterministic with a closed-form solution.
Paper uses deep imitation learning to predict aircraft trajectories accurately.
problem Inefficient and costly Air Traffic Management system limits predictability.
method Generative Adversarial Imitation Learning framework with trajectory clustering and classification.
result Accurate predictions for entire trajectory stages, pre- and tactical.
SLT reveals how grokking occurs via basin selection in training.
problem Understanding grokking in machine learning models.
method Singular Learning Theory (SLT) to analyze the loss landscape and local learning coefficient (LLC).
result LLC ranks basins by statistical preference, leading to grokking.
Pattern ensembling fills in missing or inaccurate trajectory data.
problem Incompleteness, missing information, and inaccuracies in geolocation data.
method Probabilistically ensemble similar trajectory patterns from the vicinity.
result Reconstructs missing or unreliable trajectory segments effectively.
Study reveals 2020 stock crashes were mostly endogenous, not exogenous.
problem Identifying the cause of the 2020 global stock market crash.
method Applied log-periodic power law singularity (LPPLS) methodology to analyze stock market indexes.
result The 2020 stock market crashes were mostly endogenous, driven by systemic instability.
Study shows magnetic trajectories in Berger spheres are homogeneous.
problem Homogeneity of contact magnetic trajectories in Berger spheres.
method Proved every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
result Contact magnetic trajectories in Berger spheres are homogeneous.
Improved trajectory prediction for team sports using sparse outputs.
problem Challenging to train deep learning models for player trajectory prediction.
method Sparse trajectory prediction and constant acceleration interpolation.
result Interpolation improves performance for all tested models.
WS-II algorithm segments trajectories with high accuracy.
problem Trajectory segmentation for diverse data sources.
method Supervised learning with sliding window analysis.
result WS-II outperforms other methods in all datasets.
A framework clusters vehicle motion trajectories efficiently.
problem Costly manual annotation of vehicle motion data.
method Five-stage framework: align, embed, extract, embed, cluster.
result Framework achieves promising results on real-world dataset.
In this paper we propose a new parameter-free method for trajectory classification which finds the best trajectory partition and dimension combination for robust trajectory classification. Preliminary experiments show that our approach is very promising.
Generative model learns vehicle trajectory distributions for better data generalization.
problem Data sparsity and privacy issues in urban vehicle trajectory analysis.
method Generative adversarial imitation learning framework for urban vehicle trajectory generation.
result TrajGAIL model produces synthetic trajectories similar to real ones, achieving significant performance gains.
Many AI problems, in robotics and other domains, are goal-directed, essentially seeking a trajectory leading to some goal state. In such problems, the way we choose to represent a trajectory underlies algorithms for trajectory prediction and optimization. Interestingly, most all prior work in imitation and reinforcemen…
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.
Analyzing the urban trajectory in cities has become an important topic in data mining. How can we model the human mobility consisting of stay and travel from the raw trajectory data? How can we infer such a mobility model from the single trajectory information? How can we further generalize the mobility inference to ac…
Study on billiard trajectories with fixed bounces.
problem Counting periodic trajectories with specific bounces.
method Analyzes two-dimensional dispersive billiard systems.
result Asymptotic growth of primitive periodic trajectories.
Representation learning of pedestrian trajectories transforms variable-length timestamp-coordinate tuples of a trajectory into a fixed-length vector representation that summarizes spatiotemporal characteristics. It is a crucial technique to connect feature-based data mining with trajectory data. Trajectory representati…
Method predicts NBA players' multi-modal movement trajectories.
problem Understanding NBA players' decision-making during games.
method LSTM-based architecture with multi-modal loss function.
result Method outperforms state-of-the-art in predicting realistic trajectories.
We address the problem of abnormal event detection from trajectory data. In this paper, a new adversarial approach is proposed for building a deep neural network binary classifier, trained in an unsupervised fashion, that can distinguish normal from abnormal trajectory-based events without the need for setting manual d…
Physics-informed diffusion model detects anomalous trajectories in GPS data.
problem Detecting fake GPS trajectories in international waters.
method Physics-informed diffusion model integrating kinematic constraints.
result Higher prediction accuracy and lower error rate for anomaly detection.
Develops MENT for interpreting and detecting changes in network trajectories.
problem Distortion of network geometry and invalidation of temporal comparisons in dynamic network analysis.
method Develops Multiscale Euclidean Network Trajectories (MENT) framework based on second-moment geometry.
result Validates and interprets network trajectories through isotropic normalization and orthogonal transformations.
Gaussian processes model geospatial trajectories with uncertainty.
problem Interpolating and predicting complex spatiotemporal data.
method Gaussian process models trajectories as multidimensional Gaussian distributions.
result Gaussian processes provide a flexible and probabilistic way to interpolate geospatial data.
The paper defines and analyzes conformal trajectories in 3D space forms.
problem Understanding trajectories in curved 3D spaces.
method Defined conformal trajectories and studied their properties in R3, S3, and H3. result Conformal trajectories in S3 and H3 have constant curvature and torsion. Trajectory owner prediction is the basis for many applications such as personalized recommendation, urban planning. Although much effort has been put on this topic, the results archived are still not good enough. Existing methods mainly employ RNNs to model trajectories semantically due to the inherent sequential attri…
We present CoverNet, a new method for multimodal, probabilistic trajectory prediction for urban driving. Previous work has employed a variety of methods, including multimodal regression, occupancy maps, and 1-step stochastic policies. We instead frame the trajectory prediction problem as classification over a diverse s…
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.
Algorithm learns mixtures of Markov chains and MDPs from short trajectories.
problem Learning mixtures of Markov chains and MDPs from short unlabeled trajectories.
method Subspace estimation, spectral clustering, EM algorithm, model estimation, classification.
result 96.6% average accuracy on a mixture of two MDPs in gridworld, outperforming EM algorithm with random initialization.
DETECT clusters mobility behaviors from trajectories using deep learning.
problem Clustering similar mobility behaviors in large, complex trajectory data.
method DETECT uses deep learning to cluster mobility behaviors from trajectories, transforming and summarizing them to identify similar behaviors.
result DETECT effectively clusters mobility behaviors from real-world datasets.