Study quantifies motion dynamics of ankle sprains using biosensor data.
problem Diagnosing chronic ankle instability (CAI) based on objective biomechanical measures.
method Developed a nonlinear subspace clustering method to learn motion patterns from multi-joint coordination.
result Classification accuracy of >70% on CAI vs. normal controls using leave-one-subject-out cross validation.
Agents learn and control complex mechanical systems through shared memories.
problem Controlling multi-joint dynamical systems.
method Coupled autoregressive active inference agents using Bayesian filtering and minimizing expected free energy.
result Demonstrated learning and control of a double mass-spring-damper system.
Proposes a CVaR-based method to learn robust options in uncertain environments.
problem Learning options from inaccurate models with uncertain parameters.
method Extends CVaR-based policy gradient method to robust Markov decision processes.
result Produces options that perform well in both average and worst cases.
This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single…
Coordinate descent methods usually minimize a cost function by updating a random decision variable (corresponding to one coordinate) at a time. Ideally, we would update the decision variable that yields the largest decrease in the cost function. However, finding this coordinate would require checking all of them, which…
New findings on Kähler manifolds restrict orthogonal coordinates existence.
problem Existence of orthogonal coordinates on Kähler manifolds.
method Algebraic and geometric techniques applied to Kähler manifolds.
result No nontrivial self-dual Kähler 4-manifolds or Ricci-flat Kähler 4-manifolds support orthogonal coordinates.
Defines Fenchel-Nielsen coordinates for SL(3,C) representations.
problem No specific problem stated; coordinates defined for a new context.
method Introduced Fenchel-Nielsen coordinates for mSL(3,C) representations. result Relates to classical and generalized Fenchel-Nielsen coordinates.
Submanifolds of coordinate finite-type were introduced in HV1. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of Δ. In the present study we consider coordinate finite-type surfaces in E^4. We give necessary and sufficient conditions for g…
We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely…
The Dynnikov coordinate system puts global coordinates on the boundary of Teichmüller space of an n--punctured disk. We survey the Dynnikov coordinate system, and investigate how we use this coordinate system to study pseudo--Anosov braids making use of results from Thurston's theory on surface homeomorphisms.
Method constructs orthogonal curvilinear coordinates in constant curvature spaces.
problem Creating orthogonal coordinates in spaces of constant curvature.
method Modification of Krichever's method for Euclidean space, applied to constant curvature spaces.
result Examples of orthogonal coordinate systems on the sphere and hyperbolic plane constructed.
In a previous paper, we parametrized boundary-unipotent representations of a 3-manifold group into SL(n,C) using Ptolemy coordinates, which were inspired by A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, we parametrize representations into PGL(n,C) using shape coordinates which are …
Study on conformal harmonic coordinates on manifolds, proving existence and properties.
problem Existence and properties of conformal harmonic coordinates on Riemannian manifolds.
method Solutions to the conformal Laplace equation, proving up to boundary regularity results, elliptic regularity, and unique continuation results.
result Proves conformal harmonic coordinates are a close conformal analogue of harmonic coordinates.
Symplectic coordinates found on a Hitchin component for a hyperbolic surface.
problem Parametrizing the PSL3(R)-Hitchin component with canonical coordinates. method Proved global Darboux coordinates with half canonical Goldman coordinates.
result Global Darboux coordinates exist for the PSL3(R)-Hitchin component. Accelerates coordinate descent methods for machine learning problems.
problem Slowness of coordinate descent methods in machine learning.
method Extrapolation-based accelerated coordinate descent.
result Significant speed-up in practice compared to existing methods.
We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinat…
Invariants of braids found using shear coordinates in hyperbolic geometry.
problem Finding invariants of braids.
method Using shear coordinates in hyperbolic geometry.
result Developed a method for calculating braids invariants.
Extends Dynnikov coordinates to punctured torus.
problem No specific problem stated; extending coordinates.
method Generalized Dynnikov coordinate system to punctured torus.
result Coordinates defined on punctured torus.
We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are rational curves the construction procedure reduces to solving systems of linear equatio…
Normalizes pseudo-Einstein contact forms for easier analysis.
problem Understanding pseudo-Einstein contact forms.
method Constructing intrinsic CR normal coordinates using parabolic normal coordinates.
result Normal form for pseudo-Einstein contact forms.
DP-SGD can update fewer coordinates while maintaining privacy.
problem How to update fewer coordinates in DP-SGD without losing optimization signal.
method TP-TopK (Two-Phase TopK DP-SGD), a two-phase method for coordinate-sparse private training.
result Private training can update fewer coordinates without losing optimization signal, scaling noise with active dimension \(k\) instead of full dimension \(d\).
Novel deep learning method predicts reaction coordinates and future MD trajectories.
problem Identifying optimal reaction coordinates for chemical reactions.
method Regularized Sparse Autoencoder (RSE) for discovering reaction coordinates and predicting MD trajectory evolution.
result RSE helps in choosing a small but important set of reaction coordinates.
Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat spacetime due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordin…
The study examines the regularity of branched immersions using special coordinate systems.
problem Understanding the regularity of branched immersions and their fundamental elements.
method Development and use of special coordinate systems to express maps with branch points, proving existence and regularity conditions for mean curvature vectors.
result Characterization and existence of special coordinate systems for branch immersions, proving regularity conditions for mean curvature vectors.
This monograph presents a class of algorithms called coordinate descent algorithms for mathematicians, statisticians, and engineers outside the field of optimization. This particular class of algorithms has recently gained popularity due to their effectiveness in solving large-scale optimization problems in machine lea…
This study develops methods to coordinate travel routes to reduce congestion.
problem Coordination of travel routes to reduce urban traffic congestion.
method Developed mathematical approaches to quantify coordination potential and adaptive centroid-based clustering algorithm (ACCA).
result ACCA efficiently forms proper coordination groups for CB-CRM, improving efficiency with minimal performance loss.
Flat coordinates found for algebraic Frobenius manifolds in low dimensions.
problem Understanding algebraic Frobenius manifolds in small dimensions.
method Using reflection representations of finite Coxeter groups, finding flat coordinates of the Frobenius metric.
result Explicit relations between flat coordinates of the Frobenius metric and intersection form for most known examples up to dimension 4.
The article proves Kähler 4-manifolds can only be products of 2 surfaces.
problem Finding orthogonal coordinates on Kähler manifolds.
method Using geometric and algebraic methods to construct orthogonal coordinates.
result Only Kähler 4-manifolds can be products of 2 surfaces.
This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the…
Given a finite collection of C1 vector fields on a C2 manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are real analytic. We give necessary and sufficient, coordinate-free conditions for the existence of such a…
We find necessary and sufficient conditions under which the complex coordinates on a flag manifold of a classical group described in [2] are Bochner coordinates.
Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.
problem Analyzing actions of Dehn twists in geometric group theory.
method Application of Dynnikov coordinates to describe orbits and dynamics of Dehn twists in a thrice-punctured disc.
result The action of Dehn twists has a geometric meaning as a piecewise linear Z2-automorphism. Constructs coordinate systems from spectral curve sheaves.
problem Creating coordinate systems from spectral curve sheaves.
method Finite-gap integration methods for orthogonal curvilinear coordinates.
result Constructs coordinate systems over reducible spectral curves.
Study curvature and torsion in Gaussian distribution's dual coordinate system.
problem Characterize geometric invariants of Gaussian distribution.
method Investigate Riemannian curvature and torsion in a dual coordinate system of Gaussian distribution.
result Explicitly give Amari formulas in the new coordinate system.
Develops DP-SCD for stochastic coordinate descent, making it differentially private.
problem Privacy leak in auxiliary information during stochastic coordinate descent training.
method Develops DP-SCD, leveraging independent noise addition and decoupling/parallelizing coordinate updates.
result Demonstrates competitive performance against DP-SGD with less tuning.
Carnot groups can be polarized if they have specific coordinate systems.
problem Understanding when Carnot groups can be polarized.
method Proving Carnot groups with certain coordinate systems are polarizable.
result Carnot groups with suitable horizontal polar coordinates are polarizable.
It is shown that, in four dimensions, it is possible to introduce coordinates so that an analytic metric locally takes block diagonal form. i.e. one can find coordinates such that gαβ=0 for (α,β)∈S where S=(1,3),(1,4),(2,3),(2,4). We call a coordinate system in which the metric takes this for…
Robustly computes intrinsic coordinates on point clouds using resampling and averaging.
problem Computing intrinsic coordinates on noisy or outlier-prone point clouds.
method Subsample data, vary hyperparameters, cluster candidate embeddings, identify representative embeddings, and average them using Procrustes analysis.
result Robust to noise and outliers, validated on synthetic and real data.
Paper proposes a method to improve circular coordinate representation for detecting changes in high-dimensional datasets.
problem Detecting changes in high-dimensional datasets with preserved topological structures.
method Adapt circular coordinate framework using a generalized penalty function instead of an L2 penalty.
result Circular coordinates with generalized penalty can detect changes in high-dimensional datasets under different sampling schemes.
This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, sm…
Enhanced Teichmüller space for surfaces with decorations and enhancements.
problem Parameterizing and understanding Teichmüller spaces with enhancements and decorations.
method Introduced a new variation of Teichmüller space, constructed parameterization, and introduced lamination space.
result Compatibility of shear coordinates and λ-length coordinates in the new deformation space.
Generalizes Riemann's results on flat coordinates for non-symmetric bilinear forms.
problem Finding flat coordinates for non-symmetric bilinear forms.
method Provides explicit necessary and sufficient conditions for a tensor field of type (0,2) to be flat.
result Explicit conditions for a tensor field to have constant entries in local coordinates.
The study uses isothermic coordinates to analyze space-like surfaces with constant curvature.
problem Global properties of space-like surfaces with constant mean curvature in Lorentz-Minkowski space.
method Isothermic coordinate systems
result Global properties of space-like surfaces with constant mean curvature explored.
Some results on existence of global Chebyshev coordinates on a Riemannian manifold or, more generally, on Aleksandrov surface are proved. For instance, if the positive and the negative parts of integral curvature of a Riemannian manifold M are less than 2πeach, then there exist global Chebyshev coordinates on M. These …
The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it …
Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
Harmonic coordinates for Finsler manifolds prove a theorem but not optimal regularity.
problem Proving the Myers--Steenrod theorem for Finsler manifolds.
method Existence of harmonic coordinates for nonlinear Finsler Laplacian.
result Partial results on optimal regularity for Berwald metrics.
Constructs orthogonal coordinates in curved spaces.
problem Separating variables in curved spaces.
method Explicit construction of orthogonal coordinates and transformations.
result Explicit formulas for Killing tensors and Stäckel matrices.