A new method improves coordinate descent by adaptively selecting coordinates.
problem Coordinate descent's inefficiency due to checking all coordinates.
method Adaptive multi-armed bandit algorithm to select coordinates.
result Improves convergence of coordinate descent methods.
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.
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.
Differentially private random block coordinate descent improves utility in machine learning.
problem Lack of privacy in classical CD methods when handling sensitive information.
method Proposes a differentially private random block coordinate descent method using sketch matrices and importance sampling.
result Demonstrates improved convergence rates and utility guarantees compared to non-private methods.
Unified analysis of matching pursuit and coordinate descent methods.
problem Optimization of linear spaces using first-order methods.
method Unified analysis of matching pursuit and coordinate descent, providing rates for smooth and strongly convex objectives.
result Unified analysis leading to tightest known rates for steepest coordinate descent and accelerated convergence for matching pursuit.
The paper develops efficient algorithms for solving complex problems using coordinate updates.
problem Solving large or high-dimensional datasets with linear and nonlinear mappings.
method Develops coordinate-friendly operators and algorithms for various applications.
result New algorithms for machine learning, image processing, and optimization problems.
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.
New methods find eigenvectors faster than Lanczos's method.
problem Finding the leading eigenvector efficiently.
method Coordinate-wise methods combining shift-and-invert with linear regression.
result Global convergence with runtime guarantees better than Lanczos's method.
Paper analyzes Hit-and-Run's convergence rates and applies similar methods to randomized Kaczmarz.
problem Quantifying advantages of Hit-and-Run's coordinate-free property.
method Sharp estimates via coupling methods and mixing time bounds.
result Ballistic and superdiffusive convergence rates in certain settings.
Distributed block coordinate descent solves large kernel problems.
problem Large scale kernel learning problems.
method Distributed block coordinate descent for kernel regression and classification.
result Nyström method generally achieves better statistical accuracy than random features.
Coordinate descent (CD) algorithms have become the method of choice for solving a number of optimization problems in machine learning. They are particularly popular for training linear models, including linear support vector machine classification, LASSO regression, and logistic regression. We consider general CD with …
New method improves convergence rates for convex optimization problems.
problem Convex optimization with nonsmooth objectives.
method Randomized coordinate descent with smoothing, acceleration, homotopy, and non-uniform sampling.
result First convergence rate guarantees for coordinate descent methods.
We propose and analyze a new parallel coordinate descent method---`NSync---in which at each iteration a random subset of coordinates is updated, in parallel, allowing for the subsets to be chosen non-uniformly. We derive convergence rates under a strong convexity assumption, and comment on how to assign probabilities t…
We propose a new stochastic coordinate descent method for minimizing the sum of convex functions each of which depends on a small number of coordinates only. Our method (APPROX) is simultaneously Accelerated, Parallel and PROXimal; this is the first time such a method is proposed. In the special case when the number of…
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.
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 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.
A new algorithm solves bilinear saddle-point problems efficiently.
problem Solving bilinear saddle-point problems in optimization.
method Doubly stochastic primal-dual coordinate method.
result The method converges linearly and has lower complexity than existing methods.
New method speeds up optimization over probability measures.
problem High computational overhead in optimizing probability measures.
method Randomized coordinate descent on Wasserstein space.
result Significant speedups over full-gradient methods.
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.
In this paper we develop and analyze Hydra: HYbriD cooRdinAte descent method for solving loss minimization problems with big data. We initially partition the coordinates (features) and assign each partition to a different node of a cluster. At every iteration, each node picks a random subset of the coordinates from tho…
This paper explores coordinates adapted to vector fields on smooth manifolds.
problem Finding coordinates where vector fields are smoother.
method Quantitative techniques from ODEs and PDEs.
result Results on smoothness and regularity of coordinates.
Proposes a neural network method to correct residual distortions in coordinate transformations.
problem Nonlinear and spatially dependent distortions in coordinate transformation models.
method Residual-based neural network approach focusing on systematic distortions.
result The method improves accuracy and stability in challenging conditions.
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 \(d\).
Deep autoencoder finds linear PDE coordinates for nonlinear equations.
problem Discovering linear coordinates for nonlinear PDEs.
method Residual network architecture for finding intrinsic coordinates.
result Deep learning autoencoder transforms nonlinear PDEs into linear ones.
Two policy regularization methods improve coordination in multi-agent reinforcement learning.
problem Challenging to discover successful collective behaviors in multi-agent reinforcement learning.
method Proposes TeamReg and CoachReg methods to promote coordination.
result Improved performance across cooperative multi-agent problems.
Optimal Coordinate Ascent (OCA) improves feature selection in machine learning.
problem Efficiently selecting features in machine learning models.
method Optimal Coordinate Ascent (OCA) for feature selection.
result OCA outperforms previous methods in feature selection and model performance.
New DP-CD method outperforms DP-SGD in solving composite DP-ERM problems.
problem Privacy-preserving machine learning with differential privacy.
method Differentially Private proximal Coordinate Descent (DP-CD) for composite Empirical Risk Minimization (ERM).
result DP-CD outperforms DP-SGD due to larger step sizes and better gradient exploitation.
Method selects interpretable circular coordinates from data.
problem Abstract circular coordinates are hard to interpret.
method Minimum-weight basis problem in vector matroid for selecting interpretable circle-valued coordinates.
result Proves consistency of cochain inner product estimator.
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.
A new sampling method reduces computational cost for high-dimensional log-concave distributions.
problem High computational cost of ULMC in high dimensions.
method Random Coordinate ULMC (RC-ULMC) selects a single coordinate per iteration.
result RC-ULMC is cheaper than classical ULMC, especially in highly skewed and high-dimensional problems.
Efficient method solves large-scale saddle point problems with parallel updates.
problem Large-scale convex-concave saddle point problems with separable structure.
method Stochastic parallel block coordinate descent with adaptive primal-dual updates.
result Significantly better performance than state-of-the-art methods in various applications.
The paper tackles automatic interpretation of manifold coordinates.
problem Finding physical meaning of abstract manifold coordinates.
method Proposes a method to explain embedding coordinates as compositions of functions from a dictionary.
result Demonstrates the effectiveness of the method on data.
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.
CD methods tackle nonconvex optimization with three terms, achieving critical points.
problem Minimizing nonconvex functions with specific structure.
method Developed randomized CD, randomly permuted CD, and accelerated CD methods.
result CD methods converge to critical points with sublinear complexity.
Optimized GPRNN reduces model complexity and overfitting, improving performance.
problem Overfitting in neural networks and high model complexity.
method Gaussian Process Regression - Neural Network hybrid with optimized redundant coordinates.
result Optimized GPRNN achieves lower test set error with fewer terms/neurons.
New method deflates manifolds to visualize high-dimensional data.
problem Failure of nonlinear dimensionality reduction methods on simple manifolds.
method Iterative deflation of differential operators using single-coordinate estimates.
result Empirically, recovers novel embeddings on real-world and synthetic datasets.
New algorithm learns coordinated decisions in loosely-coupled multi-agent systems.
problem Learning coordinated decisions in multi-agent systems with sparse interactions.
method Multi-Agent Thompson Sampling (MATS) for multi-agent multi-armed bandits.
result MATS achieves sublinear regret and outperforms MAUCE on synthetic and real benchmarks.
A new method for integrating without coordinates in geometric calculus.
problem Evaluating integrals in geometric calculus without using coordinates.
method Using the fundamental theorem of calculus repeatedly and cutting manifolds to create boundaries.
result A direct generalization of integration on real variables, leading to practical applications and new connections.
A general class of Newton algorithms on Graßmann and Lagrange-Graßmann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergenc…
Stochastic Gradient Descent (SGD) has become popular for solving large scale supervised machine learning optimization problems such as SVM, due to their strong theoretical guarantees. While the closely related Dual Coordinate Ascent (DCA) method has been implemented in various software packages, it has so far lacked go…
A new dynamical method calculates shear-bend coordinates for surfaces.
problem Computing shear-bend coordinates for twisted SL2C local systems.
method Dynamics-based approach to abelianization of local systems.
result A dynamical recipe for shear-bend parameterization.
The paper extends a proposal for effective twisted superpotentials to higher rank.
problem Describing effective twisted superpotentials from class S theories geometrically.
method Introducing higher rank analogues of spectral networks and spectral coordinates, and finding generating functions.
result The generating functions of the effective twisted superpotentials agree with known results.
In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, by using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surf…
Isometry pursuit identifies orthonormal submatrices from wide matrices.
problem Identifying isometric embeddings from wide matrices.
method A convex algorithm combining normalization and multitask basis pursuit.
result The method identifies isometric embeddings from interpretable dictionaries.
Stochastic dual coordinate ascent (SDCA) is an effective technique for solving regularized loss minimization problems in machine learning. This paper considers an extension of SDCA under the mini-batch setting that is often used in practice. Our main contribution is to introduce an accelerated mini-batch version of SDC…
The paper introduces a method for multi-agent reinforcement learning to coordinate exploration.
problem Sparse rewards in multi-agent settings lead to independent exploration.
method Designing intrinsic rewards that encourage coordination and developing a hierarchical policy.
result The approach accelerates and improves exploration in cooperative multi-agent settings.
A new method uses Coordinate Descent to optimize ResNet networks for private inference.
problem Reducing ReLU count in ResNet networks for private inference.
method Directly optimizing in the discrete domain using Coordinate Descent.
result Our method yields a sparse solution and is state-of-the-art.