New algorithm reduces sample complexity for sparse linear regression.
problem Sparse linear regression with correlated covariates and approximate dependencies.
method Polynomial-time algorithm that adapts the Lasso to tolerate approximate dependencies.
result Achieves near-optimal sample complexity for constant sparsity and ill-conditioned covariates.
Several methods have been recently proposed for estimating sparse Gaussian graphical models using ℓ1 regularization on the inverse covariance matrix. Despite recent advances, contemporary applications require methods that are even faster in order to handle ill-conditioned high dimensional modern day datasets. I…
Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. They have many applications in the field of Computer Experiments, in particular to perform sensitivity analysis, adaptive design of experiments and global optimization. Nearly all of the applications of GPs require the …
Developed shrinkage methods for Poisson regression models with experts to handle multicollinearity.
problem Multicollinearity in Poisson regression models with experts.
method Ridge and Liu-type shrinkage methods.
result Shrinkage methods offer more reliable estimates for coefficients in multicollinearity.
New method for private linear regression under privacy constraints, achieving optimal rates.
problem Statistical complexity of private linear regression under unknown, ill-conditioned covariates.
method Information-Weighted Regression method
result Optimal convergence rates for both central and local privacy models.
The paper addresses ill-conditioning in large spatial data, proposing solutions for prediction and likelihood estimation.
problem Ill-conditioning of the kernel matrix in large spatial data sets.
method Introduction of various optimality criteria and solutions for managing large spatial data.
result Solutions for managing large spatial data, addressing ill-conditioning and improving prediction and likelihood estimation.
With the advent of massive data sets much of the computational science and engineering community has moved toward data-intensive approaches in regression and classification. However, these present significant challenges due to increasing size, complexity and dimensionality of the problems. In particular, covariance mat…
New methods improve portfolio risk minimization by estimating covariance matrix more accurately.
problem Uncertainty in estimating covariance matrix leads to unreliable hedge trades.
method Proposes two new estimators of the inverse covariance matrix using l2 and l1 norms.
result Portfolio formed using proposed estimators achieves substantial risk reduction and improved returns.
The global minimum-variance portfolio is a typical choice for investors because of its simplicity and broad applicability. Although it requires only one input, namely the covariance matrix of asset returns, estimating the optimal solution remains a challenge. In the presence of high-dimensionality in the data, the samp…
Scalable method completes ill-conditioned matrices from few samples.
problem Matrix completion from few samples for ill-conditioned matrices.
method Iterative algorithm combining IRLS, smoothing Newton, and proximal gradient methods.
result Local quadratic convergence rate and well-conditioned linear systems.
We orthogonalize the NSS model to condition and diagnose its ill-conditioned parameters.
problem The ill-conditioning of the NSS model's design matrix.
method Exact orthogonal reparametrization via QR decomposition.
result Orthogonalization isolates the conditioning structure and maintains fit uncertainty.
New findings on optimization landscape of Toeplitz covariance estimation.
problem Understanding the geometry of the Gaussian maximum-likelihood objective for Toeplitz covariance estimation.
method Overparameterized Carathéodory representation of positive definite Toeplitz covariance matrices, focusing on both amplitudes and frequencies.
result Joint optimization of amplitudes and frequencies leads to a benign population landscape, allowing for global recovery of the true Toeplitz covariance.
Improves numerical solution of ill-conditioned linear systems for machine learning.
problem Wastefulness and instability in solving ill-conditioned linear systems.
method autonugget combines Richardson extrapolation to determine the solution of the ill-conditioned system, improving accuracy over a single nugget.
result Improves accuracy of numerical solution of ill-conditioned linear systems.
Paper explores challenges in training PINNs and loss landscape effects.
problem Challenges in training Physics-Informed Neural Networks (PINNs) due to loss landscape issues.
method Examined gradient-based optimizers Adam, L-BFGS, and their combination Adam+L-BFGS, and introduced NysNewton-CG (NNCG).
result Adam+L-BFGS outperforms other optimizers, and NysNewton-CG significantly improves PINN performance.
SAPPHIRE tackles ill-conditioned rERM problems with faster convergence.
problem Ill-conditioned objectives and non-smooth regularizers in rERM.
method Sketch-based preconditioning and scaled proximal mapping.
result Achieves condition-number-free linear convergence.
New algorithm clusters Gaussian mixtures with unknown covariance efficiently.
problem Clustering data from a mixture of Gaussians with unknown covariance.
method Developed an efficient spectral algorithm based on a Max-Cut integer program.
result Achieves optimal misclassification rate with quadratic sample size.
Improved LDA using a nonlinear covariance estimator for better performance.
problem Inefficient LDA when data covariance is ill-conditioned.
method Regularized LDA with a positive semidefinite ridge-type estimator of the inverse covariance matrix.
result The proposed NL-RLDA classifier outperforms state-of-the-art methods across multiple datasets.
This paper analyzes adaptive gradient algorithms for better performance in ill-conditioned problems.
problem Poor performance of standard stochastic gradient algorithms in ill-conditioned problems.
method Non-asymptotic analysis of adaptive gradient algorithms (Adagrad and Stochastic Newton) for strongly convex objectives.
result Theoretical analysis and adaptation to practical applications like linear regression and regularized GLM.
Improved convergence for overparameterized low-rank matrix sensing.
problem Overparameterized low-rank matrix sensing with unknown rank and ill-conditioning.
method ScaledGD(λ) - preconditioned gradient descent method. result ScaledGD(λ) converges at a constant linear rate after a logarithmic number of iterations. Non-Gaussian component analysis (NGCA) is an unsupervised linear dimension reduction method that extracts low-dimensional non-Gaussian "signals" from high-dimensional data contaminated with Gaussian noise. NGCA can be regarded as a generalization of projection pursuit (PP) and independent component analysis (ICA) to mu…
Scaled gradient descent improves matrix recovery for ill-conditioned matrices with optimal sampling complexity.
problem Recovering low-rank matrices from limited measurements efficiently and accurately.
method Scaled gradient descent (ScaledGD) with optimal sample complexity and improved iteration complexity.
result ScaledGD achieves optimal sample complexity and improved iteration complexity for ill-conditioned matrices.
ScaledGD accelerates ill-conditioned low-rank estimation.
problem Slow convergence of gradient descent in ill-conditioned problems.
method Scaled gradient descent (ScaledGD) with preconditioning.
result Linear convergence rate independent of condition number.
Paper proposes an efficient online Newton method with Nesterov's acceleration for streaming data.
problem Efficient inference of online Newton methods with robustness to noise and ill-conditioning.
method Online Newton method with Hessian averaging and Nesterov's accelerated sketch-and-project solver.
result Global almost-sure convergence and asymptotic normality of the last iterate with non-asymptotic convergence guarantees.
New framework uses score-based priors to solve ill-conditioned polynomial equations, improving signal recovery from noisy data.
problem Recovering signals from low-order moments in inverse problems, especially ill-conditioned polynomial equations.
method Integrates score-based diffusion priors with moment-based estimators to regularize and solve nonlinear inverse problems.
result Diffusion priors improve recovery from third-order moments and make super-resolution MTD feasible.
Preconditioned SGD accelerates convergence for ill-conditioned huge-scale matrix completion.
problem Recovering a low-rank matrix from incomplete data with high condition number.
method Preconditioned Stochastic Gradient Descent (SGD) for huge-scale online optimization.
result Preconditioned SGD converges to ε-accuracy in O(log(1/ε)) iterations, compared to O(κlog(1/ε)) for unpreconditioned SGD.
Proposes an efficient shrinkage path for ridge regression.
problem Ill-conditioned data in linear models.
method A new generalized ridge regression shrinkage path that minimizes MSE risk.
result The path is as short as possible while maintaining optimal trade-off.
Recently, Neural networks have seen a huge surge in its adoption due to their ability to provide high accuracy on various tasks. On the other hand, the existence of adversarial examples have raised suspicions regarding the generalization capabilities of neural networks. In this work, we focus on the weight matrix learn…
Kaczmarz++ accelerates convergence for ill-conditioned systems.
problem Solving ill-conditioned linear systems efficiently.
method Adaptive momentum acceleration, Tikhonov-regularized projections, and memoization.
result Kaczmarz++ converges faster than Krylov methods on ill-conditioned systems.
This paper investigates the control of an ML component within the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) devoted to black-box optimization. The known CMA-ES weakness is its sample complexity, the number of evaluations of the objective function needed to approximate the global optimum. This weakness is…
Regularized LAEs learn principal components efficiently.
problem Learning optimal linear representations with LAEs.
method Proper regularization schemes (non-uniform ℓ2 and nested dropout).
result Convergence to optimal representation is slow due to ill-conditioning.
RL accelerates portfolio optimization and option pricing by dynamically adjusting preconditioner sizes.
problem Large linear systems in portfolio optimization and option pricing lead to slow convergence.
method Reinforcement Learning (RL) dynamically adjusts block-preconditioner sizes to accelerate convergence.
result RL-driven solver significantly reduces computational cost and accelerates convergence.
Improved subgradient method tackles ill-conditioned composite optimization problems.
problem Slow convergence of subgradient method for composite optimization problems.
method Preconditioned subgradient method with Levenberg-Marquardt approach.
result Linear convergence rate for composite optimization problems under mild conditions.
New lower bounds show sparse recovery is hard even with multiple preconditioners.
problem Sparse recovery with ill-conditioned designs is hard for certain algorithms.
method Constructing a single signal distribution that multiple preconditioned Lasso programs fail on.
result Standard sparse random designs are robust to erasures, aiding sparse recovery.
Dual-space sampling tackles ill-conditioned inverse problems with Bayesian methods.
problem Bayesian inference in constrained inverse problems with ill-conditioned solutions.
method Dual-space posterior sampling using ADMM and SVGD.
result Well-calibrated uncertainty estimates and posterior contraction with increasing data.
New research shows how preconditioning can solve sparse linear regression problems efficiently.
problem Efficiently solving sparse linear regression problems without restrictive conditions.
method Preconditioned Lasso approach to solve sparse linear regression problems.
result Preconditioning can solve a large class of sparse linear regression problems nearly optimally.
This paper accelerates distributed convex optimization by mitigating ill-conditioning issues.
problem Distributed convex optimization with ill-conditioned aggregate cost functions.
method Iterative pre-conditioning technique to improve convergence rate and stability.
result The proposed algorithm converges linearly with improved convergence rate and superlinearly under certain conditions.
Paper shows linear convergence of ISTA and FISTA for ill-conditioned images.
problem Solving linear inverse problems with sparse representation in signal and image processing.
method Revisits iterative shrinkage-thresholding algorithms (ISTA) and improves their convergence properties.
result Linear convergence of ISTA and FISTA for strongly convex smooth parts, even in ill-conditioned cases.
A new optimization method reduces memory and compute requirements for deep learning.
problem Memory and compute constraints in second-order stochastic optimizers for deep learning.
method Proposes KrAD, a novel factorization to approximate inverse Fisher matrix without inversion, leading to KrADagrad.
result Improves performance over Shampoo for 32-bit precision and comparable/generalization on real datasets.
AI-driven framework optimizes MCMC-based preconditioners for faster linear system solving.
problem Slow convergence of Krylov subspace solvers for ill-conditioned matrices.
method Graph neural surrogate and Bayesian optimization for AI-tuned MCMC parameters.
result 50% reduction in iterations to convergence on unseen system.
New algorithms solve nonconvex-nonconcave minimax optimization problems.
problem Solving minimax optimization problems in machine learning.
method Two novel Newton-type algorithms for nonconvex-nonconcave minimax optimization.
result Proved local convergence at strict local minimax points.
Bayesian regularization tackles collinearity in large-scale systems with correlated inputs.
problem Collinearity in large-scale linear systems identification due to correlated inputs.
method Bayesian regularization with stable spline covariance and Markov chain Monte Carlo scheme.
result Efficient reconstruction of impulse responses with high correlation among inputs.
ScaledGD improves gradient descent for ill-conditioned low-rank matrix estimation.
problem Efficiently solving ill-conditioned low-rank matrix estimation problems.
method Scaled Gradient Descent (ScaledGD) with adaptive pre-conditioners.
result Linear convergence rate independent of condition number, low per-iteration cost.
Paper improves TD(0) convergence rate with LFA, i.i.d. samples, and averaging.
problem Improving convergence rate of TD(0) with linear function approximation.
method Polyak-Juditsky averaging, i.i.d. samples, strong mixing assumption.
result Established a new convergence rate for Mean-Square Error (MSE) of approximated function.
The total variation (TV) penalty, as many other analysis-sparsity problems, does not lead to separable factors or a proximal operatorwith a closed-form expression, such as soft thresholding for the ℓ_1 penalty. As a result, in a variational formulation of an inverse problem or statisticallearning estimation, it l…
Gaussian processes (GPs) provide a powerful non-parametric framework for reasoning over functions. Despite appealing theory, its superlinear computational and memory complexities have presented a long-standing challenge. State-of-the-art sparse variational inference methods trade modeling accuracy against complexity. H…
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.
In this paper, we study large-scale convex optimization algorithms based on the Newton method applied to regularized generalized self-concordant losses, which include logistic regression and softmax regression. We first prove that our new simple scheme based on a sequence of problems with decreasing regularization para…
A new method for robust matrix completion overcomes limitations of existing approaches.
problem Robust matrix completion from corrupted entries, especially under overparameterization and ill-conditioning.
method Factorization-based iterative algorithm combining Gauss-Newton linearization and outlier removal.
result Theoretical guarantees of exact recovery for suitable assumptions.