New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.
problem Comparative analysis of regularization norms in ill-posed problems.
method Small noise analysis framework for Tikhonov and RKHS regularizations.
result Optimal convergence rates achieved with adaptive fractional RKHS, but hyper-parameters decay too fast.
A new method detects small holes in noisy data.
problem Detecting small holes in high-density regions from noise.
method Robust Density-Aware Distance (RDAD) filtration, incorporating distance-to-measure concept.
result The RDAD filtration prolongs the persistences of small holes, making them distinguishable from noise.
Principal Component Analysis (PCA) has wide applications in machine learning, text mining and computer vision. Classical PCA based on a Gaussian noise model is fragile to noise of large magnitude. Laplace noise assumption based PCA methods cannot deal with dense noise effectively. In this paper, we propose Cauchy Princ…
Paper tackles label noise in large datasets, purifying noisy data with a nonparametric framework.
problem Label noise in large-scale datasets with coarse labels.
method Develops a model-agnostic nonparametric framework for classification.
result Framework purifies noisy data using a small clean dataset and manages ambiguous samples.
Given a matrix of observed data, Principal Components Analysis (PCA) computes a small number of orthogonal directions that contain most of its variability. Provably accurate solutions for PCA have been in use for a long time. However, to the best of our knowledge, all existing theoretical guarantees for it assume that …
Supervised training of deep learning models requires large labeled datasets. There is a growing interest in obtaining such datasets for medical image analysis applications. However, the impact of label noise has not received sufficient attention. Recent studies have shown that label noise can significantly impact the p…
Optimizes variance reduction in Heston model using large and moderate deviations.
problem Improving variance reduction in stochastic volatility models.
method Large and moderate deviations theory applied to Heston model.
result Derives closed-form solutions for optimal change of measure.
This paper aims to address two fundamental challenges arising in eigenvector estimation and inference for a low-rank matrix from noisy observations: (1) how to estimate an unknown eigenvector when the eigen-gap (i.e. the spacing between the associated eigenvalue and the rest of the spectrum) is particularly small; (2) …
New SDP algorithm recovers large clusters in SBM with small clusters of any size.
problem Graph clustering in SBM with large and small clusters.
method Semidefinite programming (SDP) with novel techniques to handle small clusters.
result Proves exact recovery of large clusters regardless of small cluster sizes.
Quantum neural networks improve causal inference in biomedical studies, especially for small samples.
problem Addressing selection bias in comparing surgical techniques using observational data.
method Developed QNN-based propensity score models focusing on four key covariates (Age, Sex, Stage, BMI). Employed a linear ZFeatureMap for data encoding, SummedPaulis for predictions, and CMA-ES for optimization. Integrated noise modeling to enhance predictive stability.
result QNNs, particularly with noise-aware strategies, outperformed classical models in small samples, achieving AUC up to 0.750 for n=100.
We present a theoretical analysis of the training process for a single-layer GAN fed by high-dimensional input data. The training dynamics of the proposed model at both microscopic and macroscopic scales can be exactly analyzed in the high-dimensional limit. In particular, we prove that the macroscopic quantities measu…
Developed LQ MFG theory with common noise, proving existence and uniqueness.
problem Linear-quadratic mean field games with common noise.
method Coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces.
result Existence and uniqueness of solutions for small and arbitrary finite time horizons.
SGD with machine learning noise converges to global minimum exponentially fast.
problem Optimizing machine learning models with stochastic gradient descent.
method Analysis of SGD with machine learning noise, focusing on energy landscapes and gradient noise.
result SGD converges to the global minimum exponentially fast under certain conditions.
SGD noise has no bias advantage in online learning, contrary to offline learning.
problem The role of SGD noise in online learning.
method Extensive empirical analysis of image and language data.
result Small batch sizes do not confer any implicit bias advantages in online learning.
Noise makes learning linear thresholds hard, but algorithms can still learn near-optimal thresholds.
problem Learning linear thresholds in noisy data.
method Exploiting natural assumptions on data-generating process.
result Efficient learning of near-optimal linear thresholds is still possible with small data even in the presence of noise.
Neuroimaging studies produce gigabytes of spatio-temporal data for a small number of participants and stimuli. Rarely do researchers attempt to model and examine how individual participants vary from each other -- a question that should be addressable even in small samples given the right statistical tools. We propose …
A new data-adaptive prior stabilizes kernel learning in operators.
problem Learning kernels in operators from data is ill-posed due to nonlocal dependence.
method Introduces a data-adaptive prior to stabilize the Bayesian posterior mean.
result The data-adaptive prior achieves a stable posterior with small noise limits.
Bayesian SSR on graphs improves regression with noisy labels.
problem Estimating function values on graphs from noisy labeled data.
method Bayesian approach using graph Laplacian and Gaussian prior.
result Rates of contraction of posterior measure around ground truth.
New method improves nonlinear filtering accuracy with reduced computation.
problem Complex nonlinear filtering with small system noise.
method Asymptotic expansion with ordinary differential equations and Edgeworth-type correction.
result Significantly lower computational cost with improved accuracy.
Label noise in SGD helps converge to flatter minima.
problem Improving generalization in overparametrized models.
method Analyzes SGD with label noise, showing convergence to regularized minima.
result SGD with label noise converges to flatter minima, improving generalization.
Research reveals simplicity bias in random logistic map, impacting data analysis and forecasting.
problem Simplicity bias in dynamical systems and its impact on data analysis and prediction.
method Examined the logistic map and random logistic map, focusing on simplicity bias and noise effects.
result Simplicity bias is observable in the random logistic map, persisting even with small noise levels.
Study revisits AdaGrad convergence with relaxed noise assumptions.
problem Non-convex smooth optimization problems with general noise.
method General noise model with function value gap and gradient magnitude control.
result Probabilistic convergence rate of ( ilde{\mathcal{O}}(1/\sqrt{T})) under general noise.
Momentum methods such as Polyak's heavy ball (HB) method, Nesterov's accelerated gradient (AG) as well as accelerated projected gradient (APG) method have been commonly used in machine learning practice, but their performance is quite sensitive to noise in the gradients. We study these methods under a first-order stoch…
Stochastic gradient descent (SGD) has been widely used in machine learning due to its computational efficiency and favorable generalization properties. Recently, it has been empirically demonstrated that the gradient noise in several deep learning settings admits a non-Gaussian, heavy-tailed behavior. This suggests tha…
We study the problem of recovering an incomplete m×n matrix of rank r with columns arriving online over time. This is known as the problem of life-long matrix completion, and is widely applied to recommendation system, computer vision, system identification, etc. The challenge is to design provable algorithms…
Noise enhancement improves generalization in training.
problem Improving generalization in training with controlled noise.
method Noise enhancement method to control SGD noise without changing learning rate or minibatch size.
result Noise enhancement improves generalization for real datasets.
Large learning rates prevent memorization in denoising score matching.
problem Memorization of training data in diffusion-based generative models.
method Investigating the role of large learning rates in the small-noise regime, proving that they prevent convergence to the empirical optimal score.
result Large learning rates prevent memorization by making it impossible for the learned score to be arbitrarily close to the empirical optimal score.
Stable topological summary captures evolving dependency structure in dynamic Bayesian networks.
problem Missing larger-scale patterns in evolving dependency structures in dynamic Bayesian networks.
method Topological approach using Dynamic Bayesian Graphs and persistent homology.
result Stable topological summary (barcodes) captures evolving dependency structure in DBNs.
Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as…
We analyze anomaly detection class imbalance using a solvable model.
problem Class imbalance hampers anomaly detection performance.
method We use an exact solution of the teacher-student perceptron model through replica theory.
result Optimal train imbalance is often different from 50%, influenced by intrinsic imbalance and data abundance.
VAEs analyzed using harmonic analysis, showing how variance controls frequency content and robustness.
problem Understanding and optimizing VAEs for robustness and frequency control.
method Viewing VAE latent space as Gaussian space, deriving results on variance and frequency content, and demonstrating soft Lipschitz constraints.
result Increasing encoder variance reduces high frequency content and improves adversarial robustness.
Noise injection before gradient steps helps in regularization for neural networks.
problem Improving generalization in overparametrized neural networks.
method Injecting small noise perturbations before computing gradient steps, especially in layer-wise fashion.
result Small noise perturbations can explicitly regularize neural networks without variance explosion.
Blind source separation algorithms such as independent component analysis (ICA) are widely used in the analysis of neuroimaging data. In order to leverage larger sample sizes, different data holders/sites may wish to collaboratively learn feature representations. However, such datasets are often privacy-sensitive, prec…
New methods estimate curvature, tangent spaces, and dimension of noisy data.
problem Estimating geometric properties of noisy or sparse data.
method Diffusion geometry tools for Riemannian manifold analysis.
result Significantly outperforms existing methods in noisy or sparse data.
Paper addresses eigenvector perturbation in small eigen-gap scenarios.
problem Fine-grained behavior of eigenvectors in the presence of small eigen-gaps.
method Develops de-biased estimators for linear functions of an unknown eigenvector.
result Achieves minimax lower bounds for a family of scenarios, even with small eigen-gaps.
Variable clustering is important for explanatory analysis. However, only few dedicated methods for variable clustering with the Gaussian graphical model have been proposed. Even more severe, small insignificant partial correlations due to noise can dramatically change the clustering result when evaluating for example w…
Paper proposes Sp-GD for sparse max-affine regression with theoretical guarantees.
problem Sparse max-affine regression model selection and estimation.
method Sparse Gradient Descent (Sp-GD) initialization using sparse PCA and covering search.
result Sp-GD provides ε-accurate estimates with optimal number of observations.
Image classifiers often suffer from adversarial examples, which are generated by strategically adding a small amount of noise to input images to trick classifiers into misclassification. Over the years, many defense mechanisms have been proposed, and different researchers have made seemingly contradictory claims on the…
Despite their impressive performance, deep convolutional neural networks (CNNs) have been shown to be sensitive to small adversarial perturbations. These nuisances, which one can barely notice, are powerful enough to fool sophisticated and well performing classifiers, leading to ridiculous misclassification results. In…
New method uses untrusted data for more precise causal analysis.
problem Causal questions with limited trusted data.
method Incorporates untrusted data and trains richer models.
result Tighter, sounder prediction intervals.
The paper analyzes the implicit bias of SGD near loss manifold and provides new insights.
problem Understanding the implicit bias of SGD near loss manifolds in overparametrized models.
method Adapting ideas from Katzenberger (1991) to analyze SGD dynamics using a stochastic differential equation (SDE).
result SGD with label noise locally decreases the sharpness of loss, leading to a global analysis of implicit bias.
Adding noise controls capacity of function compositions.
problem Large capacity of function compositions with bounded capacity classes.
method Adding Gaussian noise to the output of F before composing with H. result Noise effectively controls the capacity of H∘F, offering a general recipe for modular design. The correlation matrix formalism is used to study temporal aspects of the stock market evolution. This formalism allows to decompose the financial dynamics into noise as well as into some coherent repeatable intraday structures. The present study is based on the high-frequency Deutsche Aktienindex (DAX) data over the t…
Derives effective continuous dynamics for adaptive SGD methods.
problem Analyzing noise in adaptive SGD methods.
method Stochastic modified equations framework and Malladi's scaling rules.
result Sampling-induced noise in SGD limits to independent Brownian motions.
The paper tackles efficient exploration in MDPs to learn accurate models.
problem Efficient exploration in MDPs to learn accurate models.
method Formalizes the problem, introduces an algorithm for ε-accurate model estimation, and proposes a heuristic-based algorithm. result Heuristic-based algorithm outperforms original algorithm in small sample regime.
Noise can affect the overparametrization of QNNs, enabling new directions but also suppressing sensitivity.
problem The overparametrization of QNNs in the presence of noise.
method Analyzing the Quantum Fisher Information Matrix (QFIM) to understand how noise affects the rank of QFIM.
result Noise can turn previously-zero eigenvalues of the QFIM to non-zero, enabling exploration of new directions.
Study on SA with heavy-tailed and LRD noise, establishing finite-time bounds.
problem Analyzing stochastic approximation under heavy-tailed and LRD noise.
method Noise-averaging argument to regularize impact of non-classical noise.
result Established first finite-time moment bounds for SA under heavy-tailed and LRD noise.
Graph-based semi-supervised learning is the problem of propagating labels from a small number of labelled data points to a larger set of unlabelled data. This paper is concerned with the consistency of optimization-based techniques for such problems, in the limit where the labels have small noise and the underlying unl…