We establish a new theoretical framework for learning under multi-class, instance-dependent label noise. This framework casts learning with label noise as a form of domain adaptation, in particular, domain adaptation under posterior drift.

The random features method, proposed by Rahimi and Recht [2008], maps the data to a finite dimensional feature space as a random approximation to the feature space of RBF kernels. With explicit finite dimensional feature vectors available, the original KSVM is converted to a linear support vector machine (LSVM), that can be trained by faster algorithms (Shalev-Shwartz et al.

Class imbalance significantly degrades classification performance, yet its effects are rarely analyzed from a unified theoretical perspective. We propose a principled framework based on three fundamental scales: the imbalance coefficient $η$, the sample--dimension ratio $κ$, and the intrinsic separability $Δ$. Starting from the Gaussian Bayes classifier, we derive closed-form Bayes errors and show how imbalance shifts the discriminant boundary, yielding a deterioration slope that predicts four regimes: Normal, Mild, Extreme, and Catastrophic. Using a balanced high-dimensional genomic dataset, we vary only $η$ while keeping $κ$ and $Δ$ fixed. Across parametric and non-parametric models, empirical degradation closely follows theoretical predictions: minority Recall collapses once $\log(η)$ exceeds $Δ\sqrtκ$, Precision increases asymmetrically, and F1-score and PR-AUC decline in line with the predicted regimes. These results show that the triplet $(η,κ,Δ)$ provides a model-agnostic, geometrically grounded explanation of imbalance-induced deterioration.

We introduce Smart Bayes, a new classification framework that bridges generative and discriminative modeling by integrating likelihood-ratio-based generative features into a logistic-regression-style discriminative classifier. From the generative perspective, Smart Bayes relaxes the fixed unit weights of Naive Bayes by allowing data-driven coefficients on density-ratio features. From a discriminative perspective, it constructs transformed inputs as marginal log-density ratios that explicitly quantify how much more likely each feature value is under one class than another, thereby providing predictors with stronger class separation than the raw covariates. To support this framework, we develop a spline-based estimator for univariate log-density ratios that is flexible, robust, and computationally efficient. Through extensive simulations and real-data studies, Smart Bayes often outperforms both logistic regression and Naive Bayes. Our results highlight the potential of hybrid approaches that exploit generative structure to enhance discriminative performance.

Neural Information Processing Systems http://nips.cc/

We establish a new theoretical framework for learning under multi-class, instance-dependent label noise. This framework casts learning with label noise as a form of domain adaptation, in particular, domain adaptation under posterior drift.

Neural Information Processing Systems http://nips.cc/

Machine learning (ML) is often viewed as a powerful data analysis tool that is easy to learn because of its black-box nature. Yet this very nature also makes it difficult to quantify confidence in predictions extracted from ML models, and more fundamentally, to understand how such models are mathematical abstractions of training data. The goal of this paper is to unravel these issues and their connections to uncertainty quantification (UQ) by pursuing a line of reasoning motivated by diagnostics. In such settings, prevalence - i.e. the fraction of elements in class - is often of inherent interest. Here we analyze the many interpretations of prevalence to derive a level-set theory of classification, which shows that certain types of self-consistent ML models are equivalent to class-conditional probability distributions. We begin by studying the properties of binary Bayes optimal classifiers, recognizing that their boundary sets can be reinterpreted as level-sets of pairwise density ratios. By parameterizing Bayes classifiers in terms of the prevalence, we then show that they satisfy important monotonicity and class-switching properties that can be used to deduce the density ratios without direct access to the boundary sets. Moreover, this information is sufficient for tasks such as constructing the multiclass Bayes-optimal classifier and estimating inherent uncertainty in the class assignments. In the multiclass case, we use these results to deduce normalization and self-consistency conditions, the latter being equivalent to the law of total probability for classifiers. We also show that these are necessary conditions for arbitrary ML models to have valid probabilistic interpretations. Throughout we demonstrate how this analysis informs the broader task of UQ for ML via an uncertainty propagation framework.

Bayesian decision theory advocates the Bayes classifier as the optimal approach for minimizing the risk in machine learning problems. Current deep learning algorithms usually solve for the optimal classifier by \emph{implicitly} estimating the posterior probabilities, \emph{e.g.}, by minimizing the Softmax cross-entropy loss. This simple methodology has been proven effective for meticulously balanced academic benchmark datasets. However, it is not applicable to the long-tailed data distributions in the real world, where it leads to the gradient imbalance issue and fails to ensure the Bayes optimal decision rule. To address these challenges, this paper presents a novel approach (BAPE) that provides a more precise theoretical estimation of the data distributions by \emph{explicitly} modeling the parameters of the posterior probabilities and solving them with point estimation. Consequently, our method directly learns the Bayes classifier without gradient descent based on Bayes' theorem, simultaneously alleviating the gradient imbalance and ensuring the Bayes optimal decision rule. Furthermore, we propose a straightforward yet effective \emph{distribution adjustment} technique. This method enables the Bayes classifier trained from the long-tailed training set to effectively adapt to the test data distribution with an arbitrary imbalance factor, thereby enhancing performance without incurring additional computational costs. In addition, we demonstrate the gains of our method are orthogonal to existing learning approaches for long-tailed scenarios, as they are mostly designed under the principle of \emph{implicitly} estimating the posterior probabilities. Extensive empirical evaluations on CIFAR-10-LT, CIFAR-100-LT, ImageNet-LT, and iNaturalist demonstrate that our method significantly improves the generalization performance of popular deep networks, despite its simplicity.

The goal of this paper is to show how different machine learning tools on the Riemannian manifold $\mathcal{P}_d$ of Symmetric Positive Definite (SPD) matrices can be united under a probabilistic framework. For this, we will need several Gaussian distributions defined on $\mathcal{P}_d$. We will show how popular classifiers on $\mathcal{P}_d$ can be reinterpreted as Bayes Classifiers using these Gaussian distributions. These distributions will also be used for outlier detection and dimension reduction. By showing that those distributions are pervasive in the tools used on $\mathcal{P}_d$, we allow for other machine learning tools to be extended to $\mathcal{P}_d$.