Binary Classification

We revisit High-Resolution Range Profile (HRRP) classification with aspect-angle conditioning. While prior work often assumes that aspect-angle information is incomplete during training or unavailable at inference, we study a setting where angles are available for all training samples and explicitly provided to the classifier. Using three datasets and a broad range of conditioning strategies and model architectures, we show that both single-profile and sequential classifiers benefit consistently from aspect-angle awareness, with an average accuracy gain of about 7% and improvements of up to 10%, depending on the model and dataset. In practice, aspect angles are not directly measured and must be estimated. We show that a causal Kalman filter can estimate them online with a median error of 5{\textdegree}, and that training and inference with estimated angles preserves most of the gains, supporting the proposed approach in realistic conditions.

We examine the concentration of uniform generalization errors around their expectation in binary linear classification problems via an isoperimetric argument. In particular, we establish Poincaré and log-Sobolev inequalities for the joint distribution of the output labels and the label-weighted input vectors, which we apply to derive concentration bounds. The derived results improve upon existing bounds obtained from general unbounded empirical processes, as well as that tailored specifically to logistic regression. In asymptotic analysis, we also show that almost sure convergence of uniform generalization errors to their expectation occurs in very broad settings, such as proportionally high-dimensional regimes. Using this convergence, we establish uniform laws of large numbers under dimension-free conditions.

Weak supervision enables machine learning models to learn from limited or noisy labels, but it introduces challenges in reliability and semantic clarity, particularly in multi-instance partial label learning (MI-PLL), where models must resolve both ambiguous supervision signals and uncertain instance-label mappings. This paper proposes a semantics for a neuro-symbolic framework that integrates inductive logic programming (ILP) to structure MI-PLL through relational constraints. In this formulation, ILP defines a hypothesis space over label transitions, formalizes the semantics of per-instance classifiers and provides a relational scaffold for reasoning about weak supervision. Two inductive tasks are studied in this framework: inferring the transition predicate from the observed and classifier predicates, and inferring instance-level classifier assignments from the observed and transition predicates. This formal semantics facilitates constraint specification, consistency checking and the diagnosis of semantic failure modes that bag-level accuracy alone may conceal.

We study the fundamental problem of calibrating a linear binary classifier of the form $σ(\hat{w}^\top x)$, where the feature vector $x$ is Gaussian, $σ$ is a link function, and $\hat{w}$ is an estimator of the true linear weight $w^\star$. By interpolating with a noninformative $\textit{chance classifier}$, we construct a well-calibrated predictor whose interpolation weight depends on the angle $\angle(\hat{w}, w_\star)$ between the estimator $\hat{w}$ and the true linear weight $w_\star$. We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate. The angle $\angle(\hat{w}, w_\star)$ can be consistently estimated. Furthermore, the resulting predictor is uniquely $\textit{Bregman-optimal}$, minimizing the Bregman divergence to the true label distribution within a suitable class of calibrated predictors. Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions. Additionally, we identify conditions under which a classical Platt-scaling predictor converges to our Bregman-optimal calibrated solution. Thus, Platt-scaling also inherits these desirable properties provably in high dimensions.

The practical success of deep learning has led to the discovery of several surprising phenomena. One of these phenomena, that has spurred intense theoretical research, is ``benign overfitting'': deep neural networks seem to generalize well in the over-parametrized regime even though the networks show a perfect fit to noisy training data. It is now known that benign overfitting also occurs in various classical statistical models. For linear maximum margin classifiers, benign overfitting has been established theoretically in a class of mixture models with very strong assumptions on the covariate distribution. However, even in this simple setting, many questions remain open. For instance, most of the existing literature focuses on the noiseless case where all true class labels are observed without errors, whereas the more interesting noisy case remains poorly understood. We provide a comprehensive study of benign overfitting for linear maximum margin classifiers. We discover a phase transition in test error bounds for the noisy model which was previously unknown and provide some geometric intuition behind it. We further considerably relax the required covariate assumptions in both the noisy and noiseless cases. Our results demonstrate that benign overfitting of maximum margin classifiers holds in a much wider range of scenarios than was previously known and provide new insights into the underlying mechanisms.

Federated learning (FL) is a machine learning paradigm where clients with different behaviors and preferences can learn collaboratively without compromising data privacy. Typical FL methods often rely on fixed weighting for parameter aggregation, thereby neglecting the mutual influence among clients. In practice, clients with similar preferences or backgrounds may provide more useful knowledge to each other, which can be leveraged to improve local performance. However, how to quantify such cross-client influence and how to exploit it for personalized aggregation remain underexplored. To address this gap, we propose an influence-oriented Federated learning framework which quantitatively measures Client-level and Class-level Influence to realize adaptive parameter aggregation for each client (FedC^2I for short). Our core idea is to explicitly model the inter-client influence within an FL system via the well-crafted influence vector and influence matrix. Specifically, FedC^2I incorporate influence vectors to quantify client-level influence, enables clients to selectively acquire knowledge from others, and guides the aggregation of feature representation layers. Meanwhile, the influence matrix captures class-level influence in a more fine-grained manner to achieve personalized classifier aggregation. We evaluate the performance of FedC^2I against existing federated learning methods under non-IID settings, and the results demonstrate the superiority of our method in terms of effectiveness, robustness, and interpretability.

This tutorial explains Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) as two fundamental classification methods in statistical and probabilistic learning. We start with the optimization of decision boundary on which the posteriors are equal. Then, LDA and QDA are derived for binary and multiple classes. The estimation of parameters in LDA and QDA are also covered. Then, we explain how LDA and QDA are related to metric learning, kernel principal component analysis, Mahalanobis distance, logistic regression, Bayes optimal classifier, Gaussian naive Bayes, and likelihood ratio test. We also prove that LDA and Fisher discriminant analysis are equivalent. We finally clarify some of the theoretical concepts with simulations we provide.

Home-based physiotherapy is performed without supervision, which leads to incorrect execution and motivates systems that assess movement automatically from inertial measurement units (IMUs). Such systems assign each repetition to a category, yet a relevant share of repetitions falls near a class boundary, where even trained raters disagree. Classifiers trained with one-hot labels collapse these borderline repetitions onto a single class and discard this ambiguity. We address this with a method that automatically generates a label distribution per repetition without a large rater pool. We train a network to reproduce the full distribution with a Kullback-Leibler objective, the ambiguity approach, and compare it against a one-hot cross-entropy baseline on four IMU exercise datasets. From the network output we further determine whether a repetition is ambiguous and which classes are relevant to it. The ambiguity approach matched or exceeded the baseline classification on all four datasets, and detected ambiguity and the relevant classes more reliably. Representing the label distribution in the training target therefore adds information about ambiguity at no cost to classification.