Classification is a core topic in functional data analysis. A large number of functional classifiers have been proposed in the literature, most of which are based on functional principal component analysis or functional regression. In contrast, we investigate this topic from the perspective of manifold learning. It is assumed that functional data lie on an unknown low-dimensional manifold, and we expect that better classifiers can be built upon the manifold structure. To this end, we propose a novel proximity measure that takes the label information into account to learn the low-dimensional representations, also known as the supervised manifold learning outcomes. When the outcomes are coupled with multivariate classifiers, the procedure induces a family of new functional classifiers. In theory, we show that our functional classifier induced by the $k$-NN classifier is asymptotically optimal. In practice, we show that our method, coupled with several classical multivariate classifiers, achieves outstanding classification performance compared to existing functional classifiers in both synthetic and real data examples.

We analyze the behavior of nearest neighbor classification in metric spaces and provide finite-sample, distribution-dependent rates of convergence under minimal assumptions. These are more general than existing bounds, and enable us, as a by-product, to establish the universal consistency of nearest neighbor in a broader range of data spaces than was previously known. We illustrate our upper and lower bounds by introducing a new smoothness class customized for nearest neighbor classification. We find, for instance, that under the Tsybakov margin condition the convergence rate of nearest neighbor matches recently established lower bounds for nonparametric classification.

Transfer learning enhances prediction accuracy on a target distribution by leveraging data from a source distribution, demonstrating significant benefits in various applications. This paper introduces a novel dissimilarity measure that utilizes vicinity information, i.e., the local structure of data points, to analyze the excess error in classification under covariate shift, a transfer learning setting where marginal feature distributions differ but conditional label distributions remain the same. We characterize the excess error using the proposed measure and demonstrate faster or competitive convergence rates compared to previous techniques. Notably, our approach is effective in situations where the non-absolute continuousness assumption, which often appears in real-world applications, holds. Our theoretical analysis bridges the gap between current theoretical findings and empirical observations in transfer learning, particularly in scenarios with significant differences between source and target distributions.

We analyze the behavior of nearest neighbor classification in metric spaces and provide finite-sample, distribution-dependent rates of convergence under minimal assumptions. These are more general than existing bounds, and enable us, as a by-product, to establish the universal consistency of nearest neighbor in a broader range of data spaces than was previously known. We illustrate our upper and lower bounds by introducing a new smoothness class customized for nearest neighbor classification. We find, for instance, that under the Tsybakov margin condition the convergence rate of nearest neighbor matches recently established lower bounds for nonparametric classification.

Traditional deep learning models implicity encode knowledge limiting their transparency and ability to adapt to data changes. Yet, this adaptability is vital for addressing user data privacy concerns. We address this limitation by storing embeddings of the underlying training data independently of the model weights, enabling dynamic data modifications without retraining. Specifically, our approach integrates the $k$-Nearest Neighbor ($k$-NN) classifier with a vision-based foundation model, pre-trained self-supervised on natural images, enhancing interpretability and adaptability. We share open-source implementations of a previously unpublished baseline method as well as our performance-improving contributions. Quantitative experiments confirm improved classification across established benchmark datasets and the method's applicability to distinct medical image classification tasks. Additionally, we assess the method's robustness in continual learning and data removal scenarios. The approach exhibits great promise for bridging the gap between foundation models' performance and challenges tied to data privacy. The source code is available at https://github.com/TobArc/privacy-aware-image-classification-with-kNN.

We continue to investigate the $k$ nearest neighbour learning rule in separable metric spaces. Thanks to the results of C\'erou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Gy\"{o}rfi, Krzy\.{z}ak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (that is, those of Nagata dimension zero). Combining the theorem of C\'erou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the $k$-NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the $k$-NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Kor\'anyi and Reimann (1995) and Sawyer and Wheeden (1992).

Data augmentation is a widely used technique and an essential ingredient in the recent advance in self-supervised representation learning. By preserving the similarity between augmented data, the resulting data representation can improve various downstream analyses and achieve state-of-the-art performance in many applications. Despite the empirical effectiveness, most existing methods lack theoretical understanding under a general nonlinear setting. To fill this gap, we develop a statistical framework on a low-dimension product manifold to model the data augmentation transformation. Under this framework, we introduce a new representation learning method called augmentation invariant manifold learning and design a computationally efficient algorithm by reformulating it as a stochastic optimization problem. Compared with existing self-supervised methods, the new method simultaneously exploits the manifold's geometric structure and invariant property of augmented data and has an explicit theoretical guarantee. Our theoretical investigation characterizes the role of data augmentation in the proposed method and reveals why and how the data representation learned from augmented data can improve the $k$-nearest neighbor classifier in the downstream analysis, showing that a more complex data augmentation leads to more improvement in downstream analysis. Finally, numerical experiments on simulated and real datasets are presented to demonstrate the merit of the proposed method.

This paper addresses challenges in robust transfer learning stemming from ambiguity in Bayes classifiers and weak transferable signals between the target and source distribution. We introduce a novel quantity called the ''ambiguity level'' that measures the discrepancy between the target and source regression functions, propose a simple transfer learning procedure, and establish a general theorem that shows how this new quantity is related to the transferability of learning in terms of risk improvements. Our proposed ''Transfer Around Boundary'' (TAB) model, with a threshold balancing the performance of target and source data, is shown to be both efficient and robust, improving classification while avoiding negative transfer. Moreover, we demonstrate the effectiveness of the TAB model on non-parametric classification and logistic regression tasks, achieving upper bounds which are optimal up to logarithmic factors. Simulation studies lend further support to the effectiveness of TAB. We also provide simple approaches to bound the excess misclassification error without the need for specialized knowledge in transfer learning.