Out-of-Distribution (OoD) detection is vital for the reliability of Deep Neural Networks (DNNs).Existing works have shown the insufficiency of Principal Component Analysis (PCA) straightforwardly applied on the features of DNNs in detecting OoD data from In-Distribution (InD) data.The failure of PCA suggests that the network features residing in OoD and InD are not well separated by simply proceeding in a linear subspace, which instead can be resolved through proper non-linear mappings.In this work, we leverage the framework of Kernel PCA (KPCA) for OoD detection, and seek suitable non-linear kernels that advocate the separability between InD and OoD data in the subspace spanned by the principal components.Besides, explicit feature mappings induced from the devoted task-specific kernels are adopted so that the KPCA reconstruction error for new test samples can be efficiently obtained with large-scale data.Extensive theoretical and empirical results on multiple OoD data sets and network structures verify the superiority of our KPCA detector in efficiency and efficacy with state-of-the-art detection performance.

Self supervised learning (SSL) is a machine learning paradigm where models learn to understand the underlying structure of data without explicit supervision from labeled samples. The acquired representations from SSL have demonstrated useful for many downstream tasks including clustering, and linear classification, etc. To ensure smoothness of the representation space, most SSL methods rely on the ability to generate pairs of observations that are similar to a given instance. However, generating these pairs may be challenging for many types of data. Moreover, these methods lack consideration of uncertainty quantification and can perform poorly in out-of-sample prediction settings. To address these limitations, we propose Gaussian process self supervised learning (GPSSL), a novel approach that utilizes Gaussian processes (GP) models on representation learning. GP priors are imposed on the representations, and we obtain a generalized Bayesian posterior minimizing a loss function that encourages informative representations. The covariance function inherent in GPs naturally pulls representations of similar units together, serving as an alternative to using explicitly defined positive samples. We show that GPSSL is closely related to both kernel PCA and VICReg, a popular neural network-based SSL method, but unlike both allows for posterior uncertainties that can be propagated to downstream tasks. Experiments on various datasets, considering classification and regression tasks, demonstrate that GPSSL outperforms traditional methods in terms of accuracy, uncertainty quantification, and error control.

Nonlinear component analysis such as kernel Principle Component Analysis (KPCA) and kernel Canonical Correlation Analysis (KCCA) are widely used in machine learning, statistics and data analysis, but they cannot scale up to big datasets. Recent attempts have employed random feature approximations to convert the problem to the primal form for linear computational complexity. However, to obtain high quality solutions, the number of random features should be the same order of magnitude as the number of data points, making such approach not directly applicable to the regime with millions of data points. We propose a simple, computationally efficient, and memory friendly algorithm based on the "doubly stochastic gradients" to scale up a range of kernel nonlinear component analysis, such as kernel PCA, CCA and SVD. Despite the non-convex nature of these problems, our method enjoys theoretical guarantees that it converges at the rate O (1 /t) to the global optimum, even for the top k eigen subspace. Unlike many alternatives, our algorithm does not require explicit orthogonaliza-tion, which is infeasible on big datasets. We demonstrate the effectiveness and scalability of our algorithm on large scale synthetic and real world datasets.

Detecting domain shifts in myoelectric activations poses a significant challenge due to the inherent non-stationarity of electromyography (EMG) signals. This paper explores the detection of domain shifts using data stream (DS) learning techniques, focusing on the DB6 dataset from the Ninapro database. We define domains as distinct time-series segments based on different subjects and recording sessions, applying Kernel Principal Component Analysis (KPCA) with a cosine kernel to pre-process and highlight these shifts. By evaluating multiple drift detection methods such as CUSUM, Page-Hinckley, and ADWIN, we reveal the limitations of current techniques in achieving high performance for real-time domain shift detection in EMG signals. Our results underscore the potential of streaming-based approaches for maintaining stable EMG decoding models, while highlighting areas for further research to enhance robustness and accuracy in real-world scenarios.

Out-of-Distribution (OoD) detection is vital for the reliability of Deep Neural Networks (DNNs).Existing works have shown the insufficiency of Principal Component Analysis (PCA) straightforwardly applied on the features of DNNs in detecting OoD data from In-Distribution (InD) data.The failure of PCA suggests that the network features residing in OoD and InD are not well separated by simply proceeding in a linear subspace, which instead can be resolved through proper non-linear mappings.In this work, we leverage the framework of Kernel PCA (KPCA) for OoD detection, and seek suitable non-linear kernels that advocate the separability between InD and OoD data in the subspace spanned by the principal components.Besides, explicit feature mappings induced from the devoted task-specific kernels are adopted so that the KPCA reconstruction error for new test samples can be efficiently obtained with large-scale data.Extensive theoretical and empirical results on multiple OoD data sets and network structures verify the superiority of our KPCA detector in efficiency and efficacy with state-of-the-art detection performance.

Deep learning models often function as black boxes, providing no straightforward reasoning for their predictions. This is particularly true for computer vision models, which process tensors of pixel values to generate outcomes in tasks such as image classification and object detection. To elucidate the reasoning of these models, class activation maps (CAMs) are used to highlight salient regions that influence a model's output. This research introduces KPCA-CAM, a technique designed to enhance the interpretability of Convolutional Neural Networks (CNNs) through improved class activation maps. KPCA-CAM leverages Principal Component Analysis (PCA) with the kernel trick to capture nonlinear relationships within CNN activations more effectively. By mapping data into higher-dimensional spaces with kernel functions and extracting principal components from this transformed hyperplane, KPCA-CAM provides more accurate representations of the underlying data manifold. This enables a deeper understanding of the features influencing CNN decisions. Empirical evaluations on the ILSVRC dataset across different CNN models demonstrate that KPCA-CAM produces more precise activation maps, providing clearer insights into the model's reasoning compared to existing CAM algorithms. This research advances CAM techniques, equipping researchers and practitioners with a powerful tool to gain deeper insights into CNN decision-making processes and overall behaviors.

The remarkable success of transformers in sequence modeling tasks, spanning various applications in natural language processing and computer vision, is attributed to the critical role of self-attention. Similar to the development of most deep learning models, the construction of these attention mechanisms rely on heuristics and experience. In our work, we derive self-attention from kernel principal component analysis (kernel PCA) and show that self-attention projects its query vectors onto the principal component axes of its key matrix in a feature space. We then formulate the exact formula for the value matrix in self-attention, theoretically and empirically demonstrating that this value matrix captures the eigenvectors of the Gram matrix of the key vectors in self-attention. Leveraging our kernel PCA framework, we propose Attention with Robust Principal Components (RPC-Attention), a novel class of robust attention that is resilient to data contamination. We empirically demonstrate the advantages of RPC-Attention over softmax attention on the ImageNet-1K object classification, WikiText-103 language modeling, and ADE20K image segmentation task.

Given i.i.d. observations uniformly distributed on a closed manifold $\mathcal{M}\subseteq \mathbb{R}^p$, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of $\mathcal{M}$. In our analysis, we connect the empirical graph Laplacian to kernel principal component analysis, and consider the heat kernel of $\mathcal{M}$ as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.

Unlabeled data is a key component of modern machine learning. In general, the role of unlabeled data is to impose a form of smoothness, usually from the similarity information encoded in a base kernel, such as the $\epsilon$-neighbor kernel or the adjacency matrix of a graph. This work revisits the classical idea of spectrally transformed kernel regression (STKR), and provides a new class of general and scalable STKR estimators able to leverage unlabeled data. Intuitively, via spectral transformation, STKR exploits the data distribution for which unlabeled data can provide additional information. First, we show that STKR is a principled and general approach, by characterizing a universal type of "target smoothness", and proving that any sufficiently smooth function can be learned by STKR. Second, we provide scalable STKR implementations for the inductive setting and a general transformation function, while prior work is mostly limited to the transductive setting. Third, we derive statistical guarantees for two scenarios: STKR with a known polynomial transformation, and STKR with kernel PCA when the transformation is unknown. Overall, we believe that this work helps deepen our understanding of how to work with unlabeled data, and its generality makes it easier to inspire new methods.

Unsupervised and self-supervised representation learning has become popular in recent years for learning useful features from unlabelled data. Representation learning has been mostly developed in the neural network literature, and other models for representation learning are surprisingly unexplored. In this work, we introduce and analyze several kernel-based representation learning approaches: Firstly, we define two kernel Self-Supervised Learning (SSL) models using contrastive loss functions and secondly, a Kernel Autoencoder (AE) model based on the idea of embedding and reconstructing data. We argue that the classical representer theorems for supervised kernel machines are not always applicable for (self-supervised) representation learning, and present new representer theorems, which show that the representations learned by our kernel models can be expressed in terms of kernel matrices. We further derive generalisation error bounds for representation learning with kernel SSL and AE, and empirically evaluate the performance of these methods in both small data regimes as well as in comparison with neural network based models.