Generative approaches to clustering provide information on geometric properties of clusters, whereas discriminative approaches provide boundaries between clusters. Ideas from both approaches are incorporated to present a fully unsupervised, probabilistic, and discriminative clustering method via a regularized mutual information objective function, wherein a mixture of mixtures of Gaussian and uniform distributions is used for formulation of the conditional model. Automatic selection of the number of components is established with the introduction of the regularizing term and a merge step, similar to those applied in reversible jump Markov chain Monte Carlo methods used in Bayesian clustering. Consequently, the turtle shell method -- a fully unsupervised clustering method capable of estimating non-linear boundary lines, automatically selecting the number of components, and capturing intuitive clusters in the presence of data abnormalities such as noise and/or irregular cluster shapes -- is introduced. We test this method on various simulated and real datasets commonly explored in clustering research, and extend the analysis to datasets arising from flow cytometry experiments.

Existing large-dimensional theory for spectral algorithms resolves either the optimally tuned point or the interpolation limit, but leaves the under-regularized regime unexplored. We study the learning curve and benign overfitting of spectral algorithms in the largedimensional setting where the sample size and dimension are of comparable order, i.e., n dγ for some γ > 0. We first consider inner-product kernels on the sphere Sd 1 and establish a sharp asymptotic characterization of the excess risk across the full regularization path under various source conditions s 0, where smeasures the relative smoothness of the regression function. Our results reveal that the learning curve is not simply U-shaped but instead consists of three distinct regimes: over-regularized, under-regularized, and interpolation regimes. This characterization allows us to fully capture the benign overfitting phenomenon, demonstrating that benign overfitting arises consistently across both the under-regularized and interpolation regimes whenever sis positive but no larger than a critical threshold. We further show that, in the sufficiently regularized regime, the kernel learning curve is recovered by an associated sequence model. Finally, we extend the learning-curve analysis to large-dimensional KRR for a class of kernels on general domains in Rd whose low-degree eigenspaces satisfy spectral-scaling and hyper-contractivity conditions. Keywords: Spectral algorithms, learning curves, high dimension, benign overfitting. 1 Introduction Nonparametric regression studies the estimation of an unknown function f: Rd R from ni.i.d.

Convolutional neural networks (CNNs) have become increasingly difficult to deploy in resource-constrained environments due to their large memory and computational requirements. Although low-rank compression methods can reduce this burden, most existing approaches compress spatial and channel redundancy independently and therefore do not fully exploit the localised structure within convolutional feature maps. This paper proposes a hierarchical spatio-channel low-rank compression framework for CNNs that exploits redundancy across spatial regions and channel activations. Unlike conventional methods, which apply a uniform decomposition across an entire layer, the proposed approach first partitions feature maps into spatial regions, then groups channels according to their co-activation patterns within each region, and finally applies rank-adaptive SVD to each resulting spatio-channel cluster. The method is evaluated on an AlexNet-based brain tumour MRI classification model and compared with Global SVD and Tucker decomposition under \(3\times\) and \(6\times\) compression budgets. Our method outperforms both baselines, reducing FLOPs from \(8.21\,\mathrm{G}\) to \(1.55\,\mathrm{G}\) (\(81.1\%\) reduction), achieving a \(1.38\times\) inference speed-up, and increasing classification accuracy from \(87.76\%\) to \(89.80\%\). The method also improves the macro \(F_1\)-score and performance on challenging classes such as meningioma. A hyper-parameter trade-off analysis demonstrates that the framework provides Pareto-optimal configurations, enabling control over the balance between compression and predictive performance. Moderate clustering with adaptive rank selection yields strong results. Bootstrap standard errors are reported for all classification metrics.

Over decades, Markov chain Monte Carlo (MCMC) methods have been widely studied, with a typical application being the quantification of posterior uncertainties in Bayesian system identification of structural dynamic models. To address the issue of excessively low sampling efficiency in generic MCMC methods when applied to specific problems, researchers developed several MCMC algorithms that integrate trainable neural networks to replace and enhance their critical components. Later, meta-learning MCMC methods emerged to reduce training time. However, they require considerable similarity between test and training tasks, while their sampling efficiency is constrained by trade-off-simplified network designs. This paper proposes the Adaptive Principal-Component (PC) Meta-learning Stochastic Gradient Hamiltonian Monte Carlo (APM-SGHMC) algorithm. It adaptively rotates coordinate axes in the parameter space to align with the PC directions of the current posterior samples, ensuring rotation-invariance of sampling performance with respect to the posterior distribution. By incorporating translation-invariance, scale-invariance, and rotation-invariance in a unified framework, APM-SGHMC enables universal samplers to acquire generalizable knowledge across diverse Bayesian system identification tasks using minimalistic tasks while eliminating the constraints imposed by network design trade-offs on sampling efficiency. Practical feasibility issues are also addressed. Two Bayesian system identification case studies demonstrate its effectiveness and universality: our method overcomes the case-by-case limitations of traditional data-driven approaches, achieving zero-shot generalization across structurally distinct models without retraining and maintaining consistent superior performance across all scenarios.

Reliable decision-making with streaming data requires principled uncertainty quantification of online methods. While first-order methods enable efficient iterate updates, their inference procedures still require updating proper (covariance) matrices, incurring $O(d^2)$ time and memory complexity, and are sensitive to ill-conditioning and noise heterogeneity of the problem. This costly inference task offers an opportunity for more robust second-order methods, which are, however, bottlenecked by solving Newton systems with $O(d^3)$ complexity. In this paper, we address this gap by studying an online Newton method with Hessian averaging, where the Newton direction at each step is approximately computed using a sketch-and-project solver with Nesterov's acceleration, matching $O(d^2)$ complexity of first-order methods. For the proposed method, we quantify its uncertainty arising from both random data and randomized computation. Under standard smoothness and moment conditions, we establish global almost-sure convergence, prove asymptotic normality of the last iterate with a limiting covariance characterized by a Lyapunov equation, and develop a fully online covariance estimator with non-asymptotic convergence guarantees. We also connect the resulting uncertainty quantification to that of exact and sketched Newton methods without Nesterov's acceleration. Extensive experiments on regression models demonstrate the superiority of the proposed method for online inference.

The central limit theorem (CLT) is a foundation of statistical inference: it provides the asymptotic distribution needed for confidence intervals, hypothesis tests, and efficiency comparisons [24, 42]. For iterate-averaged stochastic gradient methods, it specifies both a Gaussian limit and its sandwich covariance in a single theorem statement. This foundation now underpins inference in streaming and online settings--online A/B testing, continual monitoring of treatment effects, and streaming M-estimation, for example--where the estimator is updated one observation at a time and inference must be performed in real time. A line of recent work develops online inference procedures for averaged SGD [10, 23, 46]. In practice, one-pass stochastic optimization is routinely combined with adaptive preconditioning, which improves computational efficiency and is believed to sharpen the resulting Gaussian approximation in finite samples. If the CLT fails or the asymptotic variance is altered by the adaptive preconditioning, all downstream inference-- coverage of confidence intervals, size of hypothesis tests, consistency of plug-in covariance estimators--is compromised. A rigorous understanding of when adaptive preconditioning preserves the CLT is, therefore, a prerequisite for reliable inference in these settings.

Semi-supervised classification, where unlabeled data are massive but labeled data are limited, often arises in machine learning applications. We address this challenge under high-dimensional data by leveraging the manifold and cluster assumptions. Based on the Fermat distance, a density-sensitive metric that naturally encodes the cluster assumption, we propose the weighted $k$-nearest neighbors (NN) classifier and multidimensional scaling (MDS)-induced classifiers. The use of MDS with a large target dimension allows the effective application of linear classifiers to complex manifold data. Theoretically, we derive a sharp lower bound for the expected excess risk within clusters and prove that the weighted $k$-NN classifier utilizing the true Fermat distance is minimax optimal. Furthermore, we explicitly quantify the utility of unlabeled data by showing that the error arising from estimating the Fermat distance decays exponentially with the pooled sample size. Such a rate is much faster than the related rates in the literature. Extensive experiments on synthetic and real datasets demonstrate competitive or superior performance of our approaches compared to state-of-the-art graph-based semi-supervised classifiers.

Learning low-dimensional representations from multi-view relational data is challenging when underlying geometries differ across views. We propose Bary-GWMDS, a Gromov-Wasserstein-based method that operates directly on distance matrices to learn a consensus embedding preserving shared relational structure. By leveraging intrinsic distances, the approach naturally handles nonlinear distortions across views. We also introduce Mean-GWMDS-C, a clustering-oriented formulation that averages distance matrices and learns reduced-support representations via a consensus Gromov-Wasserstein transport. Experiments on synthetic and real-world datasets show that the proposed framework yields stable and geometrically meaningful embeddings.

When, in terms of the number of data points, the size of a dataset exceeds available computing resources, or when labeling is expensive, an attractive solution consists of selecting only some of the data points (subdata) for further consideration. A central question for selecting subdata of size $n$ from $N$ available data points is which $n$ points to select. While an answer to this question depends on the objective, one approach for a parametric model and a focus on parameter estimation is to select subdata that retains maximal information. Identifying such subdata is a classical NP-hard problem due to its inherent discreteness. Based on optimal approximate design theory, we develop a new methodology for information-based subdata selection, resulting in subdata that approaches the optimal solution. To achieve this, we develop a novel algorithm that applies to a general model, accommodates arbitrary choices of $N$ and $n$, and supports multiple optimality criteria, and we prove its convergence. Moreover, the new methodology facilitates an assessment of the efficiency of subdata selected by any method by obtaining tight lower and upper bounds for the efficiency. We show that the subdata obtained through the new methodology is highly efficient and outperforms all existing methods.

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time trajectory increments, state-space partitioning, or repeated simulation of candidate models, which become unreliable or computationally expensive for high-dimensional systems, coarse temporal sampling, or unevenly sampled data. We introduce a data-driven calibration method based on a novel relationship between the coefficients of a stochastic reduced model and the conditional score of the finite-time transition density, defined as the gradient of the logarithm of the transition density with respect to the initial state. The resulting identity expresses derivatives of lagged correlation functions as stationary expectations over observed lagged pairs involving this conditional score and the unknown model coefficients. This formulation allows the drift and diffusion structure to be constrained directly from finite-lag statistics, without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration, yielding a least-squares fitting problem over stationary lagged pairs. We validate the approach on analytically tractable and data-driven nonequilibrium diffusions, demonstrating that the inferred models preserve the invariant statistics while accurately reproducing finite-lag dynamical correlations. The framework provides a scalable route for learning stochastic reduced-order models from data that reproduce prescribed statistical and dynamical properties.