Estimating singular subspaces from noisy matrices is a fundamental problem with wide-ranging applications across various fields. Driven by the challenges of data integration and multi-view analysis, this study focuses on estimating shared (left) singular subspaces across multiple matrices within a low-rank matrix denoising framework. A common approach for this task is to perform singular value decomposition on the stacked matrix (Stack-SVD), which is formed by concatenating all the individual matrices. We establish that Stack-SVD achieves minimax rate-optimality when the true (left) singular subspaces of the signal matrices are identical. Our analysis reveals some phase transition phenomena in the estimation problem as a function of the underlying signal-to-noise ratio, highlighting how the interplay among multiple matrices collectively determines the fundamental limits of estimation. We then tackle the more complex scenario where the true singular subspaces are only partially shared across matrices. For various cases of partial sharing, we rigorously characterize the conditions under which Stack-SVD remains effective, achieves minimax optimality, or fails to deliver consistent estimates, offering theoretical insights into its practical applicability. To overcome Stack-SVD's limitations in partial sharing scenarios, we propose novel estimators and an efficient algorithm to identify shared and unshared singular vectors, and prove their minimax rate-optimality. Extensive simulation studies and real-world data applications demonstrate the numerous advantages of our proposed approaches.

Harnessing the local topography of the loss landscape is a central challenge in advanced optimization tasks. By accounting for the effect of potential parameter changes, we can alter the model more efficiently. Contrary to standard assumptions, we find that the Hessian does not always approximate loss curvature well, particularly near gradient discontinuities, which commonly arise in deep learning architectures. We present a new conceptual framework to understand how curvature of expected changes in loss emerges in architectures with many rectified linear units. Each ReLU creates a parameter boundary that, when crossed, induces a pseudorandom gradient perturbation. Our derivations show how these discontinuities combine to form a glass-like structure, similar to amorphous solids that contain microscopic domains of strong, but random, atomic alignment. By estimating the density of the resulting gradient variations, we can bound how the loss may change with parameter movement. Our analysis includes the optimal kernel and sample distribution for approximating glass density from ordinary gradient evaluations. We also derive the optimal modification to quasi-Newton steps that incorporate both glass and Hessian terms, as well as certain exactness properties that are possible with Nesterov-accelerated gradient updates. Our algorithm, Alice, tests these techniques to determine which curvature terms are most impactful for training a given architecture and dataset. Additional safeguards enforce stable exploitation through step bounds that expand on the functionality of Adam. These theoretical and experimental tools lay groundwork to improve future efforts (e.g., pruning and quantization) by providing new insight into the loss landscape.

Classical kernel machines have historically faced significant challenges in scaling to large datasets and model sizes--a key ingredient that has driven the success of neural networks. In this paper, we present a new methodology for building kernel machines that can scale efficiently with both data size and model size. Our algorithm introduces delayed projections to Preconditioned Stochastic Gradient Descent (PSGD) allowing the training of much larger models than was previously feasible, pushing the practical limits of kernel-based learning. They have also served as the foundation 2024) leverage the Nyström Approximation (NA) in combination for understanding many significant phenomena in with other strategies to enhance performance. Despite these advantages, ASkotch combines it with block coordinate descent, the scalability of kernel methods has remained a persistent whereas Falkon combines it with the Conjugate Gradient challenge, particularly when applied to large datasets. However, this limitation is critical for expanding the utility these strategies are limited by model size due to memory of kernel-based techniques in modern machine learning applications.

Targeting and personalization policies can be used to improve outcomes beyond the uniform policy that assigns the best performing treatment in an A/B test to everyone. Personalization relies on the presence of heterogeneity of treatment effects, yet, as we show in this paper, heterogeneity alone is not sufficient for personalization to be successful. We develop a statistical model to quantify "actionable heterogeneity," or the conditions when personalization is likely to outperform the best uniform policy. We show that actionable heterogeneity can be visualized as crossover interactions in outcomes across treatments and depends on three population-level parameters: within-treatment heterogeneity, cross-treatment correlation, and the variation in average responses. Our model can be used to predict the expected gain from personalization prior to running an experiment and also allows for sensitivity analysis, providing guidance on how changing treatments can affect the personalization gain. To validate our model, we apply five common personalization approaches to two large-scale field experiments with many interventions that encouraged flu vaccination. We find an 18% gain from personalization in one and a more modest 4% gain in the other, which is consistent with our model. Counterfactual analysis shows that this difference in the gains from personalization is driven by a drastic difference in within-treatment heterogeneity. However, reducing cross-treatment correlation holds a larger potential to further increase personalization gains. Our findings provide a framework for assessing the potential from personalization and offer practical recommendations for improving gains from targeting in multi-intervention settings.

Deep learning has seen substantial achievements, with numerical and theoretical evidence suggesting that singularities of statistical models are considered a contributing factor to its performance. From this remarkable success of classical statistical models, it is naturally expected that quantum singular models will play a vital role in many quantum statistical tasks. However, while the theory of quantum statistical models in regular cases has been established, theoretical understanding of quantum singular models is still limited. To investigate the statistical properties of quantum singular models, we focus on two prominent tasks in quantum statistical inference: quantum state estimation and model selection. In particular, we base our study on classical singular learning theory and seek to extend it within the framework of Bayesian quantum state estimation. To this end, we define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods. The key idea of the proof is the introduction of a quantum analog of the likelihood function using classical shadows. Consequently, we construct an asymptotically unbiased estimator of the quantum generalization loss, the quantum widely applicable information criterion (QWAIC), as a computable model selection metric from given measurement outcomes.

This document is a replication of the original "Rank-N-Contrast" (arXiv:2210.01189v2) paper published in 2023. This evaluation is done for academic purposes. Deep regression models often fail to capture the continuous nature of sample orders, creating fragmented representations and suboptimal performance. To address this, we reproduced the Rank-N-Contrast (RNC) framework, which learns continuous representations by contrasting samples by their rankings in the target space. Our study validates RNC's theoretical and empirical benefits, including improved performance and robustness. We extended the evaluation to an additional regression dataset and conducted robustness tests using a holdout method, where a specific range of continuous data was excluded from the training set. This approach assessed the model's ability to generalise to unseen data and achieve state-of-the-art performance. This replication study validates the original findings and broadens the understanding of RNC's applicability and robustness.

Probabilistic predictions are probability distributions over the set of possible outcomes. Such predictions quantify the uncertainty in the outcome, making them essential for effective decision making. By combining multiple predictions, the information sources used to generate the predictions are pooled, often resulting in a more informative forecast. Probabilistic predictions are typically combined by linearly pooling the individual predictive distributions; this encompasses several ensemble learning techniques, for example. The weights assigned to each prediction can be estimated based on their past performance, allowing more accurate predictions to receive a higher weight. This can be achieved by finding the weights that optimise a proper scoring rule over some training data. By embedding predictions into a Reproducing Kernel Hilbert Space (RKHS), we illustrate that estimating the linear pool weights that optimise kernel-based scoring rules is a convex quadratic optimisation problem. This permits an efficient implementation of the linear pool when optimally combining predictions on arbitrary outcome domains. This result also holds for other combination strategies, and we additionally study a flexible generalisation of the linear pool that overcomes some of its theoretical limitations, whilst allowing an efficient implementation within the RKHS framework. These approaches are compared in an application to operational wind speed forecasts, where this generalisation is found to offer substantial improvements upon the traditional linear pool.

Tensor network diagram (graphical notation) is a useful tool that graphically represents multiplications between multiple tensors using nodes and edges. Using the graphical notation, complex multiplications between tensors can be described simply and intuitively, and it also helps to understand the essence of tensor products. In fact, most of matrix/tensor products including inner product, outer product, Hadamard product, Kronecker product, and Khatri-Rao product can be written in graphical notation. These matrix/tensor operations are essential building blocks for the use of matrix/tensor decompositions in signal processing and machine learning. The purpose of this lecture note is to learn the very basics of tensors and how to represent them in mathematical symbols and graphical notation. Many papers using tensors omit these detailed definitions and explanations, which can be difficult for the reader. I hope this note will be of help to such readers.

This study introduces a debiasing method for regression estimators, including high-dimensional and nonparametric regression estimators. For example, nonparametric regression methods allow for the estimation of regression functions in a data-driven manner with minimal assumptions; however, these methods typically fail to achieve $\sqrt{n}$-consistency in their convergence rates, and many, including those in machine learning, lack guarantees that their estimators asymptotically follow a normal distribution. To address these challenges, we propose a debiasing technique for nonparametric estimators by adding a bias-correction term to the original estimators, extending the conventional one-step estimator used in semiparametric analysis. Specifically, for each data point, we estimate the conditional expected residual of the original nonparametric estimator, which can, for instance, be computed using kernel (Nadaraya-Watson) regression, and incorporate it as a bias-reduction term. Our theoretical analysis demonstrates that the proposed estimator achieves $\sqrt{n}$-consistency and asymptotic normality under a mild convergence rate condition for both the original nonparametric estimator and the conditional expected residual estimator. Notably, this approach remains model-free as long as the original estimator and the conditional expected residual estimator satisfy the convergence rate condition. The proposed method offers several advantages, including improved estimation accuracy and simplified construction of confidence intervals.

This study focuses on Intelligent Fault Diagnosis (IFD) in rotating machinery utilizing a single microphone and a data-driven methodology, effectively diagnosing 42 classes of fault types and severities. The research leverages sound data from the imbalanced MaFaulDa dataset, aiming to strike a balance between high performance and low resource consumption. The testing phase encompassed a variety of configurations, including sampling, quantization, signal normalization, silence removal, Wiener filtering, data scaling, windowing, augmentation, and classifier tuning using XGBoost. Through the analysis of time, frequency, mel-frequency, and statistical features, we achieved an impressive accuracy of 99.54% and an F-Beta score of 99.52% with just 6 boosting trees at an 8 kHz, 8-bit configuration. Moreover, when utilizing only MFCCs along with their first- and second-order deltas, we recorded an accuracy of 97.83% and an F-Beta score of 97.67%. Lastly, by implementing a greedy wrapper approach, we obtained a remarkable accuracy of 96.82% and an F-Beta score of 98.86% using 50 selected features, nearly all of which were first- and second-order deltas of the MFCCs.