This note responds to "Promises and Pitfalls of Deep Kernel Learning" (Ober et al., 2021). The marginal likelihood of a Gaussian process can be compartmentalized into a data fit term and a complexity penalty. Ober et al. (2021) shows that if a kernel can be multiplied by a signal variance coefficient, then reparametrizing and substituting in the maximized value of this parameter sets a reparametrized data fit term to a fixed value. They use this finding to argue that the complexity penalty, a log determinant of the kernel matrix, then dominates in determining the other values of kernel hyperparameters, which can lead to data overcorrelation. By contrast, we show that the reparametrization in fact introduces another data-fit term which influences all other kernel hyperparameters. Thus, a balance between data fit and complexity still plays a significant role in determining kernel hyperparameters.

With the increasing popularity of large language models, concerns over content authenticity have led to the development of myriad watermarking schemes. These schemes can be used to detect a machine-generated text via an appropriate key, while being imperceptible to readers with no such keys. The corresponding detection mechanisms usually take the form of statistical hypothesis testing for the existence of watermarks, spurring extensive research in this direction. However, the finer-grained problem of identifying which segments of a mixed-source text are actually watermarked, is much less explored; the existing approaches either lack scalability or theoretical guarantees robust to paraphrase and post-editing. In this work, we introduce a unique perspective to such watermark segmentation problems through the lens of epidemic change-points. By highlighting the similarities as well as differences of these two problems, we motivate and propose WISER: a novel, computationally efficient, watermark segmentation algorithm. We theoretically validate our algorithm by deriving finite sample error-bounds, and establishing its consistency in detecting multiple watermarked segments in a single text. Complementing these theoretical results, our extensive numerical experiments show that WISER outperforms state-of-the-art baseline methods, both in terms of computational speed as well as accuracy, on various benchmark datasets embedded with diverse watermarking schemes. Our theoretical and empirical findings establish WISER as an effective tool for watermark localization in most settings. It also shows how insights from a classical statistical problem can lead to a theoretically valid and computationally efficient solution of a modern and pertinent problem.

We study best-of-$N$ for large language models (LLMs) where the selection is based on majority voting. In particular, we analyze the limit $N \to \infty$, which we denote as Best-of-$\infty$. While this approach achieves impressive performance in the limit, it requires an infinite test-time budget. To address this, we propose an adaptive generation scheme that selects $N$ based on answer agreement, thereby efficiently allocating inference-time computation. Beyond adaptivity, we extend the framework to weighted ensembles of multiple LLMs, showing that such mixtures can outperform any individual model. The optimal ensemble weighting is formulated and efficiently computed as a mixed-integer linear program. Extensive experiments demonstrate the effectiveness of our approach.

Probabilistic forecasting of multivariate time series is challenging due to non-stationarity, inter-variable dependencies, and distribution shifts. While recent diffusion and flow matching models have shown promise, they often ignore informative priors such as conditional means and covariances. In this work, we propose Conditionally Whitened Generative Models (CW-Gen), a framework that incorporates prior information through conditional whitening. Theoretically, we establish sufficient conditions under which replacing the traditional terminal distribution of diffusion models, namely the standard multivariate normal, with a multivariate normal distribution parameterized by estimators of the conditional mean and covariance improves sample quality. Guided by this analysis, we design a novel Joint Mean-Covariance Estimator (JMCE) that simultaneously learns the conditional mean and sliding-window covariance. Building on JMCE, we introduce Conditionally Whitened Diffusion Models (CW-Diff) and extend them to Conditionally Whitened Flow Matching (CW-Flow). Experiments on five real-world datasets with six state-of-the-art generative models demonstrate that CW-Gen consistently enhances predictive performance, capturing non-stationary dynamics and inter-variable correlations more effectively than prior-free approaches. Empirical results further demonstrate that CW-Gen can effectively mitigate the effects of distribution shift.

Scaling laws, a defining feature of deep learning, reveal a striking power-law improvement in model performance with increasing dataset and model size. Yet, their mathematical origins, especially the scaling exponent, have remained elusive. In this work, we show that scaling laws can be formally explained as redundancy laws. Using kernel regression, we show that a polynomial tail in the data covariance spectrum yields an excess risk power law with exponent alpha = 2s / (2s + 1/beta), where beta controls the spectral tail and 1/beta measures redundancy. This reveals that the learning curve's slope is not universal but depends on data redundancy, with steeper spectra accelerating returns to scale. We establish the law's universality across boundedly invertible transformations, multi-modal mixtures, finite-width approximations, and Transformer architectures in both linearized (NTK) and feature-learning regimes. This work delivers the first rigorous mathematical explanation of scaling laws as finite-sample redundancy laws, unifying empirical observations with theoretical foundations.

In scientific applications, predictive modeling is often of limited use without accurate uncertainty quantification (UQ) to indicate when a model may be extrapolating or when more data needs to be collected. Bayesian Neural Networks (BNNs) produce predictive uncertainty by propagating uncertainty in neural network (NN) weights and offer the promise of obtaining not only an accurate predictive model but also accurate UQ. However, in practice, obtaining accurate UQ with BNNs is difficult due in part to the approximations used for practical model training and in part to the need to choose a suitable set of hyperparameters; these hyperparameters outnumber those needed for traditional NNs and often have opaque effects on the results. We aim to shed light on the effects of hyperparameter choices for BNNs by performing a global sensitivity analysis of BNN performance under varying hyperparameter settings. Our results indicate that many of the hyperparameters interact with each other to affect both predictive accuracy and UQ. For improved usage of BNNs in real-world applications, we suggest that global sensitivity analysis, or related methods such as Bayesian optimization, should be used to aid in dimensionality reduction and selection of hyperparameters to ensure accurate UQ in BNNs.

We address the problem of efficiently computing Wasserstein distances for multiple pairs of distributions drawn from a meta-distribution. To this end, we propose a fast estimation method based on regressing Wasserstein distance on sliced Wasserstein (SW) distances. Specifically, we leverage both standard SW distances, which provide lower bounds, and lifted SW distances, which provide upper bounds, as predictors of the true Wasserstein distance. To ensure parsimony, we introduce two linear models: an unconstrained model with a closed-form least-squares solution, and a constrained model that uses only half as many parameters. We show that accurate models can be learned from a small number of distribution pairs. Once estimated, the model can predict the Wasserstein distance for any pair of distributions via a linear combination of SW distances, making it highly efficient. Empirically, we validate our approach on diverse tasks, including Gaussian mixtures, point-cloud classification, and Wasserstein-space visualizations for 3D point clouds. Across various datasets such as MNIST point clouds, ShapeNetV2, MERFISH Cell Niches, and scRNA-seq, our method consistently provides a better approximation of Wasserstein distance than the state-of-the-art Wasserstein embedding model, Wasserstein Wormhole, particularly in low-data regimes. Finally, we demonstrate that our estimator can also accelerate Wormhole training, yielding \textit{RG-Wormhole}.

Large language models frequently generate confident but incorrect outputs, requiring formal uncertainty quantification with abstention guarantees. We develop information-lift certificates that compare model probabilities to a skeleton baseline, accumulating evidence into sub-gamma PAC-Bayes bounds valid under heavy-tailed distributions. Across eight datasets, our method achieves 77.2\% coverage at 2\% risk, outperforming recent 2023-2024 baselines by 8.6-15.1 percentage points, while blocking 96\% of critical errors in high-stakes scenarios vs 18-31\% for entropy methods. Limitations include skeleton dependence and frequency-only (not severity-aware) risk control, though performance degrades gracefully under corruption.

A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of functions and their derivatives required by differential operators. This paper establishes the first simultaneous approximation theory for deep neural networks on manifolds. We prove that a constant-depth $\mathrm{ReLU}^{k-1}$ network with bounded weights--a property that plays a crucial role in controlling generalization error--can approximate any function in the Sobolev space $\mathcal{W}_p^{k}(\mathcal{M}^d)$ to an error of $\varepsilon$ in the $\mathcal{W}_p^{s}(\mathcal{M}^d)$ norm, for $k\geq 3$ and $s

ECG biometrics offer a unique, secure authentication method, yet their deployment on wearable devices faces real-time processing, privacy, and spoofing vulnerability challenges. This paper proposes a lightweight deep learning model (MobileNetV1+GRU) for ECG-based authentication, injection of 20dB Gaussian noise & custom preprocessing. We simulate wearable conditions and edge deployment using the ECGID, MIT-BIH, CYBHi, and PTB datasets, achieving accuracies of 99.34%, 99.31%, 91.74%, and 98.49%, F1-scores of 0.9869, 0.9923, 0.9125, and 0.9771, Precision of 0.9866, 0.9924, 0.9180 and 0.9845, Recall of 0.9878, 0.9923, 0.9129, and 0.9756, equal error rates (EER) of 0.0009, 0.00013, 0.0091, and 0.0009, and ROC-AUC values of 0.9999, 0.9999, 0.9985, and 0.9998, while under FGSM adversarial attacks, accuracy drops from 96.82% to as low as 0.80%. This paper highlights federated learning, adversarial testing, and the need for diverse wearable physiological datasets to ensure secure and scalable biometrics.