Uncertainty estimation is essential for robust decision-making in the presence of ambiguous or out-of-distribution inputs. Gaussian Processes (GPs) are classical kernel-based models that offer principled uncertainty quantification and perform well on small- to medium-scale datasets. Alternatively, formulating the weight space learning problem under tensor network assumptions yields scalable tensor network kernel machines. However, these assumptions break Gaussianity, complicating standard probabilistic inference. This raises a fundamental question: how can tensor network kernel machines provide principled uncertainty estimates? We propose a novel Bayesian Tensor Network Kernel Machine (LA-TNKM) that employs a (linearized) Laplace approximation for Bayesian inference. A comprehensive set of numerical experiments shows that the proposed method consistently matches or surpasses Gaussian Processes and Bayesian Neural Networks (BNNs) across diverse UCI regression benchmarks, highlighting both its effectiveness and practical relevance.

We show that if the conditional distribution p(C | T) factors through a sufficient statistic ϕ(T), then the Information Bottleneck (IB) problem for (T, C) is exactly equivalent to the IB problem for (ϕ(T), C). The reduction is loss-free: it preserves the full IB curve, the Lagrangian optimum at every trade-off parameter \b{eta}, and the optimal representations up to pullback through ϕ. As a result, the computational complexity of solving the IB problem is governed by the dimension of the sufficient statistic rather than the ambient dimension of the source. This identifies an exact structural condition under which the generic IB problem becomes tractable, and gives a formal bridge between the discrete and linear-Gaussian regimes. We then show that the classical Gaussian IB solution of Chechik, Globerson, Tishby and Weiss is an immediate corollary of this reduction, and we state a nonlinear-Gaussian generalisation. A small numerical example illustrates the practical consequence: when a low-dimensional sufficient statistic is available, the exact IB curve can be computed on the reduced problem at a cost determined by the statistic rather than by the ambient source dimension.

The transformer architecture [52], which underlies present-day Large Language Models, has been one of the main drivers of recent advances in machine learning and artificial intelligence. At each layer, the hidden state of the network is updated by sequentially applying two distinct operations: attention modules [3], which capture long-range interactions in the input sequence, and classical MultiLayer Perceptrons (MLPs), acting separately on each element of that sequence. Despite their empirical success, the mechanisms governing information propagation through depth, and the way attention and MLP blocks jointly shape internal representations, remain only partially understood from a theoretical viewpoint. Recent progress has come from viewing transformers in suitable scaling limits as deterministic mean-field interacting particle systems modeling the evolution of N tokens1 through the layers of the neural network architecture (the so-called residual stream dynamics), see, among others, [46, 26, 27, 45]. In these descriptions, depth plays the role of a continuous time variable, and, in the large-context regime (N), the evolution of token representations is encoded by a PDE for their empirical distribution. This viewpoint is closely connected to the literature on scaling laws, where the effect of various scaling exponents controlling the relative size of the network's hyperparameters (e.g., depth, width, context length) on the effective dynamics of the model

Learning curves are a fundamental primitive in supervised learning, describing how an algorithm's performance improves with more data and providing a quantitative measure of its generalization ability. Formally, a learning curve plots the decay of an algorithm's error for a fixed underlying distribution as a function of the number of training samples. Prior work on revenue-maximizing learning algorithms, starting with the seminal work of Cole and Roughgarden [STOC, 2014], adopts a distribution-free perspective, which parallels the PAC learning framework in learning theory. This approach evaluates performance against the hardest possible sequence of valuation distributions, one for each sample size, effectively defining the upper envelope of learning curves over all possible distributions, thus leading to error bounds that do not capture the shape of the learning curves. In this work we initiate the study of learning curves for revenue maximization and provide a near-complete characterization of their rate of decay in the basic setting of a single item and a single buyer. In the absence of any restriction on the valuation distribution, we show that there exists a Bayes-consistent algorithm, meaning that its learning curve converges to zero for any arbitrary valuation distribution as the number of samples $n \to \infty$. However, this convergence must be arbitrarily slow, even if the optimal revenue is finite. In contrast, if the optimal revenue is achieved by a finite price, then the optimal rate of decay is roughly $1/\sqrt{n}$. Finally, for distributions supported on discrete sets of values, we show that learning curves decay almost exponentially fast, a rate unattainable under the PAC framework.

In Orabona and Pál [2016], we introduced the shifted KT potentials, to remove the $\ln \ln T$ factor in the parameter-free learning with expert bound. In this short technical note, I show that this is equivalent to changing the prior in the Krichevsky--Trofimov algorithm. Then, I show how to use the same idea to remove the $\ln \ln T$ factor in the data-independent bound for the Squint algorithm.

We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.

Emerging neural reconstruction techniques based on tomography (e.g., NeRF, NeAT, and NeRP) have started showing unique capabilities in medical imaging. In this work, we present a novel Polychromatic neural representation (Polyner) to tackle the challenging problem of CT imaging when metallic implants exist within the human body. CT metal artifacts arise from the drastic variation of metal's attenuation coefficients at various energy levels of the X-ray spectrum, leading to a nonlinear metal effect in CT measurements. Recovering CT images from metal-affected measurements hence poses a complicated nonlinear inverse problem where empirical models adopted in previous metal artifact reduction (MAR) approaches lead to signal loss and strongly aliased reconstructions.

Automated seizure detection is of great importance to epilepsy diagnosis and treatment. An emerging method used in seizure detection, stereoelectroencephalography (SEEG), can provide detailed and stereoscopic brainwave information. However, modeling SEEG in clinical scenarios will face challenges like huge domain shift between different patients and dramatic pattern evolution among different brain areas. In this study, we propose a Pretraining-based model for Patient-independent seizure detection (PPi) to address these challenges. Firstly, we design two novel self-supervised tasks which can extract rich information from abundant SEEG data while preserving the unique characteristics between brain signals recorded from different brain areas. Then two techniques, channel background subtraction and brain region enhancement, are proposed to effectively tackle the domain shift problem. Extensive experiments show that PPi outperforms the SOTA baselines on two public datasets and a real-world clinical dataset collected by us, which demonstrates the effectiveness and practicability of PPi. Finally, visualization analysis illustrates the rationality of the two domain generalization techniques.

The introduction of neural radiance fields has greatly improved the effectiveness of view synthesis for monocular videos. However, existing algorithms face difficulties when dealing with uncontrolled or lengthy scenarios, and require extensive training time specific to each new scenario. To tackle these limitations, we propose DynPoint, an algorithm designed to facilitate the rapid synthesis of novel views for unconstrained monocular videos. Rather than encoding the entirety of the scenario information into a latent representation, DynPoint concentrates on predicting the explicit 3D correspondence between neighboring frames to realize information aggregation. Specifically, this correspondence prediction is achieved through the estimation of consistent depth and scene flow information across frames. Subsequently, the acquired correspondence is utilized to aggregate information from multiple reference frames to a target frame, by constructing hierarchical neural point clouds. The resulting framework enables swift and accurate view synthesis for desired views of target frames. The experimental results obtained demonstrate the considerable acceleration of training time achieved - typically an order of magnitude - by our proposed method while yielding comparable outcomes compared to prior approaches. Furthermore, our method exhibits strong robustness in handling long-duration videos without learning a canonical representation of video content.

Recalibrating probabilistic classifiers is vital for enhancing the reliability and accuracy of predictive models. Despite the development of numerous recalibration algorithms, there is still a lack of a comprehensive theory that integrates calibration and sharpness (which is essential for maintaining predictive power). In this paper, we introduce the concept of minimum-risk recalibration within the framework of mean-squared-error (MSE) decomposition, offering a principled approach for evaluating and recalibrating probabilistic classifiers. Using this framework, we analyze the uniform-mass binning (UMB) recalibration method and establish a finite-sample risk upper bound of order O(B/n+1/B2) where Bis the number of bins and nis the sample size. By balancing calibration and sharpness, we further determine that the optimal number of bins for UMB scales with n1/3, resulting in a risk bound of approximately O(n 2/3). Additionally, we tackle the challenge of label shift by proposing a two-stage approach that adjusts the recalibration function using limited labeled data from the target domain. Our results show that transferring a calibrated classifier requires significantly fewer target samples compared to recalibrating from scratch. We validate our theoretical findings through numerical simulations, which confirm the tightness of the proposed bounds, the optimal number of bins, and the effectiveness of label shift adaptation.