Bayesian filtering of partially and noisily observed dynamical systems seeks to infer the evolving conditional distribution of the state of a dynamical system, given observations, in an online fashion. This Bayesian filtering distribution is the natural object for uncertainty quantification, but it is rarely available as a supervised learning target. However, one can often use the forecast model to generate synthetic system trajectories, along with synthetic observations. We introduce the proper scoring ensemble filter (PSEF), an ensemble data assimilation method based on training an analysis map to approximate the filtering distribution using only synthetic state--observation trajectories. The analysis step is represented as a permutation-invariant, transformer-based map that takes as input a forecast ensemble and observations, producing an analysis ensemble. Training is based on strictly proper scoring rules -- with the energy score used in our implementation -- so that probabilistic accuracy is rewarded over the whole probability distribution. We prove that, under a realizability assumption, the population objective is minimized by the true Bayesian filtering distribution. We also derive the finite-ensemble empirical objective used in training and relate its single state--observation trajectory form to the population objective, using a mean-field consistency argument. Numerical experiments show that the learned filter accurately approximates challenging filtering distributions, including nonlinear, non-Gaussian, and multi-modal posteriors, and achieves stronger performance in data assimilation tasks than classical methods or learning-based methods with mean-squared-error objectives. For close-to-Gaussian problems, learning a correction to the EnKF is the best approach, while for highly non-Gaussian problems an end-to-end approach that discards this inductive bias is superior.

Neural Information Processing Systems http://nips.cc/

W = 2for = 10 2and = 10 4, omitted architectures.

Motivated by the recent successes of neural networks that have the ability to fit the data perfectlyand generalize well, we study the noiseless model in the fundamental least-squares setup.

Extrapolation methods use the last few iterates of an optimization algorithm to produce a better estimate of the optimum. They were shown to achieve optimal convergence rates in a deterministic setting using simple gradient iterates. Here, we study extrapolation methods in a stochastic setting, where the iterates are produced by either a simple or an accelerated stochastic gradient algorithm.

Neural Information Processing Systems http://nips.cc/

Artificial Intelligence (AI) for music generation is undergoing rapid developments, with recent symbolic models leveraging sophisticated deep learning and diffusion model algorithms. One drawback with existing models is that they lack structural cohesion, particularly on harmonic-melodic structure. Furthermore, such existing models are largely "black-box" in nature and are not musically interpretable. This paper addresses these limitations via a novel generative music framework that incorporates concepts of Schenkerian analysis (SchA) in concert with a diffusion modeling framework. This framework, which we call ProGress (Prolongation-enhanced DiGress), adapts state-of-the-art deep models for discrete diffusion (in particular, the DiGress model of Vignac et al., 2023) for interpretable and structured music generation. Concretely, our contributions include 1) novel adaptations of the DiGress model for music generation, 2) a novel SchA-inspired phrase fusion methodology, and 3) a framework allowing users to control various aspects of the generation process to create coherent musical compositions. Results from human experiments suggest superior performance to existing state-of-the-art methods.

Kernel methods are powerful tools in statistical learning, but their cubic complexity in the sample size n limits their use on large-scale datasets. In this work, we introduce a scalable framework for kernel regression with O(n log n) complexity, fully leveraging GPU acceleration. The approach is based on a Fourier representation of kernels combined with non-uniform fast Fourier transforms (NUFFT), enabling exact, fast, and memory-efficient computations. We instantiate our framework in three settings: Sobolev kernel regression, physics-informed regression, and additive models. When known, the proposed estimators are shown to achieve minimax convergence rates, consistent with classical kernel theory. Empirical results demonstrate that our methods can process up to tens of billions of samples within minutes, providing both statistical accuracy and computational scalability. These contributions establish a flexible approach, paving the way for the routine application of kernel methods in large-scale learning tasks.

Song generation is regarded as the most challenging problem in music AIGC; nonetheless, existing approaches have yet to fully overcome four persistent limitations: controllability, generalizability, perceptual quality, and duration. We argue that these shortcomings stem primarily from the prevailing paradigm of attempting to learn music theory directly from raw audio, a task that remains prohibitively difficult for current models. To address this, we present Bar-level AI Composing Helper (BACH), the first model explicitly designed for song generation through human-editable symbolic scores. BACH introduces a tokenization strategy and a symbolic generative procedure tailored to hierarchical song structure. Consequently, it achieves substantial gains in the efficiency, duration, and perceptual quality of song generation. Experiments demonstrate that BACH, with a small model size, establishes a new SOTA among all publicly reported song generation systems, even surpassing commercial solutions such as Suno. Human evaluations further confirm its superiority across multiple subjective metrics.

Characterizing the convergence of real-valued or vector-v alued sequences is a key theoretical problem in data science, where the sequence index typically correspon ds to the number of iterations of an iterative algorithm (such as in optimization and signal processing) o r the number of observations (as in statistics and machine learning). This characterization can be done in mostly two ways, asymptotically or non-asymptotically. In an asymptotic analysis, an asymptotic e quivalent of the sequence is identified, which readily allows comparisons with other algorithms; however, without further analysis, the behavior at any finite time cannot be controlled. This is exactly what non-as ymptotic analysis aims to achieve, by providing bounds that are valid even for a finite index, but then only pro viding bounds that cannot always be compared. While the two approaches have their own merits, in this paper, we focus on asymptotic analysis and sequences that tend to their limit at a sub-exponential r ate that is a power of the sequence index. The main goal of this paper is to show how a classical tool from signal processing, control theory, and electrical engineering ( Oppenheim et al., 1996), the z -transform method ( Jury, 1964), can be used in this context with a striking efficiency at obtaining asymptotic eq uivalents for the class of algorithms that can be seen as iterations of potentially random linear operators i n a Hilbert space. This includes gradient descent for quadratic optimization problems as well as its accelera ted and stochastic variants ( Nesterov, 2018), 1 Landweber iterations in inverse problems ( Benning and Burger, 2018), or gossip algorithms in distributed computing ( Boyd et al., 2006).