PCWorld anticipates Nvidia's N1X launch at Computex, featuring an Arm-based APU with 20 CPU cores and Blackwell graphics that could match RTX 5060 laptop performance. The article highlights growing concerns about PC hardware affordability, with examples like Steam Deck price increases suggesting higher costs may become the norm. This trend matters for consumers as powerful new hardware from Nvidia, AMD, and Intel may deliver impressive performance but potentially at premium prices that limit accessibility. The PC industry is once again on the brink of a pivotal moment in history--or so appears to be the case, given the rumors about Computex next week. In particular, the internet anticipates the launch of Nvidia's N1X, an Arm-based APU expected to marry ferocious CPU performance with equally knockout GPU chops.

Flow and diffusion generative models have established themselves as widely adopted density estimators for simulation-based inference (SBI), extending naturally from neural posterior estimation to likelihood and joint density estimation. Their principled optimization objectives and freedom from architectural constraints have driven rapid adoption across the natural sciences. Yet the most widely used SBI libraries remain PyTorch-based, leaving researchers who develop their forward models and analysis pipelines in JAX without a native option. We present GenSBI, an open-source library that implements flow matching, score matching, and denoising diffusion entirely in JAX. The library offers three transformer-based architectures -- SimFormer, Flux1, and a novel Flux1Joint that extends gate-modulated transformer blocks to joint density estimation -- all interchangeable through a unified interface that decouples generative method, neural backbone, and inference mode. GenSBI provides an end-to-end workflow from training through posterior calibration (SBC, TARP, LC2ST) and supports custom architectures with domain-specific embedding networks.

Many problems in computational science and engineering become one-to-many after coarse graining, partial observation, or inverse reconstruction: a resolved state may not determine a unique subgrid forcing, a structural descriptor may not determine a unique effective response, and a low-resolution observation may correspond to many plausible high-resolution fields. In such settings, deterministic surrogates may learn a well-defined mathematical object while still missing application-relevant uncertainty. This tutorial develops a self-contained module centered on the conditional-mean barrier: the point at which a squared-loss predictor has reached the conditional mean and the remaining error is irreducible aleatoric variance. We give two diagnostics for locating this barrier, residual-feature orthogonality and the coefficient of determination against its explained-variance ceiling, and prove that adding latent randomness to a squared-loss predictor collapses it back to the conditional mean. Crossing the barrier therefore requires a loss that scores distributions rather than point predictions. We briefly organize common distributional objectives, including negative log-likelihood, moment and observable matching, variational objectives, adversarial divergences, and score matching, by the feature of the conditional law each targets. The emphasis is the boundary itself and a finite-data procedure for recognizing it, rather than a survey of methods beyond it. CPU-based demonstrations on a two-branch law and a two-scale Lorenz-96 closure problem show how the diagnostics distinguish deterministic underfitting from residual distributional variability.

Directed acyclic graphs (DAGs) constitute a central modeling tool to enable principled reasoning about cause-effect interactions in complex systems. However, since the causal structure underlying a group of variables is often unknown and interventions may be infeasible or ethically challenging to implement, there is a need to address the task of inferring DAGs from observational data. However, most classical structure identification approaches face two key obstacles: the combinatorial challenge of enforcing acyclicity, which severely limits scalability, and identifiability challenges arising from latent confounding or heterogeneous noise. This tutorial offers an overview of recent signal processing and optimization advances that address these issues by recasting DAG structure learning as a continuous, score-based estimation problem over adjacency matrices. We begin with a didactic introduction to structural equation models and the formulation of causal graph recovery, followed by a historical survey of score-based methods ranging from early combinatorial search schemes and greedy heuristics to modern continuous frameworks that leverage smooth characterizations of acyclicity. Building on this foundation, we describe concomitant DAG estimation methods that jointly infer sparse causal structure and exogenous noise levels, improving robustness under heteroscedasticity and distribution shifts by rendering the estimator noise adaptive. All in all, the tutorial introduces readers to challenges and opportunities for signal processing research at the crossroads of causal inference, high-dimensional statistics, and scalable graph learning, while outlining emerging directions including online, nonlinear, and neural causal discovery.

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Diffusion and flow-based models are ubiquitously used for generative modelling and density estimation. They admit a deterministic probability flow ordinary differential equation (PF-ODE), analogous to continuous normalizing flows (CNFs), which describes the transport of the probability mass. Obtaining the likelihood from these models is of interest to many workflows, especially Bayesian analysis, and requires solving the trace of the Jacobian to compute the divergence of the learned PF-ODE, which is either $\mathcal{O}(D^2)$ to compute exactly or $\mathcal{O}(D)$ with a noisy estimate. We introduce StAD, a new distillation method to predict and learn the divergence of the PF-ODE using the Langevin-Stein operator without ever computing the Jacobian. We show that our method is competitive with the Hutchinson and Hutch++ on CIFAR-10, ImageNet and other density estimation tasks, consistently improving the variance and speed of the likelihood predictions compared to the Hutchinson. We additionally show our method will generalize to a varied class of generative models, and show that under some regularity conditions these learned vector fields can be made to satisfy the Stein class.

We revisit the problem of robust linear regression under Gaussian covariates with an unknown covariance matrix of condition number $κ$. For this fundamental problem, significant gaps remain in our understanding of the trade-offs among sample complexity, condition number, runtime, and prediction error for efficient algorithms. Our first result is a near-linear-time algorithm that uses $\widetilde{O}(d/ε^4)$ samples, where $d$ is the dimension and $ε$ is the corruption rate, and achieves prediction error $O(\sqrt{εκ})$ under the condition $εκ\lesssim 1$, improving over all prior works. We complement this result with a Statistical Query (SQ) lower bound showing that efficient SQ algorithms achieving error $o(\sqrt{εκ})$ when $εκ\lesssim 1$ require queries that take $Ω(d^2)$ samples to simulate. Finally, we prove a low-degree polynomial lower bound that gives fine-grained evidence that, without assumptions such as $εκ\lesssim 1$, efficient algorithms may require $\tildeΩ\left(\min\{dε^{2}κ^{2},\ ε^{2}d^{2}\}\right)$ samples to significantly outperform the trivial estimator that always guesses $0$.

This paper studies sampling error bounds for denoising diffusion probabilistic models (DDPMs) in the 2-Wasserstein distance. Our contributions are threefold. (i) Under general Lipschitz-type conditions on the score function and for a broad class of variance schedules, including the cosine schedule, we establish sharp upper bounds that are optimal in both the dimension and the number of steps, and recover several sharp error bounds previously obtained in the literature. (ii) We prove that the same Lipschitz-type conditions, which encompass those commonly imposed on the (learned) score, imply a logarithmic Sobolev inequality and hence a quadratic transportation cost inequality for the DDPM. As a consequence, in settings covered by existing work, an optimal Wasserstein bound, up to a logarithmic factor, follows from the recently obtained sharp error bound in the Kullback-Leibler divergence under geometric-type variance schedules. (iii) We show that for general log-concave target distributions, the optimal Wasserstein error bound remains attainable even without a quadratic transportation cost inequality for the target. Our analysis is based on viewing the DDPM sampler as a discretization of the Föllmer process rather than the conventional reverse Ornstein-Uhlenbeck process.

We study the classical problem of community recovery in stochastic block models with a fixed number of communities, with a twist: We seek algorithms that are stable with respect to node-wise changes in the graph structure, formally defined as a differential privacy constraint. The algorithms we develop are based on spectral clustering, where we introduce privacy to the community recovery pipeline in the form of directly privatizing the adjacency matrix; private PCA; private convex optimization; private low-rank matrix estimation; and private approximate subspace estimation. Straightforward applications of existing private algorithms lead to a rapid increase in the privacy parameter $ε$ in order to ensure consistent estimation under node differential privacy, in contrast with the simpler setting of edge privacy. To alleviate these issues, we develop novel algorithms based on (1) sampling from an exponential mechanism with a Lipschitz extension and (2) a general framework for constructing smooth projections from the space of undirected graphs to the space of bounded-degree graphs, which can then be combined with various edge-private algorithms. Importantly, the methods we develop are all computable in polynomial-time as a function of the number of nodes in the graph. We also develop novel lower bounds on the growth rate of $ε$ required in order to achieve consistent community estimation under node privacy. On a technical note, our paper highlights the complications that arise when analyzing private algorithms under the non-standard scaling $ε\rightarrow \infty$ and proposes some solutions. We also provide a novel application of the HGR maximal correlation from information theory in the context of accuracy amplification in PAC learning, which may be of independent interest.

This paper investigates the learning theory of Transformer networks for regression tasks on the compact Euclidean domain $[0,1]^d$ and $d$-dimensional compact Riemannian manifolds. We propose a novel constructive approximation framework for Transformers that builds local approximations of the target function and aggregates them into a global approximation via softmax partition of unity. This approach leverages the attention mechanism to achieve spatial localization through affine transformations of the input. The softmax activation plays a crucial role in aggregating local approximations to a global output. From an approximation perspective, we prove that a dense Transformer equipped with only two encoder blocks and standard single-hidden-layer point-wise feed-forward networks can achieve a uniform $\varepsilon$-approximation error for $α$-Hölder continuous functions with $α\in (0,1]$ using $\mathcal{O}(\varepsilon^{-d/α})$ total parameters. Building upon this approximation guarantee, we establish a near minimax-optimal generalization error bound of order $\mathcal{O}\big(n^{-\frac{2α}{2α+d}} \log n\big)$ for the empirical risk minimizer, where $n$ is the training data size. The Transformer architecture studied in this paper is dense, shallow and wide, and employs softmax activation and sinusoidal positional encodings, closely reflecting practical implementations.