Diffusion models generate samples by iteratively querying learned score estimates. A rapidly growing literature focuses on accelerating sampling by minimizing the number of score evaluations, yet the information-theoretic limits of such acceleration remain unclear. In this work, we establish the first score query lower bounds for diffusion sampling. We prove that for $d$-dimensional distributions, given access to score estimates with polynomial accuracy $\varepsilon=d^{-O(1)}$ (in any $L^p$ sense), any sampling algorithm requires $\widetildeΩ(\sqrt{d})$ adaptive score queries. In particular, our proof shows that any sampler must search over $\widetildeΩ(\sqrt{d})$ distinct noise levels, providing a formal explanation for why multiscale noise schedules are necessary in practice.

Pairwise comparisons are widely used in decision analysis, preference modeling, and evaluation problems. In many practical situations, the observed comparison matrix is not reciprocal. This lack of reciprocity is often treated as a defect to be corrected immediately. In this article, we adopt a different point of view: part of the nonreciprocity may reflect a genuine variation in the evaluation scale, while another part is due to random perturbations. We introduce an additive model in which the unknown underlying comparison matrix is consistent but not necessarily reciprocal. The reciprocal component carries the global ranking information, whereas the symmetric component describes possible scale variation. Around this structured matrix, we add a random perturbation and show how to estimate the noise level, assess whether the scale variation remains moderate, and assign probabilities to admissible ranking regions in the sense of strict ranking by pairwise comparisons. We also compare this approach with the brutal projection onto reciprocal matrices, which suppresses all symmetric information at once. The Gaussian perturbation model is used here not because human decisions are exactly Gaussian, but because observed judgment errors often result from the accumulation of many small effects. In such a context, the central limit principle provides a natural heuristic justification for Gaussian noise. This makes it possible to derive explicit estimators and probability assessments while keeping the model interpretable for decision problems.

We study privacy-preserving sparse linear regression in the high-dimensional regime, focusing on the LASSO estimator. We analyze two widely used mechanisms for differential privacy: output perturbation, which injects noise into the estimator, and objective perturbation, which adds a random linear term to the loss function. Using approximate message passing (AMP), we characterize the typical behavior of these estimators under random design and privacy noise. To quantify privacy, we adopt typical-case measures, including the on-average KL divergence, which admits a hypothesis-testing interpretation in terms of distinguishability between neighboring datasets. Our analysis reveals that sparsity plays a central role in shaping the privacy-accuracy trade-off: stronger regularization can improve privacy by stabilizing the estimator against single-point data changes. We further show that the two mechanisms exhibit qualitatively different behaviors. In particular, for objective perturbation, increasing the noise level can have non-monotonic effects, and excessive noise may destabilize the estimator, leading to increased sensitivity to data perturbations. Our results demonstrate that AMP provides a powerful framework for analyzing privacy-accuracy trade-offs in high-dimensional sparse models.

We study rank selection for low-rank tensor regression under random covariates design. Under a Gaussian random-design model and some mild conditions, we derive population expressions for the expected training-testing discrepancy (optimism) for both CP and Tucker decomposition. We further demonstrate that the optimism is minimized at the true tensor rank for both CP and Tucker regression. This yields a prediction-oriented rank-selection rule that aligns with cross-validation and extends naturally to tensor-model averaging. We also discuss conditions under which under- or over-ranked models may appear preferable, thereby clarifying the scope of the method. Finally, we showcase its practical utility on a real-world image regression task and extend its application to tensor-based compression of neural network, highlighting its potential for model selection in deep learning.

Reinforcement learning (RL) has become a standard technique for post-training diffusion-based image synthesis models, as it enables learning from reward signals to explicitly improve desirable aspects such as image quality and prompt alignment. In this paper, we propose an online RL variant that reduces the variance in the model updates by sampling paired trajectories and pulling the flow velocity in the direction of the more favorable image. Unlike existing methods that treat each sampling step as a separate policy action, we consider the entire sampling process as a single action. We experiment with both high-quality vision language models and off-the-shelf quality metrics for rewards, and evaluate the outputs using a broad set of metrics. Our method converges faster and yields higher output quality and prompt alignment than previous approaches.

Inthis supplemental material, wefirst present thedetailed networkarchitecture andparameters of the proposed approach in Sec. A. We further provide more analysis of the proposed method and ablation studies in Sec. B. Section C shows some qualitative results for potential applications of the proposed approach on medical imaging and imaging in astronomy. Figure 6: Illustration of learned deep features.(a) The blurry input and ground truth are shown in Figure 1(a)-(b). However, on may actually wonder whether the feature extraction network acts as a denoiser, leading to the observed robustness of the proposed method to various noise levels.

Data normalisation, a common and often necessary preprocessing step in engineering and scientific applications, can severely distort the discovery of governing equations by magnitudebased sparse regression methods. This issue is particularly acute for the Sparse Identification of Nonlinear Dynamics (SINDy) framework, where the core assumption of sparsity is undermined by the interaction between data scaling and measurement noise. The resulting discovered models can be dense, uninterpretable, and physically incorrect. To address this critical vulnerability, we introduce the Sequential Thresholding of Coefficient of Variation (STCV), a novel, computationally efficient sparse regression algorithm that is inherently robust to data scaling. STCV replaces conventional magnitude-based thresholding with a dimensionless statistical metric, the Coefficient Presence (CP), which assesses the statistical validity and consistency of candidate terms in the model library. This shift from magnitude to statistical significance makes the discovery process invariant to arbitrary data scaling. Through comprehensive benchmarking on canonical dynamical systems and practical engineering problems, including a physical mass-spring-damper experiment, we demonstrate that STCV consistently and significantly outperforms standard Sequential Thresholding Least Squares (STLSQ) and Ensemble-SINDy (E-SINDy) on normalised, noisy datasets. The results show that STCV-based methods can successfully identify the correct, sparse physical laws even when other methods fail. By mitigating the distorting effects of normalisation, STCV makes sparse system identification a more reliable and automated tool for real-world applications, thereby enhancing model interpretability and trustworthiness.

Guidance is a crucial technique for extracting the best performance out of image-generating diffusion models. Traditionally, a constant guidance weight has been applied throughout the sampling chain of an image.