We investigate a forecasting framework based on a simple discrete-time dynamic model with coefficients varying in time. The parameters of the model are recovered within a deep learning framework, which makes it possible to retain a transparent parametric structure while simultaneously accounting for complex and nonstationary patterns in the observed phenomenon. Our analysis covers two specifications of the noise process. Besides the standard Gaussian setting, we also consider Laplace-distributed noise, which can offer a more adequate description in the presence of heavier tails and sharper local fluctuations. For both cases, we formulate the predictive scheme of the model and analyze the associated uncertainty quantification, including the construction of prediction intervals. The results illustrate that a relatively simple model, when combined with time-dependent parameter estimation, can serve as a mathematically tractable and practically flexible tool for forecasting complex dynamics under different noise assumptions. The general model is stated for TVAR($p$), while the prediction-interval formulas and the numerical experiments are developed for the TVAR(1) case.

In this paper, we study Nyström subsampling for vector-valued regression in vector-valued reproducing kernel Hilbert spaces. Standard kernel methods often suffer from prohibitive computational costs due to the construction and inversion of large kernel matrices, which limits their scalability to large datasets. To overcome this bottleneck, we propose an efficient operator learning algorithm based on Nyström subsampling that accommodates functional outputs. Under general source conditions characterized by index functions-extending beyond the classical Hölder-type and operator-monotone frameworks-we establish minimax-optimal convergence rates for the proposed estimator. As an application of the proposed framework, we consider function denoising problems. Unlike classical denoising methods, which are typically tailored to specific signal representations or noise models, our approach formulates denoising within a general operator learning framework. Numerical experiments on signal denoising, real-time audio denoising, image denoising, inverse Radon transform reconstruction, and energy-efficiency prediction confirm that the proposed method achieves performance comparable to full kernel methods while substantially reducing computational cost.

Directed acyclic graph (DAG) learning is a central task in structure discovery and causal inference. Although the field has witnessed remarkable advances over the past few years, it remains statistically and computationally challenging to learn a single (point estimate) DAG from data, let alone provide uncertainty quantification. We address the difficult task of quantifying graph uncertainty by developing a Bayesian variational inference framework based on novel, provably valid distributions that have support directly on the space of sparse DAGs. These distributions, which we use to define our prior and variational posterior, are induced by a projection operation that maps an arbitrary continuous distribution onto the space of sparse weighted acyclic adjacency matrices. While this projection is combinatorial, it can be solved efficiently using recent continuous reformulations of acyclicity constraints. We empirically demonstrate that our method, ProDAG, can outperform state-of-the-art alternatives in both accuracy and uncertainty quantification.

Leveraging the powerful generation capability of large-scale pretrained text-to-image models, training-free methods have demonstrated impressive image editing results. Conventional diffusion-based methods, as well as recent rectified flow (RF)-based methods, typically reverse synthesis trajectories by gradually adding noise to clean images, during which the noisy latent at the current timestep is used to approximate that at the next timesteps, introducing accumulated drift and degrading reconstruction accuracy. Considering the fact that in RF the noisy latent is estimated through direct interpolation between Gaussian noises and clean images at each timestep, we propose Direct Noise Alignment (DNA), which directly refines the desired Gaussian noise in the noise domain, significantly reducing the error accumulation in previous methods. Specifically, DNA estimates the velocity field of the interpolated noised latent at each timestep and adjusts the Gaussian noise by computing the difference between the predicted and expected velocity field. We validate the effectiveness of DNA and reveal its relationship with existing RF-based inversion methods.

We propose Whitened Score (WS) diffusion models, a novel framework based on stochastic differential equations that learns the Whitened Score function instead of the standard score. This approach circumvents covariance inversion, extending score-based DMs by enabling stable training of DMs on arbitrary Gaussian forward noising processes. WS DMs establish equivalence with flow matching for arbitrary Gaussian noise, allow for tailored spectral inductive biases, and provide strong Bayesian priors for imaging inverse problems with structured noise. We experiment with a variety of computational imaging tasks using the CIFAR, CelebA ($64\times64$), and CelebA-HQ ($256\times256$) datasets and demonstrate that WS diffusion priors trained on anisotropic Gaussian noising processes consistently outperform conventional diffusion priors based on isotropic Gaussian noise.

A.1 Training dataset pre-processing We used 40000publicly available videos from YouTube which were available in a spatial resolution of at least 1920 1080 pixels. In an attempt not to skew the distribution of content too far from what may inform biological representation learning, we excluded most artificial content such as screenshots and videos of computer games. To reduce video compression artifacts and prevent systematic downsampling artifacts, each segment was then spatially downsampled to a randomized height between 128 and 160. Each segment was then separated into 15 pairs of neighboring frames, and a randomly placed, but spatially colocated patch of 64 64 pixels was cropped out of each frame pair. The order of the frame pairs was then randomized in a running buffer, and all RGB pixel values were normalized to the range between 0 and 1 before being fed into the model.

A.1 Implementation of the GPs We use the GPyTorch4 package for the computations of GPs and their kernels. The NN linear kernel is implemented in all experiments as a 1-layer MLP with ReLU activations and hidden dimension 16. For the Spectral Mixture Kernel, we use 4 mixtures. A.2 Sines Dataset For the first experiments on sines functions, we use the dataset from [9]. For each task, the input points x are sampled from the range [ 5,5], and the target values y are obtained by applying y = Asin(x ')+, where the amplitude A and phase ' are drawn uniformly at random from ranges [0.1,5] and [0, ], respectively.

The query-based black-box attacks have raised serious threats to machine learning models in many real applications. In this work, we study a lightweight defense method, dubbed Random Noise Defense (RND), which adds proper Gaussian noise to each query. We conduct the theoretical analysis about the effectiveness of RND against query-based black-box attacks and the corresponding adaptive attacks. Our theoretical results reveal that the defense performance of RND is determined by the magnitude ratio between the noise induced by RND and the noise added by the attackers for gradient estimation or local search. The large magnitude ratio leads to the stronger defense performance of RND, and it's also critical for mitigating adaptive attacks. Based on our analysis, we further propose to combine RND with a plausible Gaussian augmentation Fine-tuning (RND-GF). It enables RND to add larger noise to each query while maintaining the clean accuracy to obtain a better trade-off between clean accuracy and defense performance. Additionally, RND can be flexibly combined with the existing defense methods to further boost the adversarial robustness, such as adversarial training (AT). Extensive experiments on CIFAR-10 and ImageNet verify our theoretical findings and the effectiveness of RND and RND-GF.

In this section, we provide additional implementation details, which are useful to reproduce our experiments (note that the code is also provided). For each band i J0,c 1K, the standard deviation of the Gaussian noise is defined as: σi = βexp " 1 4η2 i c 1 2 A basic centering step is used for each input patch of our model. More precisely, for the first layer, each band of the input hyperspectral image is centered independently prior to patches extraction, and means are added back after decoding. For the second layer, patches are centered independently for each band (and similarly, the means are added back after decoding). Code and patch sizes The hyperparameters of our model are presented in Table 1.