Score-based generative models (SGMs) synthesize new data samples from Gaussian white noise by running a time-reversed Stochastic Differential Equation (SDE) whose drift coefficient depends on some probabilistic score. The discretization of such SDEs typically requires a large number of time steps and hence a high computational cost. This is because of ill-conditioning properties of the score that we analyze mathematically. Previous approaches have relied on multiscale generation to considerably accelerate SGMs. We explain how this acceleration results from an implicit factorization of the data distribution into a product of conditional probabilities of wavelet coefficients across scales. The resulting Wavelet Score-based Generative Model (WSGM) synthesizes wavelet coefficients with the same number of time steps at all scales, and its time complexity therefore grows linearly with the image size. This is proved mathematically for Gaussian distributions, and shown numerically for physical processes at phase transition and natural image datasets.

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

We present a novel approach for nonparametric regression using wavelet basis functions.

We consider the framework of non-stationary stochastic optimization [Besbes et al., 2015] with squared error losses and noisy gradient feedback where the dynamic regret of an online learner against a time varying comparator sequence

We consider the framework of non-stationary stochastic optimization [Besbes et al., 2015] with squared error losses and noisy gradient feedback where the dynamic regret of an online learner against a time varying comparator sequence

Accurate tropical cyclone (TC) short-term intensity forecasting with a 24-hour lead time is essential for disaster mitigation in the Atlantic TC basin. Since most TCs evolve far from land-based observing networks, satellite imagery is critical to monitoring these storms; however, these complex and high-resolution spatial structures can be challenging to qualitatively interpret in real time by forecasters. Here we propose a concise, interpretable, and descriptive approach to quantify fine TC structures with a multi-resolution analysis (MRA) by the discrete wavelet transform, enabling data analysts to identify physically meaningful structural features that strongly correlate with rapid intensity change. Furthermore, deep-learning techniques can build on this MRA for short-term intensity guidance.

Understanding signal behavior across scales is vital in areas such as natural phenomena analysis and financial modeling. A key property is self-similarity, quantified by the Hurst exponent (H), which reveals long-term dependencies. Wavelet-based methods are effective for estimating H due to their multi-scale analysis capability, but additive noise in real-world measurements often degrades accuracy. We propose Noise-Controlled ALPHEE (NC-ALPHEE), an enhancement of the Average Level-Pairwise Hurst Exponent Estimator (ALPHEE), incorporating noise mitigation and generating multiple level-pairwise estimates from signal energy pairs. A neural network (NN) combines these estimates, replacing traditional averaging. This adaptive learning maintains ALPHEE's behavior in noise-free cases while improving performance in noisy conditions. Extensive simulations show that in noise-free data, NC-ALPHEE matches ALPHEE's accuracy using both averaging and NN-based methods. Under noise, however, traditional averaging deteriorates and requires impractical level restrictions, while NC-ALPHEE consistently outperforms existing techniques without such constraints. NC-ALPHEE offers a robust, adaptive approach for H estimation, significantly enhancing the reliability of wavelet-based methods in noisy environments.

Time series forecasting has various applications, such as meteorological rainfall prediction, traffic flow analysis, financial forecasting, and operational load monitoring for various systems. Due to the sparsity of time series data, relying solely on time-domain or frequency-domain modeling limits the model's ability to fully leverage multi-domain information. Moreover, when applied to time series forecasting tasks, traditional attention mechanisms tend to over-focus on irrelevant historical information, which may introduce noise into the prediction process, leading to biased results. We proposed WDformer, a wavelet-based differential Transformer model. This study employs the wavelet transform to conduct a multi-resolution analysis of time series data. By leveraging the advantages of joint representation in the time-frequency domain, it accurately extracts the key information components that reflect the essential characteristics of the data. Furthermore, we apply attention mechanisms on inverted dimensions, allowing the attention mechanism to capture relationships between multiple variables. When performing attention calculations, we introduced the differential attention mechanism, which computes the attention score by taking the difference between two separate softmax attention matrices. This approach enables the model to focus more on important information and reduce noise. WDformer has achieved state-of-the-art (SOTA) results on multiple challenging real-world datasets, demonstrating its accuracy and effectiveness. Code is available at https://github.com/xiaowangbc/WDformer.

Implicit neural representations (INRs) have emerged as a compact and parametric alternative to discrete array-based data representations, encoding information directly in neural network weights to enable resolution-independent representation and memory efficiency. However, existing INR approaches, when constrained to compact network sizes, struggle to faithfully represent the multi-scale structures, high-frequency information, and fine textures that characterize the majority of scientific datasets. To address this limitation, we propose WIEN-INR, a wavelet-informed implicit neural representation that distributes modeling across different resolution scales and employs a specialized kernel network at the finest scale to recover subtle details. This multi-scale architecture allows for the use of smaller networks to retain the full spectrum of information while preserving the training efficiency and reducing storage cost. Through extensive experiments on diverse scientific datasets spanning different scales and structural complexities, WIEN-INR achieves superior reconstruction fidelity while maintaining a compact model size. These results demonstrate WIEN-INR as a practical neural representation framework for high-fidelity scientific data encoding, extending the applicability of INRs to domains where efficient preservation of fine detail is essential.

Extragalactic foregrounds in cosmic microwave background (CMB) observations are both a source of cosmological and astrophysical information and a nuisance to the CMB. Effective field-level modeling that captures their non-Gaussian statistical distributions is increasingly important for optimal information extraction, particularly given the precise and low-noise observations from current and upcoming experiments. We explore the use of Wavelet Flow (WF) models to tackle the novel task of modeling the field-level probability distributions of multi-component CMB secondaries and foreground. Specifically, we jointly train correlated CMB lensing convergence ($κ$) and cosmic infrared background (CIB) maps with a WF model and obtain a network that statistically recovers the input to high accuracy -- the trained network generates samples of $κ$ and CIB fields whose average power spectra are within a few percent of the inputs across all scales, and whose Minkowski functionals are similarly accurate compared to the inputs. Leveraging the multiscale architecture of these models, we fine-tune both the model parameters and the priors at each scale independently, optimizing performance across different resolutions. These results demonstrate that WF models can accurately simulate correlated components of CMB secondaries, supporting improved analysis of cosmological data. Our code and trained models can be found here (https://github.com/matiwosm/HybridPriorWavletFlow.git).