The first order derivative of a data density can be estimated efficiently by denoising score matching, and has become an important component in many applications, such as image generation and audio synthesis. Higher order derivatives provide additional local information about the data distribution and enable new applications. Although they can be estimated via automatic differentiation of a learned density model, this can amplify estimation errors and is expensive in high dimensional settings. To overcome these limitations, we propose a method to directly estimate high order derivatives (scores) of a data density from samples. We first show that denoising score matching can be interpreted as a particular case of Tweedie's formula. By leveraging Tweedie's formula on higher order moments, we generalize denoising score matching to estimate higher order derivatives. We demonstrate empirically that models trained with the proposed method can approximate second order derivatives more efficiently and accurately than via automatic differentiation. We show that our models can be used to quantify uncertainty in denoising and to improve the mixing speed of Langevin dynamics via Ozaki discretization for sampling synthetic data and natural images.

Flow-based generative models synthesize data by integrating a learned velocity field from a reference distribution to the target data distribution. Prior work has focused on endpoint metrics (e.g., fidelity, likelihood, perceptual quality) while overlooking a deeper question: what do the sampling trajectories reveal? Motivated by classical mechanics, we introduce kinetic path energy (KPE), a simple yet powerful diagnostic that quantifies the total kinetic effort along each generation path of ODE-based samplers. Through comprehensive experiments on CIFAR-10 and ImageNet-256, we uncover two key phenomena: ({i}) higher KPE predicts stronger semantic quality, indicating that semantically richer samples require greater kinetic effort, and ({ii}) higher KPE inversely correlates with data density, with informative samples residing in sparse, low-density regions. Together, these findings reveal that semantically informative samples naturally reside on the sparse frontier of the data distribution, demanding greater generative effort. Our results suggest that trajectory-level analysis offers a physics-inspired and interpretable framework for understanding generation difficulty and sample characteristics.

We propose a general, theoretically justified mechanism for processing missing data by neural networks. Our idea is to replace typical neuron's response in the

Joint Embedding Predictive Architectures (JEPAs) learn representations able to solve numerous downstream tasks out-of-the-box. JEPAs combine two objectives: (i) a latent-space prediction term, i.e., the representation of a slightly perturbed sample must be predictable from the original sample's representation, and (ii) an anti-collapse term, i.e., not all samples should have the same representation. While (ii) is often considered as an obvious remedy to representation collapse, we uncover that JEPAs' anti-collapse term does much more--it provably estimates the data density. In short, any successfully trained JEPA can be used to get sample probabilities, e.g., for data curation, outlier detection, or simply for density estimation. Our theoretical finding is agnostic of the dataset and architecture used--in any case one can compute the learned probabilities of sample $x$ efficiently and in closed-form using the model's Jacobian matrix at $x$. Our findings are empirically validated across datasets (synthetic, controlled, and Imagenet) and across different Self Supervised Learning methods falling under the JEPA family (I-JEPA and DINOv2) and on multimodal models, such as MetaCLIP. We denote the method extracting the JEPA learned density as {\bf JEPA-SCORE}.

A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an It\^o stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps, a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.

The first order derivative of a data density can be estimated efficiently by denoising score matching, and has become an important component in many applications, such as image generation and audio synthesis. Higher order derivatives provide additional local information about the data distribution and enable new applications. Although they can be estimated via automatic differentiation of a learned density model, this can amplify estimation errors and is expensive in high dimensional settings. To overcome these limitations, we propose a method to directly estimate high order derivatives (scores) of a data density from samples. We first show that denoising score matching can be interpreted as a particular case of Tweedie's formula.

Ray tracing is widely employed to model the propagation of radio-frequency (RF) signal in complex environment. The modelling performance greatly depends on how accurately the target scene can be depicted, including the scene geometry and surface material properties. The advances in computer vision and LiDAR make scene geometry estimation increasingly accurate, but there still lacks scalable and efficient approaches to estimate the material reflectivity in real-world environment. In this work, we tackle this problem by learning the material reflectivity efficiently from the path loss of the RF signal from the transmitters to receivers. Specifically, we want the learned material reflection coefficients to minimize the gap between the predicted and measured powers of the receivers. We achieve this by translating the neural reflectance field from optics to RF domain by modelling both the amplitude and phase of RF signals to account for the multipath effects. We further propose a differentiable RF ray tracing framework that optimizes the neural reflectance field to match the signal strength measurements. We simulate a complex real-world environment for experiments and our simulation results show that the neural reflectance field can successfully learn the reflection coefficients for all incident angles. As a result, our approach achieves better accuracy in predicting the powers of receivers with significantly less training data compared to existing approaches.

Due to the increasing complexity of technical systems, accurate first principle models can often not be obtained. Supervised machine learning can mitigate this issue by inferring models from measurement data. Gaussian process regression is particularly well suited for this purpose due to its high data-efficiency and its explicit uncertainty representation, which allows the derivation of prediction error bounds. These error bounds have been exploited to show tracking accuracy guarantees for a variety of control approaches, but their direct dependency on the training data is generally unclear. We address this issue by deriving a Bayesian prediction error bound for GP regression, which we show to decay with the growth of a novel, kernel-based measure of data density. Based on the prediction error bound, we prove time-varying tracking accuracy guarantees for learned GP models used as feedback compensation of unknown nonlinearities, and show to achieve vanishing tracking error with increasing data density. This enables us to develop an episodic approach for learning Gaussian process models, such that an arbitrary tracking accuracy can be guaranteed. The effectiveness of the derived theory is demonstrated in several simulations.

Inverse problems, i.e., estimating parameters of physical models from experimental data, are ubiquitous in science and engineering. The Bayesian formulation is the gold standard because it alleviates ill-posedness issues and quantifies epistemic uncertainty. Since analytical posteriors are not typically available, one resorts to Markov chain Monte Carlo sampling or approximate variational inference. However, inference needs to be rerun from scratch for each new set of data. This drawback limits the applicability of the Bayesian formulation to real-time settings, e.g., health monitoring of engineered systems, and medical diagnosis. The objective of this paper is to develop a methodology that enables real-time inference by learning the Bayesian inverse map, i.e., the map from data to posteriors. Our approach is as follows. We represent the posterior distribution using a parameterization based on deep neural networks. Next, we learn the network parameters by amortized variational inference method which involves maximizing the expectation of evidence lower bound over all possible datasets compatible with the model. We demonstrate our approach by solving examples a set of benchmark problems from science and engineering. Our results show that the posterior estimates of our approach are in agreement with the corresponding ground truth obtained by Markov chain Monte Carlo. Once trained, our approach provides the posterior parameters of observation just at the cost of a forward pass of the neural network.

Estimating the data density is one of the challenging problems in deep learning. In this paper, we present a simple yet effective method for estimating the data density using a deep neural network and the Donsker-Varadhan variational lower bound on the KL divergence. We show that the optimal critic function associated with the Donsker-Varadhan representation on the KL divergence between the data and the uniform distribution can estimate the data density. We also present the deep neural network-based modeling and its stochastic learning. The experimental results and possible applications of the proposed method demonstrate that it is competitive with the previous methods and has a lot of possibilities in applied to various applications.