arXiv subjects

Sebastian Hoffmann

Publications and source records attributed to Sebastian Hoffmann.

7 recordsLinked to original sources

arXiv2026

Drifting Fields are not Conservative

Drifting models have recently gained attention for generating high-quality samples in a single forward pass. During training, they learn a push-forward map by following a vector-valued field, the drift field. We ask whether this procedure is equivalent to optimizing a scalar loss and find that, in general, it is not: drift fields are not conservative and cannot be written as the gradient of any scalar potential. We identify the position-dependent normalization as the source of non-conservatism, with the Gaussian kernel as the unique radial exception. Guided by this, we introduce the sharp kernel $k^\#$ and a sharp-normalized drift field that is conservative for general radial kernels. The resulting vector field is the gradient of a scalar potential that can be optimized directly using stochastic gradient descent. Moreover, the field has the form of a score difference of kernel density estimates, and gives exact equilibrium identifiability. Thus, sharp normalization closes the gap to related literature, such as Wasserstein gradient-flows and denoising score matching, also for non-Gaussian kernels. Empirically, sharp normalization preserves the performance of the original drifting objective, suggesting that the non-conservative flexibility is not required for high-quality generation.

cs.LG↗

arXiv2026

SoftJAX & SoftTorch: Empowering Automatic Differentiation Libraries with Informative Gradients

Automatic differentiation (AD) frameworks such as JAX and PyTorch have enabled gradient-based optimization for a wide range of scientific fields. Yet, many "hard" primitives in these libraries such as thresholding, Boolean logic, discrete indexing, and sorting operations yield zero or undefined gradients that are not useful for optimization. While numerous "soft" relaxations have been proposed that provide informative gradients, the respective implementations are fragmented across projects, making them difficult to combine and compare. This work introduces SoftJAX and SoftTorch, open-source, feature-complete libraries for soft differentiable programming. These libraries provide a variety of soft functions as drop-in replacements for their hard JAX and PyTorch counterparts. This includes (i) elementwise operators such as clip or abs, (ii) utility methods for manipulating Booleans and indices via fuzzy logic, (iii) axiswise operators such as sort or rank -- based on optimal transport or permutahedron projections, and (iv) offer full support for straight-through gradient estimation. Overall, SoftJAX and SoftTorch make the toolbox of soft relaxations easily accessible to differentiable programming, as demonstrated through benchmarking and a practical case study. Code is available at github.com/a-paulus/softjax and github.com/a-paulus/softtorch.

cs.LG↗

arXiv2022

AtmoDist: Self-supervised Representation Learning for Atmospheric Dynamics

Representation learning has proven to be a powerful methodology in a wide variety of machine learning applications. For atmospheric dynamics, however, it has so far not been considered, arguably due to the lack of large-scale, labeled datasets that could be used for training. In this work, we show that the difficulty is benign and introduce a self-supervised learning task that defines a categorical loss for a wide variety of unlabeled atmospheric datasets. Specifically, we train a neural network on the simple yet intricate task of predicting the temporal distance between atmospheric fields from distinct but nearby times. We demonstrate that training with this task on ERA5 reanalysis leads to internal representations capturing intrinsic aspects of atmospheric dynamics. We do so by introducing a data-driven distance metric for atmospheric states. When employed as a loss function in other machine learning applications, this Atmodist distance leads to improved results compared to the classical $\ell_2$-loss. For example, for downscaling one obtains higher resolution fields that match the true statistics more closely than previous approaches and for the interpolation of missing or occluded data the AtmoDist distance leads to results that contain more realistic fine scale features. Since it is derived from observational data, AtmoDist also provides a novel perspective on atmospheric predictability.

physics.ao-ph↗

arXiv2021

Towards Representation Learning for Atmospheric Dynamics

The prediction of future climate scenarios under anthropogenic forcing is critical to understand climate change and to assess the impact of potentially counter-acting technologies. Machine learning and hybrid techniques for this prediction rely on informative metrics that are sensitive to pertinent but often subtle influences. For atmospheric dynamics, a critical part of the climate system, no well established metric exists and visual inspection is currently still often used in practice. However, this "eyeball metric" cannot be used for machine learning where an algorithmic description is required. Motivated by the success of intermediate neural network activations as basis for learned metrics, e.g. in computer vision, we present a novel, self-supervised representation learning approach specifically designed for atmospheric dynamics. Our approach, called AtmoDist, trains a neural network on a simple, auxiliary task: predicting the temporal distance between elements of a randomly shuffled sequence of atmospheric fields (e.g. the components of the wind field from reanalysis or simulation). The task forces the network to learn important intrinsic aspects of the data as activations in its layers and from these hence a discriminative metric can be obtained. We demonstrate this by using AtmoDist to define a metric for GAN-based super resolution of vorticity and divergence. Our upscaled data matches both visually and in terms of its statistics a high resolution reference closely and it significantly outperform the state-of-the-art based on mean squared error. Since AtmoDist is unsupervised, only requires a temporal sequence of fields, and uses a simple auxiliary task, it has the potential to be of utility in a wide range of applications.

cs.LG↗

arXiv2020

Inpainting-based Video Compression in FullHD

Compression methods based on inpainting are an evolving alternative to classical transform-based codecs for still images. Attempts to apply these ideas to video compression are rare, since reaching real-time performance is very challenging. Therefore, current approaches focus on simplified frame-by-frame reconstructions that ignore temporal redundancies. As a remedy, we propose a highly efficient, real-time capable prediction and correction approach that fully relies on partial differential equations (PDEs) in all steps of the codec: Dense variational optic flow fields yield accurate motion-compensated predictions, while homogeneous diffusion inpainting is applied for intra prediction. To compress residuals, we introduce a new highly efficient block-based variant of pseudodifferential inpainting. Our novel architecture outperforms other inpainting-based video codecs in terms of both quality and speed. For the first time in inpainting-based video compression, we can decompress FullHD (1080p) videos in real-time with a fully CPU-based implementation, outperforming previous approaches by roughly one order of magnitude.

eess.IV↗

arXiv2015

Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods.

cs.CV↗

arXiv2015

Pseudo-inverses of difference matrices and their application to sparse signal approximation

We derive new explicit expressions for the components of Moore-Penrose inverses of symmetric difference matrices. These generalized inverses are applied in a new regularization approach for scattered data interpolation based on partial differential equations. The columns of the Moore-Penrose inverse then serve as elements of a dictionary that allow a sparse signal approximation. In order to find a set of suitable data points for signal representation we apply the orthogonal patching pursuit (OMP) method.

math.NA↗