V olumes were then downsampled to D = 128 .

Knowledge of the primordial matter density field from which the large-scale structure of the Universe emerged over cosmic time is of fundamental importance for cosmology. However, reconstructing these cosmological initial conditions from late-time observations is a notoriously difficult task, which requires advanced cosmological simulators and sophisticated statistical methods to explore a multi-million-dimensional parameter space. We show how simulation-based inference (SBI) can be used to tackle this problem and to obtain data-constrained realisations of the primordial dark matter density field in a simulation-efficient way with general non-differentiable simulators. Our method is applicable to full high-resolution dark matter $N$-body simulations and is based on modelling the posterior distribution of the constrained initial conditions to be Gaussian with a diagonal covariance matrix in Fourier space. As a result, we can generate thousands of posterior samples within seconds on a single GPU, orders of magnitude faster than existing methods, paving the way for sequential SBI for cosmological fields. Furthermore, we perform an analytical fit of the estimated dependence of the covariance on the wavenumber, effectively transforming any point-estimator of initial conditions into a fast sampler. We test the validity of our obtained samples by comparing them to the true values with summary statistics and performing a Bayesian consistency test.

Since its introduction, the transformers architecture has seen great adoption in NLP applications, but it also has limitations. Although the self-attention mechanism allows for generating very rich representations of the input text, its effectiveness may be limited in specialized domains such as legal, where, for example, language models often have to process very long texts. In this paper, we explore alternatives to replace the attention-based layers with simpler token-mixing mechanisms: Hartley and Fourier transforms. Using these non-parametric techniques, we train models with long input documents from scratch in the legal domain setting. We also introduce a new hybrid Seq2Seq architecture, a no-attention-based encoder connected with an attention-based decoder, which performs quite well on existing summarization tasks with much less compute and memory requirements. We believe that similar, if not better performance, as in the case of long correlations of abstractive text summarization tasks, can be achieved by adopting these simpler infrastructures. This not only makes training models from scratch accessible to more people, but also contributes to the reduction of the carbon footprint during training.

In most convolution neural networks (CNNs), downsampling hidden layers is adopted for increasing computation efficiency and the receptive field size. Such operation is commonly so-called pooling. Maximation and averaging over sliding windows (max/average pooling), and plain downsampling in the form of strided convolution are popular pooling methods. Since the pooling is a lossy procedure, a motivation of our work is to design a new pooling approach for less lossy in the dimensionality reduction. Inspired by the Fourier spectral pooling(FSP) proposed by Rippel et. al. [1], we present the Hartley transform based spectral pooling method in CNNs. Compared with FSP, the proposed spectral pooling avoids the use of complex arithmetic for frequency representation and reduces the computation. Spectral pooling preserves more structure features for network's discriminability than max and average pooling. We empirically show that Hartley spectral pooling gives rise to the convergence of training CNNs on MNIST and CIFAR-10 datasets.