Unlike the prior results on fast spherical harmonic transform, our proposed algorithm works efficiently using a nearly optimal number of samples in any dimension d. Furthermore, we illustrate the empirical performance of our algorithm on numerical examples.

Neural operators effectively solve PDE problems from data without knowing the explicit equations, which learn the map from the input sequences of observed samples to the predicted values. Most existing works build the model in the original geometric space, leading to high computational costs when the number of sample points is large. We present the Latent Neural Operator (LNO) solving PDEs in the latent space. In particular, we first propose Physics-Cross-Attention (PhCA) transforming representation from the geometric space to the latent space, then learn the operator in the latent space, and finally recover the real-world geometric space via the inverse PhCA map. Our model retains flexibility that can decode values in any position not limited to locations defined in the training set, and therefore can naturally perform interpolation and extrapolation tasks particularly useful for inverse problems. Moreover, the proposed LNO improves both prediction accuracy and computational efficiency. Experiments show that LNO reduces the GPU memory by 50%, speeds up training 1.8 times, and reaches state-of-the-art accuracy on four out of six benchmarks for forward problems and a benchmark for inverse problem.

Unpaired image-to-image translation aims to translate an input image to another domain such that the output image looks like an image from another domain while important semantic information are preserved. Inferring the optimal mapping with unpaired data is impossible without making any assumptions. In this paper, we make a density changing assumption where image patches of high probability density should be mapped to patches of high probability density in another domain. Then we propose an efficient way to enforce this assumption: we train the flows as density estimators and penalize the variance of density changes. Despite its simplicity, our method achieves the best performance on benchmark datasets and needs only 56 86% of training time of the existing state-of-the-art method. The training and evaluation code are avaliable at https://github.com/Mid-Push/

This paper identifies a structural property of data distributions that enables deep neural networks to learn hierarchically. We define the "staircase" property for functions over the Boolean hypercube, which posits that high-order Fourier coefficients are reachable from lower-order Fourier coefficients along increasing chains. We prove that functions satisfying this property can be learned in polynomial time using layerwise stochastic coordinate descent on regular neural networks - a class of network architectures and initializations that have homogeneity properties. Our analysis shows that for such staircase functions and neural networks, the gradient-based algorithm learns high-level features by greedily combining lower-level features along the depth of the network. We further back our theoretical results with experiments showing that staircase functions are learnable by more standard ResNet architectures with stochastic gradient descent. Both the theoretical and experimental results support the fact that the staircase property has a role to play in understanding the capabilities of gradient-based learning on regular networks, in contrast to general polynomial-size networks that can emulate any Statistical Query or PAC algorithm, as recently shown.

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National security and military analyst Dr. Rebecca Grant joins'Fox & Friends First' to discuss President Donald Trump's historic business-focused trip to the Middle East and why a Trump-Putin meeting could be essential for peace in Ukraine. President Donald Trump secured 2 trillion worth of deals with Saudi Arabia, Qatar and the UAE during his trip to the Middle East last week in what some have argued is a move to counter China's influence in the region. While China has increasingly bolstered its commercial ties with top Middle Eastern nations who have remained steadfast in their refusal to pick sides amid growing geopolitical tension between Washington and Beijing, Trump may have taken steps to give the U.S. an edge over its chief competitor. But concern has mounted after Trump reversed a Biden-era policy – which banned the sale of AI-capable chips to the UAE and Saudi Arabia – that highly coveted U.S. technologies could potentially fall into the hands of Chinese companies, and in extension, the Chinese Communist Party (CCP). U.S. President Donald Trump walks with Saudi Crown Prince Mohammed Bin Salman during a welcoming ceremony in Riyadh, Saudi Arabia, May 13, 2025.

Fox News anchor Bret Baier has the latest on the Murdoch Children's Research Institute's partnership with the Gladstone Institutes for the "Decoding Broken Hearts" initiative on "Special Report." Imagine having a tireless guardian watching over you during your subway commute. New York City's subway system is testing artificial intelligence to boost security and reduce crime. Michael Kemper, a 33-year NYPD veteran and the chief security officer for the Metropolitan Transportation Authority (MTA), which is the largest transit agency in the United States, is leading the rollout of AI software designed to spot suspicious behavior as it happens. The MTA says this technology represents the future of subway surveillance and reassures riders that privacy concerns are being taken seriously.

Saliency methods have been widely used to highlight important input features in model predictions. Most existing methods use backpropagation on a modified gradient function to generate saliency maps. Thus, noisy gradients can result in unfaithful feature attributions.

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