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.

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.

OpenAI started as a non-profit dedicated to building safe A.I. Now, they're obsessed with building artificial general intelligence by any means necessary - even if they don't quite know what that is. Subscribe to Slate Plus to access ad-free listening to the whole What Next family and all your favorite Slate podcasts. Subscribe today on Apple Podcasts by clicking "Try Free" at the top of our show page. Sign up now at slate.com/whatnextplus to get access wherever you listen.

We introduce a "learning-based" algorithm for the low-rank decomposition problem: given an n d matrix A, and a parameter k, compute a rank-k matrix A

In this work, we observe that model trained on vast general images via masking strategy, has been naturally embedded with their distribution knowledge, thus spontaneously attains the underlying potential for strong image denoising. Based on this observation, we propose a novel zero-shot denoising paradigm, i.e., Masked Pre-train then Iterative fill (MPI). MPI first trains model via masking and then employs pre-trained weight for high-quality zero-shot image denoising on a single noisy image. Concretely, MPI comprises two key procedures: 1) Masked Pre-training involves training model to reconstruct massive natural images with random masking for generalizable representations, gathering the potential for valid zero-shot denoising on images with varying noise degradation and even in distinct image types.

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inference rules that infer new inequalities from the premises. These proof systems are known to be very powerful, but searching for proofs remains a major difficulty. In this work, we introduce a machine learning based method to search for a dynamic proof within these proof systems. We propose a deep reinforcement learning framework that learns an embedding of the polynomials and guides the choice of inference rules, taking the inherent symmetries of the problem as an inductive bias. We compare our approach with powerful and widely-studied linear programming hierarchies based on static proof systems, and show that our method reduces the size of the linear program by several orders of magnitude while also improving performance. These results hence pave the way towards augmenting powerful and well-studied semi-algebraic proof systems with machine learning guiding strategies for enhancing the expressivity of such proof systems.

A.1 Dataset Samples We show different kinds of perturbations in our benchmarks in Fig.5. Specifically, our benchmarks include 9 basic types of perturbations, including Gaussian blur, Gaussian noise, radial distortion, and RGB and HSV channels. Another type of datasets include multiple perturbations, where we create multiple random combinations of the basic perturbations. We also include 7 types of previously unseen perturbations (during training) from ImageNet-C [20], which are snow, fog, frost, motion blur, zoom blur, pixelate, and jpeg compression. For each type of perturbation, we generate 5 or 10 levels of varying intensity based on sensitivity analysis in the FID-MA space.