The folds in the tilted rock layers and differences in their erosion rate gives the limestone the step-like appearance we see today. Breakthroughs, discoveries, and DIY tips sent six days a week. While Ireland's natural landscape is known for every shade of green imaginable, a different color dominates one part of Ireland. Along the Burren Region on the country's western coast, gray limestone pavement covers the rocky and treeless landscape. NASA's Operational Land Imager (OLI) on the Landsat 8 satellite captured a view of Burren, showing the rocky landscape and an 860-foot-tall limestone hill called Moneen Mountain.

The performance of neural networks on high-dimensional data distributions suggests that it may be possible to parameterize a representation of a given high-dimensional function with controllably small errors, potentially outperforming standard interpolation methods. We demonstrate, both theoretically and numerically, that this is indeed the case. We map the parameters of a neural network to a system of particles relaxing with an interaction potential determined by the loss function. We show that in the limit that the number of parameters $n$ is large, the landscape of the mean-squared error becomes convex and the representation error in the function scales as $O(n^{-1})$. In this limit, we prove a dynamical variant of the universal approximation theorem showing that the optimal representation can be attained by stochastic gradient descent, the algorithm ubiquitously used for parameter optimization in machine learning. In the asymptotic regime, we study the fluctuations around the optimal representation and show that they arise at a scale $O(n^{-1})$. These fluctuations in the landscape identify the natural scale for the noise in stochastic gradient descent. Our results apply to both single and multi-layer neural networks, as well as standard kernel methods like radial basis functions.

The sophisticated network crossed mountains in Peru and pre-dates the Inca Empire. Breakthroughs, discoveries, and DIY tips sent six days a week. Ancient parrots really got around. A new analysis of their DNA found that humans transported living Amazonian macaw parrots across the Andes mountains to coastal Peru hundreds of years before the Inca Empire. The findings are detailed in a study published today in the journal and reveal a highly sophisticated and long-distance bird trading network across deserts, highlands, and rainforests.

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science.

By analyzing the loss landscape of a single Transformer layer using Softmax and Gaussian attention kernels, our work provides concrete answers to these questions.

Addressing these challenges requires scale. To that end we introduce ProteinGym, a large-scale and holistic set of benchmarks specifically designed for protein fitness prediction and design.

Related mixed matrix-tensor models have also been studied in the context of text-analysis applications in [20, 19].