This paper introduces the sigma flow model for the prediction of structured labelings of data observed on Riemannian manifolds, including Euclidean image domains as special case. The approach combines the Laplace-Beltrami framework for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the assignment flow approach introduced and studied by the authors. The sigma flow arises as Riemannian gradient flow of generalized harmonic energies and thus is governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher-Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof of concept experiments demonstrate the expressivity of the sigma flow model and prediction performance. Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning.

Hackers are having a field day, and weak authentication is a major cause. The vast majority of cyberattacks -- some 80%, statistics show -- have their roots in compromised passwords that hackers get hold of. All it takes is one stolen password for hackers to wreak havoc; and according to experts, that single password breach can cost enterprise firms over $7 million. Many schemes have been tried to build up password security, including increased education and 2FA. But despite that, password compromise statistics remain stubbornly high, cybersecurity education programs, although widespread, don't seem to work, and 2FA has its own security issues.

For those who don't have the patience to wait for fish to take the bait, a robotics firm has unveiled a device that finds and attracts them to you. Called the PowerRay, the underwater drone uses a precision remote bait drop and blue-hued fishing lures that basically attaches the fish to the dangling hook. The unmanned vehicle also transmits data, fish alerts and live video feed and photos of the underwater world to the fisherman's smartphone via a companion app. PowerRay uses a precision remote bait drop and blue-hued fishing lures that basically attaches the fish to the dangling hook. The unmanned vehicle also transmits data, alters and live video feed and photos of the underwater world to the fisherman's smartphone via a companion app The sonar system can detect fish up to 131ft below the swimming robot and can dive up to 98 feet.