Tree tensor networks (TTNs) are widely used in low-rank approximation and quantum many-body simulation. In this work, we present a formal analysis of the differential geometry underlying TTNs. Building on this foundation, we develop efficient first- and second-order optimization algorithms that exploit the intrinsic quotient structure of TTNs. Additionally, we devise a backpropagation algorithm for training TTNs in a kernel learning setting. We validate our methods through numerical experiments on a representative machine learning task.

OpenAI's trust and safety lead, Dave Willner, has left the position, as announced via a Linkedin post. Willner is staying on in an "advisory role" but has asked Linkedin followers to "reach out" for related opportunities. The former OpenAI project lead states that the move comes after a decision to spend more time with his family. Yes, that's what they always say, but Willner follows it up with actual details. "In the months following the launch of ChatGPT, I've found it more and more difficult to keep up my end of the bargain," he writes.