Transformers, which are state-of-the-art in most machine learning tasks, represent the data as sequences of vectors called tokens. This representation is then exploited by the attention function, which learns dependencies between tokens and is key to the success of Transformers. However, the iterative application of attention across layers induces complex dynamics that remain to be fully understood. To analyze these dynamics, we identify each input sequence with a probability measure and model its evolution as a Vlasov equation called Transformer PDE, whose velocity field is non-linear in the probability measure. Our first set of contributions focuses on compactly supported initial data. We show the Transformer PDE is well-posed and is the mean-field limit of an interacting particle system, thus generalizing and extending previous analysis to several variants of self-attention: multi-head attention, L2 attention, Sinkhorn attention, Sigmoid attention, and masked attention--leveraging a conditional Wasserstein framework. In a second set of contributions, we are the first to study non-compactly supported initial conditions, by focusing on Gaussian initial data. Again for different types of attention, we show that the Transformer PDE preserves the space of Gaussian measures, which allows us to analyze the Gaussian case theoretically and numerically to identify typical behaviors. This Gaussian analysis captures the evolution of data anisotropy through a deep Transformer. In particular, we highlight a clustering phenomenon that parallels previous results in the non-normalized discrete case.

Caloza was once a community organizer for President Obama and a Los Angeles Board of Public Works commissioner. She also served in the Obama administration's Department of Education and as a staffer to former L.A. Mayor Eric Garcetti. Her main priorities are protecting reproductive health and access to abortion by fully funding Planned Parenthood and making it easier for the organization to open more locations across California. She also plans to focus on how artificial intelligence is replacing jobs and making sure public education in the state is fully funded. The presidential race between Democratic Vice President Kamala Harris and Republican former President Trump is at the top of the ticket, but Californians will vote on a number of other races.

A hotel room automatically adjusting to the tastes of each guest, virtual reality headsets as brochures: the tourism sector is starting to embrace new technologies, hoping to benefit from lucrative personal data. In a prototype of the hotel of the future on display at Madrid's Fitur tourism fair, receptionists have disappeared and customers are checked-in via a mirror equipped with facial recognition. Once the client is identified, the room adapts itself automatically to all demands made at reservation: temperature, lighting, Picasso or Van Gogh in the digital frames hanging on the walls. "Technology will allow us to know what the client needs before he even knows he wants it," says Alvaro Carrillo de Albornoz, head of Spain's Hotel Technology Institute. Some hotels already offer such experiences at a more basic level.

In a prototype of the hotel of the future on display at Madrid's Fitur tourism fair, receptionists have disappeared and customers are checked-in via a mirror equipped with facial recognition. Once the client is identified, the room adapts itself automatically to all demands made at reservation: temperature, lighting, Picasso or Van Gogh in the digital frames hanging on the walls. "Technology will allow us to know what the client needs before he even knows he wants it," says Alvaro Carrillo de Albornoz, head of Spain's Hotel Technology Institute, which promotes innovation in the sector. Some hotels already offer such experiences at a more basic level. But the room prototype put on show by French technology consultancy Altran, aimed at luxury hotels, has incorporated cutting-edge speech recognition technology, allowing for instance a guest to order a pizza in 40 languages.

MADRID (AFP) - A hotel room automatically adjusting to the tastes of each guest, virtual reality headsets as brochures: the tourism sector is starting to embrace new technologies, hoping to benefit from lucrative personal data. In a prototype of the hotel of the future on display at Madrid's Fitur tourism fair, receptionists have disappeared and customers are checked-in via a mirror equipped with facial recognition. Once the client is identified, the room adapts itself automatically to all demands made at reservation: temperature, lighting, Picasso or Van Gogh in the digital frames hanging on the walls. "Technology will allow us to know what the client needs before he even knows he wants it," says Alvaro Carrillo de Albornoz, head of Spain's Hotel Technology Institute, which promotes innovation in the sector. Some hotels already offer such experiences at a more basic level.