A related situation arises when an auxiliary process is introduced to aid training but modelling its dynamics at generation time is unnecessary or difficult, as in Billera et al. [2025b] and Kim et al. [2025]. In each of these works, the projection result and its associated loss are derived on a case-by-case basis, and all theorems are restricted to marginalization over a discrete component of the extended state space. We introduce a general framework that removes these restrictions: given a time-inhomogeneous Feller process (Yt)0 t 1 on an arbitrary state space Y and a map Φ: Y X, one may learn a linear parametrisation of the generator of a Feller process on X whose one-time marginals coincide with those of (Φ(Yt))0 t 1. For Y = X Z and Φthe projection onto the first coordinate, this subsumes these prior works as special cases, allowing for a general class of latent processes (Zt)0 t 1 in a nearly arbitrary state space Z, using the formalism of generator matching to allow for continuous, discrete, or manifold-valued processes. In particular, the learnt process at t = 1 samples from the distribution of Φ(Y1), which is the desired data distribution. We give sufficient conditions for a loss function to be valid in this general setting, recovering the results of the works cited above as corollaries. This result has broad applicability, enabling the construction of a wide array of new flow matching schemes by allowing for a more general class of latent spaces. As a concrete new application, we outline a non-projection Φ: Y X with manifold-valued latents for protein structure generation that separates chain-level rigid-body motion from internal flexibility ( 4), where the particular chain-level versus residue-level or internal state is latent, and the model only sees the world state, which we plan to implement in future work. 2 EARLIERWORK Several recent generative models train with the aid of a latent stochastic process that is marginalised out at generation time.

When it comes to using computers to steal money, few can come close to matching the success of Russian hacker Evgeniy Bogachev. The $3 million bounty the FBI has offered for Bogachev's capture is larger than any that has ever been offered for a cybercriminal--but that sum represents only a tiny fraction of the money he has stolen through his botnet GameOver ZeuS.1 At its height in 2012 and 2013, GameOver ZeuS, or GOZ, comprised between 500,000 and 1 million compromised computers all over the world that Bogachev could control remotely. For years, Bogachev used these machines to spread malware that allowed him to steal banking credentials and perpetrate online extortion.2 No one knows exactly how much money Bogachev stole from his thousands of victims using GOZ, but the FBI conservatively estimates that it was well over $100 million.2 Meanwhile, Bogachev has spent lavishly on a fleet of luxury cars, two French villas, and a large yacht.1 Bogachev lives in the resort town of Anapa on the Black Sea, where Russian officials have declined for years to arrest him or extradite him to the United States.

The three college-age defendants behind the creation of the Mirai botnet--an online tool that wreaked destruction across the internet in the fall of 2016 with unprecedentedly powerful distributed denial of service attacks--will stand in an Alaska courtroom Tuesday and ask for a novel ruling from a federal judge: They hope to be sentenced to work for the FBI. Josiah White, Paras Jha, and Dalton Norman, who were all between 18 and 20 years old when they built and launched Mirai, pleaded guilty last December to creating the malware that hijacked hundreds of thousands of Internet of Things devices, uniting them as a digital army that began as a way to attack rival Minecraft video game hosts, and evolved into an online tsunami of nefarious traffic that knocked entire web hosting companies offline. At the time, the attacks raised fears amid the presidential election targeted online by Russia that an unknown adversary was preparing to lay waste to the internet. The original creators, panicking as they realized their invention was more powerful than they had imagined, released the code--a common tactic by hackers to ensure that if and when authorities catch them, they don't possess any code that isn't already publicly known that can help finger them as the inventors. That release in turn lead to attacks by others throughout the fall, including one that made much of the internet unusable for the East Coast of the United States on an October Friday.