The proliferation of Large Language Models (LLMs) necessitates efficient mechanisms to distinguish machine-generated content from human text. While statistical watermarking has emerged as a promising solution, existing methods suffer from two critical limitations: the lack of a principled approach for selecting sampling distributions and the reliance on fixed-horizon hypothesis testing, which precludes valid early stopping. In this paper, we bridge this gap by developing the first e-value-based watermarking framework, Anchored E-Watermarking, that unifies optimal sampling with anytime-valid inference. Unlike traditional approaches where optional stopping invalidates Type-I error guarantees, our framework enables valid, anytime-inference by constructing a test supermartingale for the detection process. By leveraging an anchor distribution to approximate the target model, we characterize the optimal e-value with respect to the worst-case log-growth rate and derive the optimal expected stopping time. Our theoretical claims are substantiated by simulations and evaluations on established benchmarks, showing that our framework can significantly enhance sample efficiency, reducing the average token budget required for detection by 13-15% relative to state-of-the-art baselines.

We study one-sided and $α$-correct sequential hypothesis testing for data generated by an ergodic Markov chain. The null hypothesis is that the unknown transition matrix belongs to a prescribed set $P$ of stochastic matrices, and the alternative corresponds to a disjoint set $Q$. We establish a tight non-asymptotic instance-dependent lower bound on the expected stopping time of any valid sequential test under the alternative. Our novel analysis improves the existing lower bounds, which are either asymptotic or provably sub-optimal in this setting. Our lower bound incorporates both the stationary distribution and the transition structure induced by the unknown Markov chain. We further propose an optimal test whose expected stopping time matches this lower bound asymptotically as $α\to 0$. We illustrate the usefulness of our framework through applications to sequential detection of model misspecification in Markov Chain Monte Carlo and to testing structural properties, such as the linearity of transition dynamics, in Markov decision processes. Our findings yield a sharp and general characterization of optimal sequential testing procedures under Markovian dependence.

A new artificial intelligence (AI) model developed by the Chinese company behind TikTok rocked Hollywood this week - not just because of what it can do, but what it could mean for creative industries. Created by tech giant ByteDance, Seedance 2.0 can generate cinema-quality video, complete with sound effects and dialogue, from just a few written prompts. Many of the clips said to have been made using Seedance, and featuring popular characters like Spider-Man and Deadpool, went viral. What is Seedance - and why the stir? Seedance was launched to little fanfare in June 2025 but it is the second version that came eight months later that has caused a major stir.

The United States trade deficit has widened sharply in December amid a surge in imports, and the goods shortfall in 2025 was the highest on record despite US President Donald Trump's tariffs on foreign-manufactured merchandise. The second straight monthly deterioration in the trade deficit reported by the US Commerce Department on Thursday suggested that trade made little or no contribution to gross domestic product (GDP) in the fourth quarter. The US deficit in the trade of goods widened 2 percent to a record $1.24 trillion last year as American companies boosted imports of computer chips and other tech goods from Taiwan to support massive investments in artificial intelligence. Amid continuing tensions with Beijing, the deficit in the goods trade with China plunged nearly 32 percent to $202bn in 2025 on a sharp drop in both exports to and imports from the world's second-biggest economy. But trade was diverted away from China.

We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation.

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Then we introduce a semantic mapper that produces the proxy masks from various input conditions including semantic masks.

Results averaged over ResNet18 and ResNet50.