For example, while prior work has suggested that theglobally optimal VAEsolution canlearn thecorrect manifold dimension, anecessary (butnotsufficient)condition forproducing samplesfrom the true data distribution, this has never been rigorously proven. Moreover, it remains unclear how such considerations would change when various types of conditioning variablesare introduced, or when the data support is extended to a union of manifolds (e.g., as is likely the case for MNIST digits and related). In this work, we address these points by first proving that VAE global minima are indeed capable of recovering the correct manifold dimension.

Previous ablation studies showedthatadding WD to normalized nets indeed leads to better generalization [126, 72, 125].

In this paper, we consider various tree constructions and examine how the choice of parameters affects the generalization error of the resulting random forests as the sample size goes to infinity.

FAA closes El Paso airspace citing cartel drone incursions near military facilities, revealing over 60,000 cartel drone flights along the border in 2024.

Chinese open models are spreading fast, from Hugging Face to Silicon Valley. In this photo illustration, the DeepSeek apps is seen on a phone in front of a flag of China on January 28, 2025 in Hong Kong, China. The past year has marked a turning point for Chinese AI. Since DeepSeek released its R1 reasoning model in January 2025, Chinese companies have repeatedly delivered AI models that match the performance of leading Western models at a fraction of the cost. Just last week the Chinese firm Moonshot AI released its latest open-weight model, Kimi K2.5, which came close to top proprietary systems such as Anthropic's Claude Opus on some early benchmarks. The difference: K2.5 is roughly one-seventh Opus's price.

We consider learning problems in which examples are dependent and their dependency relation is characterized by a graph.

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