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Diffusion models have become the most popular approach to deep generative modeling of images, largely due to their empirical performance and reliability. From a theoretical standpoint, a number of recent works~\cite{chen2022,chen2022improved,benton2023linear} have studied the iteration complexity of sampling, assuming access to an accurate diffusion model. In this work, we focus on understanding the \emph{sample complexity} of training such a model; how many samples are needed to learn an accurate diffusion model using a sufficiently expressive neural network? Prior work~\cite{BMR20} showed bounds polynomial in the dimension, desired Total Variation error, and Wasserstein error. We show an \emph{exponential improvement} in the dependence on Wasserstein error and depth, along with improved dependencies on other relevant parameters. 

The landscape of privacy laws and regulations around the world is complex and ever-changing. National and super-national laws, agreements, decrees, and other government-issued rules form a patchwork that companies must follow to operate internationally. To examine the status and evolution of this patchwork, we introduce the Privacy Law Corpus, of 1,043 privacy laws, regulations, and guidelines, covering 183 jurisdictions. This corpus enables a large-scale quantitative and qualitative examination of legal focus on privacy. We examine the temporal distribution of when privacy laws were created and illustrate the dramatic increase in privacy legislation over the past 50 years, although a finer-grained examination reveals that the rate of increase varies depending on the personal data types that privacy laws address. Our exploration also demonstrates that most privacy laws respectively address relatively few personal data types. Additionally, topic modeling results show the prevalence of common themes in privacy laws, such as finance, healthcare, and telecommunications. Finally, we release the corpus to the research community to promote further study. 

Abstract The radiation mechanism underlying the prompt emission remains unresolved and can be resolved using a systematic and uniform time-resolved spectro-polarimetric study. In this paper, we investigated the spectral, temporal, and polarimetric characteristics of five bright gamma-ray bursts (GRBs) using archival data from AstroSat CZTI, Swift Burst Alert Telescope, and Fermi/GBM. These bright GRBs were detected by CZTI in its first year of operation, and their average polarization characteristics have been published in Chattopadhyay et al. In the present work, we examined the time-resolved (in 100–600 keV) and energy-resolved polarization measurements of these GRBs with an improved polarimetric technique such as increasing the effective area and bandwidth (by using data from low-gain pixels), using an improved event selection logic to reduce noise in the double events and extend the spectral bandwidth. In addition, we also separately carried out detailed time-resolved spectral analyses of these GRBs using empirical and physical synchrotron models. By these improved time-resolved and energy-resolved spectral and polarimetric studies (not fully coupled spectro-polarimetric fitting), we could pin down the elusive prompt emission mechanism of these GRBs. Our spectro-polarimetric analysis reveals that GRB 160623A, GRB 160703A, and GRB 160821A have Poynting flux-dominated jets. On the other hand, GRB 160325A and GRB 160802A have baryonic-dominated jets with mild magnetization. Furthermore, we observe a rapid change in polarization angle by ∼90° within the main pulse of very bright GRB 160821A, consistent with our previous results. Our study suggests that the jet composition of GRBs may exhibit a wide range of magnetization, which can be revealed by utilizing spectro-polarimetric investigations of the bright GRBs.