We propose a class of structured diffusion models, in which the prior distribution is chosen as a mixture of Gaussians, rather than a standard Gaussian distribution. The specific mixed Gaussian distribution, as prior, can be chosen to incorporate certain structured information of the data. We develop a simple-to-implement training procedure that smoothly accommodates the use of mixed Gaussian as prior. Theory is provided to quantify the benefits of our proposed models, compared to the classical diffusion models. Numerical experiments with synthetic, image and operational data are conducted to show comparative advantages of our model. Our method is shown to be robust to mis-specifications and in particular suits situations where training resources are limited or faster training in real time is desired.

The problem of symbolic music generation presents unique challenges due to the combination of limited data availability and the need for high precision in note pitch. To overcome these difficulties, we introduce Fine-grained Textural Guidance (FTG) within diffusion models to correct errors in the learned distributions. By incorporating FTG, the diffusion models improve the accuracy of music generation, which makes them well-suited for advanced tasks such as progressive music generation, improvisation and interactive music creation. We derive theoretical characterizations for both the challenges in symbolic music generation and the effect of the FTG approach. We provide numerical experiments and a demo page for interactive music generation with user input to showcase the effectiveness of our approach.

Many decision making problems naturally give rise to setting where there are a number of different policies (or systems, designs) each with unknown expected performances, from which the decision maker wants to select the policy with the best expected performance. Even though the expected performances are unknown, the decision maker generally has access to observe independent noisy samples of the expected performance for each policy. The statistically principled way of identifying the best policy through the noisy samples has been a fundamental research problem in several research areas. Some early statistical work includes Bechhofer (1954) and Bechhofer et al. (1995). In the stochastic simulation literature, the research problem is called ranking and selection (R&S); see Hong et al. (2021), Hunter and Nelson (2017), Chick (2006) and Kim and Nelson (2006) for reviews. In the multi-armed bandit literature, the research problem is called best arm identification (BAI); see Audibert et al. (2010), Garivier and Kaufmann (2016), Kaufmann et al. (2016), fore references. Ma and Henderson (2017) and Glynn and Juneja (2015) have discussed some connections between the two literature. The R&S literature and BAI literature differ in assumptions and analysis tools.