Deep Tensor Network

In this paper, we delve into the foundational principles of tensor categories, harnessing the universal property of the tensor product to pioneer novel methodologies in deep network architectures. Our primary contribution is the introduction of the Tensor Attention and Tensor Interaction Mechanism, a groundbreaking approach that leverages the tensor category to enhance the computational efficiency and the expressiveness of deep networks, and can even be generalized into the quantum realm

Multivariate Probit Models for Interval-Censored Failure Time Data

Survival analysis is an important branch of statistics that analyzes the time to event data. The events of interest can be death, disease occurrence, the failure of a machine part, etc.. One important feature of this type of data is censoring: information on time to event is not observed exactly due to loss to follow-up or non-occurrence of interested event before the trial ends. Censored data are commonly observed in clinical trials and epidemiological studies, since monitoring a person’s health over time after treatment is often required in medical or health studies. In this dissertation we focus on studying multivariate interval-censored data, a special type of survival data. By saying multivariate interval-censored data, we mean that there are multiple failure time events of interest, and these failure times are known only to lie within certain intervals instead of being observed exactly. These events of interest can be associated because of sharing some common characteristics. Multivariate interval-censored data draw more and more attention in epidemiological, social-behavioral and medical studies, in which subjects are examined multiple times and several events of interest are tested at the observation times. There are some existing methods available in literatures for analyzing multivariate interval-censored failure time data. Various models were developed for regression analysis. However, due to the complicated correlation structure between events, analyzing such type of survival data is much more difficult and new efficient methodologies are needed. Chapter 1 of this dissertation illustrates the important concepts of interval-censored data with several real data examples. A literature review of existing regression models and approaches is included as well. Chapter 2 introduces a new normal-frailty multivariate probit model for regression analysis of interval-censored failure time data and proposes an efficient Bayesian approach to get parameter estimates. Simulations and an analysis on a real data set are conducted to evaluate and illustrate the performance of this new method. This new approach is proved efficient and has accurate estimations on both the regression parameters and the baseline survival function. Several appealing properties of the model are discussed here. Chapter 3 proposes a more general multivariate probit model for multivariate interval-censored data. This new model allows arbitrary correlation among the correlated survival times. A new Gibbs sampler is proposed for the joint estimation of the regression parameters, the baseline CDF, and the correlation parameters. Chapter 4 extends the normal frailty multivariate probit model to allow arbitrary pairwise correlations. Simulation studies are conducted to explore the underlying relationship between the normal frailty multivariate probit model and the general multivariate probit model