Statistical Machine Learning for Large High Dimensional Complex Data

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1. Machine Learning Theory As the core of the modern artificial intelligence, deep learning methods are reshaping our world on a daily basis. However, a general and unified framework still has not been established. Consequently, the powerful deep neural network method still remains as a black box. This lack of theoretical understanding also created difficulties to further improve an effective deep learning method by significantly reducing the computational complexity and extend it to handle more complex situations such as high dimensional non-stationary time series data. I would like to establish a theoretical framework and then investigate the theoretical properties of deep learning by building a mathematical and statistical model to describe the behaviors of the deep learning method. In the past, we have been successful in establishing a unified theoretical framework and derive theoretical properties such as convergence and stability for a class of dynamic clustering methods including the famous mean-shift algorithm that has been widely used in image analysis. We will extend our framework to investigate some fundamental issues of machine learning. Our approach is based on statistical mechanics and information theory. Method from physics such as quantum field theory will also be employed. 2. High Dimensional Statistical Inference One of the common challenges for analyzing high dimensional complex data is to reduce the dimensionality efficiently to a manageable level. Common methods include regularization type of method such as Lasso and Bayesian inference which usually reduce the problem into a subspace. Monte Carlo Markov Chain is also an essential tool for Bayesian inference for high dimensional data when the posterior is not analytically trackable. We have developed a very effective high dimensional optimization method for searching global optimum for non-convex objective functions. Our method is based on the idea of geodesics or exponential map which is an essential concept in general relativity. We will extend our method to improve the numerical performance of various dimensionality reduction methods and also apply it to high dimensional Bayesian including MCMC. 3.Statistical Analysis of Brain Data We are developing some novel and effective mathematical and statistical models for highly oscillatory non-stationary time series. The new model will take a drastically different approach than most of the traditional spatial-temporal methods which often integrates deterministic and stochastic component in either an additive or certain very restrictive fashion. Our approach will consider the underlying biological mechanism first and then consider other factors that can be best modelled by statistical models. We will also consider a hidden dimension approach which take into account the fact that the information we obtained could be incomplete.