Statistical Machine Learning for Large High Dimensional Complex Data

1.机器学习理论作为现代人工智能的核心，深度学习方法每天都在重塑我们的世界。然而，还没有建立一个普遍和统一的框架。因此，强大的深度神经网络方法仍然是一个黑匣子。这种对理论认识的缺乏也给进一步改进一种有效的深度学习方法带来了困难，因为它大大降低了计算复杂度，并将其扩展到处理更复杂的情况，如高维非平稳时间序列数据。*我想要建立一个理论框架，然后通过建立一个描述深度学习方法行为的数学和统计模型来研究深度学习的理论属性。在过去，我们已经成功地建立了统一的理论框架，并推导了一类动态聚类方法的收敛和稳定性等理论性质，其中包括著名的Mean-Shift算法，该算法在图像分析中得到了广泛的应用。我们将扩展我们的框架来研究机器学习的一些基本问题。我们的方法是基于统计力学和信息论的。还将采用量子场论等物理学方法。*2.高维统计推断*分析高维复杂数据的常见挑战之一是将维度有效地降维到可管理的水平。常用的方法包括正则化方法，如Lasso和贝叶斯推理，它们通常将问题归结为一个子空间。蒙特卡罗马尔可夫链也是高维数据贝叶斯推断的重要工具，当后验数据不可分析跟踪时。*我们发展了一种非常有效的高维优化方法，用于搜索非凸目标函数的全局最优。我们的方法是基于测地线或指数映射的思想，这是广义相对论中的一个基本概念。我们将扩展我们的方法，以提高各种降维方法的数值性能，并将其应用于包括MCMC在内的高维贝叶斯。*3.脑数据的统计分析*我们正在为高振荡的非平稳时间序列开发一些新的、有效的数学和统计模型。新的模型将采取一种与大多数传统时空方法截然不同的方法，后者通常以相加或某些非常受限的方式集成确定性和随机性分量。我们的方法将首先考虑潜在的生物学机制，然后再考虑其他可以用统计模型最佳建模的因素。我们还将考虑隐藏维度方法，该方法考虑到我们获得的信息可能是不完整的。

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.