Changes from Classical Statistics to Modern Statistics and Data Science

A coordinate system is a foundation for every quantitative science, engineering, and medicine. Classical physics and statistics are based on the Cartesian coordinate system. The classical probability and hypothesis testing theory can only be applied to Euclidean data. However, modern data in the real world are from natural language processing, mathematical formulas, social networks, transportation and sensor networks, computer visions, automations, and biomedical measurements. The Euclidean assumption is not appropriate for non Euclidean data. This perspective addresses the urgent need to overcome those fundamental limitations and encourages extensions of classical probability theory and hypothesis testing , diffusion models and stochastic differential equations from Euclidean space to non Euclidean space. Artificial intelligence such as natural language processing, computer vision, graphical neural networks, manifold regression and inference theory, manifold learning, graph neural networks, compositional diffusion models for automatically compositional generations of concepts and demystifying machine learning systems, has been rapidly developed. Differential manifold theory is the mathematic foundations of deep learning and data science as well. We urgently need to shift the paradigm for data analysis from the classical Euclidean data analysis to both Euclidean and non Euclidean data analysis and develop more and more innovative methods for describing, estimating and inferring non Euclidean geometries of modern real datasets. A general framework for integrated analysis of both Euclidean and non Euclidean data, composite AI, decision intelligence and edge AI provide powerful innovative ideas and strategies for fundamentally advancing AI. We are expected to marry statistics with AI, develop a unified theory of modern statistics and drive next generation of AI and data science.Comment: 37 page

Semi-Supervised Manifold Alignment Using Parallel Deep Autoencoders

The aim of manifold learning is to extract low-dimensional manifolds from high-dimensional data. Manifold alignment is a variant of manifold learning that uses two or more datasets that are assumed to represent different high-dimensional representations of the same underlying manifold. Manifold alignment can be successful in detecting latent manifolds in cases where one version of the data alone is not sufficient to extract and establish a stable low-dimensional representation. The present study proposes a parallel deep autoencoder neural network architecture for manifold alignment and conducts a series of experiments using a protein-folding benchmark dataset and a suite of new datasets generated by simulating double-pendulum dynamics with underlying manifolds of dimensions 2, 3 and 4. The dimensionality and topological complexity of these latent manifolds are above those occurring in most previous studies. Our experimental results demonstrate that the parallel deep autoencoder performs in most cases better than the tested traditional methods of semi-supervised manifold alignment. We also show that the parallel deep autoencoder can process datasets of different input domains by aligning the manifolds extracted from kinematics parameters with those obtained from corresponding image data