Power System Anomaly Detection And Image Recognition Based On Random Matrix Theory And Deep Neural Networks

Data sets arising in modern science and engineering,e.g.deep neural networks and smart grids,are often extremely large.The main challenge in modern data analysis is the explosion of the data in terms of its dimension,i.e.,the data dimension is often of the same order as,or possibly larger than the data size.In this thesis,we are trying to address high dimensional problems from the perspective of random matrix theory,a vital methodology for high dimensional statistics,which are summarized as follows.In the first part of this thesis,we revisit the weight initialization of deep residual networks(Res Nets)by introducing a novel analytical tool in free probability to the community of deep learning.This tool deals with the limiting spectral distribution of Non-Hermitian random matrices,rather than their conventional Hermitian counterparts in the literature.This new tool enables us to evaluate the singular value spectrum of the input-output Jacobian of a fullyconnected deep Res Net in both linear and nonlinear cases.With the powerful tool of free probability,we conduct an asymptotic analysis of the(limiting)spectrum on the single-layer case,and then extend this analysis to the multi-layer case of an arbitrary number of layers.The asymptotic analysis illustrates the necessity and university of re-scaling the classical random initialization by the number of residual units ,so that the squared singular value of the associated Jacobian remains of order (1),when compared with the large width and depth of the network.We empirically demonstrate that the proposed initialization scheme learns at a speed of orders of magnitudes faster than the classical ones,and thus attests a strong practical relevance of this investigation.In the second part of this thesis,we propose a novel approach of exploiting self-adjoint matrix polynomials of large random matrices for anomaly detection and fault location in smart grid.Synchronized measurements of a large power grid enable an unprecedented opportunity to study the spatial-temporal correlations.Statistical analytics for those massive datasets start with high-dimensional data matrices.Uncertainty is ubiquitous in a future's power grid.These data matrices are recognized as random matrices.This new point of view is fundamental in our theoretical analysis since true covariance matrices cannot be estimated accurately in a high-dimensional regime.As an alternative,we consider large-dimensional sample covariance matrices in the asymptotic regime to replace the true covariance matrices.The self-adjoint polynomials of large-dimensional random matrices are studied as statistics for big data analysis.The calculation of the asymptotic spectrum distribution for such a matrix polynomial is understandably challenging.This task is made possible by a recent breakthrough in free probability,an active research branch in random matrix theory.This is the very reason why the work of this work is inspired initially.The new approach is interesting in many aspects.The mathematical reason may be most critical.The real-world problems can be solved using this approach,however.The other focus of this thesis,we propose a new deep learning framework for the location of broken insulators(in particular the self-blast glass insulator)in aerial images.We address the broken insulators location problem in a low signal-noise-ratio(SNR)setting.We deal with two modules: 1)object detection based on Faster R-CNN,and 2)classification of pixels based on U-net.For the first time,we combine the above two modules.This combination is motivated as follows: Faster R-CNN is used to improve SNR,while the U-net is used for classification of pixels.A diverse aerial image set measured by a power grid in China is tested to validate the proposed approach.Furthermore,a comparison is made among different methods and the result shows that our approach is accurate in real time.