Font Size: a A A

Research On Theory And Applications Of Sparsity Reconstruction Of High-Dimension Data From Quantized Measurements

Posted on:2022-10-30Degree:DoctorType:Dissertation
Country:ChinaCandidate:J Y HouFull Text:PDF
GTID:1488306734450834Subject:Statistics
Abstract/Summary:PDF Full Text Request
With the wide application of modern information processing technologies such as big data and artificial intelligence,reconstruction methods for sparse high-dimensional data rep-resented by compressed sensing,low-rank matrix recovery,and low-rank tensor recovery are developing rapidly in recent years.The proposed methods have been widely used in many fields,such as medical imaging,computer vision,pattern recognition,meteorological moni-toring and so on.However,most of the existing research is based on the ideal mathematical hypothesis and does not take into account the quantization usually encountered in practical applications.In short,quantization is the process of transforming the analog signal into a dig-ital signal,which is the basis of digital communication.In fact,signal sampling,transmission,and processing are inseparable from quantization,so it is particularly necessary to study the sparse recovery problem combined with quantization.Based on binary quantization and low-level quantization(collectively referred to as extreme quantization)as the core clue,we explore non-convex optimization methods for one-bit compressed sensing,low-tubal-rank tensor sens-ing from binary measurements,low-tubal-rank tensor completion from binary observations,and tensor robust principal component analysis from multi-level quantized observations.The research about tensors is under the framework of the recently developed(generalized)tensor singular value decomposition framework.The main work can be summarized into four aspects as follows:Firstly,in order to achieve a better balance between”low sampling rate”and”easy to solve”in one-bit compressed sensing,anlp(0<p<1)-minimization program is proposed,with some theoretical results on robust recovery established.Especially,the lower bound es-timation of the number of measurements is obtained.The theoretical results show that under the same conditions,our methods can robustly reconstruct the underlying signal at a lower cost than thel1-minimization method.Meanwhile,we also propose an optimization algorith-m for solving the proposed model and verify the effectiveness of the theory and algorithm through some numerical experiments.The results in this part provide a useful reference for the introduction of more non-convex penalty functions in one-bit compressed sensing and the design of measurement matrices.Secondly,we consider the problem of low-rank tensor sensing from binary measurements.We first define the tensor restricted isometry property based on thel1-norm,which shows what kind of linear measurement operators can achieve the robust recovery of low-rank ten-sors.At the same time,by introducing the adaptive disturbance vector selection strategy,we propose two efficient reconstruction algorithms.Then we verify the theoretical results and test the performance of the proposed algorithms through the applications on face recovery and multispectral image restoration.The results significantly broaden the application scenarios of low-rank tensor recovery technology and make theoretical and algorithmic preparation for the further research of quantization-based low-rank tensor recovery technology.Thirdly,we study the problem of low-rank tensor completion from binary observations.In the framework of generalized tensor singular value decomposition,we integrate the binary quantization technique into the tensor completion problem and propose a recovery method based on maximum likelihood estimation.The upper bound of the error estimates between the original and the recovered tensors obtained by solving the proposed model is given.And the lower bound of the minimax error that can be achieved under binary observation,ignoring specific methods,is also obtained,which shows that the upper bound of the error estimates obtained has approximately reached the optimal.Numerical experiments on simulation data,grey video sequence,and multispectral images demonstrate the effectiveness of the proposed method.These results provide theoretical support for more methods about one-bit tensor completion in the framework of tensor singular value decomposition based on more reversible linear transformations.The tensor robust principal component analysis is one of the effective methods to re-duce the dimension of data.Fourthly,we combine low-level quantization with tensor robust principal component analysis and study tensor robust principal component analysis based on multi-level quantization observation in the framework of tensor singular value decompo-sition.Two recovery strategies based on maximum likelihood estimation are proposed,and the corresponding model theoretical results are established by using probability concentration inequality and Fano inequality.It is proved that the optimal solutions of the two models can effectively approximate the low-rank tensor and sparse tensor under fairly mild assumptions.Finally,the theoretical results are verified by numerical experiments,and the effectiveness of the methods is verified by experiments on multispectral image restoration and color image denoising.
Keywords/Search Tags:Binary quantization, Tensor singular value decomposition, Tensor sensing, Tensor completion, Tensor robust principal component analysis
PDF Full Text Request
Related items