Multidimensional Signal Recovery Using Tensor Network Decompositions

With the development of era of big data,tensor provides an effective representation for multi-dimensional data.To extract the hidden structure or pattern,tensor decomposi-tion emerges as a common rank-revealing algebra which decomposes a tensor into several small(and often explainable and meaningful)tensors.Given a fraction of tensor compo-nents as measurement,tensor completion interpolates the missing components by exploit-ing the low rank structure of multi-dimensional array.The recently proposed tensor ring(TR)decomposition,which is a quantum inspired method,shows better performance than existing ones in the task of low-level computer vi-sion problems.In this thesis,we show that by solving a convex optimization model based on this TR decomposition,O(I[D/2]R2 ln7(I[D/2]))samples suffice to exactly recover a D-order tensor of size I×…× I and TR rank[R;…;R]with high probability.To further deal with sensitivity of the algorithm to sparse component we propose the robust TR com-pletion,which separates latent low-rank tensor component from sparse component with limited number of measurements.Specifically,the original model is regularized by a l1 norm of sparse tensor.We also extend the aforementioned exact recovery guarantee to this robust completion model.As to optimization,we use the alternating direction method of multipliers to divide them into several sub-problems with fast solutions.By considering a priori knowledge from side information,we then propose a coupled tensor decomposition which reveals the joint structure of multi-modal data.This problem is developed with a nonlinear least square model,which is solved by an accelerated block coordinate descent algorithm with linear convergence rate.We also derive an excess risk bound for this optimization model,and it shows the theoretical performance enhancement in comparison with other coupled completion methods.Compared with TT and TR,the projected entangled pair state(PEPS)which we also call tensor grid(TG),allows more interactions between different dimensions,and may lead to more compact representation.Thus we propose a novel tensor completion model based on this decomposition.We develop a two-stage density matrix renormalization group algorithm for initialization of TG decomposition.We use the alternating minimization to solve the completion problem.In order to improve the computational efficiency,we also propose a parallel matrix factorization method.To further validate the theory and the proposed algorithms,we conduct two groups of experiments for each algorithm.We generate synthetic data to verify the theory and com-pare our method with the state-of-the-art methods on the real-world data,including color images and videos,light field images,YaleB face dataset,social data,short wave-near infrared spectrums and hyperspectral images,etc.The experimental results demonstrate the superiority of our methods over the existing ones.