Yicong He

Yicong He, George K. Atia

Tensor ring (TR) decomposition has recently received increased attention due to its superior expressive performance for high-order tensors. However, the applicability of traditional TR decomposition algorithms to real-world applications is hindered by prevalent large data sizes, missing entries, and corruption with outliers. In this work, we propose a scalable and robust TR decomposition algorithm capable of handling large-scale tensor data with missing entries and gross corruptions. We first develop a novel auto-weighted steepest descent method that can adaptively fill the missing entries and identify the outliers during the decomposition process. Further, taking advantage of the tensor ring model, we develop a novel fast Gram matrix computation (FGMC) approach and a randomized subtensor sketching (RStS) strategy which yield significant reduction in storage and computational complexity. Experimental results demonstrate that the proposed method outperforms existing TR decomposition methods in the presence of outliers, and runs significantly faster than existing robust tensor completion algorithms.

Yicong He, George K. Atia

Tensor completion is the problem of estimating the missing values of high-order data from partially observed entries. Data corruption due to prevailing outliers poses major challenges to traditional tensor completion algorithms, which catalyzed the development of robust algorithms that alleviate the effect of outliers. However, existing robust methods largely presume that the corruption is sparse, which may not hold in practice. In this paper, we develop a two-stage robust tensor completion approach to deal with tensor completion of visual data with a large amount of gross corruption. A novel coarse-to-fine framework is proposed which uses a global coarse completion result to guide a local patch refinement process. To efficiently mitigate the effect of a large number of outliers on tensor recovery, we develop a new M-estimator-based robust tensor ring recovery method which can adaptively identify the outliers and alleviate their negative effect in the optimization. The experimental results demonstrate the superior performance of the proposed approach over state-of-the-art robust algorithms for tensor completion.

Yicong He, George K. Atia

Tensor completion aims to recover the missing entries of a partially observed tensor by exploiting its low-rank structure, and has been applied to visual data recovery. In applications where the data arrives sequentially such as streaming video completion, the missing entries of the tensor need to be dynamically recovered in a streaming fashion. Traditional streaming tensor completion algorithms treat the entire visual data as a tensor, which may not work satisfactorily when there is a big change in the tensor subspace along the temporal dimension, such as due to strong motion across the video frames. In this paper, we develop a novel patch tracking-based streaming tensor ring completion framework for visual data recovery. Given a newly incoming frame, small patches are tracked from the previous frame. Meanwhile, for each tracked patch, a patch tensor is constructed by stacking similar patches from the new frame. Patch tensors are then completed using a streaming tensor ring completion algorithm, and the incoming frame is recovered using the completed patch tensors. We propose a new patch tracking strategy that can accurately and efficiently track the patches with missing data. Further, a new streaming tensor ring completion algorithm is proposed which can efficiently and accurately update the latent core tensors and complete the missing entries of the patch tensors. Extensive experimental results demonstrate the superior performance of the proposed algorithms compared with both batch and streaming state-of-the-art tensor completion methods.

Yicong He, George K. Atia

The goal of tensor completion is to recover a tensor from a subset of its entries, often by exploiting its low-rank property. Among several useful definitions of tensor rank, the low-tubal-rank was shown to give a valuable characterization of the inherent low-rank structure of a tensor. While some low-tubal-rank tensor completion algorithms with favorable performance have been recently proposed, these algorithms utilize second-order statistics to measure the error residual, which may not work well when the observed entries contain large outliers. In this paper, we propose a new objective function for low-tubal-rank tensor completion, which uses correntropy as the error measure to mitigate the effect of the outliers. To efficiently optimize the proposed objective, we leverage a half-quadratic minimization technique whereby the optimization is transformed to a weighted low-tubal-rank tensor factorization problem. Subsequently, we propose two simple and efficient algorithms to obtain the solution and provide their convergence and complexity analysis. Numerical results using both synthetic and real data demonstrate the robust and superior performance of the proposed algorithms.