In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4-D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD) [11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].
Abstract-In this paper we focus on the problem of completion of multidimensional arrays (also referred to as tensors) from limited sampling. Our approach is based on a recently proposed tensor-Singular Value Decomposition (t-SVD) [1]. Using this factorization one can derive notion of tensor rank, referred to as the tensor tubal rank, which has optimality properties similar to that of matrix rank derived from SVD. As shown in [2] some multidimensional data, such as panning video sequences exhibit low tensor tubal rank and we look at the problem of completing such data under random sampling of the data cube. We show that by solving a convex optimization problem, which minimizes the tensor nuclear norm obtained as the convex relaxation of tensor tubal rank, one can guarantee recovery with overwhelming probability as long as samples in proportion to the degrees of freedom in t-SVD are observed. In this sense our results are order-wise optimal. The conditions under which this result holds are very similar to the incoherency conditions for the matrix completion, albeit we define incoherency under the algebraic set-up of t-SVD. We show the performance of the algorithm on some real data sets and compare it with other existing approaches based on tensor flattening and Tucker decomposition.
In this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. We consider two settings: output noise models where the noise enters after the projection and input noise models where the noise enters before the projection. We consider two types of distortion for reconstruction: support errors and mean-squared errors. Our goal is to relate the number of measurements, m, and SNR, to signal sparsity, k, distortion level, d, and signal dimension, n.We consider support errors in a worst-case setting. We employ different variations of Fano's inequality to derive necessary conditions on the number of measurements and SNR required for exact reconstruction. To derive sufficient conditions we develop new insights on max-likelihood analysis based on a novel superposition property. In particular this property implies that small support errors are the dominant error events. Consequently, our ML analysis does not suffer the conservatism of the union bound and leads to a tighter analysis of max-likelihood. These results provide order-wise tight bounds. For output noise models we show that asymptotically an SNR of Θ(log(n)) together with Θ(k log(n/k)) measurements is necessary and sufficient for exact support recovery. Furthermore, if a small fraction of support errors can be tolerated, a constant SNR turns out to be sufficient in the linear sparsity regime. In contrast for input noise models we show that support recovery fails if the number of measurements scales as o(n log(n)/SN R) implying poor compression performance for such cases.Motivated by the fact that the worst-case setup requires significantly high SNR and substantial number of measurements for input and output noise models, we consider a Bayesian setup. To derive necessary conditions we develop novel extensions to Fano's inequality to handle continuous domains and arbitrary distortions. We then develop a new max-likelihood analysis over the set of rate distortion quantization points to characterize tradeoffs between mean-squared distortion and the number of measurements using rate-distortion theory. We show that with constant SNR the number of measurements scales linearly with the rate-distortion function of the sparse phenomena.
Using the matrix product state (MPS) representation of the recently proposed tensor ring decompositions, in this paper we propose a tensor completion algorithm, which is an alternating minimization algorithm that alternates over the factors in the MPS representation. This development is motivated in part by the success of matrix completion algorithms that alternate over the (low-rank) factors. In this paper, we propose a spectral initialization for the tensor ring completion algorithm and analyze the computational complexity of the proposed algorithm. We numerically compare it with existing methods that employ a low rank tensor train approximation for data completion and show that our method outperforms the existing ones for a variety of real computer vision settings, and thus demonstrate the improved expressive power of tensor ring as compared to tensor train. I. INTRODUCTIONTensor decompositions for representing and storing data have recently attracted considerable attention due to their effectiveness in compressing data for statistical signal processing [1]- [5]. In this paper we focus on Tensor Ring (TR) decomposition [6] and in particular its relation to Matrix Product States (MPS) [7] representation for tensor representation and use it for completing data from missing entries. In this context our algorithm is motivated by recent work in matrix completion where under a suitable initialization an alternating minimization algorithm [8], [9] over the low rank factors is able to accurately predict the missing data.Recently, tensor networks, considered as the generalization of tensor decompositions, have emerged as the potentially powerful tools for analysis of large-scale tensor data [7]. The most popular tensor network is the Tensor Train (TT) representation, which for a order-d tensor with each dimension of size n requires O(dnr 2 ) parameters, where r is the rank of each of the factors, and thus allows for the efficient data representation [10]. Tensor completion based on tensor train decompositions have been recently considered in [11], [12]. The authors of [11] considered the completion of data based on the alternating least square method.Although the TT format has been widely applied in numerical analysis, its applications to image classification and completion are rather limited [4], [11], [12]. As outlined in [6], TT decomposition suffers from the following limitations. Namely, (i) TT model requires rank-1 constraints on the border factors, (ii) TT ranks are typically small for near-border factors and large for the middle factors, and (iii) the multiplications of the TT factors are not permutation invariant. In order to alleviate those drawbacks, a tensor ring (TR) decomposition has been proposed in [6]. TR decomposition removes the unit rank constraints for the boundary tensor factors and utilizes a
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