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We propose a sparse and low-rank tensor regression model to relate a univariate outcome to a feature tensor, in which each unit-rank tensor from the CP decomposition of the coefficient tensor is assumed to be sparse. This structure is both parsimonious and highly interpretable, as it implies that the outcome is related to the features through a few distinct pathways, each of which may only involve subsets of feature dimensions. We take a divide-and-conquer strategy to simplify the task into a set of sparse unit-rank tensor regression problems. To make the computation efficient and scalable, for the unit-rank tensor regression, we propose a stagewise estimation procedure to efficiently trace out its entire solution path. We show that as the step size goes to zero, the stagewise solution paths converge exactly to those of the corresponding regularized regression. The superior performance of our approach is demonstrated on various real-world and synthetic examples.
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This paper studies a tensor-structured linear regression model with a scalar response variable and tensor-structured predictors, such that the regression parameters form a tensor of order $d$ (i.e., a $d$-fold multiway array) in $mathbb{R}^{n_1 times n_2 times cdots times n_d}$. It focuses on the task of estimating the regression tensor from $m$ realizations of the response variable and the predictors where $mll n = prod olimits_{i} n_i$. Despite the seeming ill-posedness of this problem, it can still be solved if the parameter tensor belongs to the space of sparse, low Tucker-rank tensors. Accordingly, the estimation procedure is posed as a non-convex optimization program over the space of sparse, low Tucker-rank tensors, and a tensor variant of projected gradient descent is proposed to solve the resulting non-convex problem. In addition, mathematical guarantees are provided that establish the proposed method linearly converges to an appropriate solution under a certain set of conditions. Further, an upper bound on sample complexity of tensor parameter estimation for the model under consideration is characterized for the special case when the individual (scalar) predictors independently draw values from a sub-Gaussian distribution. The sample complexity bound is shown to have a polylogarithmic dependence on $bar{n} = max big{n_i: iin {1,2,ldots,d } big}$ and, orderwise, it matches the bound one can obtain from a heuristic parameter counting argument. Finally, numerical experiments demonstrate the efficacy of the proposed tensor model and estimation method on a synthetic dataset and a collection of neuroimaging datasets pertaining to attention deficit hyperactivity disorder. Specifically, the proposed method exhibits better sample complexities on both synthetic and real datasets, demonstrating the usefulness of the model and the method in settings where $n gg m$.
Low-rank tensor completion recovers missing entries based on different tensor decompositions. Due to its outstanding performance in exploiting some higher-order data structure, low rank tensor ring has been applied in tensor completion. To further deal with its sensitivity to sparse component as it does in tensor principle component analysis, we propose robust tensor ring completion (RTRC), which separates latent low-rank tensor component from sparse component with limited number of measurements. The low rank tensor component is constrained by the weighted sum of nuclear norms of its balanced unfoldings, while the sparse component is regularized by its l1 norm. We analyze the RTRC model and gives the exact recovery guarantee. The alternating direction method of multipliers is used to divide the problem into several sub-problems with fast solutions. In numerical experiments, we verify the recovery condition of the proposed method on synthetic data, and show the proposed method outperforms the state-of-the-art ones in terms of both accuracy and computational complexity in a number of real-world data based tasks, i.e., light-field image recovery, shadow removal in face images, and background extraction in color video.
Multitask learning, i.e. taking advantage of the relatedness of individual tasks in order to improve performance on all of them, is a core challenge in the field of machine learning. We focus on matrix regression tasks where the rank of the weight matrix is constrained to reduce sample complexity. We introduce the common mechanism regression (CMR) model which assumes a shared left low-rank component across all tasks, but allows an individual per-task right low-rank component. This dramatically reduces the number of samples needed for accurate estimation. The problem of jointly recovering the common and the local components has a non-convex bi-linear structure. We overcome this hurdle and provide a provably beneficial non-iterative spectral algorithm. Appealingly, the solution has favorable behavior as a function of the number of related tasks and the small number of samples available for each one. We demonstrate the efficacy of our approach for the challenging task of remote river discharge estimation across multiple river sites, where data for each task is naturally scarce. In this scenario sharing a low-rank component between the tasks translates to a shared spectral reflection of the water, which is a true underlying physical model. We also show the benefit of the approach on the markedly different setting of image classification where the common component can be interpreted as the shared convolution filters.
Tensor completion refers to the task of estimating the missing data from an incomplete measurement or observation, which is a core problem frequently arising from the areas of big data analysis, computer vision, and network engineering. Due to the multidimensional nature of high-order tensors, the matrix approaches, e.g., matrix factorization and direct matricization of tensors, are often not ideal for tensor completion and recovery. In this paper, we introduce a unified low-rank and sparse enhanced Tucker decomposition model for tensor completion. Our model possesses a sparse regularization term to promote a sparse core tensor of the Tucker decomposition, which is beneficial for tensor data compression. Moreover, we enforce low-rank regularization terms on factor matrices of the Tucker decomposition for inducing the low-rankness of the tensor with a cheap computational cost. Numerically, we propose a customized ADMM with enough easy subproblems to solve the underlying model. It is remarkable that our model is able to deal with different types of real-world data sets, since it exploits the potential periodicity and inherent correlation properties appeared in tensors. A series of computational experiments on real-world data sets, including internet traffic data sets, color images, and face recognition, demonstrate that our model performs better than many existing state-of-the-art matricization and tensorization approaches in terms of achieving higher recovery accuracy.
Tensor completion estimates missing components by exploiting the low-rank structure of multi-way data. The recently proposed methods based on tensor train (TT) and tensor ring (TR) show better performance in image recovery than classical ones. Compared with TT and TR, the projected entangled pair state (PEPS), which is also called tensor grid (TG), allows more interactions between different dimensions, and may lead to more compact representation. In this paper, we propose to perform image completion based on low-rank tensor grid. A two-stage density matrix renormalization group algorithm is used for initialization of TG decomposition, which consists of multiple TT decompositions. The latent TG factors can be alternatively obtained by solving alternating least squares problems. To further improve the computational efficiency, a multi-linear matrix factorization for low rank TG completion is developed by using parallel matrix factorization. Experimental results on synthetic data and real-world images show the proposed methods outperform the existing ones in terms of recovery accuracy.
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