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Register a new userAlternating minimization algorithms for graph regularized tensor completion
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Added by
Yu Guan
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We consider a low-rank tensor completion (LRTC) problem which aims to recover a tensor from incomplete observations. LRTC plays an important role in many applications such as signal processing, computer vision, machine learning, and neuroscience. A widely used approach is to combine the tensor completion data fitting term with a regularizer based on a convex relaxation of the multilinear ranks of the tensor. For the data fitting function, we model the tensor variable by using the Canonical Polyadic (CP) decomposition and for the low-rank promoting regularization function, we consider a graph Laplacian-based function which exploits correlations between the rows of the matrix unfoldings. For solving our LRTC model, we propose an efficient alternating minimization algorithm. Furthermore, based on the Kurdyka-{L}ojasiewicz property, we show that the sequence generated by the proposed algorithm globally converges to a critical point of the objective function. Besides, an alternating direction method of multipliers algorithm is also developed for the LRTC model. Extensive numerical experiments on synthetic and real data indicate that the proposed algorithms are effective and efficient.
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Existing tensor completion formulation mostly relies on partial observations from a single tensor. However, tensors extracted from real-world data are often more complex due to: (i) Partial observation: Only a small subset (e.g., 5%) of tensor elements are available. (ii) Coarse observation: Some tensor modes only present coarse and aggregated patterns (e.g., monthly summary instead of daily reports). In this paper, we are given a subset of the tensor and some aggregated/coarse observations (along one or more modes) and seek to recover the original fine-granular tensor with low-rank factorization. We formulate a coupled tensor completion problem and propose an efficient Multi-resolution Tensor Completion model (MTC) to solve the problem. Our MTC model explores tensor mode properties and leverages the hierarchy of resolutions to recursively initialize an optimization setup, and optimizes on the coupled system using alternating least squares. MTC ensures low computational and space complexity. We evaluate our model on two COVID-19 related spatio-temporal tensors. The experiments show that MTC could provide 65.20% and 75.79% percentage of fitness (PoF) in tensor completion with only 5% fine granular observations, which is 27.96% relative improvement over the best baseline. To evaluate the learned low-rank factors, we also design a tensor prediction task for daily and cumulative disease case predictions, where MTC achieves 50% in PoF and 30% relative improvements over the best baseline.
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