We present an alternative approach to decompose non-negative tensors, called many-body approximation .

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Wethen update our estimates, and optimize overadifferent set offactors, continuing inthis fashion until convergence.

Given taxi-ride counts information between departure and destination locations, how can we forecast their future demands?

Matrix completion studies the problem of reconstructing a matrix from a (typically random) subset of its entries by exploiting prior structural assumptions such as low rank and incoherence. Roughly speaking, when the underlying n n matrix has low rank and its eigenvectors are sufficiently incoherent, observing Ω(n log n) entries sampled uniformly at random is sufficient for exact recovery via efficient optimization methods [Keshavan et al., 2009, 2010, Candes and Tao, 2010, Candes and Plan, 2010, Recht, 2011, Candes and Recht, 2012, Jain et al., 2013]. This sample complexity is nearly optimal, since specifying a rank-r matrix requires only O(n) degrees of freedom. Tensor completion generalizes this problem to higher-order arrays, aiming to recover a low-rank tensor from a limited set of observed entries, for example, under uniform random sampling. As a natural higher-order analogue of matrix completion, tensor completion has found broad applications in areas such as recommendation systems [Frolov and Oseledets, 2017], signal and image processing [Govindu, 2005, Liu et al., 2012], and data science [Song et al., 2019]. Despite this close analogy, tensor completion behaves fundamentally differently from its matrix counterpart. In contrast to the classical matrix setting, tensor completion exhibits a pronounced trade-off between computational and statistical complexity: while information-theoretic considerations suggest that relatively few samples suffice for recovery, all currently known polynomial-time algorithms require substantially more observations than this optimal limit. Polynomial-time methods A widely used polynomial-time approach to tensor completion is to reduce the problem to matrix completion via matricization.

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Coupled norms have emerged as a convex method to solve coupled tensor completion. A limitation with coupled norms is that they only induce low-rankness using the multilinear rank of coupled tensors. In this paper, we introduce a new set of coupled norms known as coupled nuclear norms by constraining the CP rank of coupled tensors. We propose new coupled completion models using the coupled nuclear norms as regularizers, which can be optimized using computationally efficient optimization methods. We derive excess risk bounds for proposed coupled completion models and show that proposed norms lead to better performance. Through simulation and real-data experiments, we demonstrate that proposed norms achieve better performance for coupled completion compared to existing coupled norms.