3,962 research outputs found
A dual framework for low-rank tensor completion
One of the popular approaches for low-rank tensor completion is to use the
latent trace norm regularization. However, most existing works in this
direction learn a sparse combination of tensors. In this work, we fill this gap
by proposing a variant of the latent trace norm that helps in learning a
non-sparse combination of tensors. We develop a dual framework for solving the
low-rank tensor completion problem. We first show a novel characterization of
the dual solution space with an interesting factorization of the optimal
solution. Overall, the optimal solution is shown to lie on a Cartesian product
of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian
optimization framework for proposing computationally efficient trust region
algorithm. The experiments illustrate the efficacy of the proposed algorithm on
several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing
Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on
Synergies in Geometric Data Analysis 201
Recovery Guarantees for Quadratic Tensors with Limited Observations
We consider the tensor completion problem of predicting the missing entries
of a tensor. The commonly used CP model has a triple product form, but an
alternate family of quadratic models which are the sum of pairwise products
instead of a triple product have emerged from applications such as
recommendation systems. Non-convex methods are the method of choice for
learning quadratic models, and this work examines their sample complexity and
error guarantee. Our main result is that with the number of samples being only
linear in the dimension, all local minima of the mean squared error objective
are global minima and recover the original tensor accurately. The techniques
lead to simple proofs showing that convex relaxation can recover quadratic
tensors provided with linear number of samples. We substantiate our theoretical
results with experiments on synthetic and real-world data, showing that
quadratic models have better performance than CP models in scenarios where
there are limited amount of observations available
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