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

Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and System

On the Power of Adaptivity in Matrix Completion and Approximation

We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures. For exact recovery of a low-rank matrix, our algorithm judiciously selects a few columns to observe in full and, with few additional measurements, projects the remaining columns onto their span. This algorithm exactly recovers an $n \times n$ rank $r$ matrix using $O(nr\mu_0 \log^2(r))$ observations, where $\mu_0$ is a coherence parameter on the column space of the matrix. In addition to completely eliminating any row space assumptions that have pervaded the literature, this algorithm enjoys a better sample complexity than any existing matrix completion algorithm. To certify that this improvement is due to adaptive sampling, we establish that row space coherence is necessary for passive sampling algorithms to achieve non-trivial sample complexity bounds. For constructing a low-rank approximation to a high-rank input matrix, we propose a simple algorithm that thresholds the singular values of a zero-filled version of the input matrix. The algorithm computes an approximation that is nearly as good as the best rank-$r$ approximation using $O(nr\mu \log^2(n))$ samples, where $\mu$ is a slightly different coherence parameter on the matrix columns. Again we eliminate assumptions on the row space