181,098 research outputs found
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 rank matrix using observations,
where 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- approximation using
samples, where is a slightly different coherence parameter on the matrix
columns. Again we eliminate assumptions on the row space
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