A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

Guo, Chen; Lin, Wen‐Wei; Liu, Ching-Sung

doi:10.48550/arxiv.1705.07487

Cited by 2 publications

(6 citation statements)

References 10 publications

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“…Here we derive a Newton-based method for computing the eigenpairs of symmetric tensors. Recently, [12] considered a similar approach for finding some nonnegative eigenpairs of a nonnegative tensor.…”

Section: Newton Correction Methodsmentioning

confidence: 99%

“…By definition, the solution y * to (12) satisfies x + y * = x * . However, solving (12) exactly for y * is as difficult as finding the eigenpair (x * , λ * ) of the tensor T we started from. Instead, we devise an iterative Newton correction method (NCM) that solves (12) only approximately.…”

Section: Newton Correction Methodsmentioning

confidence: 99%

“…However, solving (12) exactly for y * is as difficult as finding the eigenpair (x * , λ * ) of the tensor T we started from. Instead, we devise an iterative Newton correction method (NCM) that solves (12) only approximately. Given the approximation x (k) of x * at the k th iteration, NCM computes a new approximation x (k+1) by neglecting the high order terms ∆(x, y * ) in (12).…”

Section: Newton Correction Methodsmentioning

confidence: 99%

“…Instead, we devise an iterative Newton correction method (NCM) that solves (12) only approximately. Given the approximation x (k) of x * at the k th iteration, NCM computes a new approximation x (k+1) by neglecting the high order terms ∆(x, y * ) in (12). This amounts to solving the system of n linear equations…”

Section: Newton Correction Methodsmentioning

confidence: 99%

“…Hence, it suffices to bound y * − y (0) . To this end, we view the exact system of non-linear equations (12), whose solution is y * , as a perturbation of the approximate system of linear equations ( 14), whose solution is y (0) . Consider the matrix A of Eq.…”

Section: Lemma 2 Consider One Update Step Of Algorithm 1 As In Equati...mentioning

confidence: 99%

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Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors

Jaffe

Weiss

Nadler

2018

SIAM J. Matrix Anal. & Appl.

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Real eigenpairs of symmetric tensors play an important role in multiple applications. In this paper we propose and analyze a fast iterative Newton-based method to compute real eigenpairs of symmetric tensors. We derive sufficient conditions for a real eigenpair to be a stable fixed point for our method, and prove that given a sufficiently close initial guess, the convergence rate is quadratic. Empirically, our method converges to a significantly larger number of eigenpairs compared to previously proposed iterative methods, and with enough random initializations typically finds all real eigenpairs. In particular, for a generic symmetric tensor, the sufficient conditions for local convergence of our Newton-based method hold simultaneously for all its real eigenpairs.

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Section: Newton Correction Methodsmentioning

confidence: 99%

Section: Newton Correction Methodsmentioning

confidence: 99%

Section: Newton Correction Methodsmentioning

confidence: 99%

Section: Newton Correction Methodsmentioning

confidence: 99%

Section: Lemma 2 Consider One Update Step Of Algorithm 1 As In Equati...mentioning

confidence: 99%

See 3 more Smart Citations

Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors

Jaffe

Weiss

Nadler

2018

SIAM J. Matrix Anal. & Appl.

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Learning Binary Latent Variable Models: A Tensor Eigenpair Approach

Jaffe,

Weiss,

Carmi

et al. 2018

Preprint

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Latent variable models with hidden binary units appear in various applications. Learning such models, in particular in the presence of noise, is a challenging computational problem. In this paper we propose a novel spectral approach to this problem, based on the eigenvectors of both the second order moment matrix and third order moment tensor of the observed data. We prove that under mild non-degeneracy conditions, our method consistently estimates the model parameters at the optimal parametric rate. Our tensor-based method generalizes previous orthogonal tensor decomposition approaches, where the hidden units were assumed to be either statistically independent or mutually exclusive. We illustrate the consistency of our method on simulated data and demonstrate its usefulness in learning a common model for population mixtures in genetics.

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A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

Cited by 2 publications

References 10 publications

Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors

Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors

Learning Binary Latent Variable Models: A Tensor Eigenpair Approach

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