Some convergence results on the Regularized Alternating Least-Squares method for tensor decomposition

Li, Na; Kindermann, Stefan; Navasca, Carmeliza

doi:10.1016/j.laa.2011.12.002

Cited by 66 publications

(64 citation statements)

References 34 publications

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“…Even so, we do not know which columns are activated. So, we can use the standard trilinear decomposition algorithm, such as alternating least squares (ALS) [22] and regularized alternating least squares (RALS) [23], [24], where some sufficient conditions for uniqueness up to permutation and scalings of the decomposition are provided. After the decomposition, we can obtain A, B, and S (f ).…”

Section: A Algorithm Based On Trilinear Decompositionmentioning

confidence: 99%

Joint DOA and frequency estimation with sub-Nyquist sampling

Liu

Wei

2019

Signal Processing

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In this paper, to jointly estimate the frequency and the direction-of-arrival(DOA) of the narrowband far-field signals, a novel array receiver architecture is presented by the concept of the sub-Nyquist sampling techniques. In particular, our contribution is threefold. i) First, we propose a timespace union signal reception model for receiving array signals, where the sub-Nyquist sampling techniques and arbitrary array geometries are employed to decrease the time-domain sampling rate and improve the DOA estimation accuracy. A better joint estimation is obtained in the higher time-space union space. ii) Second, two joint estimation algorithms are proposed for the receiving model. One is based on a trilinear decomposition from the third-order tensor theory and the other is based on subspace decomposition. iii) Third, we derive the corresponding Cramér-Rao Bound (CRB) for frequency and DOA estimates. In the case of the branch number of our architecture is equal to the reduction factor of the sampling rate, it is observed that the CRB is robust in terms of the number of signals, while the CRB based on the Nyquist sampling scheme will increase with respect to the number of signals. In addition, the new steer vectors of the union time-space model are completely uncorrelated under the limited number of sensors, which improves the estimation performance. Furthermore, the simulation results demonstrate that our estimates via the receiver architecture associated with the proposed algorithms closely match the CRB according to the noise levels, the branch number and the source number as well.

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Section: A Algorithm Based On Trilinear Decompositionmentioning

confidence: 99%

Joint DOA and frequency estimation with sub-Nyquist sampling

Liu

Wei

2019

Signal Processing

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“…The assumption A2 corresponds to the persistency of excitation from the data and is a key ingredient in the convergence. When using ALS for a system identification problem and assuming A2 enables to avoid rank-deficiencies in the matrix F and therefore swamps as observed for tensor decomposition in [35] do not occur.…”

Section: B Convergence Proof For the Normalized Alsmentioning

confidence: 99%

QUARKS: Identification of Large-Scale Kronecker Vector-AutoRegressive Models

Sinquin

Verhaegen

2018

IEEE Trans. Automat. Contr.

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In this paper, we address the identification of twodimensional spatial-temporal dynamical systems described by the Vector Auto-Regressive (VAR) form. The coefficient matrices of the VAR model are parametrized as sums of Kronecker products. When the number of terms in the sum is small compared to the size of the matrices, such a Kronecker representation efficiently models large-scale VAR models. Estimating the coefficient matrices in least-squares sense gives rise to a bilinear estimation problem which is tackled using an Alternating Least Squares (ALS) algorithm. Regularization or parameter constraints on the coefficient matrices allows to induce temporal network properties such as stability as well as spatial properties such as sparsity or Toeplitz structure. Convergence of a particular formulation of ALS which features some normalization is proved using fixedpoint theory. A numerical example demonstrates the advantages of the new modeling paradigm. It leads to comparable variance of the prediction error with the unstructured least-squares estimation of VAR models. However, the number of parameters grows only linearly with respect to the number of nodes in the 2D sensor network instead of quadratically in the case of fully unstructured coefficient matrices.

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“…13,[17][18][19][20][21][22][23][24][25][26] Regularized ALS scheme was considered in other works. 27,28 Recovering a tensor using only few of its entries is called tensor completion. Tensor completion has been widely studied in the literature.…”

Section: Introductionmentioning

confidence: 99%

Quantized CP approximation and sparse tensor interpolation of function‐generated data

Khoromskij

Naraparaju

Schneider

2019

Numerical Linear Algebra App

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Summary In this article, we consider the iterative schemes to compute the canonical polyadic (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the quantics‐tensor train (QTT) method (“O(d log N)‐quantics approximation of N‐d tensors in high‐dimensional numerical modeling” in Constructive Approximation, 2011) developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach, the target vector of length 2L is reshaped to a Lth‐order tensor with two entries in each mode (quantized representation) and then approximated by the QTT tensor including 2r2L parameters, where r is the maximal TT rank. In what follows, we consider the alternating least squares (ALS) iterative scheme to compute the rank‐r CP approximation of the quantized vectors, which requires only 2rL≪2L parameters for storage. In the earlier papers (“Tensors‐structured numerical methods in scientific computing: survey on recent advances” in Chemom Intell Lab Syst, 2012), such a representation was called QCan format, whereas in this paper, we abbreviate it as the QCP (quantized canonical polyadic) representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases, we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank‐r QCP format using the reduced number of the functional samples, calculated only at O(2rL) grid points, is presented. The special version of the ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP‐interpolation method that allows to recover all 2rL representation parameters of the rank‐r QCP tensor. Numerical examples show the efficiency of the QCP‐ALS‐type iteration and indicate the exponential convergence rate in r.

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Some convergence results on the Regularized Alternating Least-Squares method for tensor decomposition

Cited by 66 publications

References 34 publications

Joint DOA and frequency estimation with sub-Nyquist sampling

Joint DOA and frequency estimation with sub-Nyquist sampling

QUARKS: Identification of Large-Scale Kronecker Vector-AutoRegressive Models

Quantized CP approximation and sparse tensor interpolation of function‐generated data

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