ilmenau university of technology communications research laboratory 1 a new multi-dimensional model...

Ilmenau University of TechnologyCommunications Research Laboratory

1

A new multi-dimensional model order selection technique called closed-form PARAFAC based model order selection (CFP-MOS) scheme based on the multiple estimates of the closed-form PARAFAC [4] suitable for applications with PARAFAC data model

For the estimation of spatial frequencies, we propose to apply the closed-form PARAFAC based parameter estimator (CFP-PE) for arrays without shift invariance property

• in conjunction with the Peak Search based estimator due to the decoupling of dimensions

• robust against modeling errors increase of the maximum model order

• via merging of dimensions

• separation via Least Squares Khatri-Rao Factorization (LSKRF) [8]

Main ContributionsMain Contributions


2RR-D Parameter Estimation-D Parameter Estimation

Encountered in a variety of applications mobile communications, spectroscopy, multi-dimensional medical imaging, and the estimation of the parameters of the dominant multipath components from

MIMO channel sounder measurements Traditional approaches require

stacking the dimensions into one highly structured matrix, since the measured data is multi-dimensional

For the R-D parameter estimation the model order selection, i.e., the estimation of the number of principal

components, is required parameters can be extracted from the main components using the estimated

model order and assuming some structure of the data

• For instance, in MIMO applications, the main components are represented by the superposition of undamped complex exponentials, where each vector of complex exponentials is mapped to a certain spatial frequency.


3

In [1], we have proposed the R-Dimensional Exponential Fitting Test (R-D EFT) a multi-dimensional extension of the Modified Exponential Fitting Test (M-EFT) is based on the HOSVD of the measurement tensor is superior to other schemes in the literature [2,3] restricted to applications in the presence of white Gaussian noise

Since colored noise is common in many applications, we propose the closed-form PARAFAC based model order selection (CFP-MOS) scheme.

Once the model order is estimated, the extraction of the spatial frequencies from the main components can be performed.

In general, for this task, closed-form schemes like R-D ESPRIT-type algorithms [5] are applied, since their performance is close to the Cramér-Rao lower bound (CRLB).

In [6], the 3-D and 4-D versions of the Multi-linear Alternating Least Squares (MALS) decompose the measurement tensor into factors Easy to obtain the spatial frequencies via

• shift invariance or peak search based estimator We propose to apply a closed-form PARAFAC based parameter estimator (CFP-PE).

State of the ArtState of the Art


4Tensor AlgebraTensor Algebra

3-D tensor = 3-way array

n-mode products between and

Unfoldings

M1

M2

M3

“1-mode vectors”



i.e., all the n-mode vectors multiplied from the left-hand-side by

11 22


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For the estimation of the spatial frequencies, we assume a data model where

Data ModelData Model

Noiseless data representationNoiseless data representation

ProblemProblem

where is the colored noise tensor. Therefore, our objective is to estimate the model order d and the spatial frequencies.

the elements of the vector can be mapped into a certain spatial frequency for and


6SVD and PARAFACSVD and PARAFAC

+ += + +=

Another way to look at the SVD PARAFAC Decomposition

The task of the PARAFAC analysis: Given (noisy) measurements

and the model order d, findsuch that

Here is the higher-order Frobenius norm (square root of the sum of the squared magnitude of all elements).


7Closed-form PARAFAC basedClosed-form PARAFAC basedModel Order SelectionModel Order Selection

Our approach: based on simultaneous matrix diagonalizations (“closed-form”). By applying the closed-form PARAFAC (CFP) [4]

R*(R-1) simultaneous matrix diagonalizations (SMD) are possible; R*(R-1) estimates for each factor are possible; selection of the best solution by different heuristics (residuals of the SMD) is done

Res

idua

lsR

esid

uals

((kk,,ll,,ii))R

esid

uals

Res

idua

ls

kk and and l l are the tuples of the SMD and are the tuples of the SMD and ii indicates left or right factor matrix to be estimated [4]. indicates left or right factor matrix to be estimated [4]. bb

Sorting based onthe residuals

Residuals the Frobenius norm of the off-diagonal elements of the diagonalized matrices. In practice, the small residuals means small error in the estimated parameters.

The index b sorts the factors based on the residuals, which gives us

11 33 55 997722 44 66 88 101011111212BBlimlim


8Closed-form PARAFAC basedClosed-form PARAFAC basedModel Order SelectionModel Order Selection

For P = 2, i.e., P < d

We assume d = 3 and we consider only solutions with the two smallest residuals of the SMD, i.e., b = 1 and 2.

Due to the permutation ambiguities, the components of different tensors are ordered using the amplitude based approach proposed in [7].

For P = 4, i.e., P > d

+=

+=

= +

= +

PP11 22 33 44 55

+ +

+ +


9Closed-form PARAFAC basedClosed-form PARAFAC basedModel Order SelectionModel Order Selection For instance, let us consider the estimated factor

where P is the candidate value for the model order d and b is the index of the ordered multiple factors according to the assumed heuristics. We define the following error function

Taking into account all the components in the cost function

As proposed in the paper, another expression for the cost function is possible by

using the spatial frequencies instead of the angular distances.


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Comparing the performance of CFP-MOSComparing the performance of CFP-MOS

SimulationsSimulations

White Gaussian noise

Colored Gaussian noise


11Closed-form PARAFAC basedClosed-form PARAFAC basedParameter EstimationParameter Estimation

Let us consider

Merging dimensionsMerging dimensions

We can stack dimensions and obtain

where

With merging

Least Squares Khatri-RaoLeast Squares Khatri-RaoFactorization Factorization [8]

Given , we desire and

Reshaping the merged vector

Since the product should be a

rank-one matrix, we can apply the SVD-based rank-one

approximation

Therefore,

Without merging

in both cases, we assume that


12Closed-form PARAFAC basedClosed-form PARAFAC basedParameter EstimationParameter Estimation

Peak search based estimatorPeak search based estimator Given from the CFP decomposition, we can compute the respective spatial frequency via



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Comparing the performance of CFP-PEComparing the performance of CFP-PE



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Comparing the performance of CFP-PEComparing the performance of CFP-PE



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[1] J. P. C. L. da Costa, M. Haardt, F. Roemer, and G. Del Galdo, “Enhanced model order estimation using higher-order arrays”, in Proc. 41-st Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, USA, Nov. 2007, invited paper.

[2] J. P. C. L. da Costa, M. Haardt, and F. Roemer, “Robust methods based on the HOSVD for estimating the model order in PARAFAC models,” in Proc. 5-th IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM 2008), (Darmstadt, Germany), pp. 510 - 514, July 2008.

[3] J. P. C. L. da Costa, A. Thakre, F. Roemer, and M. Haardt, “Comparison of model order selection techniques for high-resolution parameter estimation algorithms,” in Proc. 54th International Scientific Colloquium (IWK), (Ilmenau, Germany), Sept. 2009.

[4] F. Roemer and M. Haardt, “A closed-form solution for multilinear PARAFAC decompositions,” in Proc. 5-th IEEE Sensor Array and Multichannel Signal Processing Workshop, (Darmstadt, Germany), pp. 487 - 491, July 2008.

[5] M. Haardt, F. Roemer, and G. Del Galdo, “Higher-order SVD based subspace estimation to improve the parameter estimation accuracy in multi-dimensional harmonic retrieval problems,” IEEE Trans. Signal Processing, vol. 56, pp. 3198-3213, July 2008.

[6] X. Liu and N. Sidiropoulos, “PARAFAC techniques for high-resolution array processing”, in High-Resolution and Robust Signal Processing, Y. Hua, A. Gershman, and Q. Chen, Eds., Marcel Dekker, 2004, Chapter 3.

[7] M. Weis, F. Roemer, M. Haardt, D. Jannek, and P. Husar, “Multi-dimensional Space-Time-Frequency component analysis of event-related EEG data using closed-form PARAFAC”, in Proc. IEEE Int. Conf. Acoustic, Speech, and Signal Processing (ICASSP), Taipei, Taiwan, pp. 349-352, Apr. 2009.

[8] F. Roemer and M. Haardt, “Tensor-Based channel estimation (TENCE) for Two-Way relaying with multiple antennas and spatial reuse”, in Proc. IEEE Int. Conf. Acoust. Speech, and Signal Processing (ICASSP), Taipei, Taiwan, pp. 3641-3644, Apr. 2009, invited paper.

[9] E. Radoi and A. Quinquis, “A new method for estimating the number of harmonic components in noise with application in high resolution RADAR”, EURASIP Journal on App. Sig. Proc., pp. 1177-1188, 2004.

ReferencesReferences

ilmenau university of technology communications research laboratory 1 a new multi-dimensional model...

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