ilmenau university of technology communications research laboratory 1 a new multi-dimensional model...
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Ilmenau University of TechnologyCommunications Research Laboratory
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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
Ilmenau University of TechnologyCommunications Research Laboratory
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.
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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
Ilmenau University of TechnologyCommunications Research Laboratory
4Tensor AlgebraTensor Algebra
3-D tensor = 3-way array
n-mode products between and
Unfoldings
M1
M2
M3
“1-mode vectors”
“2-mode vectors”
“3-mode vectors”
i.e., all the n-mode vectors multiplied from the left-hand-side by
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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
Ilmenau University of TechnologyCommunications Research Laboratory
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).
Ilmenau University of TechnologyCommunications Research Laboratory
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
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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
+ +
+ +
Ilmenau University of TechnologyCommunications Research Laboratory
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
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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
Ilmenau University of TechnologyCommunications Research Laboratory
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
SimulationsSimulations
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Comparing the performance of CFP-PEComparing the performance of CFP-PE
SimulationsSimulations
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Comparing the performance of CFP-PEComparing the performance of CFP-PE
SimulationsSimulations
Ilmenau University of TechnologyCommunications Research Laboratory
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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