(1)Faculty of Mech.

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Semi-nonnegative INDSCAL analysis
Ahmad Karfoul (1), Julie Coloigner (2,3), Laurent
Albera (2,3), Pierre Comon (4,5)
(1)Faculty of Mech. Elec. Engineering,
University AL-Baath, Syria
(2)Laboratory LTSI - INSERM U642, France
(3)University of Rennes 1, France
(4)Laboratory I3S - CNRS, France
(5)University of Nice Sophia - Antipolis, France
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Outlines

• Preliminaries and problem formulation

• Global line search

• Optimization methods

• A compact matrix form of derivatives

• Numerical results

• Conclusion

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Preliminaries and problem formulation
Outer product
Ex. Order 3
?
Ex. Order q
?
Outer product of q-vectors ? rank-one q-th order
tensor
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Preliminaries and problem formulation
Tensor to rectangular matrix transformation

(unfolding according to the i-th mode)
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Preliminaries and problem formulation
CANonical Decomposition (CAND) Hitchcock 1927,
Carroll Chang 1970, Harshman 1970
CAND Linear combinantion of minimal number of
rank -1 terms
?P
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Preliminaries and problem formulation
INDSCAL decomposition Carroll Chang 1970
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Preliminaries and problem formulation
CANonical Decomposition (CAND)
INDSCAL decomposition
INDSCAL CAND of 3-order tensor symmetric in two
of three modes
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Preliminaries and problem formulation
(Semi-) nonnegative INDSCAL decomposition for
(semi-) nonnegative BSS
Example
Diagonalizing a set of covariance matrices
the (N ? P) mixing matrix
s zero-mean random vector of P statistically
independent components
Covariance matrix
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Preliminaries and problem formulation
Problem at hand
Constrained problem
Unconstrained problem
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Preliminaries and problem formulation

• Solution minimizing the following cost
  function

Some iterative algorithms

• Steepest Descent

First second order derivatives of ?

• Newton

• Levenberg Marquardt

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Optimization methods
Global line search (1/2)

• Looking for the global optimum in a given
  direction

Update rules
Directions given by the iterative algorithm
with respect to A and C, respectively.
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Optimization methods
Global line search (2/2)
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Optimization methods
Steepest Descent (SD)

• Optimization by searching for stationary points
  of ? based on first-order approximation (i.e. the
  gradient)

Update rules
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Optimization methods
Steepest Descent (SD)
Then
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Compact matrix form of derivatives
Gradient computation of ?(A,C)
Then
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Optimization methods
Newton

• Optimization by including the second-order
  approximation to accelerate the convergence

Update rules
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Optimization methods
Newton

• Convergence requirement Hessians are positive
  definite matrices

• Solution Necessity to regularization (i.e.
  Eigen-Value Decomposition (EVD) - based
  technique )

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Optimization methods
Levenberg-Marquardt (LM)
Update rules
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Numerical results
Convergence speed VS SNR

• Noise-free random 3-order tensor

• Noisy 3-way array

• Results averaged over 200 Monte Carlos
  realizations.

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Numerical results
Convergence speed VS SNR
SNR 0 dB
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Numerical results
Convergence speed VS SNR
SNR 15 dB
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Numerical results
Convergence speed VS SNR
SNR 30 dB
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Conclusion

• Solving an unconstrained semi-nonnegative
  INDSCAL problem .

• Differential concept ? Powerful tool for
  compact matrix derivations forms

• Global line search for symmetric case ? global
  optimum in the considered direction

• Iterative algorithms with global line search ?
  suitable step to reach the global
  optimum