2

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A Taxonomy of Emerging Multilinear Discriminant Analysis Solutions

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CONCLUSIONS

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This chapter provides a comprehensive introduction to the area of multilinear learning algorithms, in particular the multilinear discriminant analysis (MLDA) algorithms, for the recognition of biometric signals, most of which are naturally tensor objects. Three typical projections are introduced rst: the vector-to-vector projections (VVP), the tensor-to-tensor projections (TTP) and the tensor-to-vector projections (TVP), and two general MLDA solutions are formulated: the MLDA-TTP and the MLDA-TVP. The choices of the separation criteria and the initialization methods are then presented and the relationships between LDA, MLDA-TTP, and MLDA-TVP are discussed. A taxonomy of MLDA variants is subsequently suggested; and it not only helps us to understand the existing mutlilinear algorithms, but also bene ts us in the development of new multilinear algorithms. Finally, the MLDA variants are experimentally evaluated on the CMU PIE database and the extended Yale database B to demonstrate their performance on the popular face recognition problem. The experimental results indicate that the MLDA solutions, and multilinear learning algorithms in general, are promising emerging areas for research and applications.

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ACKNOWLEDGMENTS

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We would like to thank Cai Deng from the University of Illinois at Urbana-Champaign for making the standard-size face data available. We also would like to acknowledge all those who contributed to the research described in this chapter.

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APPENDIX: MULTILINEAR DECOMPOSITIONS

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There are two types of decompositions used most in multilinear applications: the canonical decomposition (CANDECOMP) [21, 22, 53], which is also known as the parallel factors (PARAFAC) decomposition [21, 22, 54], and the TUCKER decomposition [21, 22, 55]. With the CANDECOMP decomposition, a tensor A can be decomposed into a linear combination of P rank-1 tensors:

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(1) (2) (N) p u p u p u p ,

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(B.1)

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where P N In . With the TUCKER decomposition, a tensor A can be expressed n=1 as the product:

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P1 P2 PN

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p1 =1 p2 =1

pN =1 (1)

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(1) (2) (N) S(p1 , p2 , . . . , pN )up1 up2 upN

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= S 1 U

2 U(2) N U(N) ,

(B.2)

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References

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T T T

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where Pn In for n = 1, . . . , N, S = A 1 U(1) 2 U(2) . . . N U(N) , and (n) (n) (n) U(n) = u1 u2 . . . uPn is an In Pn matrix with orthonormal column vectors. The CONDECOMP decomposition is in fact a special case of the TUCKER decomposition.

REFERENCES

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