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The multilinear extension of the LDA using the TVP is named MLDA-TVP and the formal de nition of the problem to be solved in MLDA-TVP is described below: A set of M training tensor objects {X1 , X2 , . . . , XM } is available. Each tensor object Xm RI1 I2 IN assumes values in the tensor space RI1 RI2 RIN , where In is the n-mode dimension of the tensor. The objective of MLDA-TVP is to (n) nd a set of P EMPs {up RIn 1 , n = 1, . . . , N}P mapping from the original p=1
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2.3 Multilinear Discriminant Analysis
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Input: A set of tensor samples {Xm RI1 I2 IN , m = 1, . . . , M} with class labels c RM , Pn for n = 1, . . . , N. Output: Low-dimensional representations {Ym RP1 P2 PN , m = 1, . . . , M} of the input tensor samples maximizing a separation criterion. Algorithm: Step 1: Initialize U(n) for n = 1, . . . , N. 0 Step 2 (Local optimization):
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For n = 1 : N T T Calculate {Ym = Xm 1 U(1) n 1 U(n 1) n+1 k k T T U(n+1) N U(N) , m = 1, . . . , M}. k 1 k 1 (n) Calculate SBY and S(n) . WY
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Set the matrix U(n) to optimize a separation criterion. k If k > 2 and U(n) converges for all n, set U(n) = U(n) and break. k k Step 3 (Projection): The feature tensor after projection is obtained as T T T {Ym = Xm 1 U(1) 2 U(2) N U(N) , m = 1, . . . , M}. Figure 2.5. The pseudo-code implementation of a general MLDA-TTP.
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tensor space RI1 RI2 . . . RIN into a vector subspace RP (with P <
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(n) ym = Xm N n=1 up , n = 1, . . . , N
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based on the optimization of a certain separation criteria, such that an enhanced separability between different classes is achieved. The MLDA-TVP objective is to determine the P projection bases in each mode (n) RIn 1 , n = 1, . . . , N, p = 1, . . . , P that maximize a class separation criup terion. In MLDA-TVP, since the projected space is a vector space, the de nition of scatter matrices in classical LDA can be followed. For the samples projected by the pth EMP {ymp , m = 1, . . . , M}, where ymp is the projection of the mth sample by the pth EMP, the between-class scatter matrix and the within-class scatter matrix are de ned as
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1 1 respectively, where yp = M m ymp , ycp = Nc m,cm =c ymp . Figure 2.6 is the pseudo-code implementation of a general MLDA-TVP algorithm. To solve the problem, the alternating projection principal is again employed. In each iteration k, for
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Input: A set of tensor samples {Xm RI1 I2 IN , m = 1, . . . , M} with class labels c RM , the projected feature dimension P . Output: Low-dimensional representations {ym RP , m = 1, . . . , M} of the input tensor samples maximizing a separation criterion. Algorithm: Step 1 (Stepwise optimization): For p = 1 : P
r For n = 1, . . . , N, initialize u(n) RIn . p r For k = 1 : K
For n = 1 : N (1)T (n 1)T n+1 * Calculate {ym = Xmp 1 upk n 1 upk
(n+1) (N) upk 1 N upk 1 , m = 1, . . . , M}. * Calculate the between-class and the within-class scatter matrices by treating {ym } as the input vector samples, as in classical LDA. (n) * Compute the vector upk that optimizes a separation criterion. (n) (n) (n) If k >2 and upk converges for all n, set up = upk and break.
Step 2 (Projection): The feature vector after projection is obtained as (1)T (N)T {ym (p) = Xm 1 up N up , p = 1, . . . , P, m = 1, . . . , M}. Figure 2.6. The pseudo-code implementation of a general MLDA-TVP.
mode n, the input tensor samples are projected using the current projection vectors in all modes except n to obtain a set of vector samples and the problem is then converted to a number of classical LDA problems.