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This constraint merges the sample points with the same label together in the projection space. SPP-CGE The affinity matrix is obtained by sparse representation as showed in Eq. (11). In this way, the affinity matrix can be obtained without setting any parameters. We also want to keep the sparse representation in the projection space. e. let y = Uz. Then, Eq. (22) becomes: (23) where the i-th item of ei is 1, 0 otherwise. With simple algebraic manipulations, the objective function can be rewritten as: (24) where = S + ST- ST S.

Pd]. The projected samples are obtained by Y = PT X. Sparsity Preserving Projection SPP is similar to NPE, using sparse representation instead of the linear representation of k nearest neighbors to get the weight matrix. For xi, the representative coefficients are obtained by solving a L1 problem: (11) where si = [si1, . . , si(i-1),0,si(i+1), . . ,sin] The problem of SPP is: (12) With the constraint pT XXT p = 1, Eq. (12) becomes: (13) where = S + ST- ST S, and S = [s1, . . , sn]T . The corresponding eigenvalue problem is: (14) The eigenvectors p1, .

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