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Locality preserving projection (LPP) is a classical tool for dimensionality reduction problems. However, it is sensitive to outliers because of utilizing the ℓ₂-norm-based distance criterion. In this paper, we propose a new approach, termed Euler-LPP, by preserving the local structures of data under the distance criterion of the cosine-based dissimilarity. Euler-LPP is robust to outliers in that the cosine-based dissimilarity suppresses the influence of outliers more efficiently than the ℓ₂ -norm. An explicit mapping, defined by a complex kernel (euler kernel) is adopted to map the data from the input space to complex reproducing kernel Hilbert spaces (CRKHSs), in which the distance of the data pairs under the ℓ₂-norm is equal to that in the input space under the cosine-based dissimilarity. Thus, the robust dimensionality problem can be directly solved in CRKHS, where the solution is guaranteed to converge to a global minimum. In addition, Euler-LPP is easy to implement without significantly increasing computational complexity. Experiment results on several benchmark databases confirm the effectiveness of the proposed method.