0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 5 10 15 20 25 30 35 40 45 50 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 35 40 45 50 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Fig. 1: A synthetic data set (top left) created by mixing the local basis vectors (top right) with strengths determined by random walks. Bottom, local basis vectors blindly recovered from the mixture using our algorithm. ABSTRACT In some applications a local or 'parts based' representation

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... preferable to global basis functions such as those used in Fourier and principal component analysis. In applications that require human understanding and editing of the data, it is also desirable that the basis functions be in some sense as "simple" as possible. This means, for example, that the basis functions should not have Gabor-like ripples if such ripples are not a prominent feature of the data to be represented. This paper introduces a direct local basis construction. Specifically, we show that local bases result from maximizing an appropriate redefinition of pairwise orthogonality while maintaining the ability to represent the data. The resulting basis functions are competitive with (and for some applications superior to) those obtained from existing algorithms, and the construction does not require that the basis coefficients be statistically non-Gaussian or independent, as would be the case with an independent component analysis approach. Fig. 2: The 100th and 150th eigenvectors of a collection of faces. It is difficult to intuitively estimate the needed contribution of these images in a task such as enlarging the nose on a given face.