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Given a set of data W={w1,…,wN}∈RD drawn from a union of subspaces, we focus on determining a nonlinear model of the form U=⋃i∈ISi, where {Si⊂RD}i∈I is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our approach is based on the binary reduced row echelon form of data matrix, combined with an iterative scheme based on a non-linear approximation method. We prove that, in absence of noise, our approach can find the number of subspaces, their dimensions, and an orthonormal basis for each subspace Si. We provide a comprehensive analysis of our theory and determine its limitations and strengths in presence of outliers and noise.