Analysis of alignment algorithms with mixed dimensions for dimensionality reduction

We consider an alignment algorithm for reconstructing global coordinates of a given data set from coordinates constructed for data points in small local neighborhoods through computing a spectral subspace of an alignment matrix. We show that, under certain conditions, the null space of the alignment matrix recovers global coordinates even when local point sets have different dimensions. This result generalizes a previous analysis to allow alignment of local coordinates of mixed dimensions. We also extend this result to the setting of a semi-supervised learning problem, and we present several examples to illustrate our results.