Fuzzy Granule Manifold Alignment Preserving Local Topology

Granular computing has the advantage of discovering complex data knowledge, and manifold alignment has proven of great value in a lot of areas of machine learning. We propose a novel algorithm of fuzzy granule manifold alignment (FGMA), where we define some new operations, measurements, and local topology of fuzzy granular vectors in fuzzy granular space. Furthermore, the algorithm is very different from Semi-supervised and Procrustes algorithm because predetermining correspondence is not necessary. A projection is learned that can map instances described by two types of features to a low-dimensional space. Meanwhile, the local topology of the fuzzy granular vector induced by the instance is also preserved and matched within each set in lower dimensional space. This approach makes it possible to directly compare between data instances in different spaces. We convert an alignment problem of data in feature space into fuzzy granular manifold alignment problem of granular space. Specifically, we first define fuzzy granule, fuzzy granular vector, operations, and measurements in fuzzy granular space and gave proofs of theorems and deductions. Next, the local topology around the fuzzy granular vector is introduced and the optimal local topology matching can be achieved by minimizing their Frobenius norm. Finally, two manifolds are connected and the optimal mapping can be calculated to obtain dimensionality reduction of the joint structure. Thus, the corresponding relationship between two data instances can be got. We verified this algorithm in Oxford image and Alzheimer's disease voice dataset. Theoretical analysis and experiments demonstrate the algorithm proposed is robust and effective.