Random Projections for Manifold Learning
We propose a novel method for {\em linear} dimensionality reduction of manifold modeled data. First, we show that with a small number $M$ of {\em random projections} of sample points in $\reals^N$ belonging to an unknown $K$-dimensional Euclidean manifold, the intrinsic dimension (ID) of the sample set can be estimated to high accuracy. Second, we rigorously prove that using only this set of random projections, we can estimate the structure of the underlying manifold. In both cases, the number random projections required is linear in $K$ and logarithmic in $N$, meaning that $K<M\ll N$. To handle practical situations, we develop a greedy algorithm to estimate the smallest size of the projection space required to perform manifold learning. Our method is particularly relevant in distributed sensing systems and leads to significant potential savings in data acquisition, storage and transmission costs.
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