Provable Dimension Detection Using Principal Component Analysis

We analyze an algorithm based on principal component analysis (PCA) for detecting the dimension k of a smooth manifold from a set P of point samples. The best running time so far is O(d 2O(k7log k)) by Giesen and Wagner after the adaptive neighborhood graph is constructed. Given the adaptive neighborhood graph, the PCA-based algorithm outputs the true dimension in O(d2O(k)) time, provided that P satisfies a standard sampling condition as in previous results. Our experimental results validate the effectiveness of the approach. A further advantage is that both the algorithm and its analysis can be generalized to the noisy case, in which small perturbations of the samples and a small portion of outliers are allowed.

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