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Abstract:
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Data lying in a high dimensional ambient space are commonly thought to have a much lower intrinsic dimension. In particular, the data may be concentrated near a lower dimensional manifold. There is an immense literature focused on approximating the unknown manifold and in exploiting such approximations in clustering, data compression, and building of predictive models. Most of the literature relies on approximating manifolds using a locally linear, and potentially multiscale, dictionary. In this article, we propose a simple and general alternative, which instead uses pieces of spheres, or spherelets, to locally approximate the unknown manifold. Theory is developed showing that spherelets can produce lower covering numbers and MSEs for many manifolds. We develop spherical principal components analysis (SPCA). Results relative to state-of-the-art competitors show gains in ability to accurately approximate the manifold with fewer components. In addition, unlike most competitors, our approach can be used for data denoising and can embed new data without retraining. The methods are illustrated with standard toy manifold learning examples, and applications to multiple real data sets.
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