We characterize the intrinsic complexity of a set in a metric space by the least dimension of a linear space that can approximate the set to a given tolerance. This is dual to the characterization using Kolmogorov n-width, the distance from the set to the best n-dimensional linear space. We characterize the intrinsic complexity of a set of random vectors (via principal component analysis a.k.a. singular value decomposition) and random fields (via Karhunen–Loève expansion) as well as solutions to partial differential equations of various type.
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