Intrinsic Complexity and its Scaling Law: From Approximation of Random Vectors and Random Fields to High Frequency Waves

Start: 04/18/2018 - 4:15pm
End  : 04/18/2018 - 5:15pm


Hongkai Zhao (UC Irvine)


 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 start with approximate embedding of a set of random vectors (principal component analysis a.k.a. singular value decomposition), then study the approximation of random fields and high frequency waves.  We provide lower bounds and upper bounds for the intrinsic complexity and its explicit asymptotic scaling laws in terms of the total number of random vectors, the correlation length for random fields, and the wave length for high frequency waves respectively. 

Freeberg Forum, LC 62, Kravis Center CMC

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