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1 year ago in Computational Geometry , Mathematical Analysis By Srajan

How can the complexity of a function be measured?

Reading papers on machine learning and high-dimensional analysis, I constantly see references to function complexity affecting model behavior and computational cost. Beyond vague intuition, what are the concrete, measurable properties we're actually referring to?

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By Rani Answered 1 year ago

In my research on approximation algorithms, we measure complexity to predict how hard a function is to learn, optimize, or integrate. It's not one number. Practically, I look at several axes: smoothness, via bounds on derivatives or the Lipschitz constant (how wildly it can oscillate); intrinsic dimensionality, meaning it might live in a high-dimensional space but actually vary along only a few key directions; and geometric complexity, like the number of local minima or the curvature of its level sets. For learning theory, we use Vapnik-Chervonenkis dimension to measure its "flexibility." Each metric informs different computational strategies.

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