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Interpolation, the Rudimentary Geometry of Spaces of Lipschitz Functions, and Geometric Complexity

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Abstract

We consider seriously the analogy between interpolation of nonlinear functions and manifold learning from samples, and examine the results of transferring ideas from each of these domains to the other. Illustrative examples are given in approximation theory, variational calculus (closed geodesics), and quantitative cobordism theory.

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Notes

  1. Or geometry; I shall not distinguish between these.

  2. Alternatively, if a connected submanifold M didn’t separate, one would construct a closed curve transverse to M lying in a neighborhood of M, and then argue that this would contradict the simple connectivity of the sphere (by homotopy invariance of intersection numbers).

  3. In some cases the connection between the homotopy theory and the geometric problem is sufficiently tight that any new geometric complexity statement would readily imply some, although perhaps not optimal, result about the complexity of homotopies.

  4. Many classical problems in analysis are also solved by homotopy methods. Typically, one starts, though, with an explicit homotopy to apply them to. I hope that the ideas discussed in later sections can be of use in that setting.

  5. Unless something extraordinary is happening—say f is locally constant.

  6. This can be achieved using either of the subdivision schemes of [23] or [30].

  7. The first level for \(E = \int \langle \nabla f, \nabla f \rangle / \int \langle f, f \rangle \) where i-dimensional mod 2 homology appears in the projective space of the Sobolev space \(H^{1,2}(M)\) is in the ith positive eigenvalue of the Laplacian.

  8. The reader could well want to know what can distinguish manifolds which mutually approach one another. The simplest invariants are odd primary characteristic classes of the topological tangent bundle. However, there are additional subtle secondary invariants that are torsion analogues of the invariants used by Atiyah and Bott [1] to distinguish lens spaces from one another.

  9. The process described, though, is useful even when \(B\Gamma \) does not exist as a finite complex. The homology described can be used as a substitute that occasionally has more useful properties. See [20] for an example of this.

  10. We shall not discuss here what to do about noise or when populations have overlapping images.

  11. Of course, in the noncompact case, Morse theory in its ordinary sense requires a properness condition, or a Palais–Smale condition in the infinite-dimensional setting.

  12. Other applications to approximation theory can be found in [49].

  13. Except the global maximum, which doesn’t close any \(H_0\) bar. Thought of as being the circle, it begins the nontrivial \(H_1\) bar.

  14. See the proof for a definition, or see, e.g., the Wikipedia page on Dehn functions of groups.

  15. Here I mean a large scale idea of texture that is analogous to the differences that one would notice on the small scale if one took the graphs of (Baire generic) Hölder functions for different exponents. If you consider the persistence homology, the Hölder exponent would be apparent in the “length spectrum of the bars” as will be explained in detail in a future paper with Yuliy Baryshnikov (and is not hard to see by wrinkling a map by putting in as many bumps at small scale as permitted by dimension and the Hölder condition).

  16. This linearity conjecture implies an estimate for \(\eta \)-invariants of manifolds (spectral invariants of odd-dimensional manifolds defined by Atiyah–Patodi–Singer) in terms of volume; Cheeger and Gromov proved these directly [13]. While the quantitative cobordism work we discuss is not strong enough to give the Cheeger–Gromov inequality, a different approach—with very interesting complexity applications—was given by [14].

  17. Actually, there is a beautiful proof of Thom’s theorem [6] that avoids this. This method actually directly leads to a polynomial estimate (of degree \(2^n\)) for k(W). Unlike what we are about to discuss, this does not extend to the oriented case.

  18. To see the issue, the number of point inverses for a Lipschitz map from \(S^1\) to itself can we be infinite, even with \(L = 2\). However, the degree of a Lipschitz map is bounded by L and one can \(C^0\)-approximate such a map by one where the number inverse images grows linearly with L.

  19. The construction we are about to explain is called the Whitehead product in homotopy theory. (See, e.g., [54].)

  20. However, for \({\text {Lip}}(S^1:M)\) for M a 3-dimensional Sol manifold the diameter is \(\exp (L)\), as suggested by a worst case analysis based on covering numbers.

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    Shmuel Weinberger

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Weinberger, S. Interpolation, the Rudimentary Geometry of Spaces of Lipschitz Functions, and Geometric Complexity. Found Comput Math 19, 991–1011 (2019). https://doi.org/10.1007/s10208-019-09416-0

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