Abstract
From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding result for a Lipschitz-continuous function between two such manifolds. That is, we outline the construction of a simplicial map which recovers the induced maps on homotopy and homology groups with high confidence using only finite sampled data from the domain and range, as well as knowledge of the image of every point sampled from the domain. We provide explicit bounds on the size of the point samples required for such reconstruction in terms of intrinsic properties of the domain, the co-domain and the function. This reconstruction is robust to certain types of bounded sampling and evaluation noise.
Highlights
The use of algebraic topological methods for the analysis of nonlinear data has become a subject of considerable interest with a wide variety of promising applications [3, 5, 7, 10]
It seems reasonable that the geometry generated by the data may provide insight concerning the original system
One is interested in the geometry associated with the data, and in the action of an unknown nonlinear process on the data
Summary
The use of algebraic topological methods for the analysis of nonlinear data has become a subject of considerable interest with a wide variety of promising applications [3, 5, 7, 10]. An obvious step is to obtain bounds on the probability of reconstructing, up to homotopy, a continuous function between Riemannian manifolds from images of dense samples. This is the focus of our work with the main result as follows. Given (1) probability parameters δX, δY ∈ (0, 1], (2) radii X < τX/2 and Y < τY/2 satisfying 4κ · X < Y, and (3) finite sets X ⊂ X and Y ⊂ Y of independent and identically distributed (i.i.d.) uniformly sampled points, let N(X) and N(Y) be nerves of the covers generated by open balls of radius X and Y around X and Y respectively.
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