Homotopy-Correct Initialization for Manifold Regression

Definitions and mathematical statement

Let $M,N$ be the manifolds in a nonparametric regression problem and let $\eta:M\to N$ be the target mapping. Homotopy class is understood for continuous maps: maps $\psi,\eta$ have the same class if there exists a continuous $H:M\times[0,1]\to N$ with $H(x,0)=\psi(x)$ and $H(x,1)=\eta(x)$. For a training sample $D_k$ of size $k$, write $\phi_k(0):M\to N$ for the constructed initial mapping; the zero refers to initialization, and the index makes its possible dependence on sample size explicit. The source asks whether an initialization constructed from the training information can be guaranteed to satisfy \[\phi_k(0)\simeq\eta\qquad\text{for every sufficiently large }k.\] No direct access to the unknown target is intended to trivialize the regression task. The source does not state the manifold class, data distribution, observation/noise model, target regularity beyond what homotopy requires, or how dense the observed inputs become. It also does not define guarantee. A deterministic eventual guarantee, an almost-sure eventual guarantee, and $\Pr(\phi_k(0)\simeq\eta)\to1$ are different statements; none is silently substituted for another. The threshold in sufficiently large may depend on the target, sampling, and geometric parameters, but the permitted dependencies are not specified. Nor is uniformity over all target maps asserted. The supplied 2026-08-21 triage labels the question partially solved and reports a positive homotopy-class initialization result for maps from the circle to itself in the cited paper, while leaving the general manifold-valued question open.

Short English statement

Can sample-based initialization eventually recover the target map's homotopy class in manifold regression? The data assumptions, regularity and geometry, and deterministic versus probabilistic meaning of guarantee must be specified; the supplied assessment reports a circle-to-circle special case.

Sources

• Local UnsolvedMath registry, permanent ID 30000993, source problem catalogue entry, Homotopy-Correct Initialization for Manifold Regression, proposed year 2008; no individual proposer supplied.
• Supplied citation: Learning Theory and Approximation (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/30, pp. 1655--1706: https://doi.org/10.4171/owr/2008/30. Supplied literature label: original OWR report.
• Supplied literature label: Steinke--Hein--Scholkopf, https://doi.org/10.1137/080744189.