Consistent Noisy Single-Index Regression

Definitions and mathematical statement

Let $\varepsilon$ denote the noise in the nonlinear single-index regression problem described by the supplied record. The estimator under discussion uses tangent-field approximations and level-set partitions. Write $I_k$ for the objects occurring in that construction; neither their definition nor the meaning of their size $|I_k|$ is supplied.

Problem as supplied. Can this estimator be modified to achieve consistent estimation in the presence of nonzero noise, $\varepsilon\ne0$? The reported obstruction is that \[ |I_k|\not\to0 \] even when the widths of adjacent levels are arbitrarily small.

Unresolved qualifications. This is a source-qualified research question, not a fully specified statistical assertion. The supplied record does not define the regression model, the tangent-field construction, the level-set partitions, the objects $I_k$, or the limiting regime in the displayed expression. It also does not identify the estimand, the mode of consistency, the permitted modifications, or the assumptions on the covariates, regression function, sampling scheme, and noise. In particular, $\varepsilon\ne0$ does not specify a probability law or a precise probabilistic nondegeneracy condition. Thus a quantified consistency requirement cannot be supplied without additional source information.

Short English statement

The question asks whether a regression estimator using tangent-field approximations and level-set partitions can be modified to remain consistent with nonzero noise. The source reports that the quantities $|I_k|$ need not shrink even when adjacent level widths are arbitrarily small. The estimator, statistical target, assumptions, and meaning of consistency require clarification before this can become a precise statistical problem.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003790$, title "Consistent Noisy Single-Index Regression," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is named.
• Supplied source citation: Nonlinear Data: Theory and Algorithms (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/20, pp. 1161--1234. https://doi.org/10.4171/owr/2018/20. The record separately gives the proposed year as 2018; the relationship between these year labels is not explained.
• Supplied literature source, labelled "workshop report": https://ems.press/content/serial-article-files/46744.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open, states that the source explicitly identifies inconsistency for nonzero noise and modification for consistency as an open problem, and reports that no later primary resolution was found by that cutoff. This is supplied dated curation, not an independently verified literature assessment or a claim that the linked sources were visited for this draft.