Detecting Genuine Modes in Density Deconvolution

Definitions and mathematical statement

Let $D\subseteq{\mathbb{R}}$ denote the unspecified one-dimensional domain of the target density $f:D\to[0,\infty)$, with $\int_D f(x)\,{\,\mathrm{d}} x=1$. Write $\widehat f_\lambda$ for a regularized reconstruction, where $\lambda$ denotes its regularization choice. The record does not specify the observation model, convolution kernel, sampling assumptions, admissible densities, or reconstruction procedure.

For clarity, one possible convention calls $m\in\operatorname{int}(D)$ a strict local mode of a function $u$ if \[ \exists\delta>0\quad\forall x\in D,\qquad 0<|x-m|<\delta\ \Longrightarrow\ u(x)<u(m). \] This convention is illustrative: the source does not specify strictness, treatment of plateaus, boundary modes, or the regularity needed for pointwise modes of a density to be well defined independently of its almost-everywhere representative.

Source question. In one-dimensional nonparametric density deconvolution, suppose a regularized reconstruction exhibits an apparent additional mode that may have been introduced by regularization. Can statistical inference determine whether this feature corresponds to a genuine mode of the underlying density and whether its significance warrants rejecting the smooth reconstruction?

The source does not define the comparison smooth reconstruction, how a reconstructed mode is matched to a true mode, what makes a mode additional, or what rejection means statistically. In particular, it supplies no null and alternative hypotheses, significance level, effect-size criterion, or error guarantee. Consequently, this is an unresolved formulation of a statistical research question, rather than a fully specified testing problem. No universal quantifier over observation models or regularization methods is justified by the supplied statement.

Short English statement

Can one determine whether an extra peak in a regularized one-dimensional density deconvolution is a genuine feature of the underlying density, with sufficient statistical evidence to reject a smoother reconstruction? The supplied record leaves the model, mode convention, comparison, and significance criterion unspecified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30002211$, title Detecting Genuine Modes in Density Deconvolution, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. The proposed year is recorded as 2012; no proposer is named.
• Supplied source citation: Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory (2013), Oberwolfach Reports, pp. 3061--3127, DOI 10.4171/owr/2012/51. https://doi.org/10.4171/owr/2012/51. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "multiscale shape inference in deconvolution": https://arxiv.org/abs/1107.1404.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2012/51.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved. It reports that multiscale confidence procedures can assess modes and qualitative features in one-dimensional deconvolution, while a general interpretability and significance theory for arbitrary regularized inverse reconstructions is unavailable. This is supplied dated curation derived from the OWR curation, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement exact and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification assertion is supplied provenance and has not been independently checked here.