Regression with non-Euclidean responses ('random objects' such as probability distributions, networks, or covariance matrices measured in a metric space) is handled by global Fréchet regression, and its variable-selection counterpart, the Fréchet Ridge Selection Operator (FRISO) of Tucker–Wu–Müller as adopted and scaled up by Matabuena, currently rests on a weak asymptotic guarantee: the estimated per-covariate selection weights diverge in probability for relevant covariates and vanish in probability for irrelevant ones. The open problem, posed in Matabuena (2025, Remark 5) and rooted in Tucker–Wu–Müller (2021), is to upgrade this in-probability separation to an almost-sure selection guarantee; the technical route identified by the source is a characterization of the gradient of the Fréchet-regression objective function in the general non-Euclidean setting. This core remains open: closed-form gradients and Hessians are now available for distributional responses, and several newly proposed alternative estimators achieve asymptotic selection consistency in probability, but no work establishes almost-sure selection guarantees for Fréchet variable selection. Related theoretical directions that the source does not pose as part of the open problem — non-asymptotic finite-sample false-discovery control, extension beyond the standing convex-loss/strong-negative-type/uniqueness framework, and rigorous post-selection inference — likewise remain undeveloped; they are recorded below as background only.

Background

Many modern biomedical data sources record outcomes that are not numbers or vectors but complex objects: full probability distributions of continuous glucose-monitoring profiles, networks represented by graph Laplacians, covariance matrices, or functions. Such data are random objects, i.e., random variables taking values in a metric space (Y,d)(\mathcal{Y}, d) that carries no addition or scalar multiplication, so classical regression cannot be applied directly. Object-oriented data analysis addresses this through Fréchet regression. The Fréchet mean generalizes the expectation to metric spaces: M(Y)=arg⁡min⁡y∈YE[d2(Y,y)]M(Y)=\arg\min_{y\in\mathcal{Y}} \mathbb{E}[d^{2}(Y,y)]. Global Fréchet regression (Petersen and Müller 2019) models the conditional Fréchet mean m(x)=arg⁡min⁡y∈YE[d2(Y,y)∣X=x]m(x)=\arg\min_{y\in\mathcal{Y}}\mathbb{E}[d^{2}(Y,y)\mid X=x] of a metric-valued response YY given Euclidean predictors X∈RpX\in\mathbb{R}^{p}: fitted values are metric-weighted Fréchet means of the observed responses, with weights built from the usual linear-regression geometry. Because the model involves no coefficient vector, the standard machinery of high-dimensional variable selection, which shrinks regression coefficients, does not transfer.

Tucker, Wu and Müller (2021) proposed the first variable-selection method for this model, the Fréchet Ridge Selection Operator (FRISO). Each covariate jj receives a nonnegative weight λj\lambda_j with ∑jλj=τ\sum_{j}\lambda_j=\tau; the fitted value at a point x\mathbf{x} is R^⊕(x;λ−1)=arg⁡min⁡y∈Y∑i=1nwi(x;λ) d2(Yi,y)\widehat{R}_{\oplus}(\mathbf{x};\lambda^{-1})=\arg\min_{y\in\mathcal{Y}}\sum_{i=1}^{n} w_{i}(\mathbf{x};\lambda)\,d^{2}(Y_{i},y), where the weights wiw_{i} are ridge-type quadratic functions of the λ\lambda-scaled covariates, and the total weight budget τ\tau is tuned by validation. The selected set is I^(τ)={j:λ^j(τ)>0}\widehat{\mathcal{I}}(\tau)=\{j:\widehat{\lambda}_{j}(\tau)>0\}. Matabuena (2025) recast this ridge-type criterion inside a scalable best-subset/dual-subgradient optimization framework, extending it to univariate, multivariate, functional, and metric-valued responses (including random graphs) with runtimes reaching n=100,000n=100{,}000 in minutes, and proved the two approaches equivalent.

The available statistical theory for this selection route is weak. The central result (Theorem 5 of Matabuena, imported from Tucker–Wu–Müller) states only that, under regularity conditions and with τ=τn→∞\tau=\tau_{n}\to\infty, the weights satisfy λ^j(τn)→p∞\widehat{\lambda}_{j}(\tau_{n})\xrightarrow{p}\infty for relevant covariates j∈Ij\in\mathcal{I} — where I⊆{1,…,p}\mathcal{I}\subseteq\{1,\dots,p\} denotes the true relevant set, the covariates on which the conditional Fréchet mean m(x)m(x) genuinely depends — and λ^j′(τn)→p0\widehat{\lambda}_{j'}(\tau_{n})\xrightarrow{p}0 for irrelevant j′∉Ij'\notin\mathcal{I}. This in-probability separation provides no convergence rates, no almost-sure statements, and no control of false discoveries at finite sample sizes; moreover, later work notes it does not even imply correct recovery of the support. The theory is also restricted in scope: convex loss functions, and responses lying in convex subsets of separable Hilbert spaces or in metric spaces of strong negative type — spaces whose distance itself admits a squared-Hilbert-space representation, d(y,y′)=∥ϕ(y)−ϕ(y′)∥H2d(y,y')=\|\phi(y)-\phi(y')\|^{2}_{\mathcal{H}} for a map ϕ\phi into a separable Hilbert space (Schoenberg's characterization), with the qualifier "strong" additionally requiring that the induced energy distance separates distinct probability measures on Y\mathcal{Y}; the space of univariate probability distributions equipped with the 2-Wasserstein metric, where d(μ,ν)=∥Qμ−Qν∥L2d(\mu,\nu)=\|Q_{\mu}-Q_{\nu}\|_{L^{2}} in terms of the quantile functions Qμ,QνQ_{\mu},Q_{\nu}, is the canonical example — together with uniqueness of the conditional Fréchet mean. Strengthening the guarantee is hard because the objective is a highly complex, generally non-convex function of λ\lambda: closed-form derivative expressions derived for distributional responses (Coulter et al. 2024) show the Hessian can have negative eigenvalues. For inferential use at scale, the source points to stability selection (selecting covariates by their subsampling selection frequencies, for which generic bounds on the expected number of false selections exist) and grid-search tuning, but develops no rigorous post-selection inference; post-selection inference here means valid uncertainty quantification — tests or confidence statements with proven error control — for the covariates selected by the data-driven procedure, a problem already delicate in Euclidean regression.

Problem Statement

The problem is to strengthen the selection guarantee for variable selection in global Fréchet regression with metric-space-valued responses, as posed by Matabuena (2025, Remark 5) following Tucker–Wu–Müller (2021).

The central result (Theorem 5 of Matabuena, imported from Tucker–Wu–Müller) states that, under its regularity conditions and with τ=τn→∞\tau=\tau_{n}\to\infty, the FRISO solution λ^(τn)\widehat{\lambda}(\tau_{n}) of the ridge-type selection objective satisfies λ^j(τn)→p∞\widehat{\lambda}_{j}(\tau_{n})\xrightarrow{p}\infty for relevant covariates j∈Ij\in\mathcal{I} — where I⊆{1,…,p}\mathcal{I}\subseteq\{1,\dots,p\} denotes the true relevant set, the covariates on which the conditional Fréchet mean m(x)m(x) genuinely depends — and λ^j′(τn)→p0\widehat{\lambda}_{j'}(\tau_{n})\xrightarrow{p}0 for irrelevant j′∉Ij'\notin\mathcal{I}, the selected set being I^(τ)={j:λ^j(τ)>0}\widehat{\mathcal{I}}(\tau)=\{j:\widehat{\lambda}_{j}(\tau)>0\}. Remark 5 observes that this in-probability separation is only a weak conclusion and that a stronger result, such as attaining zero almost surely, would require characterizing the gradient of the objective function, which is very challenging for Fréchet regression due to the general non-Euclidean setting.

The open problem is to upgrade this in-probability separation to an almost-sure selection guarantee for the ridge-type (FRISO) selection weights — equivalently, for the best-subset/dual-subgradient formulation of Matabuena (2025), which its Proposition 4 proves equivalent: establish that, under explicitly stated regularity conditions and τ=τn→∞\tau=\tau_{n}\to\infty, the irrelevant weights attain zero almost surely, λ^j′(τn)=0\widehat{\lambda}_{j'}(\tau_{n})=0 for all sufficiently large nn almost surely for every irrelevant j′∉Ij'\notin\mathcal{I} (the strengthening explicitly contemplated by Remark 5), and the relevant weights diverge, λ^j(τn)→∞\widehat{\lambda}_{j}(\tau_{n})\to\infty almost surely for every relevant j∈Ij\in\mathcal{I}; together these give almost-sure selection consistency, I^(τn)=I\widehat{\mathcal{I}}(\tau_{n})=\mathcal{I} for all sufficiently large nn almost surely. The mere almost-sure vanishing λ^j′(τn)→0\widehat{\lambda}_{j'}(\tau_{n})\to 0 is strictly weaker and insufficient here, because the selected set I^(τ)={j:λ^j(τ)>0}\widehat{\mathcal{I}}(\tau)=\{j:\widehat{\lambda}_{j}(\tau)>0\} excludes an irrelevant covariate only when its weight is exactly zero, not merely when it vanishes.

A solution consists of precise theorems with complete proofs of such an almost-sure selection guarantee, with explicitly stated regularity conditions and a demonstration that the conditions are non-vacuous and cover the standing framework of Theorem 5 as a whole (convex losses; responses in convex subsets of separable Hilbert spaces or in metric spaces of strong negative type; unique conditional Fréchet means), rather than merely a single special instance of that framework. The empirical benchmarking of scalability at biobank scale acknowledged by the source is an engineering exercise and is not part of the completion boundary. Related directions that the source does not pose as part of this problem — non-asymptotic finite-sample false-discovery control, convergence rates, delineation of admissible losses and response spaces beyond the standing framework (including settings with non-unique conditional Fréchet means), and rigorous post-selection inference for the selected support — are discussed in Background as context only.

References

  1. Marcos Matabuena (2025). Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age. arXiv:2501.06868, DOI 10.48550/arxiv.2501.06868. https://arxiv.org/abs/2501.06868
  2. Danielle C. Tucker, Yichao Wu, Hans-Georg Müller (2021). Variable Selection for Global Fréchet Regression. Journal of the American Statistical Association 118(542):1023–1037 (online 2021; print issue 2023), DOI 10.1080/01621459.2021.1969240. https://doi.org/10.1080/01621459.2021.1969240
  3. Alexander Petersen, Hans-Georg Müller (2019). Fréchet regression for random objects with Euclidean predictors. The Annals of Statistics 47(2):691–719, DOI 10.1214/17-AOS1624. https://doi.org/10.1214/17-AOS1624
  4. Alexander Coulter, Rashmi N. Aurora, Naresh M. Punjabi, Irina Gaynanova (2024). Fast variable selection for distributional regression with application to continuous glucose monitoring data. arXiv:2403.00922, DOI 10.48550/arxiv.2403.00922. https://arxiv.org/abs/2403.00922
  5. Haoyi Yang, Satarupa Bhattacharjee, Lingzhou Xue, Bing Li (2025). Variable Selection for Additive Global Fréchet Regression. arXiv:2509.13685, DOI 10.48550/arxiv.2509.13685. https://arxiv.org/abs/2509.13685
  6. Huanlin Li, Lu Lin (2026). Adaptive feature-weighted variable selection for Fréchet regression. Statistical Papers 67(4), DOI 10.1007/s00362-026-01857-4. https://doi.org/10.1007/s00362-026-01857-4
  7. Yushen Dong, Youhan Lu, Yichao Wu, Hans-Georg Müller (2026). Variable selection for Nadaraya–Watson–Fréchet regression. Journal of Nonparametric Statistics (online 2026, pp. 1–26), DOI 10.1080/10485252.2026.2671048. https://doi.org/10.1080/10485252.2026.2671048
  8. Wookyeong Song, Paromita Dubey, Hans-Georg Müller, Alexander Petersen (2026). Inference for Fréchet Regression. arXiv:2605.19519, DOI 10.48550/arxiv.2605.19519. https://arxiv.org/abs/2605.19519
  9. Gábor Lugosi, Marcos Matabuena (2024). Uncertainty quantification in metric spaces. arXiv:2405.05110, DOI 10.48550/arxiv.2405.05110. https://arxiv.org/abs/2405.05110
  10. Lei Yan, Xin Zhang, Zhou Lan, Dipankar Bandyopadhyay, Yichao Wu, and the Alzheimer's Disease Neuroimaging Initiative (2025). Variable screening and spatial smoothing in Fréchet regression with application to diffusion tensor imaging. Annals of Applied Statistics 19(1), DOI 10.1214/24-AOAS1978. https://doi.org/10.1214/24-AOAS1978