Information-computation gap for planted linear regression

Determine whether planted linear regression has an information-computation gap in the regime d+1 \le m \ll d^{3/4}n^{1/4}, specifically whether recovering the coefficient vector \beta exactly or with error asymptotically smaller than the ordinary least-squares rate \sqrt{d/n} requires superpolynomial time despite being information-theoretically possible.

Background

Planted linear regression consists of Gaussian covariates X_i \sim N(0,I_d), with exactly m noiseless labels and the remaining labels having standard Gaussian noise. Once m \ge d+1, the true coefficient vector is information-theoretically identifiable because any hyperplane containing at least d+1 noiseless samples is almost surely the true hyperplane.

The paper gives a polynomial-time estimator above the threshold m \gg d{3/4}n{1/4} and an SQ lower bound below essentially the same threshold. The intermediate regime d+1 \le m \ll d{3/4}n{1/4} therefore remains the proposed computationally hard region, although exact recovery remains statistically possible.

References

In the regime $d + 1 \le m \ll d{3/4}n{1/4}$, recovering $\beta$ is information-theoretically possible, yet we know no polynomial-time algorithm; we conjecture that this is an information-computation gap, and we prove a statistical query (SQ) lower bound in support of this claim (\cref{thm:SQ}).

— Algorithms for adaptive and heteroskedastic linear regression at the computational threshold  (2608.18402 - Compton et al., 19 Aug 2026) in Section 1, paragraph “Planted linear regression”; also stated in the Abstract

The work of conjectures that any polynomial-time algorithm requires $m = \Omega(\sqrt{nd})$ to exactly recover $\beta$, and they provide Statistical Query (SQ) lower bounds to support their claim (the SQ lower bounds are not yet tight to their conjecture, this is left as an open problem).

— Algorithms for adaptive and heteroskedastic linear regression at the computational threshold  (2608.18402 - Compton et al., 19 Aug 2026) in Section 1, subsection “Related work,” paragraph “Oblivious contamination and noiseless linear regression”

The work of conjectures that any polynomial-time algorithm requires $m = \Omega(\sqrt{nd})$ to exactly recover $\beta$, and they provide Statistical Query (SQ) lower bounds to support their claim (the SQ lower bounds are not yet tight to their conjecture, this is left as an open problem).

— Algorithms for adaptive and heteroskedastic linear regression at the computational threshold  (2608.18402 - Compton et al., 19 Aug 2026) in Section 1, subsection “Related work,” paragraph “Oblivious contamination and noiseless linear regression”