Bandeira et al. (2022) introduced the Franz-Parisi (FP) criterion for characterizing the computational hard phases in statistical detection problems. The FP criterion, based on an annealed version of the celebrated Franz-Parisi potential from statistical physics, was shown to be equivalent to low-degree polynomial (LDP) lower bounds for Gaussian additive models, thereby connecting two distinct approaches to understanding the computational hardness in statistical inference. In this paper, we propose a refined FP criterion that aims to better capture the geometric ``overlap structure of statistical models. Our main result establishes that this optimized FP criterion is equivalent to Statistical Query (SQ) lower bounds---another foundational framework in computational complexity of statistical inference. Crucially, this equivalence holds under a mild, verifiable assumption satisfied by a broad class of statistical models, including Gaussian additive models, planted sparse models, as well as non-Gaussian component analysis (NGCA), single-index (SI) models, and convex truncation detection settings. For instance, in the case of convex truncation tasks, the assumption is equivalent with the Gaussian correlation inequality (Royen, 2014) from convex geometry. In addition to the above, our equivalence not only unifies and simplifies the derivation of several known SQ lower bounds--such as for the NGCA model (Diakonikolas et al., 2017) and the SI model (Damian et al., 2024)--but also yields new SQ lower bounds of independent interest, including for the computational gaps in mixed sparse linear regression (Arpino et al., 2023) and convex truncation (De et al., 2023).

The proof of Claim 2.3 is obtained via the following calculation, using the definition of Hermite tensor (Definition 2.2). We will use i,j for indexes in [d]. The above is equivalent to Hk(Bx) = B kHk(x). We construct the truncated distribution A as follows. We first sample x A, then we reject x unless x 2 B. Let A be the distribution of the samples we get from this process. Using Markov's inequality and union bound, we have Then it only remains to verify Ex A [Hk(x)] Ex Nm[Hk(x)] 2 for any k < d.

We study the problem of learning mixtures of linear classifiers under Gaussian covariates.

We give superpolynomial statistical query (SQ) lower bounds for learning twohidden-layer ReLU networks with respect to Gaussian inputs in the standard (noise-free)model.

We study the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query (SQ) model. Prior work developed a general methodology to prove SQ lower bounds for this task that have been applicable to a wide range of contexts. In particular, it was known that for any univariate distribution A satisfying certain conditions, distinguishing between a standard multivariate Gaussian and a distribution that behaves like A in a random hidden direction and like a standard Gaussian in the orthogonal complement, is SQ-hard. The required conditions were that (1) A matches many low-order moments with the standard univariate Gaussian, and (2) the chi-squared norm of A with respect to the standard Gaussian is finite. While the moment-matching condition is necessary for hardness, the chi-squared condition was only required for technical reasons. In this work, we establish that the latter condition is indeed not necessary. In particular, we prove near-optimal SQ lower bounds for NGCA under the moment-matching condition only. Our result naturally generalizes to the setting of a hidden subspace. Leveraging our general SQ lower bound, we obtain near-optimal SQ lower bounds for a range of concrete estimation tasks where existing techniques provide sub-optimal or even vacuous guarantees.

The statistical-query model is a well-studied framework for analyzing the sample complexity of learning problems andcaptures mostknownlearning algorithms.

Our main result is a Statistical Query (SQ) lower bound suggesting that known algorithms for this problem are essentially best possible,even for the special case of uniform mixtures.In particular, we show that the complexity of any SQ algorithm for the problem is $n^{\mathrm{poly}(1/\Delta) \log(r)}$,where $\Delta$ is a lower bound on the pairwise $\ell_2$-separation between the $\mathbf{v}_{\ell}$'s.The key technical ingredient underlying our result is a new construction of spherical designs on the unit sphere that may be of independent interest.