Efficient reductions from a Gaussian source with applications to statistical-computational tradeoffs

• URL: http://arxiv.org/abs/2510.07250v1
• Date: Wed, 08 Oct 2025 17:16:36 GMT
• Title: Efficient reductions from a Gaussian source with applications to statistical-computational tradeoffs
• Authors: Mengqi Lou, Guy Bresler, Ashwin Pananjady,
• Abstract summary: We leverage our technique to establish reduction-based computational lower bounds for several canonical high-dimensional statistical models under widely-believed conjectures in average-case complexity.<n>We show that computational lower bounds proved for spiked tensor PCA with Gaussian noise are universal, in that they extend to other non-Gaussian noise distributions within our class.
• Score: 8.162867143465382
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Given a single observation from a Gaussian distribution with unknown mean $\theta$, we design computationally efficient procedures that can approximately generate an observation from a different target distribution $Q_{\theta}$ uniformly for all $\theta$ in a parameter set. We leverage our technique to establish reduction-based computational lower bounds for several canonical high-dimensional statistical models under widely-believed conjectures in average-case complexity. In particular, we cover cases in which: 1. $Q_{\theta}$ is a general location model with non-Gaussian distribution, including both light-tailed examples (e.g., generalized normal distributions) and heavy-tailed ones (e.g., Student's $t$-distributions). As a consequence, we show that computational lower bounds proved for spiked tensor PCA with Gaussian noise are universal, in that they extend to other non-Gaussian noise distributions within our class. 2. $Q_{\theta}$ is a normal distribution with mean $f(\theta)$ for a general, smooth, and nonlinear link function $f:\mathbb{R} \rightarrow \mathbb{R}$. Using this reduction, we construct a reduction from symmetric mixtures of linear regressions to generalized linear models with link function $f$, and establish computational lower bounds for solving the $k$-sparse generalized linear model when $f$ is an even function. This result constitutes the first reduction-based confirmation of a $k$-to-$k^2$ statistical-to-computational gap in $k$-sparse phase retrieval, resolving a conjecture posed by Cai et al. (2016). As a second application, we construct a reduction from the sparse rank-1 submatrix model to the planted submatrix model, establishing a pointwise correspondence between the phase diagrams of the two models that faithfully preserves regions of computational hardness and tractability.

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