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Title: All-or-Nothing Phenomena: From Single-Letter to High Dimensions
We consider the problem of estimating a $p$ -dimensional vector $\beta$ from $n$ observations $Y=X\beta+W$ , where $\beta_{j}\mathop{\sim}^{\mathrm{i.i.d}.}\pi$ for a real-valued distribution $\pi$ with zero mean and unit variance’ $X_{ij}\mathop{\sim}^{\mathrm{i.i.d}.}\mathcal{N}(0,1)$ , and $W_{i}\mathop{\sim}^{\mathrm{i.i.d}.}\mathcal{N}(0,\ \sigma^{2})$ . In the asymptotic regime where $n/p\rightarrow\delta$ and $p/\sigma^{2}\rightarrow$ snr for two fixed constants $\delta,\ \mathsf{snr}\in(0,\ \infty)$ as $p\rightarrow\infty$ , the limiting (normalized) minimum mean-squared error (MMSE) has been characterized by a single-letter (additive Gaussian scalar) channel. In this paper, we show that if the MMSE function of the single-letter channel converges to a step function, then the limiting MMSE of estimating $\beta$ converges to a step function which jumps from 1 to 0 at a critical threshold. Moreover, we establish that the limiting mean-squared error of the (MSE-optimal) approximate message passing algorithm also converges to a step function with a larger threshold, providing evidence for the presence of a computational-statistical gap between the two thresholds.  more » « less
Award ID(s):
1750362 1718494
NSF-PAR ID:
10139968
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
Page Range / eLocation ID:
654 to 658
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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