# Detection limits in the high-dimensional spiked rectangular model

arXiv: https://arxiv.org/abs/1802.07309 · HTML: https://mlsanity.com/paper/1802.07309/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2018.

## Built on by (most cited first, 8 of 8)

- [Homophily modulates double descent generalization in graph convolution networks](https://mlsanity.com/paper/2212.13069/) (2022)
- [From Spikes to Heavy Tails: Unveiling the Spectral Evolution of Neural Networks](https://mlsanity.com/paper/2406.04657/) (2024)
- [Detection of Signal in the Spiked Rectangular Models](https://mlsanity.com/paper/2104.13517/) (2021)
- [Optimal thresholds and algorithms for a model of multi-modal learning in high dimensions](https://mlsanity.com/paper/2407.03522/) (2024)
- [Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices](https://mlsanity.com/paper/2203.00821/) (2022)
- [Subspace clustering in high-dimensions: Phase transitions & Statistical-to-Computational gap](https://mlsanity.com/paper/2205.13527/) (2022)
- [Detection problems in the spiked matrix models](https://mlsanity.com/paper/2301.05331/) (2023)
- [Spectral Rank Certification for Foundation Model Adapters](https://mlsanity.com/paper/2608.15351/) (2026)

## Abstract (arXiv)

We study the problem of detecting the presence of a single unknown spike in a rectangular data matrix, in a high-dimensional regime where the spike has fixed strength and the aspect ratio of the matrix converges to a finite limit. This setup includes Johnstone's spiked covariance model. We analyze the likelihood ratio of the spiked model against an "all noise" null model of reference, and show it has asymptotically Gaussian fluctuations in a region below---but in general not up to---the so-called BBP threshold from random matrix theory. Our result parallels earlier findings of Onatski et al. (2013) and Johnstone-Onatski (2015) for spherical spikes. We present a probabilistic approach capable of treating generic product priors. In particular, sparsity in the spike is allowed. Our approach is based on Talagrand's interpretation of the cavity method from spin-glass theory. The question of the maximal parameter region where asymptotic normality is expected to hold is left open. This region is shaped by the prior in a non-trivial way. We conjecture that this is the entire paramagnetic phase of an associated spin-glass model, and is defined by the vanishing of the replica-symmetric solution of Lesieur et al. (2015).