# Spiked Covariance Estimation from Modulo-Reduced Measurements

arXiv: https://arxiv.org/abs/2110.01150 · HTML: https://mlsanity.com/paper/2110.01150/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 30 references that are in the map.
- First posted on arXiv 4 Oct 2021.

## Builds on (most cited first, 5 of 5)

- [On Unlimited Sampling and Reconstruction](https://mlsanity.com/paper/1905.03901/) (2019)
- [Unlimited Sampling from Theory to Practice: Fourier-Prony Recovery and Prototype ADC](https://mlsanity.com/paper/2105.05818/) (2021)
- [Above the Nyquist Rate, Modulo Folding Does Not Hurt](https://mlsanity.com/paper/1903.12289/) (2019)
- [Blind Modulo Analog-to-Digital Conversion](https://mlsanity.com/paper/2108.08937/) (2021)
- [Blind Unwrapping of Modulo Reduced Gaussian Vectors: Recovering MSBs from LSBs](https://mlsanity.com/paper/1901.10396/) (2019)

## Built on by (most cited first, 1 of 1)

- [Fold First, Detect Directly: Communication Symbol Detection Without Unfolding for Low-Bitrate Modulo-ADCs](https://mlsanity.com/paper/2609.11298/) (2026)

## Abstract (arXiv)

Consider the rank-1 spiked model: bfX=√nuxi bfu+ bfZ, where nu is the spike intensity, bfu∈mathbbSk-1 is an unknown direction and xi~ N(0,1),bfZ~ N(bf0,bfI). Motivated by recent advances in analog-to-digital conversion, we study the problem of recovering bfu∈ mathbbSk-1 from n i.i.d. modulo-reduced measurements bfY=[bfX]mod Delta, focusing on the high-dimensional regime (kgg 1). We develop and analyze an algorithm that, for most directions bfu and nu=poly(k), estimates bfu to high accuracy using n=poly(k) measurements, provided that Deltagtrsim √log k. Up to constants, our algorithm accurately estimates bfu at the smallest possible Delta that allows (in an information-theoretic sense) to recover bfX from bfY. A key step in our analysis involves estimating the probability that a line segment of length ≈√nu in a random direction bfu passes near a point in the lattice Delta mathbbZk. Numerical experiments show that the developed algorithm performs well even in a non-asymptotic setting.