# Iterative Algorithm for Discrete Structure Recovery

arXiv: https://arxiv.org/abs/1911.01018 · HTML: https://mlsanity.com/paper/1911.01018/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2019.

## Built on by (most cited first, 21 of 21)

- [Optimal Full Ranking from Pairwise Comparisons](https://mlsanity.com/paper/2101.08421/) (2021)
- [An iterative clustering algorithm for the Contextual Stochastic Block Model with optimality guarantees](https://mlsanity.com/paper/2112.10467/) (2021)
- [Convergence and Recovery Guarantees of the K-Subspaces Method for Subspace Clustering](https://mlsanity.com/paper/2206.05553/) (2022)
- [Optimal Clustering in Anisotropic Gaussian Mixture Models](https://mlsanity.com/paper/2101.05402/) (2021)
- [Localization in 1D non-parametric latent space models from pairwise affinities](https://mlsanity.com/paper/2108.03098/) (2021)
- [Optimal detection of the feature matching map in presence of noise and outliers](https://mlsanity.com/paper/2106.07044/) (2021)
- [Optimal Clustering by Lloyd Algorithm for Low-Rank Mixture Model](https://mlsanity.com/paper/2207.04600/) (2022)
- [Optimal Orthogonal Group Synchronization and Rotation Group Synchronization](https://mlsanity.com/paper/2109.13491/) (2021)
- [Multiway Spherical Clustering via Degree-Corrected Tensor Block Models](https://mlsanity.com/paper/2201.07401/) (2022)
- [Seeded graph matching for the correlated Gaussian Wigner model via the projected power method](https://mlsanity.com/paper/2204.04099/) (2022)
- [SDP Achieves Exact Minimax Optimality in Phase Synchronization](https://mlsanity.com/paper/2101.02347/) (2021)
- [Universal Lower Bounds and Optimal Rates: Achieving Minimax Clustering Error in Sub-Exponential Mixture Models](https://mlsanity.com/paper/2402.15432/) (2024)
- [Consistency of Lloyd's Algorithm Under Perturbations](https://mlsanity.com/paper/2309.00578/) (2023)
- [Causal Customer Churn Analysis with Low-rank Tensor Block Hazard Model](https://mlsanity.com/paper/2405.11377/) (2024)
- [Matching Map Recovery with an Unknown Number of Outliers](https://mlsanity.com/paper/2210.13354/) (2022)
- [A Novel and Optimal Spectral Method for Permutation Synchronization](https://mlsanity.com/paper/2303.12051/) (2023)
- [Improved seeding strategies for k-means and k-GMM](https://mlsanity.com/paper/2506.21291/) (2025)
- [The Catastrophic Failure of The k-Means Algorithm in High Dimensions, and How Hartigan's Algorithm Avoids It](https://mlsanity.com/paper/2602.09936/) (2026)
- [Revisiting Instance-Optimal Cluster Recovery in the Labeled Stochastic Block Model](https://mlsanity.com/paper/2306.12968/) (2023)
- [VEC-SBM: Optimal Community Detection with Vectorial Edges Covariates](https://mlsanity.com/paper/2402.18805/) (2024)
- [Recovery Theory for Projected Power Iterations in Permutation Synchronization](https://mlsanity.com/paper/2609.09502/) (2026)

## Abstract (arXiv)

We propose a general modeling and algorithmic framework for discrete structure recovery that can be applied to a wide range of problems. Under this framework, we are able to study the recovery of clustering labels, ranks of players, signs of regression coefficients, cyclic shifts, and even group elements from a unified perspective. A simple iterative algorithm is proposed for discrete structure recovery, which generalizes methods including Lloyd's algorithm and the power method. A linear convergence result for the proposed algorithm is established in this paper under appropriate abstract conditions on stochastic errors and initialization. We illustrate our general theory by applying it on several representative problems: (1) clustering in Gaussian mixture model, (2) approximate ranking, (3) sign recovery in compressed sensing, (4) multireference alignment, and (5) group synchronization, and show that minimax rate is achieved in each case.