We study the projected power method for permutation synchronization: recovering $n$ unknown permutations of $m$ objects from pairwise relative permutations, observed independently with probability $p$ and independently corrupted by uniform permutations with probability $1-π_0$. Our main theorem establishes uniform one-step contraction for all estimates with optimally aligned block-error fraction at most $1/2-ε$, for fixed $ε>0$, including measurement-dependent estimates. When $np\ge C_0\log(n)$ and $m=o(npπ_0^2)$, one update contracts the error by a factor $O(m/(npπ_0^2))$, up to a floor of $O(e^{-c\min\{m\,npπ_0^2,\,npπ_0\}} + \log(n) / n)$. Thus, every initialization in this majority basin achieves $o(1)$ error after one update and maintains it uniformly over subsequent iterations. A reference-block spectral initializer enters the basin with probability $1-o(1)$, yielding end-to-end almost-exact recovery. If additionally $\min\{m\,npπ_0^2,\,npπ_0\} \ge C\log(n)$ for sufficiently large $C$, the method achieves exact recovery in finitely many iterations. We also establish prescribed-block recovery under weaker conditions for measurement-independent estimates and extend the guarantees to independent permutation-valued corruptions with uniform mean and to partial permutations with common visible support.
Robust subspace estimation is fundamental to many machine learning and data analysis tasks. Iteratively Reweighted Least Squares (IRLS) is an elegant and empirically effective approach to this problem, yet its theoretical properties remain poorly understood. This paper establishes that, under deterministic conditions, a variant of IRLS with dynamic smoothing regularization converges linearly to the underlying subspace from any initialization. We extend these guarantees to affine subspace estimation, a setting that lacks prior recovery theory. Additionally, we illustrate the practical benefits of IRLS through an application to low-dimensional neural network training. Our results provide the first global convergence guarantees for IRLS in robust subspace recovery and, more broadly, for nonconvex IRLS on a Riemannian manifold.
There is growing experimental evidence that $Q$-learning agents can learn to charge supracompetitive prices. We ask when such behavior persists, whether it emerges, and which strategies support it. In a repeated pricing game in which firms condition on the previous period's price profile, we give conditions on the learned action values under which supracompetitive prices become an absorbing state once experimentation ends. We then show that under softmax exploration these conditions are reached with positive and explicitly bounded probability, both from a random and from an all-zero initial $Q$-table. In the latter case firms lock in supracompetitive prices because they never learn that undercutting pays; the same mechanism can lock in the competitive price, so the long-run outcome is selected by the exploration path. Finally, we give sufficient conditions on the learned table under which the induced strategy is naive collusion, a grim trigger strategy or an increasing-price strategy; grim trigger strategies are subgame perfect under a standard discounting condition, whereas naive collusion is generally not an equilibrium.