Eventual prediction of mean-matrix diagonalizability
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For fixed dimension and an unknown distribution of iid binary random matrices, can one predict whether their mean is diagonalizable over R or C with only finitely many errors almost surely?
Why it matters and what counts as progress
Why it matters: It links an algebraic matrix property to eventual statistical predictability, beyond estimating individual matrix entries.
What counts as progress: An online predictor making finitely many diagonalizability mistakes almost surely for each fixed dimension and unknown binary-matrix distribution.
How the problem developed
- 2021 · Open problem [1]
Yihong Wu and Prasanna Santhanam Asked whether diagonalizability of the unknown mean of i.i.d. binary random matrices can be predicted with only finitely many errors almost surely.
- 11 Aug 2026 · Theorem [2]
Jinze Zhao Proved eventual prediction for fixed-dimensional bounded mean matrices and extended it to observations with a finite moment of order greater than one; showed failure for all merely integrable laws in dimension at least two.
AI contribution
Predicting Diagonalizability of a Mean Matrix
For fixed matrix dimension, whether the mean of i.i.d. binary random matrices is diagonalizable over R or C can be predicted with only finitely many errors almost surely. The argument uses eventual predictability of semialgebraic properties of bounded means.
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