The Ellipsoid Fitting Conjecture
Statement
Given n independent standard Gaussian vectors in R^d, an ellipsoid fit is a positive semidefinite S with x_i' S x_i = d for every i. Saunderson, Parrilo and Willsky conjectured that this semidefinite feasibility problem has a sharp threshold at n ~ d^2/4. Proved: below the threshold a fit exists with probability tending to one, above it none does.
Context
Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.
A conjecture with a decade of documented attack by multiple groups and a settled place in the literature on semidefinite programming and random geometry.
People
Attempts
No person has examined this. 1 attempt is published here and nothing has been checked against it at all. Saying whether the mathematics holds is the most useful thing anybody can do on this page.
The approach is the authors' own - they say so, and trace it to the dual formulation of Bandeira and Maillard. What the model did is named step by step: ChatGPT 5.4 and 5.5 were used "to explore several possible proof strategies", and then, "Given an earlier draft, GPT 5.6 helped repair and complete several arguments, including the tightened head-tail decomposition in Lemma 3.5 and the decomposition used in the proof of Proposition 4.4, which ultimately led to the completion of the proofs."
Reviews
No person has reviewed this attempt. It has not been checked at all.
Discussion of this attempt
no comments
Solve with an agent
Open the statement in a chat, with the problem and the ground rules already written into the prompt.
Discussion
Nothing has been said about this problem yet.
Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.