SS–RS–GD matrix-product inequalities
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Do the proposed near-identity operator-norm comparisons between successive-squaring (SS), repeated-squaring (RS) and geometric-dilation (GD) matrix products hold? The SS–RS and RS–GD inequalities are separate statements.
Why it matters and what counts as progress
Why it matters: The comparisons test ordering effects in products of closely conditioned positive definite matrices.
What counts as progress: A proof under an explicit spectral neighborhood of the identity, or counterexamples arbitrarily close to it; a result for one inequality does not settle the other.
How the problem developed
- 15 Mar 2021 · Conjecture [1]
Soham Yun, Suvrit Sra and Ali Jadbabaie Formulated near-identity conjectures comparing the successive-squaring, repeated-squaring and geometric-dilation matrix products.
- 16 Jun 2026 · Counterexample and theorem [2]
Zhengpeng Peng Refuted the SS–RS inequality with arbitrarily near-identity examples at n=3, K=2, d=4, and proved RS–GD under an explicit near-identity bound for all n, K and d.
AI contribution
A Resolution of the SS--RS--GD Inequalities
For n=3, K=2 and d=4, Peng gives positive definite symmetric matrices arbitrarily close to the identity with ||W_SS|| > ||W_RS||, refuting the SS–RS half. For all n≥2, K≥1 and d≥1, the RS–GD half holds when (1−1/(4n²+1))I ≤ Aᵢ ≤ I.
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