While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a bit with it, I decided to bring it to the attention of the MO community.

Among other questions, Recht and Re (RR) **[1]** pose the following *Arithmetic-Mean, Geometric-Mean* conjecture concerning symmetrized products:

RR.Let $A_1,\ldots,A_n$ be symmetric positive definite matrices. Let $\pi \in \mathfrak{S}_n$, where $\mathfrak{S}_n$ denotes the symmetric group. Let $P_{\pi} := \prod_{i=1}^n A_{\pi(i)}$. Then, $$\begin{equation*} \left\|\frac{1}{n!}\sum_{\pi \in \mathfrak{S}_n} P_{\pi}\right\| \le \left\|\left(\frac{1}{n}\sum_{i=1}^n A_i\right)^n\right\|, \end{equation*}$$ where $\|\cdot\|$ denotes the spectral norm.

Playing around a bit, it seems that in fact, the following much stronger conjecture should be true, but as of now, I don't know how to prove it---so I'm putting it out here. Hopefully, someone will be able to suggest to me potential areas of math where I could go to further educate myself about related inequalities.

(S., 2011).Let $\sigma(\cdot)$ denote the singular value map, i.e., the map that associates to a matrix the vector of its singular values (arranged in decreasing order). And let $\prec_w$ denoteweak majorization. Then, $$\begin{equation*} \sigma^{1/n}\left(\frac{1}{n!}\sum_{\pi \in \mathfrak{S}_n} P_{\pi}\right) \prec_w \sigma\left(\frac{1}{n}\sum_{i=1}^n A_i\right). \end{equation*}$$

**Notes.**

(a) The two matrix version of the my conjecture follows immediately from a stronger conjecture of Bhatia and Kittaneh that was actually recently resolved in Ref. [2].

(b) It seems that for random matrices, Recht and Re proved their conjecture (because everything just disentangles easily); see cited paper for details.

**REFERENCES**

**[1].** B. Recht and C. Re (2012). Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences.

**[2].** S. W. Drury, On a question of Bhatia and Kittaneh, Linear Algebra and its Applications, 437 (2012) pp. 1955--1960.