On ℓp-Gaussian–Grothendieck Problem

For $p\geq 1$ and $(g_{ij})_{1\leq i,j\leq n}$ being a matrix of i.i.d. standard Gaussian entries, we study the $n$-limit of the $\ell _p$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $p=\infty $, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $1\leq p<2$ and $2<p<\infty .$ For the former, we compute the limit of the $\ell _p$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $n^{-1}$.

The Free Energy of a Quantum Sherrington–Kirkpatrick Spin-Glass Model for Weak Disorder

We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington–Kirkpatrick spin-glass model without external magnetic field to the quantum case with a “transverse field” of strength $$\mathsf {b}$$ b . More precisely, if the Gaussian disorder is weak in the sense that its standard deviation $$\mathsf {v}>0$$ v > 0 is smaller than the temperature $$1/\beta $$ 1 / β , then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any $$\mathsf {b}/\mathsf {v}\ge 0$$ b / v ≥ 0 . The macroscopic annealed free energy turns out to be non-trivial and given, for any $$\beta \mathsf {v}>0$$ β v > 0 , by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For $$\beta \mathsf {v}<1$$ β v < 1 we determine this minimum up to the order $$(\beta \mathsf {v})^{4}$$ ( β v ) 4 with the Taylor coefficients explicitly given as functions of $$\beta \mathsf {b}$$ β b and with a remainder not exceeding $$(\beta \mathsf {v})^{6}/16$$ ( β v ) 6 / 16 . As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong $$\beta \mathsf {b}$$ β b -dependence even to lowest order. Our main tool for dealing with the non-commutativity of the spin-operator components is a probabilistic representation of the Boltzmann–Gibbs operator by a Feynman–Kac (path-integral) formula based on an independent collection of Poisson processes in the positive half-line with common rate $$\beta \mathsf {b}$$ β b . Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.

Deep Boltzmann Machines: Rigorous Results at Arbitrary Depth

A class of deep Boltzmann machines is considered in the simplified framework of a quenched system with Gaussian noise and independent entries. The quenched pressure of a K-layers spin glass model is studied allowing interactions only among consecutive layers. A lower bound for the pressure is found in terms of a convex combination of K Sherrington–Kirkpatrick models and used to study the annealed and replica symmetric regimes of the system. A map with a one-dimensional monomer–dimer system is identified and used to rigorously control the annealed region at arbitrary depth K with the methods introduced by Heilmann and Lieb. The compression of this high-noise region displays a remarkable phenomenon of localisation of the processing layers. Furthermore, a replica symmetric lower bound for the limiting quenched pressure of the model is obtained in a suitable region of the parameters and the replica symmetric pressure is proved to have a unique stationary point.