Sharp dynamic emergence thresholds for XOR/multilayer GMM classification

Let $u,v\in\mathbb R^d$ be orthonormal unit vectors, let $m>0$, and sample a hidden sign pair $S=(s_1,s_2)\in\{\pm1\}^2$ uniformly. Define

The observed class label is the XOR label $Y=s_1s_2\in\{\pm1\}$ (equivalently, one class has $s_1=s_2$ and the other has $s_1\ne s_2$). Work in proportional asymptotics $n,d\to\infty$ with $n/d\to\phi\in(0,\infty)$.

Use a two-layer classifier with fixed hidden width $K=O(1)$ and activation $g$:

Train by online SGD with step size $\eta$ on fresh samples. The role of fixed width is that $K$ does not scale with $d,n$, so the summary-statistic dimension remains finite; this is exactly what allows a finite-dimensional effective spectral description in the high-dimensional limit.

Let $G(x)$ be the summary-statistic Gram matrix built from first-layer weights and class means, and define the effective trajectory

As shown in Ben Arous et al. (2025), $G_t$ solves a finite-dimensional autonomous ODE $dG_t/dt=\mathsf F(G_t)$, and the first-layer Hessian/Gradient spectra at time $t$ are approximated by deterministic objects depending only on $(G_t,\beta)$.

For the Hessian block under study, let $\nu_{G_t,\beta}^H$ be the effective bulk measure (defined via its Stieltjes fixed-point equation) and define its right spectral edge by

Effective outliers are then the real roots outside the bulk of the finite-dimensional equation

where $q=K+k$ in the source corollary notation and $F^H$ is an explicit $q\times q$ matrix-valued function defined by Gaussian expectations. Equivalently, with

outliers solve $\det M(\lambda;G_t,\beta)=0$ for $\lambda>\lambda_+(t,\beta)$.