Small-SNR no-outlier regime and monotonicity in SNR

Fix integers $k,C\ge 1$, mixture weights $p_1,\dots,p_k>0$ with $\sum_{b=1}^k p_b=1$, a class map $c:[k]\to [C]$, and an aspect ratio $\phi\in(0,\infty)$. For each $d$, let class means $\mu_1,\dots,\mu_k\in\mathbb R^d$, collect $\mu=(\mu_1,\dots,\mu_k)\in\mathbb R^{d\times k}$, and let parameters $x=(x_1,\dots,x_C)\in\mathbb R^{d\times C}$. For SNR parameter $\beta>0$, data are generated by

with observed class label $c(B)\in[C]$. Define $q:=C+k$ and the summary Gram matrix

Call $G$ feasible if $G=(x,\mu)^\top(x,\mu)$ for some $(x,\mu)$ (equivalently, $G\succeq 0$ and realizable in some ambient dimension).

Fix a class index $\alpha\in[C]$. Let $\mathrm{sm}_\alpha(u):=\exp(u_\alpha)/\sum_{j=1}^C\exp(u_j)$ for $u\in\mathbb R^C$, and for $g\sim N(0,I_q)$ define

For $M\in\{H,I\}$, let $\nu_M(\beta,G)$ be the effective bulk law with Stieltjes transform $S_M$ solving

Define $F^M(z,G)\in\mathbb R^{q\times q}$ by

Effective outliers are real roots of $\det(zI_q-F^M(z,G))=0$; define the right-outlier set