A Tensor Critical Threshold in Cubic Spherical Kernel Graphs

Source abstract

We identify a strong-detection transition within the critical scale for testing cubic spherical kernel graphs against $G(n,1/2)$. Let $X_1,\ldots,X_n$ be independent uniform points on $\mathbb S^{d-1}$. Conditional on these points, edges are independent, with connection probability $$\frac{1+βZ_{3,d}(\langle X_i,X_j\rangle)}{2}.$$ Here $Z_{3,d}$ is the normalized cubic zonal harmonic, with multiplicity $N_{3,d}\sim d^3/6$. Assume $n\to\infty$, $d=d_n\to\infty$, $β=β_n$ with $|β|\le1$, and $n\le d^A$, with $A r_{\mathrm{tens}}$, the likelihood-ratio test is strongly consistent, although the centered signed triangle still has Gaussian fluctuations. In the strict regimes $r_n\to0$ and $r_n\to\infty$, total variation tends to zero and one, respectively, even when $β\to0$.