A Tensor Critical Threshold in Cubic Spherical Kernel Graphs
Abstract
We identify a strong-detection transition within the critical scale for testing cubic spherical kernel graphs against G(n,1/2). Let X1,…,Xn be independent uniform points on 𝕊d−1. Conditional on these points, edges are independent, with connection probability (1+βZ3,d(⟨ Xi,Xj⟩))/2. Here Z3,d is the normalized cubic zonal harmonic, with multiplicity N3,d∼ d3/6. Assume n→∞, d=dn→∞, β=βn with |β|≤1, and n≤ dA, with A<∞ fixed. Write rn=nβ2/(N3,dd). On the critical scale rn→ r∈(0,∞), we identify the tensor variational threshold rtens :=min0<q<1(−2{log(1−q)+q})/q6. For 0<r<rtens, the log-likelihood ratio under the null admits an asymptotically Gaussian expansion in the signed triangle, and the total-variation distance converges to 2Φ(r3/2/2)−1, where Φ is the standard normal distribution function. For r>rtens, the likelihood-ratio test is strongly consistent, although the centered signed triangle still has Gaussian fluctuations. In the strict regimes rn→0 and rn→∞, total variation tends to zero and one, respectively, even when β→0.
Related
- Sharp detection and triangle likelihood limits for quadratic spherical kernel graphs
- Critical tensor covariance at the Marchenko-Pastur threshold
- Optimal detection of geometry in random graphs and kernel matrices at critical thresholds
- Infinite-Dimensional Spherical Kernel ridge Regression
- Bayesian Tensor Train Kernel Machines with Automatic Model Complexity Inference
BibTeX
@misc{lin2026tensor,
title = {{A Tensor Critical Threshold in Cubic Spherical Kernel Graphs}},
author = {Ziqian Lin and Huangang Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24772},
eprint = {2610.24772},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24772},
}
RIS
TY - UNPB
TI - A Tensor Critical Threshold in Cubic Spherical Kernel Graphs
AU - Ziqian Lin
AU - Huangang Lu
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24772
UR - https://arxiv.org/abs/2610.24772
AB - We identify a strong-detection transition within the critical scale for testing cubic spherical kernel graphs against G(n,1/2). Let X_1,…,X_n be independent uniform points on 𝕊^d−1. Conditional on these points, edges are independent, with connection probability (1+βZ_3,d(⟨ X_i,X_j⟩))/2. Here Z_3,d is the normalized cubic zonal harmonic, with multiplicity N_3,d∼ d^3/6. Assume n→∞, d=d_n→∞, β=β_n with |β|≤1, and n≤ d^A, with A<∞ fixed. Write r_n=nβ^2/(N_3,dd). On the critical scale r_n→ r∈(0,∞), we identify the tensor variational threshold r_tens :=min_0<q<1(−2{log(1−q)+q})/q^6. For 0<r<r_tens, the log-likelihood ratio under the null admits an asymptotically Gaussian expansion in the signed triangle, and the total-variation distance converges to 2Φ(r^3/2/2)−1, where Φ is the standard normal distribution function. For r>r_tens, the likelihood-ratio test is strongly consistent, although the centered signed triangle still has Gaussian fluctuations. In the strict regimes r_n→0 and r_n→∞, total variation tends to zero and one, respectively, even when β→0.
ER -