Quaternion Gaussian Probes for Tensor Frobenius-Norm Estimation
Abstract
Large-scale tensors arising in science and engineering are often too large to form or store explicitly. Their multilinear contractions, however, can frequently be evaluated efficiently, which makes randomized Frobenius-norm estimation a practical alternative to exact computation. In this paper, we study estimation of the squared Frobenius norm of a real order-d tensor from such contractions using independent real, complex, or quaternion Gaussian probes. For a rank-one tensor, one normalized sample is a product of d−1 independent Gamma variables. This gives exact relative-variance factors 3d−1−1, 2d−1−1, and (3/2)d−1−1, respectively. Under the same budget of underlying real Gaussian vectors, the quaternion probe has smaller rank-one variance than four independent real probes for d≥3; a real-channel representation identifies the gain as modewise pooling. For orthogonal CP tensors, the exact relative variance equals the corresponding rank-one factor divided by the participation-ratio effective rank and the number of samples. For arbitrary real tensors, we prove sharp uniform moment bounds, attained at rank one, and obtain Chebyshev guarantees. A classical block-median construction then gives two-sided exponential confidence for the modified estimator. Numerical examples test the rank-one and effective-rank formulas. An implicit infinite-series kernel tensor illustrates contraction-based norm evaluation; EEG maps provide a qualitative spatial interpolation illustration.
Related
- A Fast Algorithm for Quaternion Tensor Singular Value Decomposition under the QT-Product with Applications to Color Video Processing
- A Fast Algorithm for Quaternion Tensor Singular Value Decomposition under the T-product
- Hutch#: Optimal non-adaptive Frobenius norm estimation
- Efficient estimation methods for non-Gaussian regression models in continuous time
- Federated Estimation of Loading Spaces in Tensor Factor Models under Differential Privacy
BibTeX
@misc{chen2026quaternion,
title = {{Quaternion Gaussian Probes for Tensor Frobenius-Norm Estimation}},
author = {Yannan Chen and Hailin Sun and Min Xi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12534},
eprint = {2610.12534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12534},
}
RIS
TY - UNPB
TI - Quaternion Gaussian Probes for Tensor Frobenius-Norm Estimation
AU - Yannan Chen
AU - Hailin Sun
AU - Min Xi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12534
UR - https://arxiv.org/abs/2610.12534
AB - Large-scale tensors arising in science and engineering are often too large to form or store explicitly. Their multilinear contractions, however, can frequently be evaluated efficiently, which makes randomized Frobenius-norm estimation a practical alternative to exact computation. In this paper, we study estimation of the squared Frobenius norm of a real order-d tensor from such contractions using independent real, complex, or quaternion Gaussian probes. For a rank-one tensor, one normalized sample is a product of d−1 independent Gamma variables. This gives exact relative-variance factors 3^d−1−1, 2^d−1−1, and (3/2)^d−1−1, respectively. Under the same budget of underlying real Gaussian vectors, the quaternion probe has smaller rank-one variance than four independent real probes for d≥3; a real-channel representation identifies the gain as modewise pooling. For orthogonal CP tensors, the exact relative variance equals the corresponding rank-one factor divided by the participation-ratio effective rank and the number of samples. For arbitrary real tensors, we prove sharp uniform moment bounds, attained at rank one, and obtain Chebyshev guarantees. A classical block-median construction then gives two-sided exponential confidence for the modified estimator. Numerical examples test the rank-one and effective-rank formulas. An implicit infinite-series kernel tensor illustrates contraction-based norm evaluation; EEG maps provide a qualitative spatial interpolation illustration.
ER -