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October 3, 2025Open Access

Bias-variance Tradeoff in Tensor Estimation

Key Points

The study reveals that increasing the ranks of the estimator leads to lower bias but higher variance in the tensor estimation.
Achieving a bias-variance tradeoff allows for better approximation of the ground-truth tensor while controlling noise levels.
The bias term indicates achievable approximation errors, while the variance term accounts for model complexity related to free parameters.
Our method provides a bias-variance decomposition applicable to both higher-order tensors and standard low-rank matrix estimators.

Abstract

We study denoising of a third-order tensor when the ground-truth tensor is not necessarily Tucker low-rank. Specifically, we observe Y=X^+Z R^p₁ p₂ p₃, where X^ is the ground-truth tensor, and Z is the noise tensor. We propose a simple variant of the higher-order tensor SVD estimator X. We show that uniformly over all user-specified Tucker ranks (r₁, r₂, r₃), \| X - X^* \| ₅² = O (κ² \ r₁r₂r₃+₊=₁^{3 p₊ r₊ \} \; + \; ξ (ₑ_₁, r₂, r₃) ²) with high probability. Here, the bias term ξ (ₑ䃑, ₑ䃒, ₑ䃓) corresponds to the best achievable approximation error of X^ over the class of tensors with Tucker ranks (r₁, r₂, r₃) ; κ² quantifies the noise level; and the variance term κ² \r₁r₂r₃+₊=₁^{3 p₊ r₊\} scales with the effective number of free parameters in the estimator X. Our analysis achieves a clean rank-adaptive bias--variance tradeoff: as we increase the ranks of estimator X, the bias ξ (r₁, r₂, r₃) decreases and the variance increases. As a byproduct we also obtain a convenient bias-variance decomposition for the vanilla low-rank SVD matrix estimators.

Bias-variance Tradeoff in Tensor Estimation

Key Points

Abstract

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