# Heteroskedastic Canonical Polyadic Tensor Decomposition

arXiv: https://arxiv.org/abs/2610.00498 · HTML: https://mlsanity.com/paper/2610.00498/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 59 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [An Optimal Statistical and Computational Framework for Generalized Tensor Estimation](https://mlsanity.com/paper/2002.11255/) (2020)
- [Generalized Canonical Polyadic Tensor Decomposition](https://mlsanity.com/paper/1808.07452/) (2018)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence](https://mlsanity.com/paper/2105.13346/) (2021)
- [Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance](https://mlsanity.com/paper/2012.10249/) (2020)
- [On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors](https://mlsanity.com/paper/2107.09660/) (2021)
- [HePPCAT: Probabilistic PCA for Data with Heteroscedastic Noise](https://mlsanity.com/paper/2101.03468/) (2021)
- [Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity](https://mlsanity.com/paper/2106.11950/) (2021)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods](https://mlsanity.com/paper/2405.13912/) (2024)
- [Tensor Generalized Estimating Equations for Longitudinal Imaging Analysis](https://mlsanity.com/paper/1412.6592/) (2014)
- [Towards a Theoretical Analysis of PCA for Heteroscedastic Data](https://mlsanity.com/paper/1610.03595/) (2016)
- [Statistical Inference for Low-Rank Tensors: Heteroskedasticity, Subgaussianity, and Applications](https://mlsanity.com/paper/2410.06381/) (2024)
- [Near-Universal Multiplicative Updates for Nonnegative Einsum Factorization](https://mlsanity.com/paper/2602.02759/) (2026)

## Abstract (arXiv)

When minimizing the squared-error loss, the popular CP decomposition can be interpreted as parameter inference in a Gaussian model with a low-rank mean tensor and constant variance across the tensor entries. We introduce heteroskedastic-CP (HCP), which models entrywise variability with a non-constant, low-rank precision tensor, and develop an alternating block-coordinate ascent method to recover both the low-rank mean and precision tensors from noisy observations. Our procedure is computationally competitive, with the same leading-order factor-update complexity as CP-ALS. We demonstrate HCP on synthetic experiments and an EEG application.