# Covariate-assisted Sparse Tensor Completion

arXiv: https://arxiv.org/abs/2103.06428 · HTML: https://mlsanity.com/paper/2103.06428/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 64 references that are in the map.
- First posted on arXiv 11 Mar 2021.

## Builds on (most cited first, 30 of 32)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Tensor Completion Algorithms in Big Data Analytics](https://mlsanity.com/paper/1711.10105/) (2017)
- [Nonconvex Low-Rank Tensor Completion from Noisy Data](https://mlsanity.com/paper/1911.04436/) (2019)
- [Provable Tensor Factorization with Missing Data](https://mlsanity.com/paper/1406.2784/) (2014)
- [Community Detection on Mixture Multi-layer Networks via Regularized Tensor Decomposition](https://mlsanity.com/paper/2002.04457/) (2020)
- [Multilinear tensor regression for longitudinal relational data](https://mlsanity.com/paper/1412.0048/) (2014)
- [Spectral algorithms for tensor completion](https://mlsanity.com/paper/1612.07866/) (2016)
- [On Polynomial Time Methods for Exact Low Rank Tensor Completion](https://mlsanity.com/paper/1702.06980/) (2017)
- [Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entries](https://mlsanity.com/paper/1711.04934/) (2017)
- [Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank-1 Updates](https://mlsanity.com/paper/1402.5180/) (2014)
- [Parsimonious Tensor Response Regression](https://mlsanity.com/paper/1501.07815/) (2015)
- [Factor Models for High-Dimensional Tensor Time Series](https://mlsanity.com/paper/1905.07530/) (2019)
- [Provable Sparse Tensor Decomposition](https://mlsanity.com/paper/1502.01425/) (2015)
- [Multilayer tensor factorization with applications to recommender systems](https://mlsanity.com/paper/1711.01598/) (2017)
- [Optimal computational and statistical rates of convergence for sparse nonconvex learning problems](https://mlsanity.com/paper/1306.4960/) (2013)
- [Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality](https://mlsanity.com/paper/2006.08580/) (2020)
- [Dynamic Tensor Clustering](https://mlsanity.com/paper/1708.07259/) (2017)
- [Sparse and Low-rank Tensor Estimation via Cubic Sketchings](https://mlsanity.com/paper/1801.09326/) (2018)
- [Integrating multi-source block-wise missing data in model selection](https://mlsanity.com/paper/1901.03797/) (2019)
- [Incoherent Tensor Norms and Their Applications in Higher Order Tensor Completion](https://mlsanity.com/paper/1606.03504/) (2016)
- [Cross: Efficient Low-rank Tensor Completion](https://mlsanity.com/paper/1611.01129/) (2016)
- [Covariate-Adjusted Tensor Classification in High-Dimensions](https://mlsanity.com/paper/1805.04421/) (2018)
- [Spectral norm of random tensors](https://mlsanity.com/paper/1407.1870/) (2014)
- [Sparse Nonnegative Tensor Factorization and Completion with Noisy Observations](https://mlsanity.com/paper/2007.10626/) (2020)
- [TenIPS: Inverse Propensity Sampling for Tensor Completion](https://mlsanity.com/paper/2101.00323/) (2021)
- [High Dimensional Expectation-Maximization Algorithm: Statistical Optimization and Asymptotic Normality](https://mlsanity.com/paper/1412.8729/) (2014)
- [Optimal Sparse Singular Value Decomposition for High-dimensional High-order Data](https://mlsanity.com/paper/1809.01796/) (2018)
- [Effective Tensor Sketching via Sparsification](https://mlsanity.com/paper/1710.11298/) (2017)
- [Convex Coupled Matrix and Tensor Completion](https://mlsanity.com/paper/1705.05197/) (2017)
- [Partially Observed Dynamic Tensor Response Regression](https://mlsanity.com/paper/2002.09735/) (2020)

## Built on by (most cited first, 4 of 4)

- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Statistical Inference For Noisy Matrix Completion Incorporating Auxiliary Information](https://mlsanity.com/paper/2403.14899/) (2024)
- [Causal Customer Churn Analysis with Low-rank Tensor Block Hazard Model](https://mlsanity.com/paper/2405.11377/) (2024)
- [Tensor Completion using Subspace Information](https://mlsanity.com/paper/2609.24501/) (2026)

## Abstract (arXiv)

We aim to provably complete a sparse and highly-missing tensor in the presence of covariate information along tensor modes. Our motivation comes from online advertising where users click-through-rates (CTR) on ads over various devices form a CTR tensor that has about 96% missing entries and has many zeros on non-missing entries, which makes the standalone tensor completion method unsatisfactory. Beside the CTR tensor, additional ad features or user characteristics are often available. In this paper, we propose Covariate-assisted Sparse Tensor Completion (COSTCO) to incorporate covariate information for the recovery of the sparse tensor. The key idea is to jointly extract latent components from both the tensor and the covariate matrix to learn a synthetic representation. Theoretically, we derive the error bound for the recovered tensor components and explicitly quantify the improvements on both the reveal probability condition and the tensor recovery accuracy due to covariates. Finally, we apply COSTCO to an advertisement dataset consisting of a CTR tensor and ad covariate matrix, leading to 23% accuracy improvement over the baseline. An important by-product is that ad latent components from COSTCO reveal interesting ad clusters, which are useful for better ad targeting.