# Multi-Dictionary Tensor Decomposition

arXiv: https://arxiv.org/abs/2309.09717 · HTML: https://mlsanity.com/paper/2309.09717/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 42 references that are in the map.
- First posted on arXiv 18 Sep 2023.

## Builds on (most cited first, 11 of 11)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [A survey of sparse representation: algorithms and applications](https://mlsanity.com/paper/1602.07017/) (2016)
- [Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion](https://mlsanity.com/paper/1809.02288/) (2018)
- [Generalized Canonical Polyadic Tensor Decomposition](https://mlsanity.com/paper/1808.07452/) (2018)
- [Multi-way Graph Signal Processing on Tensors: Integrative analysis of irregular geometries](https://mlsanity.com/paper/2007.00041/) (2020)
- [Temporal Graph Signal Decomposition](https://mlsanity.com/paper/2106.13517/) (2021)
- [Robust Factorization of Real-world Tensor Streams with Patterns, Missing Values, and Outliers](https://mlsanity.com/paper/2102.08466/) (2021)
- [Automatic Unsupervised Tensor Mining with Quality Assessment](https://mlsanity.com/paper/1503.03355/) (2015)
- [Dictionary-based Tensor Canonical Polyadic Decomposition](https://mlsanity.com/paper/1704.00541/) (2017)
- [Spectral Unmixing with Multiple Dictionaries](https://mlsanity.com/paper/1711.02883/) (2017)

## Built on by (most cited first, 1 of 1)

- [Multi-Dictionary Learning for Low Rank Sparse Coding](https://mlsanity.com/paper/2509.10033/) (2025)

## Abstract (arXiv)

Tensor decomposition methods are popular tools for analysis of multi-way datasets from social media, healthcare, spatio-temporal domains, and others. Widely adopted models such as Tucker and canonical polyadic decomposition (CPD) follow a data-driven philosophy: they decompose a tensor into factors that approximate the observed data well. In some cases side information is available about the tensor modes. For example, in a temporal user-item purchases tensor a user influence graph, an item similarity graph, and knowledge about seasonality or trends in the temporal mode may be available. Such side information may enable more succinct and interpretable tensor decomposition models and improved quality in downstream tasks. We propose a framework for Multi-Dictionary Tensor Decomposition (MDTD) which takes advantage of prior structural information about tensor modes in the form of coding dictionaries to obtain sparsely encoded tensor factors. We derive a general optimization algorithm for MDTD that handles both complete input and input with missing values. Our framework handles large sparse tensors typical to many real-world application domains. We demonstrate MDTD's utility via experiments with both synthetic and real-world datasets. It learns more concise models than dictionary-free counterparts and improves (i) reconstruction quality (60% fewer non-zero coefficients coupled with smaller error); (ii) missing values imputation quality (two-fold MSE reduction with up to orders of magnitude time savings) and (iii) the estimation of the tensor rank. MDTD's quality improvements do not come with a running time premium: it can decompose 19GB datasets in less than a minute. It can also impute missing values in sparse billion-entry tensors more accurately and scalably than state-of-the-art competitors.