# An AO-ADMM approach to constraining PARAFAC2 on all modes

arXiv: https://arxiv.org/abs/2110.01278 · HTML: https://mlsanity.com/paper/2110.01278/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 67 references that are in the map.
- First posted on arXiv 4 Oct 2021.

## Builds on (most cited first, 9 of 9)

- [A Flexible and Efficient Algorithmic Framework for Constrained Matrix and Tensor Factorization](https://mlsanity.com/paper/1506.04209/) (2015)
- [COPA: Constrained PARAFAC2 for Sparse & Large Datasets](https://mlsanity.com/paper/1803.04572/) (2018)
- [Stochastic Gradients for Large-Scale Tensor Decomposition](https://mlsanity.com/paper/1906.01687/) (2019)
- [Subspace Learning and Imputation for Streaming Big Data Matrices and Tensors](https://mlsanity.com/paper/1404.4667/) (2014)
- [Nonnegative PARAFAC2: a flexible coupling approach](https://mlsanity.com/paper/1802.05035/) (2018)
- [Tracing Network Evolution Using the PARAFAC2 Model](https://mlsanity.com/paper/1911.02926/) (2019)
- [A Flexible Optimization Framework for Regularized Matrix-Tensor Factorizations with Linear Couplings](https://mlsanity.com/paper/2007.09605/) (2020)
- [Nesterov Acceleration of Alternating Least Squares for Canonical Tensor Decomposition: Momentum Step Size Selection and Restart Mechanisms](https://mlsanity.com/paper/1810.05846/) (2018)
- [PARAFAC2 AO-ADMM: Constraints in all modes](https://mlsanity.com/paper/2102.02087/) (2021)

## Built on by (most cited first, 12 of 12)

- [Unsupervised EHR-based Phenotyping via Matrix and Tensor Decompositions](https://mlsanity.com/paper/2209.00322/) (2022)
- [PARAFAC2-based Coupled Matrix and Tensor Factorizations](https://mlsanity.com/paper/2210.13054/) (2022)
- [MULTIPAR: Supervised Irregular Tensor Factorization with Multi-task Learning](https://mlsanity.com/paper/2208.00993/) (2022)
- [A Time-aware tensor decomposition for tracking evolving patterns](https://mlsanity.com/paper/2308.07126/) (2023)
- [Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models](https://mlsanity.com/paper/2403.18517/) (2024)
- [SWoTTeD: An Extension of Tensor Decomposition to Temporal Phenotyping](https://mlsanity.com/paper/2310.01201/) (2023)
- [PARAFAC2-based Coupled Matrix and Tensor Factorizations with Constraints](https://mlsanity.com/paper/2406.12338/) (2024)
- [tPARAFAC2: Tracking evolving patterns in (incomplete) temporal data](https://mlsanity.com/paper/2407.01356/) (2024)
- [Toward Temporal Causal Representation Learning with Tensor Decomposition](https://mlsanity.com/paper/2507.14126/) (2025)
- [dCMF: Learning interpretable evolving patterns from temporal multiway data](https://mlsanity.com/paper/2502.19367/) (2025)
- [Variational Low-rank Tensor Decomposition for Multisubject Spatiotemporal Data Analysis](https://mlsanity.com/paper/2607.22262/) (2026)
- [Knowledge-guided Pattern Discovery via Coupled Tensor Factorizations](https://mlsanity.com/paper/2608.13234/) (2026)

## Abstract (arXiv)

Analyzing multi-way measurements with variations across one mode of the dataset is a challenge in various fields including data mining, neuroscience and chemometrics. For example, measurements may evolve over time or have unaligned time profiles. The PARAFAC2 model has been successfully used to analyze such data by allowing the underlying factor matrices in one mode (i.e., the evolving mode) to change across slices. The traditional approach to fit a PARAFAC2 model is to use an alternating least squares-based algorithm, which handles the constant cross-product constraint of the PARAFAC2 model by implicitly estimating the evolving factor matrices. This approach makes imposing regularization on these factor matrices challenging. There is currently no algorithm to flexibly impose such regularization with general penalty functions and hard constraints. In order to address this challenge and to avoid the implicit estimation, in this paper, we propose an algorithm for fitting PARAFAC2 based on alternating optimization with the alternating direction method of multipliers (AO-ADMM). With numerical experiments on simulated data, we show that the proposed PARAFAC2 AO-ADMM approach allows for flexible constraints, recovers the underlying patterns accurately, and is computationally efficient compared to the state-of-the-art. We also apply our model to two real-world datasets from neuroscience and chemometrics, and show that constraining the evolving mode improves the interpretability of the extracted patterns.