# Near-Universal Multiplicative Updates for Nonnegative Einsum Factorization

arXiv: https://arxiv.org/abs/2602.02759 · HTML: https://mlsanity.com/paper/2602.02759/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 63 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [TensorLy: Tensor Learning in Python](https://mlsanity.com/paper/1610.09555/) (2016)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [Failures of Gradient-Based Deep Learning](https://mlsanity.com/paper/1703.07950/) (2017)
- [Generalized Canonical Polyadic Tensor Decomposition](https://mlsanity.com/paper/1808.07452/) (2018)
- [Tensor-on-tensor regression](https://mlsanity.com/paper/1701.01037/) (2017)
- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness](https://mlsanity.com/paper/1404.4412/) (2014)
- [Stochastic Gradients for Large-Scale Tensor Decomposition](https://mlsanity.com/paper/1906.01687/) (2019)
- [Inference of hyperedges and overlapping communities in hypergraphs](https://mlsanity.com/paper/2204.05646/) (2022)
- [Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance](https://mlsanity.com/paper/2012.10249/) (2020)
- [A tensor factorization model of multilayer network interdependence](https://mlsanity.com/paper/2206.01804/) (2022)
- [E2M: Double Bounded α-Divergence Optimization for Tensor-based Discrete Density Estimation](https://mlsanity.com/paper/2405.18220/) (2024)
- [Many-body Approximation for Non-negative Tensors](https://mlsanity.com/paper/2209.15338/) (2022)
- [Broad Spectrum Structure Discovery in Large-Scale Higher-Order Networks](https://mlsanity.com/paper/2505.21748/) (2025)
- [Restricted Tweedie Stochastic Block Models](https://mlsanity.com/paper/2310.10952/) (2023)

## Built on by (most cited first, 2 of 2)

- [Bayesian Poisson-Randomized Gamma Tensor Factorization with Application to International Trade Flows](https://mlsanity.com/paper/2606.17267/) (2026)
- [Heteroskedastic Canonical Polyadic Tensor Decomposition](https://mlsanity.com/paper/2610.00498/) (2026)

## Abstract (arXiv)

Despite the ubiquity of multiway data across scientific domains, there are few performant and user-friendly methods that fit non-standard nonnegative tensor factorization models tailored to the data at-hand. Researchers may use gradient-based automatic differentiation, which often struggles under nonnegative constraints, choose between a limited set of methods with mature implementations, or implement their own model from scratch. As an alternative, we introduce NNEinFact, an einsum-based multiplicative update algorithm that fits any nonnegative tensor factorization expressible as a tensor contraction by minimizing one of many user-specified loss functions, including the (α,β)-divergence. To use NNEinFact, the researcher specifies their model with a string. NNEinFact converges to a stationary point of the loss, supports missing data, and fits to tensors with hundreds of millions of entries in seconds. Empirically, NNEinFact fits custom models which outperform standard ones in prediction tasks on real-world tensor data by over 37% and attains less than half the test loss of gradient-based methods while converging up to 90 times faster. Software is publicly available at https://github.com/jhood3/einfact.