# Kullback-Leibler Principal Component for Tensors is not NP-hard

arXiv: https://arxiv.org/abs/1711.07925 · HTML: https://mlsanity.com/paper/1711.07925/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2017.

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

- [Stochastic Mirror Descent for Low-Rank Tensor Decomposition Under Non-Euclidean Losses](https://mlsanity.com/paper/2104.14562/) (2021)
- [Many-body Approximation for Non-negative Tensors](https://mlsanity.com/paper/2209.15338/) (2022)
- [Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation](https://mlsanity.com/paper/2103.02898/) (2021)
- [A Latent-Variable Formulation of the Poisson Canonical Polyadic Tensor Model: Maximum Likelihood Estimation and Fisher Information](https://mlsanity.com/paper/2511.05352/) (2025)

## Abstract (arXiv)

We study the problem of nonnegative rank-one approximation of a nonnegative tensor, and show that the globally optimal solution that minimizes the generalized Kullback-Leibler divergence can be efficiently obtained, i.e., it is not NP-hard. This result works for arbitrary nonnegative tensors with an arbitrary number of modes (including two, i.e., matrices). We derive a closed-form expression for the KL principal component, which is easy to compute and has an intuitive probabilistic interpretation. For generalized KL approximation with higher ranks, the problem is for the first time shown to be equivalent to multinomial latent variable modeling, and an iterative algorithm is derived that resembles the expectation-maximization algorithm. On the Iris dataset, we showcase how the derived results help us learn the model in an unsupervised manner, and obtain strikingly close performance to that from supervised methods.