# Tensor Balancing on Statistical Manifold

arXiv: https://arxiv.org/abs/1702.08142 · HTML: https://mlsanity.com/paper/1702.08142/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2017.

## Built on by (most cited first, 6 of 6)

- [Many-body Approximation for Non-negative Tensors](https://mlsanity.com/paper/2209.15338/) (2022)
- [Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions](https://mlsanity.com/paper/2306.00904/) (2023)
- [Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation](https://mlsanity.com/paper/2103.02898/) (2021)
- [Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models](https://mlsanity.com/paper/2403.18517/) (2024)
- [Fast Rank-1 NMF for Missing Data with KL Divergence](https://mlsanity.com/paper/2110.12595/) (2021)
- [Pseudo-Nonlinear Data Augmentation: A Constrained Energy Minimization Viewpoint](https://mlsanity.com/paper/2410.00718/) (2024)

## Abstract (arXiv)

We solve tensor balancing, rescaling an Nth order nonnegative tensor by multiplying N tensors of order N - 1 so that every fiber sums to one. This generalizes a fundamental process of matrix balancing used to compare matrices in a wide range of applications from biology to economics. We present an efficient balancing algorithm with quadratic convergence using Newton's method and show in numerical experiments that the proposed algorithm is several orders of magnitude faster than existing ones. To theoretically prove the correctness of the algorithm, we model tensors as probability distributions in a statistical manifold and realize tensor balancing as projection onto a submanifold. The key to our algorithm is that the gradient of the manifold, used as a Jacobian matrix in Newton's method, can be analytically obtained using the Moebius inversion formula, the essential of combinatorial mathematics. Our model is not limited to tensor balancing, but has a wide applicability as it includes various statistical and machine learning models such as weighted DAGs and Boltzmann machines.