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.

利用牛顿法解决了张量平衡的问题，进而实现了概率分布投影，并证明了算法的正确性和适用性，能够适用于多种统计和机器学习模型，如加权 DAG 和玻尔兹曼机。