张量统计流形上的平衡
Tensor Balancing on Statistical Manifold
AI总结：
本文提出了一种高效的张量平衡算法，通过将张量视为概率分布，在统计流形上进行投影，利用Moebius inversion公式获得梯度，实现二次收敛，显著提高了计算效率。
摘要：
我们解决了张量平衡问题，通过乘以N个N-1阶张量来重新缩放N阶非负张量，使得每个纤维之和为1。这扩展了用于比较矩阵的基本过程，广泛应用于生物学和经济学等领域。我们提出了一种高效的平衡算法，使用牛顿方法并证明其具有二次收敛性，在数值实验中显示比现有方法快几个数量级。为了证明算法的正确性，我们将张量建模为统计流形上的概率分布，并将张量平衡视为对子流形的投影。我们算法的关键在于流形的梯度，作为牛顿方法中的雅可比矩阵，可以利用Moebius inversion公式解析地获得。我们的模型不仅限于张量平衡，还具有广泛应用，包括加权DAG和Boltzmann机等多种统计和机器学习模型。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
