Timezone: »

 
Poster
Tensor Balancing on Statistical Manifold
Mahito Sugiyama · Hiroyuki Nakahara · Koji Tsuda

Tue Aug 08 01:30 AM -- 05:00 AM (PDT) @ Gallery #26

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 Möbius 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.

Author Information

Mahito Sugiyama (National Institute of Informatics)
Hiroyuki Nakahara (RIKEN Brain Science Institute)
Koji Tsuda (University of Tokyo / RIKEN)

Related Events (a corresponding poster, oral, or spotlight)

More from the Same Authors