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.
|Original language||English (US)|
|Title of host publication||Conference Record of 51st Asilomar Conference on Signals, Systems and Computers, ACSSC 2017|
|Editors||Michael B. Matthews|
|Publisher||Institute of Electrical and Electronics Engineers Inc.|
|Number of pages||5|
|State||Published - Apr 10 2018|
|Event||51st Asilomar Conference on Signals, Systems and Computers, ACSSC 2017 - Pacific Grove, United States|
Duration: Oct 29 2017 → Nov 1 2017
|Name||Conference Record of 51st Asilomar Conference on Signals, Systems and Computers, ACSSC 2017|
|Other||51st Asilomar Conference on Signals, Systems and Computers, ACSSC 2017|
|Period||10/29/17 → 11/1/17|
Bibliographical noteFunding Information:
This work is supported in part by NSF IIS-1247632, IIS-1447788, and IIS-1704074.
© 2017 IEEE.
- canonical polyadic decomposition
- generalized Kullback-Leibler (KL) divergence
- latent variable modeling
- tensor factorization