This paper addresses the problem of Unbalanced Optimal Transport (UOT) in which the marginal conditions are relaxed (using weighted penalties in lieu of equality) and no additional regularization is enforced on the OT plan.

This paper addresses the problem of Unbalanced Optimal Transport (UOT) in which the marginal conditions are relaxed (using weighted penalties in lieu of equality) and no additional regularization is enforced on the OT plan.

Non-negative matrix factorization (NMF) and non-negative tensor factorization (NTF) decompose non-negative high-dimensional data into non-negative low-rank components. NMF and NTF methods are popular for their intrinsic interpretability and effectiveness on large-scale data. Recent work developed Stratified-NMF, which applies NMF to regimes where data may come from different sources (strata) with different underlying distributions, and seeks to recover both strata-dependent information and global topics shared across strata. Applying Stratified-NMF to multi-modal data requires flattening across modes, and therefore loses geometric structure contained implicitly within the tensor. To address this problem, we extend Stratified-NMF to the tensor setting by developing a multiplicative update rule and demonstrating the method on text and image data. We find that Stratified-NTF can identify interpretable topics with lower memory requirements than Stratified-NMF. We also introduce a regularized version of the method and demonstrate its effects on image data.

Topic modeling, or more broadly, dimensionality reduction, techniques provide powerful tools for uncovering patterns in large datasets and are widely applied across various domains. We investigate how Non-negative Matrix Factorization (NMF) can introduce bias in the representation of data groups, such as those defined by demographics or protected attributes. We present an approach, called Fairer-NMF, that seeks to minimize the maximum reconstruction loss for different groups relative to their size and intrinsic complexity. Further, we present two algorithms for solving this problem. The first is an alternating minimization (AM) scheme and the second is a multiplicative updates (MU) scheme which demonstrates a reduced computational time compared to AM while still achieving similar performance. Lastly, we present numerical experiments on synthetic and real datasets to evaluate the overall performance and trade-offs of Fairer-NMF

Main ideas of the submission The manuscript presents an approximation of nonnegative multi-way tensorial data (or high-order probability mass functions) based on structured energy function form that minimizes the Kullback-Leibler divergence. Comparing against other multilinear decomposition methods of nonnegative tensors, the proposal approach operates on multiplicative parameters under convex objective function and converges to a globally optimal solution. It also shows interesting connections with graphical models such as the high-order Boltzmann machines. Two optimization algorithms are developed, based upon gradient and natural gradient, respectively. The experiment shows that under the same number of parameters, the proposed approach yields smaller RMSEs than the other two baseline non-negative tensor decomposition methods.