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Unbalanced Optimal Transport through Non-negative Penalized Linear Regression
- Source :
- IEEE Transactions on Signal Processing, Advances in Neural Information Processing Systems (NeurIPS), Advances in Neural Information Processing Systems (NeurIPS), Dec 2021, Online, France, Advances in Neural Information Processing Systems (NeurIPS), Dec 2021, Virtual, Canada
- Publication Year :
- 2021
- Publisher :
- HAL CCSD, 2021.
-
Abstract
- 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. In this context, we show that the corresponding optimization problem can be reformulated as a non-negative penalized linear regression problem. This reformulation allows us to propose novel algorithms inspired from inverse problems and nonnegative matrix factorization. In particular, we consider majorization-minimization which leads in our setting to efficient multiplicative updates for a variety of penalties. Furthermore, we derive for the first time an efficient algorithm to compute the regularization path of UOT with quadratic penalties. The proposed algorithm provides a continuity of piece-wise linear OT plans converging to the solution of balanced OT (corresponding to infinite penalty weights). We perform several numerical experiments on simulated and real data illustrating the new algorithms, and provide a detailed discussion about more sophisticated optimization tools that can further be used to solve OT problems thanks to our reformulation.<br />Laetitia Chapel and R\'emi Flamary have equal contribution
- Subjects :
- FOS: Computer and information sciences
Computer Science - Machine Learning
machine learning
optimal transport
[INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG]
Optimization and Control (math.OC)
Statistics - Machine Learning
FOS: Mathematics
Machine Learning (stat.ML)
out-of-distribution samples
regularization path
Mathematics - Optimization and Control
Machine Learning (cs.LG)
Subjects
Details
- Language :
- English
- ISSN :
- 1053587X
- Database :
- OpenAIRE
- Journal :
- IEEE Transactions on Signal Processing, Advances in Neural Information Processing Systems (NeurIPS), Advances in Neural Information Processing Systems (NeurIPS), Dec 2021, Online, France, Advances in Neural Information Processing Systems (NeurIPS), Dec 2021, Virtual, Canada
- Accession number :
- edsair.doi.dedup.....3ccdbea7fdefdef048732e8f1cefb3fb