Toric Geometry of Entropic Regularization
This work provides foundational geometric insights for optimization methods, but it is incremental as it revisits and extends existing entropic regularization approaches.
The paper tackles the geometric interpretation of entropic regularization in large-scale linear programming by analyzing intersections with toric varieties and comparing it to log-barrier methods, resulting in theoretical insights such as computing the degree of toric varieties and exploring algorithms like iterative scaling.
Entropic regularization is a method for large-scale linear programming. Geometrically, one traces intersections of the feasible polytope with scaled toric varieties, starting at the Birch point. We compare this to log-barrier methods, with reciprocal linear spaces, starting at the analytic center. We revisit entropic regularization for unbalanced optimal transport, and we develop the use of optimal conic couplings. We compute the degree of the associated toric variety, and we explore algorithms like iterative scaling.