International Medical and Technological University

The push-forward operation enables one to redistribute a probability measure through a deterministic map. It plays a key role in statistics and optimization: many learning problems (notably from optimal transport, generative modeling, and algorithmic fairness) include constraints or penalties framed as push-forward conditions on the model. However, the literature lacks general theoretical insights on the (non)convexity of such constraints and its consequences on the associated learning problems. This paper aims at filling this gap. In the first part, we provide a range of sufficient and necessary conditions for the (non)convexity of two sets of functions: the maps transporting one probability measure to another and the maps inducing equal output distributions across distinct probability measures. This highlights that for most probability measures, these push-forward constraints are not convex. In the second part, we show how this result implies critical limitations on the design of convex optimization problems for learning generative models or groupwise fair predictors. This work will hopefully help researchers and practitioners have a better understanding of the critical impact of push-forward conditions onto convexity.

Two simple spectrophotometric methods (I and II) are described for the determination of rivaroxaban in bulk and tablets.Method I is based on the reaction of rivaroxaban with 4-Chloro-7-nitrobenzo-2-oxa-1,3-diazole in alkaline medium.Method II involved the reaction of rivaroxaban with p-Dimethylaminocinnamaldehyde in acidic medium.The reactions products were measured at 405 nm and 545 nm for methods I and II, respectively.Beer's law was obeyed over the concentration ranges of 2-20 μg/mL and 25-125 μg/mL with lower limits of detection of 0.110 μg/mL and 0.483 μg/mL for Methods I and II, respectively.Percent recovery of rivaroxaban for both methods I and II were found in the range of 99.74-100.51%with relative standard deviation in the range of 0.621-0.900%.The proposed methods I and II were successfully applied for quantification of rivaroxaban in tablet dosage forms with good accuracy and precision.Hence, the proposed methods I and II can be suitably adopted for routine analysis of rivaraxaban.