Title: Min-max Entropy for Weakly Supervised Pointwise Localization. Abstract: Pointwise localization allows more precise localization and accurate interpretability, compared to bounding box, in applications where objects are highly unstructured such as in medical domain. In this work, we focus on  weakly supervised localization (WSL) where a model is trained to classify an image and localize regions of interest at pixel-level using only global image annotation. Typical convolutional attentions maps are prune to high false positive regions. To alleviate this issue, we propose a new deep learning method for WSL, composed of a localizer and a classifier, where the localizer is constrained to determine relevant and irrelevant regions using conditional entropy (CE) with the aim to reduce false positive regions. Experimental results on a public medical dataset and two natural datasets, using Dice index, show that, compared to state of the art WSL methods, our proposal can provide significant improvements in terms of image-level classification and pixel-level localization (low false positive) with robustness to overfitting. A public reproducible PyTorch implementation is provided.