Weakly supervised object detection is a challenging task when provided with image category supervision but required to learn, at the same time, object locations and object detectors. The inconsistency between the weak supervision and learning objectives introduces significant randomness to object locations and ambiguity to detectors. In this paper, a min-entropy latent model (MELM) is proposed for weakly supervised object detection. Min-entropy serves as a model to learn object locations and a metric to measure the randomness of object localization during learning. It aims to principally reduce the variance of learned instances and alleviate the ambiguity of detectors. MELM is decomposed into three components including proposal clique partition, object clique discovery, and object localization. MELM is optimized with a recurrent learning algorithm, which leverages continuation optimization to solve the challenging non-convexity problem. Experiments demonstrate that MELM significantly improves the performance of weakly supervised object detection, weakly supervised object localization, and image classification, against the state-of-the-art approaches.