1. For notation simplicity, we drop the $n$ from $\mathbf{x}_i^{n}$ and $M_1(n)$ when there is no ambiguity. Thus, the problem is how to build an effective object detector to count the object instances based on the incomplete annotations $\{\mathbf{x}_i^n\}$. We name the problem as *incompletely-supervised learning* for object counting.

### Comparison with other supervised learning problems.

The differences with other supervised learning settings in the context of object detection are also shown in Fig. \[fig:isl\]. Fully-supervised learning requires that all the bounding boxes are labeled. This represents the upper bound of detection-based object counting performance, but is costly to label every instance. Weakly-supervised learning assumes that only image-level category information is available. That means, we know there are some object instances in one image, but we have no information of their locations. Another problem setting is low-shot learning, where the number of training images is small, but each training image has full annotations. This is more like the fully-supervised learning but with only a few training images.

[@c@[     ]{}c@[     ]{}c@[     ]{}c@[     ]{}c@[     ]{}c@]{} ![Intuition of the proposed label propagation through a positiveness-focused object detector to manage the incompletely-supervised labels[]{data-label="fig:intuition"}](figs/ideal.eps "fig:"){width="\intuitionwidth\linewidth"} & ![Intuition of the proposed label propagation through a positiveness-focused 