In the field of pattern mining, a negative sequential pattern is specified by means of a sequence consisting of events to occur and of other events, called negative events, to be absent. For instance, containment of the pattern $\langle a\ \neg b\ c\rangle$ arises with an occurrence of a and a subsequent occurrence of c but no occurrence of b in between. This article is to shed light on the ambiguity of such a seemingly intuitive notation and we identify eight possible semantics for the containment relation between a pattern and a sequence. These semantics are illustrated and formally studied, in particular we propose dominance and equivalence relations between them. Also we prove that support is anti-monotonic for some of these semantics. Some of the results are discussed with the aim of developing algorithms to extract efficiently frequent negative patterns.

We study the problem of learning permutation invariant representations that can capture "flexible" notions of containment. We formalize this problem via a measure theoretic definition of multisets, and obtain a theoretically-motivated learning model. We propose training this model on a novel task: predicting the size of the symmetric difference (or intersection) between pairs of multisets. We demonstrate that our model not only performs very well on predicting containment relations (and more effectively predicts the sizes of symmetric differences and intersections than DeepSets-based approaches with unconstrained object representations), but that it also learns meaningful representations.

This paper studies a challenging problem of tracking severely occluded objects in long video sequences. The proposed method reasons about the containment relations and human actions, thus infers and recovers occluded objects identities while contained or blocked by others. There are two conditions that lead to incomplete trajectories: i) Contained. The occlusion is caused by a containment relation formed between two objects, e.g., an unobserved laptop inside a backpack forms containment relation between the laptop and the backpack. ii) Blocked. The occlusion is caused by other objects blocking the view from certain locations, during which the containment relation does not change. By explicitly distinguishing these two causes of occlusions, the proposed algorithm formulates tracking problem as a network flow representation encoding containment relations and their changes. By assuming all the occlusions are not spontaneously happened but only triggered by human actions, an MAP inference is applied to jointly interpret the trajectory of an object by detection in space and human actions in time. To quantitatively evaluate our algorithm, we collect a new occluded object dataset captured by Kinect sensor, including a set of RGB-D videos and human skeletons with multiple actors, various objects, and different changes of containment relations. In the experiments, we show that the proposed method demonstrates better performance on tracking occluded objects compared with baseline methods.