  This paper is about labeling video frames with action classes under weak
supervision in training, where we have access to a temporal ordering of
actions, but their start and end frames in training videos are unknown.
Following prior work, we use an HMM grounded on a Gated Recurrent Unit (GRU)
for frame labeling. Our key contribution is a new constrained discriminative
forward loss (CDFL) that we use for training the HMM and GRU under weak
supervision. While prior work typically estimates the loss on a single,
inferred video segmentation, our CDFL discriminates between the energy of all
valid and invalid frame labelings of a training video. A valid frame labeling
satisfies the ground-truth temporal ordering of actions, whereas an invalid one
violates the ground truth. We specify an efficient recursive algorithm for
computing the CDFL in terms of the logadd function of the segmentation energy.
Our evaluation on action segmentation and alignment gives superior results to
those of the state of the art on the benchmark Breakfast Action, Hollywood
Extended, and 50Salads datasets.
