We propose a dual system for unsupervised object segmentation in video, which brings together two modules with complementary properties: a space-time graph that discovers objects in videos and a deep network that learns powerful object features. The system uses an iterative knowledge exchange policy. A novel spectral space-time clustering process on the graph produces unsupervised segmentation masks passed to the network as pseudo-labels. The net learns to segment in single frames what the graph discovers in video and passes back to the graph strong image-level features that improve its node-level features in the next iteration. Knowledge is exchanged for several cycles until convergence. The graph has one node per each video pixel, but the object discovery is fast. It uses a novel power iteration algorithm computing the main space-time cluster as the principal eigenvector of a special Feature-Motion matrix without actually computing the matrix. The thorough experimental analysis validates our theoretical claims and proves the effectiveness of the cyclical knowledge exchange. We also perform experiments on the supervised scenario, incorporating features pretrained with human supervision. We achieve state-of-the-art level on unsupervised and supervised scenarios on four challenging datasets: DAVIS, SegTrack, YouTube-Objects, and DAVSOD. We will make our code publicly available.