Slow feature analysis (SFA) is an unsupervised-learning algorithm that\nextracts slowly varying features from a multi-dimensional time series. A\nsupervised extension to SFA for classification and regression is graph-based\nSFA (GSFA). GSFA is based on the preservation of similarities, which are\nspecified by a graph structure derived from the labels. It has been shown that\nhierarchical GSFA (HGSFA) allows learning from images and other\nhigh-dimensional data. The feature space spanned by HGSFA is complex due to the\ncomposition of the nonlinearities of the nodes in the network. However, we show\nthat the network discards useful information prematurely before it reaches\nhigher nodes, resulting in suboptimal global slowness and an under-exploited\nfeature space.\n To counteract these problems, we propose an extension called hierarchical\ninformation-preserving GSFA (HiGSFA), where information preservation\ncomplements the slowness-maximization goal. We build a 10-layer HiGSFA network\nto estimate human age from facial photographs of the MORPH-II database,\nachieving a mean absolute error of 3.50 years, improving the state-of-the-art\nperformance. HiGSFA and HGSFA support multiple-labels and offer a rich feature\nspace, feed-forward training, and linear complexity in the number of samples\nand dimensions. Furthermore, HiGSFA outperforms HGSFA in terms of feature\nslowness, estimation accuracy and input reconstruction, giving rise to a\npromising hierarchical supervised-learning approach.\n