Position: A Theory of Deep Learning Must Include Compositional Sparsity
Overparametrized Deep Neural Networks (DNNs) have demonstrated remarkable success in a wide variety of domains too high-dimensional for classical shallow networks subject to the curse of dimensionality. However, open questions about fundamental principles, that govern the learning dynamics of DNNs, remain. In this position paper we argue that it is the ability of DNNs to exploit the compositionally sparse structure of the target function driving their success. As such, DNNs can leverage the property that most practically relevant functions can be composed from a small set of constituent functions, each of which relies only on a low-dimensional subset of all inputs. We show that this property is shared by all efficiently Turing-computable functions and is therefore highly likely present in all current learning problems. While some promising theoretical insights on questions concerned with approximation and generalization exist in the setting of compositionally sparse functions, several important questions on the learnability and optimization of DNNs remain. Completing the picture of the role of compositional sparsity in deep learning is essential to a comprehensive theory of artificial—and even general—intelligence.
May 1, 2025
Score and Rank Semi-monotonicity for Closeness, Betweenness and Harmonic Centrality
In the study of the behavior of centrality measures with respect to network modifications, score monotonicity means that adding an arc increases the centrality score of the target of the arc; rank monotonicity means that adding an arc improves the importance of the target of the arc relative to the remaining nodes. It is known that score and rank monotonicity hold in directed graphs for almost all the classical centrality measures. In undirected graphs one expects that the corresponding properties (where both endpoints of the new edge enjoy the increase in score/rank) hold when adding a new edge. However, recent results have shown that in undirected networks this is not true: for many centrality measures, it is possible to find situations where adding an edge reduces the rank of one of its two endpoints. In this paper we introduce a weaker condition for undirected networks, semi-monotonicity, in which just one of the endpoints of a new edge is required to enjoy score or rank monotonicity. We show that this condition is satisfied by closeness and betweenness centrality, and that harmonic centrality satisfies it in an even stronger sense.
Jan 1, 2024