Learning Expanding Graphs for Signal Interpolation

Abstract

Performing signal processing over graphs requires knowledge of the underlying fixed topology. However, graphs often grow in size with new nodes appearing over time, whose connectivity is typically unknown; hence, making more challenging the downstream tasks in applications like cold start recommendation. We address such a challenge for signal interpolation at the incoming nodes blind to the topological connectivity of the specific node. Specifically, we propose a stochastic attachment model for incoming nodes parameterized by the attachment probabilities and edge weights. We estimate these parameters in a data-driven fashion by relying only on the attachment behaviour of earlier incoming nodes with the goal of interpolating the signal value. We study the non-convexity of the problem at hand, derive conditions when it can be marginally convexified, and propose an alternating projected descent approach between estimating the attachment probabilities and the edge weights. Numerical experiments with synthetic and real data dealing in cold start collaborative filtering corroborate our findings.

Task-Aware Connectivity Learning for Incoming Nodes Over Growing Graphs

“…So, regularizers are needed for each variable. In the MSE expression, we remark that the Lth shift A L x + does not appear in (7) because of the structure of matrix A + in (1). We also remark that the MSE is ( 7) is only an approximation because for filter order L ≥ 3 the MSE expression becomes intractable due to the higher order moments of a + .…”

“…in which the last row and column represent the connectivity of v + with the nodes in V. 1 We consider v + connects independently to each existing v i ∈ V with probability p i . Thus, the attachment vector a + is random with each entry being a weighted Bernoulli random variable; i.e,…”

“…Let the incoming node v + attach to G with probability vector p and edge weight vector w, forming graph G + with the expanded adjacency matrix A + [cf. (1)]. The MSE of the interpolated signal y + at node v + by an order L graph filter H(A + ) [cf.…”

Graph Filtering Over Expanding Graphs

“…Differently, the work in [16] estimates the attachment and uses it for for subsequent classification. The works in [17], [18] deal with a stochastic attachment model for filter interpolation or a filter bank, but they are limited only to one incoming node an not a stream of nodes. Other works focus on mapping features to graph signals [19], node classification [20], and dynamic embedding [21].…”