Can random linear networks store multiple long input streams?

The short term memory of randomly connected networks has been recently studied in order to better understand the computational and predictive power of such networks. In particular, random, linear, orthogonal networks have been explored extensively in the context a single input stream driving the network. The most recent results state that a stream of length N can be recovered from a network of size O(S log⁶ (N)) assuming that the input is S-sparse in some basis. Little work, however, addresses more complex networks where multiple input streams feed into the same network. In this paper we extend the results for recovering sparse input streams the multiple input streams feeding into the same network. We find that we can recover L input streams of length N with a network that has O(S log⁵ (LN)) nodes.