Echoes of the Past: A Unified Perspective on Fading Memory and Echo States

Recurrent neural networks (RNNs) have become increasingly popular in information processing tasks involving time series & temporal data. A fundamental property of RNNs is their ability to create reliable input/output responses, often linked to how the network handles its memory of the information it processed. Various notions have been proposed to conceptualize the behavior of memory in RNNs, including steady states, echo states, state forgetting, input forgetting, & fading memory. Although these notions are often used interchangeably, their precise relationships remain unclear. This work aims to unify these notions in a common language, derive new implications & equivalences between them, & provide alternative proofs to some existing results. By clarifying the relationships between these concepts, this research contributes to a deeper understanding of RNNs & their temporal information processing capabilities.