Biological systems need to process information in real time and must trade off accuracy of presentation and coding costs. Here we operationalize this trade-off and develop an information-theoretic framework that selectively extracts information of the input past that is predictive about the output future, obtaining a generalized eigenvalue problem. Thereby, we unravel the input history in terms of structural phase transitions corresponding to additional dimensions of a state space. We elucidate the relation to canonical correlation analysis and give a numerical example. Altogether, this work relates information-theoretic optimization to the joint problem of system identification and model reduction.
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