Minimum-Information Kalman-Bucy Filtering and A Fundamental Limitation of Continuous-Time Data Compression

Motivated by the problem of streaming neuromorphic camera data, we consider the issue of encoding, compressing, and transmitting a continuous-time source signal, possibly with an event-triggered scheme. We study the fundamental trade-off between the data rate and the best achievable data quality.

After briefly reviewing the "causal" rate-distortion theory in discrete-time, in this talk, we motivate the problem of estimating a continuous-time Gauss-Markov source process observed through a vector Gaussian channel with an adjustable channel gain matrix. For a given channel gain matrix, we provide formulas to compute (i) the mean-square estimation error attainable by the classical Kalman-Bucy filter, and (ii) the mutual information between the source process and its Kalman-Bucy estimate. We then formulate a novel “optimal channel gain control problem” where the objective is to control the channel gain matrix strategically to minimize the weighted sum of these two performance metrics.

To develop insights into the optimal solution, we first consider the problem of controlling a time-varying channel gain over a finite time interval. A necessary optimality condition is derived based on Pontryagin's minimum principle. For a scalar system, we show that the optimal channel gain is a piece-wise constant signal with at most two discontinuities. We also consider the problem of designing the optimal time-invariant gain to minimize the average cost over an infinite time horizon. A novel semidefinite programming (SDP) heuristic is proposed to compute the optimal solution.