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“Adaptive Stochastic Resonance ”
by Sanya Mitaim
August 1999
This dissertation introduces and explores the new property of adaptive stochastic resonance (ASR). Stochastic resonance (SR) occurs when noise enhances an external forcing signal in a nonlinear dynamical system. ASR uses statistical learning techniques that learn the optimal level of noise to add to a nonlinear system in the sense that this level of noise will maximize the system's signal-to-noise ratio or that it will improve or extremize other measures of how well the system performs. This dissertation studies how adaptive systems can achieve ASR based on only samples from the process or based on these samples and minimal estimates of the system dynamics. The fundamental result of this research is that stochastic gradient learning can achieve ASR. A statistical learning system can learn the SR effect if it performs a stochastic gradient ascent on a system performance measure such as the system's spectral signal-to-noise ratio. But the gradient becomes so impulsive near optimality that it can destabilize the learning process. A Cauchy noise suppressor solves this problem and lets the stochastic-gradient learning laws train on noisy input-output samples to achieve stochastic resonance. This research led to new stochastic learning laws for different types of systems and signals and for different types of performance measures. Stochastic gradient ascent on the signal-to-noise ratio led to ASR for narrowband signals. But broadband forcing signals required a correlation performance measure and often required some estimate of the Jacobian structure of the dynamical system. We discovered stochastic resonance in nonlinear systems with impulsive noise that has infinite variance. An exponential law relates the SR effect or the optimal noise dispersion to the impulsiveness. We also showed that ``smart'' or black-box function approximators such as adaptive fuzzy systems can learn to induce the SR effect in many nonlinear systems. We developed new fuzzy learning laws for systems that take as input both numerical vectors and entire fuzzy sets. The appendices present these fuzzy results and apply them to the multimedia problem of teaching an intelligent agent to learn a user's preferences and to search databases on the user's behalf. This research revealed many problems with ASR. The adaptive system required a large number of input-output samples or it required at least some knowledge of the system dynamics and signals. We found no theorems to guarantee that the stochastic learning algorithms converge. This reflects a fundamental problem of research in SR and ASR. Even simple system nonlinearity can complicate or preclude a closed-form analysis and do so even if we have exact knowledge of the nonlinear signal systems. Future research may lead to new learning laws or to new ways to approximate the dynamics of nonlinear systems that stochastically resonate. This will help answer the key question that underlies ASR: Which noisy dynamical systems show what SR effects for which forcing signals and for which performance measures?