118,379 research outputs found
Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (or : How to Prove Kabashima's Replica Formula)
There has been a recent surge of interest in the study of asymptotic
reconstruction performance in various cases of generalized linear estimation
problems in the teacher-student setting, especially for the case of i.i.d
standard normal matrices. Here, we go beyond these matrices, and prove an
analytical formula for the reconstruction performance of convex generalized
linear models with rotationally-invariant data matrices with arbitrary bounded
spectrum, rigorously confirming a conjecture originally derived using the
replica method from statistical physics. The formula includes many problems
such as compressed sensing or sparse logistic classification. The proof is
achieved by leveraging on message passing algorithms and the statistical
properties of their iterates, allowing to characterize the asymptotic empirical
distribution of the estimator. Our proof is crucially based on the construction
of converging sequences of an oracle multi-layer vector approximate message
passing algorithm, where the convergence analysis is done by checking the
stability of an equivalent dynamical system. We illustrate our claim with
numerical examples on mainstream learning methods such as sparse logistic
regression and linear support vector classifiers, showing excellent agreement
between moderate size simulation and the asymptotic prediction.Comment: 19 pages,25 appendix,4 figure
System Identification for Nonlinear Control Using Neural Networks
An approach to incorporating artificial neural networks in nonlinear, adaptive control systems is described. The controller contains three principal elements: a nonlinear inverse dynamic control law whose coefficients depend on a comprehensive model of the plant, a neural network that models system dynamics, and a state estimator whose outputs drive the control law and train the neural network. Attention is focused on the system identification task, which combines an extended Kalman filter with generalized spline function approximation. Continual learning is possible during normal operation, without taking the system off line for specialized training. Nonlinear inverse dynamic control requires smooth derivatives as well as function estimates, imposing stringent goals on the approximating technique
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