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Using this free online PDF compressor, you can make sure that both uploading or sending ... Reduce the PDF file size with this online PDF compression tool. when E s2 suddenly become 10 times smaller, the ordinary learning dynamics makes the i i-th row of W become 10 times larger in order to compensate for this change and keep 2 E yi = 1. Therefore, even when W converges to the true C W , large uctuations emerge from the ine ective movements caused by changes in amplitude of si . This sometimes causes numerical instability. On the other hand, our nonholonomic dynamics is always orthogonal to C W so that such ine ective uctuations are suppressed. This is con rmed by computer simulations a few of which are presented in this chapter. There is a lot of research on nonholonomic dynamics [30]. Classical analytical dynamics uses nonholonomic bases to analyze the spinning gyro (Whittaker, 1940). It was also used in relativity. G. Kron (1952) used the nonholonomic constraints to present a general theory of electro-mechanical dynamics of generators and motors. Nonholonomic properties play a fundamental role in continuum mechanics of distributed dislocations (see, e.g., Amari, 1962). An excellent explanation is found in Brockett (1993), where the controllability of nonlinear dynamical systems is shown by using the related Lie algebra. Recently, robotics researchers have been eager to analyze dynamics with nonholonomic constraints (Suzuki and Nakamura 1995) [30]. Remark D.6 Theorem D.4 shows that the orthogonal natural gradient descent algorithm evolves along a trajectory path which doesn t include redundant (useless) components in the directions of C W . Therefore, it seems likely that the orthogonal algorithm can be more ef cient than other algorithms as has been con rmed by preliminary computer simulations. Stability Analysis In this section, we discuss the local stability of the nonholonomic orthogonal natural gradient descent algorithm de ned by W(k) = (k) F(y(k)) W(k), (D.17) free online pdf compressor trial Compress PDF - Compress your PDF document online - PDF2Go
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