Asymptotic Analysis of the LMS Algorithm with Momentum

A widely studied filtering algorithm in signal processing is the least mean square (LMS) method, due to B. Widrow and T. Hoff, 1960. A popular extension of the LMS algorithm, which is also important in deep learning, is the LMS method with momentum, originated by S. Roy and J.J. Shynk back in 1988. This is a fixed gain (or constant step-size) version of the LMS method modified by an additional momentum term that is proportional to the last correction term. Recently, a certain equivalence of the two methods has been rigorously established by K. Yuan, B. Ying and A.H. Sayed, assuming martingale difference gradient noise. The purpose of this paper is to present the outline of a significantly simpler and more transparent asymptotic analysis of the LMS algorithm with momentum under the assumption of stationary, ergodic and mixing signals

The D1-D61N Mutation in <i>Synechocystis</i> sp. PCC 6803 Allows the Observation of pH-Sensitive Intermediates in the Formation and Release of O₂ from Photosystem II

The active site of photosynthetic water oxidation by Photosystem II (PSII) is a manganese–calcium cluster (Mn₄CaO₅). A postulated catalytic base is assumed to be crucial. CP43-Arg357, which is a candidate for the identity of this base, is a second-sphere ligand of the Mn₄–Ca cluster and is located near a putative proton exit pathway, which begins with residue D1-D61. Transient absorption spectroscopy and time-resolved O₂ polarography reveal that in the D1-D61N mutant, the transfer of an electron from the Mn₄CaO₅ cluster to Y_Z^OX and O₂ release during the final step of the catalytic cycle, the S₃–S₀ transition, proceed simultaneously but are more dramatically decelerated than previously thought (<i>t</i>_1/2 of up to ∼50 ms vs a <i>t</i>_1/2 of 1.5 ms in the wild type). Using a bare platinum electrode to record the flash-dependent yields of O₂ from mutant and wild-type PSII has allowed the observation of the kinetics of release of O₂ from extracted thylakoid membranes at various pH values and in the presence of deuterated water. In the mutant, it was possible to resolve a clear lag phase prior to the appearance of O₂, indicating formation of an intermediate before the onset of O₂ formation. The lag phase and the photochemical miss factor were more sensitive to isotope substitution in the mutant, indicating that proton efflux in the mutant proceeds via an alternative pathway. The results are discussed in comparison with earlier results obtained from the substitution of CP43-Arg357 with lysine and in regard to hypotheses concerning the nature of the final steps in photosynthetic water oxidation. These considerations led to the conclusion that proton expulsion during the initial phase of the S₃–S₀ transition starts with the deprotonation of the primary catalytic base, probably CP43-Arg357, followed by efficient proton egress involving the carboxyl group of D1-D61 in a process that constitutes the lag phase immediately prior to O₂ formation chemistry