![]() The degree to which optimality is approached by simple learning rules in current use is considered, and it is found, in particular, that the algorithm adopted in the Hopfield model is more effective in minimizing G than the original Hebb law. Generalizing inequality relating Euclidean distance & Frobenius norm to Bregman divergences such as relative entropy & von Neumann divergence. Given that the quantum ergotropy can be expressed as the difference of the quantum and classical relative entropies, we identified three distinct contributions to the coherent ergotropy, of which the relative entropy of coherence and the population mismatch between thermal state and fully decohered state are the most important. We will use the convention that 0 log 00, which is easilyjustied by continuity sincexlogx0. For example, the entropy of a fair cointoss is 1 bit. The log is to the base 2and entropy is expressed in bits. (2.1) xX We also writeH(p) for the above quantity. Minimization of G subject to appropriate resource constraints leads to ‘‘optimal’’ learning rules for pairwise and higher-order neuronal interactions. DenitionThe entropy H(X) of a discrete random variable Xisdened by H(X)p(x)logp(x). The relative entropy G of the probability distribution δ x ( s + 1 ), x ’ concentrated at the desired successor state, evaluated with respect to the dynamical distribution ν( x ’‖ x ( s )), is used to quantify this criterion, by providing a measure of the distance between actual and ideal probability distributions. KL Divergence or Relative Entropy is a measure of how two distributions are different. A successful procedure for learning this pattern must modify the neuronal interactions in such a way that the dynamical successor of x ( s ) is likely to be x ( s + 1 ), with x ( l + 1 )= x ( 1 ). A prescribed memory or behavior pattern is represented in terms of an ordered sequence of network states x ( 1 ), x ( 2 ). The relative entropy (also known as the Kullback-Leibler divergence) is a measure of how different two probability distributions (over the same event space). Extensive simulation studies on challenging multimodal 1D and 2D distributions and Bayesian logistic regression on real datasets demonstrate that the REGS outperforms the state-of-the-art sampling methods included in the comparison.The dynamics of a probabilistic neural network is characterized by the distribution ν( x ’‖ x) of successor states x ’ of an arbitrary state x of the network. We propose a novel nonparametric approach to estimating the logarithmic density ratio using neural networks. Relative entropy techniques are robust, compelling, and can be applied to many physical situations. To sample with REGS, we need to estimate the density ratios and simulate the ODE system with particle evolution. It is characterized by an ODE system with velocity fields depending on the density ratios of the density of evolving particles and the unnormalized target density. This gradient flow determines a path of probability distributions that interpolates the reference distribution and the target distribution. To determine the nonlinear transforms at each iteration, we consider the Wasserstein gradient flow of relative entropy. Relative Entropy and Mutual Information in Gaussian Statistical Field Theory. REGS is a particle method that seeks a sequence of simple nonlinear transforms iteratively pushing the initial samples from a reference distribution into the samples from an unnormalized target distribution. Abstract: We propose a relative entropy gradient sampler (REGS) for sampling from unnormalized distributions. Relative entropy and the convergence of the posterior and empirical distributions under incomplete and conflicting information. ![]()
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