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- W2595431794 abstract "This technical report is extends the conference paper [1] and the abstract [2] with detailed derivations and proofs. First we recap notation that we use on the Boltzmann machine and its learning. Then we define transformations for the machine where some of its bits are flipped for all samples, and show the equivalence of the transformed model to the original one. Then we show that traditional update rules are not invariant to the transformations, propose a new update rule called the enhanced gradient, and finally show its invariance to the transformations. 1 Boltzmann Machine Boltzmann machine has a binary state column vector x, of which part is observed (visible) and part is latent (hidden) x = [ xv xh ] . The machine has parameters θ which include a square weight matrix W and a bias column vector b. The state vector x varies for each sample1, whereas parameters are constant. The probability of the machine being in state x is defined as P (x | θ) = 1 Z(θ) exp [−E(x | θ)] (1) E(x | θ) = − 2 xWx− bx, (2) whereZ(θ) = ∑ x exp [−E(x | θ)] is a normalizing constant called the partition function, and the E(x | θ) is called the energy function. The weight matrix W is symmetric (Wij = Wji) and the diagonal elements are zero (Wii = 0), that is, connections between units are undirected, and a unit is not connected to itself. 2 Traditional Gradient Parameters are typically learned to maximize likelihood. Data set or distribution d contains samples of the visible part of the state xv . The traditional gradient is obtained 1We do not use sample indices in the notation of this report." @default.
- W2595431794 created "2017-03-23" @default.
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- W2595431794 date "2011-01-01" @default.
- W2595431794 modified "2023-09-26" @default.
- W2595431794 title "Derivations of the Enhanced Gradient for the Boltzmann Machine" @default.
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- W2595431794 hasPublicationYear "2011" @default.
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