By Russ Bubley

Randomized Algorithms discusses difficulties of good pedigree: counting and new release, either one of that are of primary significance to discrete arithmetic and likelihood. whilst asking questions like "How many are there?" and "What does it seem like on average?" of households of combinatorial constructions, solutions are frequently tricky to discover - we will be able to be blocked by means of possible intractable algorithms. Randomized Algorithms indicates the right way to get round the challenge of intractability with the Markov chain Monte Carlo process, in addition to highlighting the method's ordinary limits. It makes use of the means of coupling earlier than introducing "path coupling" a brand new method which appreciably simplifies and improves upon earlier equipment within the quarter.

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**Example text**

13) can be applied with β(x) = b x and γi (x, hi ) = −hi (ci + Wi x), where Wi is the row vector corresponding to the ith row of W . , its unnormalized log-probability) can be computed eﬃciently: FreeEnergy(x) = −b x − ehi (ci +Wi x) . 12)) due to the aﬃne form of Energy(x, h) with respect to h, we readily obtain a tractable expression for the conditional probability P (h|x): exp(b x + c h + h W x) ˜ ˜ ˜ exp(b x + c h + h W x) h i exp(ci hi + hi Wi x) ˜ ˜ ˜ exp(ci hi + hi Wi x) P (h|x) = = i = i hi exp(hi (ci + Wi x)) ˜ ˜ exp(hi (ci + Wi x)) hi P (hi |x).

2. Each hidden unit creates a tworegion partition of the input space (with a linear separation). When we consider the conﬁgurations of say three hidden units, there are eight corresponding possible intersections of three half-planes (by choosing each half-plane among the two half-planes associated with the linear separation performed by a hidden unit). , code). The binary setting of the hidden units thus identiﬁes one region in input space. For all x in one of these regions, P (h|x) is maximal for the corresponding h conﬁguration.

We know from experience that a two-layer network (one hidden layer) can be well trained in general, and that from the point of view of the top two layers in a deep network, they form a shallow network whose input is the output of the lower layers. Optimizing the last layer of a deep neural network is a convex optimization problem for the training criteria commonly used. Optimizing the last two layers, although not convex, is known to be much easier than optimizing a deep network (in fact when the number of hidden units goes to inﬁnity, the training criterion of a two-layer network can be cast as convex [18]).