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Introduction to the Theory of Formal Groups by Jean A. Dieudonne

By Jean A. Dieudonne

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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]).

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