Statistical mechanics of an error correcting code using monotonic and nonmonotonic treelike multilayer perceptrons.

An error correcting code using a treelike multilayer perceptron is proposed. An original message s0 is encoded into a codeword y0 using a treelike committee machine (committee tree) or a treelike parity machine (parity tree). Based on these architectures, several schemes featuring monotonic or nonmonotonic units are introduced. The codeword y0 is then transmitted via a binary asymmetric channel where it is corrupted by noise. The analytical performance of these schemes is investigated using the replica method of statistical mechanics. Under some specific conditions, some of the proposed schemes are shown to saturate the Shannon bound at the infinite codeword length limit. The influence of the monotonicity of the units on the performance is also discussed.

Statistical mechanics of lossy compression for nonmonotonic multilayer perceptrons.

A lossy data compression scheme for uniformly biased Boolean messages is investigated via statistical mechanics techniques. We utilize a treelike committee machine (committee tree) and a treelike parity machine (parity tree) whose transfer functions are nonmonotonic. The scheme performance at the infinite code length limit is analyzed using the replica method. Both committee and parity treelike networks are shown to saturate the Shannon bound. The Almeida-Thouless stability of the replica symmetric solution is analyzed, and the tuning of the nonmonotonic transfer function is also discussed.

Dynamics of learning in multilayer perceptrons near singularities.

The dynamical behavior of learning is known to be very slow for the multilayer perceptron, being often trapped in the "plateau." It has been recently understood that this is due to the singularity in the parameter space of perceptrons, in which trajectories of learning are drawn. The space is Riemannian from the point of view of information geometry and contains singular regions where the Riemannian metric or the Fisher information matrix degenerates. This paper analyzes the dynamics of learning in a neighborhood of the singular regions when the true teacher machine lies at the singularity. We give explicit asymptotic analytical solutions (trajectories) both for the standard gradient (SGD) and natural gradient (NGD) methods. It is clearly shown, in the case of the SGD method, that the plateau phenomenon appears in a neighborhood of the critical regions, where the dynamical behavior is extremely slow. The analysis of the NGD method is much more difficult, because the inverse of the Fisher information matrix diverges. We conquer the difficulty by introducing the "blow-down" technique used in algebraic geometry. The NGD method works efficiently, and the state converges directly to the true parameters very quickly while it staggers in the case of the SGD method. The analytical results are compared with computer simulations, showing good agreement. The effects of singularities on learning are thus qualitatively clarified for both standard and NGD methods.

Dynamics of learning near singularities in layered networks.

We explicitly analyze the trajectories of learning near singularities in hierarchical networks, such as multilayer perceptrons and radial basis function networks, which include permutation symmetry of hidden nodes, and show their general properties. Such symmetry induces singularities in their parameter space, where the Fisher information matrix degenerates and odd learning behaviors, especially the existence of plateaus in gradient descent learning, arise due to the geometric structure of singularity. We plot dynamic vector fields to demonstrate the universal trajectories of learning near singularities. The singularity induces two types of plateaus, the on-singularity plateau and the near-singularity plateau, depending on the stability of the singularity and the initial parameters of learning. The results presented in this letter are universally applicable to a wide class of hierarchical models. Detailed stability analysis of the dynamics of learning in radial basis function networks and multilayer perceptrons will be presented in separate work.