A central challenge in machine learning is to distinguish genuine structure from chance correlations in high-dimensional data. In this work, we address this issue for the perceptron, a foundational model of neural computation. Specifically, we investigate the relationship between the pattern load $α$ and the variable selection ratio $ρ$ for which a simple perceptron can perfectly classify $P = αN$ random patterns by optimally selecting $M = ρN$ variables out of $N$ variables. While the Cover--Gardner theory establishes that a random subset of $ρN$ dimensions can separate $αN$ random patterns if and only if $α< 2ρ$, we demonstrate that optimal variable selection can surpass this bound by developing a method, based on the replica method from statistical mechanics, for enumerating the combinations of variables that enable perfect pattern classification. This not only provides a quantitative criterion for distinguishing true structure in the data from spurious regularities, but also yields the storage capacity of associative memory models with sparse asymmetric couplings.

Shun-ichi Amari RIKEN Frontier Research Program, RIKEN, Hirosawa 2-1, Wako-shi 351-01, Japan amari@zoo.riken.go.jp Abstract The parameter space of neural networks has a Riemannian metric structure.The natural Riemannian gradient should be used instead of the conventional gradient, since the former denotes the true steepest descent direction of a loss function in the Riemannian space. The behavior of the stochastic gradient learning algorithm is much more effective if the natural gradient is used. The present paper studies the information-geometrical structure of perceptrons and other networks, and prove that the online learning method based on the natural gradient is asymptotically as efficient as the optimal batch algorithm. Adaptive modification of the learning constant is proposed and analyzed in terms of the Riemannian measure andis shown to be efficient. The natural gradient is finally applied to blind separation of mixtured independent signal sources. 1 Introd uction Neural learning takes place in the parameter space of modifiable synaptic weights of a neural network.

The parameter space of neural networks has a Riemannian metric structure. The natural Riemannian gradient should be used instead of the conventional gradient, since the former denotes the true steepest descent direction of a loss function in the Riemannian space. The behavior of the stochastic gradient learning algorithm is much more effective if the natural gradient is used. The present paper studies the information-geometrical structure of perceptrons and other networks, and prove that the online learning method based on the natural gradient is asymptotically as efficient as the optimal batch algorithm. Adaptive modification of the learning constant is proposed and analyzed in terms of the Riemannian measure and is shown to be efficient.

The parameter space of neural networks has a Riemannian metric structure. The natural Riemannian gradient should be used instead of the conventional gradient, since the former denotes the true steepest descent direction of a loss function in the Riemannian space. The behavior of the stochastic gradient learning algorithm is much more effective if the natural gradient is used. The present paper studies the information-geometrical structure of perceptrons and other networks, and prove that the online learning method based on the natural gradient is asymptotically as efficient as the optimal batch algorithm. Adaptive modification of the learning constant is proposed and analyzed in terms of the Riemannian measure and is shown to be efficient.

Recently, several researchers have reported encouraging experimental results whenusing Gaussian or bump-like activation functions in multilayer perceptrons. Networks of this type usually require fewer hidden layers and units and often learn much faster than typical sigmoidal networks. To explain these results we consider a hyper-ridge network, which is a simple perceptron with no hidden units and a rid¥e activation function. If we are interested in partitioningp points in d dimensions into two classes then in the limit as d approaches infinity the capacity of a hyper-ridge and a perceptron is identical.

Recently, several researchers have reported encouraging experimental results when using Gaussian or bump-like activation functions in multilayer perceptrons. Networks of this type usually require fewer hidden layers and units and often learn much faster than typical sigmoidal networks. To explain these results we consider a hyper-ridge network, which is a simple perceptron with no hidden units and a rid¥e activation function. If we are interested in partitioningp points in d dimensions into two classes then in the limit as d approaches infinity the capacity of a hyper-ridge and a perceptron is identical.

Recently, several researchers have reported encouraging experimental results when using Gaussian or bump-like activation functions in multilayer perceptrons. Networks of this type usually require fewer hidden layers and units and often learn much faster than typical sigmoidal networks. To explain these results we consider a hyper-ridge network, which is a simple perceptron with no hidden units and a rid¥e activation function. If we are interested in partitioningp points in d dimensions into two classes then in the limit as d approaches infinity the capacity of a hyper-ridge and a perceptron is identical.

We use statistical mechanics to study generalization in large committee machines.For an architecture with nonoverlapping receptive fields a replica calculation yields the generalization error in the limit of a large number of hidden units.

We use statistical mechanics to study generalization in large committee machines. For an architecture with nonoverlapping receptive fields a replica calculation yields the generalization error in the limit of a large number of hidden units.

We use statistical mechanics to study generalization in large committee machines. For an architecture with nonoverlapping receptive fields a replica calculation yields the generalization error in the limit of a large number of hidden units.