In this short note, we reify the connection between work on the storage capacity problem in wide two-layer treelike neural networks and the rapidly-growing body of literature on kernel limits of wide neural networks. Concretely, we observe that the "effective order parameter" studied in the statistical mechanics literature is exactly equivalent to the infinite-width Neural Network Gaussian Process Kernel. This correspondence connects the expressivity and trainability of wide two-layer neural networks.
The storage capacity of the extremely diluted Hopfield Model is studied by using Monte Carlo techniques. In this work, instead of diluting the synapses according to a given distribution, the dilution of the synapses is obtained systematically by retaining only the synapses with dominant contributions. It is observed that by using the prescribed dilution method the critical storage capacity of the system increases with decreasing number of synapses per neuron reaching almost the value obtained from mean-field calculations. It is also shown that the increase of the storage capacity of the dilute…
Synaptic efficacy between neurons is known to change within a short time scale dynamically. Neurophysiological experiments show that high-frequency presynaptic inputs decrease synaptic efficacy between neurons. This phenomenon is called synaptic depression, a short term synaptic plasticity. Many researchers have investigated how the synaptic depression affects the memory storage capacity. However, the noise has not been taken into consideration in their analysis. By introducing "temperature", which controls the level of the noise, into an update rule of neurons, we investigate the effects of s…
I.N.F.N Milano（国家核物理研究所米兰分所）；Università degli Studi di Milano（米兰大学）
Data structure has a dramatic impact on the properties of neural networks, yet its significance in the established theoretical frameworks is poorly understood. Here we compute the Vapnik-Chervonenkis entropy of a kernel machine operating on data grouped into equally labelled subsets. At variance with the unstructured scenario, entropy is non-monotonic in the size of the training set, and displays an additional critical point besides the storage capacity. Remarkably, the same behavior occurs in margin classifiers even with randomly labelled data, as is elucidated by identifying the synaptic vol…
We examine numerically the storage capacity and the behaviour near saturation of an attractor neural network consisting of bistable elements with an adjustable coupling strength, the Bistable Gradient Network (BGN). For strong coupling, we find evidence of a first-order "memory blackout" phase transition as in the Hopfield network. For weak coupling, on the other hand, there is no evidence of such a transition and memorized patterns can be stable even at high levels of loading. The enhanced storage capacity comes, however, at the cost of imperfect retrieval of the patterns from corrupted versi…
The storage capacity of an incremental learning algorithm for the parity machine, the Tilinglike Learning Algorithm, is analytically determined in the limit of a large number of hidden perceptrons. Different learning rules for the simple perceptron are investigated. The usual Gardner-Derrida one leads to a storage capacity close to the upper bound, which is independent of the learning algorithm considered.
Upper and lower bounds for the typical storage capacity of a constructive algorithm, the Tilinglike Learning Algorithm for the Parity Machine [M. Biehl and M. Opper, Phys. Rev. A {\bf 44} 6888 (1991)], are determined in the asymptotic limit of large training set sizes. The properties of a perceptron with threshold, learning a training set of patterns having a biased distribution of targets, needed as an intermediate step in the capacity calculation, are determined analytically. The lower bound for the capacity, determined with a cavity method, is proportional to the number of hidden units. The…
Recent studies point to the potential storage of a large number of patterns in the celebrated Hopfield associative memory model, well beyond the limits obtained previously. We investigate the properties of new fixed points to discover that they exhibit instabilities for small perturbations and are therefore of limited value as associative memories. Moreover, a large deviations approach also shows that errors introduced to the original patterns induce additional errors and increased corruption with respect to the stored patterns.
Harvard University（哈佛大学）；John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Center for Brain Science, Harvard University（哈佛大学脑科学中心）
The expressive power of artificial neural networks crucially depends on the nonlinearity of their activation functions. Though a wide variety of nonlinear activation functions have been proposed for use in artificial neural networks, a detailed understanding of their role in determining the expressive power of a network has not emerged. Here, we study how activation functions affect the storage capacity of treelike two-layer networks. We relate the boundedness or divergence of the capacity in the infinite-width limit to the smoothness of the activation function, elucidating the relationship be…
Solution space and storage capacity of fully connected two-layer neural networks with generic activation functions
School of Engineering, Tohoku University（东北大学工学院）；Graduate School of Information Sciences, Tohoku University（东北大学信息科学研究科）；Department of Physics, Tokyo Institute of Technology（东京工业大学物理系）；Sigma-i Co., Ltd.（Sigma-i有限公司）
We consider an ensemble of $K$ single-layer perceptrons exposed to random inputs and investigate the conditions under which the couplings of these perceptrons can be chosen such that prescribed correlations between the outputs occur. A general formalism is introduced using a multi-perceptron costfunction that allows to determine the maximal number of random inputs as a function of the desired values of the correlations. Replica-symmetric results for $K=2$ and $K=3$ are compared with properties of two-layer networks of tree-structure and fixed Boolean function between hidden units and output. T…
We study the storage of multiple phase-coded patterns as stable dynamical attractors in recurrent neural networks with sparse connectivity. To determine the synaptic strength of existent connections and store the phase-coded patterns, we introduce a learning rule inspired to the spike-timing dependent plasticity (STDP). We find that, after learning, the spontaneous dynamics of the network replay one of the stored dynamical patterns, depending on the network initialization. We study the network capacity as a function of topology, and find that a small- world-like topology may be optimal, as a c…
We introduce and analyze a minimal network model of semantic memory in the human brain. The model is a global associative memory structured as a collection of N local modules, each coding a feature, which can take S possible values, with a global sparseness a (the average fraction of features describing a concept). We show that, under optimal conditions, the number c of modules connected on average to a module can range widely between very sparse connectivity (c/N -> 0) and full connectivity (c = N), maintaining a global network storage capacity (the maximum number p of stored and retrievable …
It is known that storage capacity per synapse increases by synaptic pruning in the case of a correlation-type associative memory model. However, the storage capacity of the entire network then decreases. To overcome this difficulty, we propose decreasing the connecting rate while keeping the total number of synapses constant by introducing delayed synapses. In this paper, a discrete synchronous-type model with both delayed synapses and their prunings is discussed as a concrete example of the proposal. First, we explain the Yanai-Kim theory by employing the statistical neurodynamics. This theor…
We investigate the maximum number of embedded patterns in the two-dimensional Hopfield model. The grand state energies of two specific network states, namely, the energies of the pure-ferromagnetic state and the state of specific one stored pattern are calculated exactly in terms of the correlation function of the ferromagnetic Ising model. We also investigate the energy landscape around them by computer simulations. Taking into account the qualitative features of the phase diagrams obtained by Nishimori, Whyte and Sherrington [Phys. Rev. E {\bf 51}, 3628 (1995)], we conclude that the network …
We study analytically the effect of metrically structured connectivity on the behavior of autoassociative networks. We focus on three simple rate-based model neurons: threshold-linear, binary or smoothly saturating units. For a connectivity which is short range enough the threshold-linear network shows localized retrieval states. The saturating and binary models also exhibit spatially modulated retrieval states if the highest activity level that they can achieve is above the maximum activity of the units in the stored patterns. In the zero quenched noise limit, we derive an analytical formula …
The Hopfield model is a pioneering neural network model with associative memory retrieval. The analytical solution of the model in mean field limit revealed that memories can be retrieved without any error up to a finite storage capacity of $O(N)$, where $N$ is the system size. Beyond the threshold, they are completely lost. Since the introduction of the Hopfield model, the theory of neural networks has been further developed toward realistic neural networks using analog neurons, spiking neurons, etc. Nevertheless, those advances are based on fully connected networks, which are inconsistent wi…
We solve the dynamics of Hopfield-type neural networks which store sequences of patterns, close to saturation. The asymmetry of the interaction matrix in such models leads to violation of detailed balance, ruling out an equilibrium statistical mechanical analysis. Using generating functional methods we derive exact closed equations for dynamical order parameters, viz. the sequence overlap and correlation- and response functions, in the thermodynamic limit. We calculate the time translation invariant solutions of these equations, describing stationary limit-cycles, which leads to a phase diagra…
In this work we study, analytically and employing Monte Carlo simulations, the influence of the competition between several activity-dependent synaptic processes, such as short-term synaptic facilitation and depression, on the maximum memory storage capacity in a neural network. In contrast with the case of synaptic depression, which drastically reduces the capacity of the network to store and retrieve "static" activity patterns, synaptic facilitation enhances the storage capacity in different contexts. In particular, we found optimal values of the relevant synaptic parameters (such as the neu…
We study the problem of determining the capacity of the binary perceptron for two variants of the problem where the corresponding constraint is symmetric. We call these variants the rectangle-binary-perceptron (RPB) and the $u-$function-binary-perceptron (UBP). We show that, unlike for the usual step-function-binary-perceptron, the critical capacity in these symmetric cases is given by the annealed computation in a large region of parameter space (for all rectangular constraints and for narrow enough $u-$function constraints, $K<K^*$). We prove this fact (under two natural assumptions) using t…
A single McCulloch-Pitts neuron, that is, the simple perceptron is studied, with focus on the region beyond storage capacity. It is shown that Parisi's hierarchical ansatz for the overlap matrix of the synaptic couplings with so called continuous replica symmetry breaking is a solution, and as we propose it is the exact one, to the equilibrium problem. We describe some of the most salient features of the theory and give results about the low temperature region. In particular, the basics of the Parisi technique and the way to calculate thermodynamical expectation values is explained. We have nu…
The primate heteromodal cortex presents an evident functional modularity at a mesoscopic level, with physiological and anatomical evidence pointing to it as likely substrate of long-term memory. In order to investigate some of its properties, a model of multimodular autoassociator is studied. Each of the many modules represents a neocortical functional ensemble of recurrently connected neurons and operates as a Hebbian autoassociator, storing a number of local features which it can recall upon cue. The global memory patterns are made of combinations of features sparsely distributed across the …
We implement a mechanism - originally proposed as a model for the large memory storage capacity of black holes - in quantum neural networks and show that an exponentially increased capacity of pattern storage and recognition is achieved in certain critically excited states, without involvement of synaptic plasticity. We consider a simple network of N interconnected quantum neurons with weak excitatory synaptic connections. We show that for frozen synaptic weights there exist the critical states of enhanced memory storage capacity. These states are achieved thanks to the high excitation levels …
Classicalization is a phenomenon of redistribution of energy - initially stored in few hard quanta - into the high occupation numbers of the soft modes, described by a final state that is approximately classical. Using an effective Hamiltonian, we first show why the transition amplitudes that increase occupation numbers are exponentially suppressed and how a very special family of classicalizing theories compensates this suppression. This is thanks to a large micro-state entropy generated by the emergent gapless modes around the final classical state. The dressing of the process by the super-s…
