Memory in reservoirs for high dimensional input

Reservoir Computing (RC) is a recently introduced scheme to employ recurrent neural networks while circumventing the difficulties that typically appear when training the recurrent weights. The ‘reservoir’ is a fixed randomly initiated recurrent network which receives input via a random mapping. Only an instantaneous linear mapping from the network to the output is trained which can be done with linear regression. In this paper we study dynamical properties of reservoirs receiving a high number of inputs. More specifically, we investigate how the internal state of the network retains fading memory of its input signal. Memory properties for random recurrent networks have been thoroughly examined in past research, but only for one-dimensional input. Here we take into account statistics which will typically occur in high dimensional signals. We find useful empirical data which expresses how memory in recurrent networks is distributed over the individual principal components of the input

The spectral radius remains a valid indicator of the echo state property for large reservoirs

In the field of Reservoir Computing, scaling the spectral radius of the weight matrix of a random recurrent neural network to below unity is a commonly used method to ensure the Echo State Property. Recently it has been shown that this condition is too weak. To overcome this problem, other more involved - sufficient conditions for the Echo State Property have been proposed. In this paper we provide a large-scale experimental verification of the Echo State Property for large recurrent neural networks with zero input and zero bias. Our main conclusion is that the spectral radius method remains a valid indicator of the Echo State Property; the probability that the Echo State Property does not hold, drops for larger networks with spectral radius below unity, which are the ones of practical interest

Optimal modularity and memory capacity of neural reservoirs

The neural network is a powerful computing framework that has been exploited by biological evolution and by humans for solving diverse problems. Although the computational capabilities of neural networks are determined by their structure, the current understanding of the relationships between a neural network's architecture and function is still primitive. Here we reveal that neural network's modular architecture plays a vital role in determining the neural dynamics and memory performance of the network of threshold neurons. In particular, we demonstrate that there exists an optimal modularity for memory performance, where a balance between local cohesion and global connectivity is established, allowing optimally modular networks to remember longer. Our results suggest that insights from dynamical analysis of neural networks and information spreading processes can be leveraged to better design neural networks and may shed light on the brain's modular organization

Optoelectronic Reservoir Computing

Reservoir computing is a recently introduced, highly efficient bio-inspired approach for processing time dependent data. The basic scheme of reservoir computing consists of a non linear recurrent dynamical system coupled to a single input layer and a single output layer. Within these constraints many implementations are possible. Here we report an opto-electronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line. Our implementation is sufficiently fast for real time information processing. We illustrate its performance on tasks of practical importance such as nonlinear channel equalization and speech recognition, and obtain results comparable to state of the art digital implementations.Comment: Contains main paper and two Supplementary Material

Product Reservoir Computing: Time-Series Computation with Multiplicative Neurons

Echo state networks (ESN), a type of reservoir computing (RC) architecture, are efficient and accurate artificial neural systems for time series processing and learning. An ESN consists of a core of recurrent neural networks, called a reservoir, with a small number of tunable parameters to generate a high-dimensional representation of an input, and a readout layer which is easily trained using regression to produce a desired output from the reservoir states. Certain computational tasks involve real-time calculation of high-order time correlations, which requires nonlinear transformation either in the reservoir or the readout layer. Traditional ESN employs a reservoir with sigmoid or tanh function neurons. In contrast, some types of biological neurons obey response curves that can be described as a product unit rather than a sum and threshold. Inspired by this class of neurons, we introduce a RC architecture with a reservoir of product nodes for time series computation. We find that the product RC shows many properties of standard ESN such as short-term memory and nonlinear capacity. On standard benchmarks for chaotic prediction tasks, the product RC maintains the performance of a standard nonlinear ESN while being more amenable to mathematical analysis. Our study provides evidence that such networks are powerful in highly nonlinear tasks owing to high-order statistics generated by the recurrent product node reservoir

Exploring Transfer Function Nonlinearity in Echo State Networks

Supralinear and sublinear pre-synaptic and dendritic integration is considered to be responsible for nonlinear computation power of biological neurons, emphasizing the role of nonlinear integration as opposed to nonlinear output thresholding. How, why, and to what degree the transfer function nonlinearity helps biologically inspired neural network models is not fully understood. Here, we study these questions in the context of echo state networks (ESN). ESN is a simple neural network architecture in which a fixed recurrent network is driven with an input signal, and the output is generated by a readout layer from the measurements of the network states. ESN architecture enjoys efficient training and good performance on certain signal-processing tasks, such as system identification and time series prediction. ESN performance has been analyzed with respect to the connectivity pattern in the network structure and the input bias. However, the effects of the transfer function in the network have not been studied systematically. Here, we use an approach tanh on the Taylor expansion of a frequently used transfer function, the hyperbolic tangent function, to systematically study the effect of increasing nonlinearity of the transfer function on the memory, nonlinear capacity, and signal processing performance of ESN. Interestingly, we find that a quadratic approximation is enough to capture the computational power of ESN with tanh function. The results of this study apply to both software and hardware implementation of ESN.Comment: arXiv admin note: text overlap with arXiv:1502.0071