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

Towards a Calculus of Echo State Networks

Reservoir computing is a recent trend in neural networks which uses the dynamical perturbations on the phase space of a system to compute a desired target function. We present how one can formulate an expectation of system performance in a simple class of reservoir computing called echo state networks. In contrast with previous theoretical frameworks, which only reveal an upper bound on the total memory in the system, we analytically calculate the entire memory curve as a function of the structure of the system and the properties of the input and the target function. We demonstrate the precision of our framework by validating its result for a wide range of system sizes and spectral radii. Our analytical calculation agrees with numerical simulations. To the best of our knowledge this work presents the first exact analytical characterization of the memory curve in echo state networks

Theory and Practice of Computing with Excitable Dynamics

Reservoir computing (RC) is a promising paradigm for time series processing. In this paradigm, the desired output is computed by combining measurements of an excitable system that responds to time-dependent exogenous stimuli. The excitable system is called a reservoir and measurements of its state are combined using a readout layer to produce a target output. The power of RC is attributed to an emergent short-term memory in dynamical systems and has been analyzed mathematically for both linear and nonlinear dynamical systems. The theory of RC treats only the macroscopic properties of the reservoir, without reference to the underlying medium it is made of. As a result, RC is particularly attractive for building computational devices using emerging technologies whose structure is not exactly controllable, such as self-assembled nanoscale circuits. RC has lacked a formal framework for performance analysis and prediction that goes beyond memory properties. To provide such a framework, here a mathematical theory of memory and information processing in ordered and disordered linear dynamical systems is developed. This theory analyzes the optimal readout layer for a given task. The focus of the theory is a standard model of RC, the echo state network (ESN). An ESN consists of a fixed recurrent neural network that is driven by an external signal. The dynamics of the network is then combined linearly with readout weights to produce the desired output. The readout weights are calculated using linear regression. Using an analysis of regression equations, the readout weights can be calculated using only the statistical properties of the reservoir dynamics, the input signal, and the desired output. The readout layer weights can be calculated from a priori knowledge of the desired function to be computed and the weight matrix of the reservoir. This formulation explicitly depends on the input weights, the reservoir weights, and the statistics of the target function. This formulation is used to bound the expected error of the system for a given target function. The effects of input-output correlation and complex network structure in the reservoir on the computational performance of the system have been mathematically characterized. Far from the chaotic regime, ordered linear networks exhibit a homogeneous decay of memory in different dimensions, which keeps the input history coherent. As disorder is introduced in the structure of the network, memory decay becomes inhomogeneous along different dimensions causing decoherence in the input history, and degradation in task-solving performance. Close to the chaotic regime, the ordered systems show loss of temporal information in the input history, and therefore inability to solve tasks. However, by introducing disorder and therefore heterogeneous decay of memory the temporal information of input history is preserved and the task-solving performance is recovered. Thus for systems at the edge of chaos, disordered structure may enhance temporal information processing. Although the current framework only applies to linear systems, in principle it can be used to describe the properties of physical reservoir computing, e.g., photonic RC using short coherence-length light

On the Effect of Heterogeneity on the Dynamics and Performance of Dynamical Networks

The high cost of processor fabrication plants and approaching physical limits have started a new wave research in alternative computing paradigms. As an alternative to the top-down manufactured silicon-based computers, research in computing using natural and physical system directly has recently gained a great deal of interest. A branch of this research promotes the idea that any physical system with sufficiently complex dynamics is able to perform computation. The power of networks in representing complex interactions between many parts make them a suitable choice for modeling physical systems. Many studies used networks with a homogeneous structure to describe the computational circuits. However physical systems are inherently heterogeneous. We aim to study the effect of heterogeneity in the dynamics of physical systems that pertains to information processing. Two particularly well-studied network models that represent information processing in a wide range of physical systems are Random Boolean Networks (RBN), that are used to model gene interactions, and Liquid State Machines (LSM), that are used to model brain-like networks. In this thesis, we study the effects of function heterogeneity, in-degree heterogeneity, and interconnect irregularity on the dynamics and the performance of RBN and LSM. First, we introduce the model parameters to characterize the heterogeneity of components in RBN and LSM networks. We then quantify the effects of heterogeneity on the network dynamics. For the three heterogeneity aspects that we studied, we found that the effect of heterogeneity on RBN and LSM are very different. We find that in LSM the in-degree heterogeneity decreases the chaoticity in the network, whereas it increases chaoticity in RBN. For interconnect irregularity, heterogeneity decreases the chaoticity in LSM while its effects on RBN the dynamics depends on the connectivity. For {K} \u3c 2, heterogeneity in the interconnect will increase the chaoticity in the dynamics and for {K} \u3e 2 it decreases the chaoticity. We find that function heterogeneity has virtually no effect on the LSM dynamics. In RBN however, function heterogeneity actually makes the dynamics predictable as a function of connectivity and heterogeneity in the network structure. We hypothesize that node heterogeneity in RBN may help signal processing because of the variety of signal decomposition by different nodes

Hierarchical Composition of Memristive Networks for Real-Time Computing

Advances in materials science have led to physical instantiations of self-assembled networks of memristive devices and demonstrations of their computational capability through reservoir computing. Reservoir computing is an approach that takes advantage of collective system dynamics for real-time computing. A dynamical system, called a reservoir, is excited with a time-varying signal and observations of its states are used to reconstruct a desired output signal. However, such a monolithic assembly limits the computational power due to signal interdependency and the resulting correlated readouts. Here, we introduce an approach that hierarchically composes a set of interconnected memristive networks into a larger reservoir. We use signal amplification and restoration to reduce reservoir state correlation, which improves the feature extraction from the input signals. Using the same number of output signals, such a hierarchical composition of heterogeneous small networks outperforms monolithic memristive networks by at least 20% on waveform generation tasks. On the NARMA-10 task, we reduce the error by up to a factor of 2 compared to homogeneous reservoirs with sigmoidal neurons, whereas single memristive networks are unable to produce the correct result. Hierarchical composition is key for solving more complex tasks with such novel nano-scale hardware

Learning, Generalization, and Functional Entropy in Random Automata Networks

It has been shown \citep{broeck90:physicalreview,patarnello87:europhys} that feedforward Boolean networks can learn to perform specific simple tasks and generalize well if only a subset of the learning examples is provided for learning. Here, we extend this body of work and show experimentally that random Boolean networks (RBNs), where both the interconnections and the Boolean transfer functions are chosen at random initially, can be evolved by using a state-topology evolution to solve simple tasks. We measure the learning and generalization performance, investigate the influence of the average node connectivity $K$, the system size $N$, and introduce a new measure that allows to better describe the network's learning and generalization behavior. We show that the connectivity of the maximum entropy networks scales as a power-law of the system size $N$. Our results show that networks with higher average connectivity $K$ (supercritical) achieve higher memorization and partial generalization. However, near critical connectivity, the networks show a higher perfect generalization on the even-odd task