Volterra Series Research Articles

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Quantification of nonlinear output frequency responses for a general input based on volterra series and conditioned spectral analysis

Quantification of nonlinear output frequency responses for a general input based on volterra series and conditioned spectral analysis

Color image sparse adversarial samples generation via color channel Volterra expansion

Color image sparse adversarial samples generation via color channel Volterra expansion

Distributed reservoir computing based nonlinear equalizer for VCSEL based optical interconnects

We propose a distributed reservoir computing based nonlinear equalizer for 106 Gbps VCSEL based optical interconnects. By comparing with neural network and Volterra series based equalizers, the computational complexity of this proposed nonlinear equalizer is significantly reduced in the training, prediction and inference process. 106 Gbps PAM4 short-range optical transmission is demonstrated by experiment, using a VCSEL-based co-packaged optical transmitter to verify the performance of the equalizer and key parameters that affect the nonlinear compensation performance are carefully investigated. We believe that by incorporating such simplified nonlinear equalizers, it is possible to compensate for nonlinear impairments while preserving the inherent advantages of VCSEL-based optical interconnects. This optical interconnect will become a promising paradigm to overcome I/O performance penalty when communication leaving packages.

Inverse Modelling Techniques as Dynamic Digital Pre-Distortion for Broadband Systems for CS-DACs Based on Volterra Series Expansion

Inverse Modelling Techniques as Dynamic Digital Pre-Distortion for Broadband Systems for CS-DACs Based on Volterra Series Expansion

Local Independence Testing for Point Processes.

Constraint-based causal structure learning for point processes require empirical tests of local independence. Existing tests require strong model assumptions, e.g., that the true data generating model is a Hawkes process with no latent confounders. Even when restricting attention to Hawkes processes, latent confounders are a major technical difficulty because a marginalized process will generally not be a Hawkes process itself. We introduce an expansion similar to Volterra expansions as a tool to represent marginalized intensities. Our main theoretical result is that such expansions can approximate the true marginalized intensity arbitrarily well. Based on this, we propose a test of local independence and investigate its properties in real and simulated data.

Biomimetic Deep Learning Networks With Applications to Epileptic Spasms and Seizure Prediction.

In this study, we present a novel biomimetic deep learning network for epileptic spasms and seizure prediction and compare its performance with state-of-the-art conventional machine learning models. Our proposed model incorporates modular Volterra kernel convolutional networks and bidirectional recurrent networks in combination with the phase amplitude cross-frequency coupling features derived from scalp EEG. They are applied to the standard CHB-MIT dataset containing focal epilepsy episodes as well as two other datasets from the Montefiore Medical Center and the University of California Los Angeles that provide data of patients experiencing infantile spasm (IS) syndrome. Overall, in this study, the networks can produce accurate predictions (100%) and significant detection latencies (10 min). Furthermore, the biomimetic network outperforms conventional ones by producing no false positives. Biomimetic neural networks utilize extensive knowledge about processing and learning in the electrical networks of the brain. Predicting seizures in adults can improve their quality of life. Epileptic spasms in infants are part of a particular seizure type that needs identifying when suspicious behaviors are noticed in babies. Predicting epileptic spasms within a given time frame (the prediction horizon) suggests their existence and allows an epileptologist to flag an EEG trace for future review.

Universal scaling between wave speed and size enables nanoscale high-performance reservoir computing based on propagating spin-waves

Physical implementation of neuromorphic computing using spintronics technology has attracted recent attention for the future energy-efficient AI at nanoscales. Reservoir computing (RC) is promising for realizing the neuromorphic computing device. By memorizing past input information and its nonlinear transformation, RC can handle sequential data and perform time-series forecasting and speech recognition. However, the current performance of spintronics RC is poor due to the lack of understanding of its mechanism. Here we demonstrate that nanoscale physical RC using propagating spin waves can achieve high computational power comparable with other state-of-art systems. We develop the theory with response functions to understand the mechanism of high performance. The theory clarifies that wave-based RC generates Volterra series of the input through delayed and nonlinear responses. The delay originates from wave propagation. We find that the scaling of system sizes with the propagation speed of spin waves plays a crucial role in achieving high performance.

Truncated stochastically switching processes.

There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation(hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.

Volterra Black-Box Models Identification Methods: Direct Collocation vs. Least Squares

The Volterra integral-functional series is the classic approach for nonlinear black box dynamical system modeling. It is widely employed in many domains including radiophysics, aerodynamics, electronic and electrical engineering and many others. Identifying the time-varying functional parameters, also known as Volterra kernels, poses a difficulty due to the curse of dimensionality. This refers to the exponential growth in the number of model parameters as the complexity of the input-output response increases. The least squares method (LSM) is widely acknowledged as the standard approach for tackling the issue of identifying parameters. Unfortunately, the LSM suffers with many drawbacks such as the sensitivity to outliers causing biased estimation, multicollinearity, overfitting and inefficiency with large datasets. This paper presents an alternative approach based on direct estimation of the Volterra kernels using the collocation method. Two model examples are studied. It is found that the collocation method presents a promising alternative for optimization, surpassing the traditional least squares method when it comes to the Volterra kernels identification including the case when input and output signals suffer from considerable measurement errors.

An Active Track and Hold Circuit With Linearity Enhancement Technique and Its Analysis Using Volterra Series

An Active Track and Hold Circuit With Linearity Enhancement Technique and Its Analysis Using Volterra Series

Learning nonlinear systems via Volterra series and Hilbert-Schmidt operators

Learning nonlinear systems via Volterra series and Hilbert-Schmidt operators

Power law in Sandwiched Volterra Volatility model

The paper presents an analytical proof demonstrating that the Sandwiched Volterra Volatility (SVV) model is able to reproduce the power-law behavior of the at-the-money implied volatility skew, provided the correct choice of the Volterra kernel. To obtain this result, the second-order Malliavin differentiability of the volatility process is assessed and the conditions that lead to explosive behavior in the Malliavin derivative are investigated. As a supplementary result, a general Malliavin product rule is proved.

Linear and non-linear thermal system identification based on the integral of non-integer order — Application to solve inverse heat conduction linear and non-linear problems

The identification of thermal systems is an approach particularly well suited to solve inverse heat conduction problems (IHCP) in the case of complex systems where many parameters are unknown and where even the sensors must be modeled in order to obtain accurate results. In this paper, we propose significant advances in the formulation of the mathematical model structure by basing it exclusively on the non-integer order integration operator. We show that this approach is consistent with the heat diffusion process and that it allows to get rid of the step of pre-filtering the experimental signals to the sensors which was necessary when using the derivation operator. Demonstrations are made on the simulated configurations through simple analytical solutions and also a complex real configuration. I generalize the use of the non-integer integration operator in the treatment of non-linear heat conduction problems based on the Volterra series decomposition. We show in particular that the use of a generalized non-integer transfer function gives encouraging and generalizable results for a large class of related problems.

Analog image processing with nonlinear nonlocal flat optics

Digital signal processing has revolutionized many fields of science and engineering, but it still shows critical limits, mainly related to the complexity, power consumption, and limited speed of analogue-to-digital converters. A long-sought solution to overcome these hurdles is optical analog computing. In this regard, flat optics has been recently unveiled as a powerful platform to perform data processing in real-time, with low power consumption and a small footprint. So far, these explorations have been mainly limited to linear optics. Arguably, significantly more impact may be garnered from pushing this operation towards nonlinear processing of the incoming signals. In this context, we demonstrate here that nonlinear phenomena combined with engineered nonlocality in flat optics devices can be leveraged to synthesize Volterra kernels able to outperform linear optical analog image processing.

Integral Models in the Form of Volterra Polynomials and Continued Fractions in the Problem of Identifying Input Signals

The paper discusses the prospect of using a combined model based on finite segments (polynomials) of the Volterra integral power series. We consider a case when the problem of identifying the Volterra kernels is solved. The predictive properties of the classic Volterra polynomial are improved by adding a linear part in the form of an equivalent continued fraction. This technique allows us to distinguish an additional parameter—the connection coefficient α, which is effective in adapting the constructed integral model to changes in technical parameters at the input of a dynamic system. In addition, this technique allows us to take into account the case of perturbing the kernel of the linear term of the Volterra polynomial in the metric C[0,T] by a given value δ, implying the ideas of Volterra regularizing procedures. The problem of choosing the connection coefficient is solved using a special extremal problem. The developed algorithms are used to solve the problem of identifying input signals of test dynamic systems, among which, in addition to mathematical ones, thermal power engineering devices are used.

A Nonlinear Functional Link Multilayer Perceptron Using Volterra Series as an Adaptive Noise Canceler for the Extraction of Fetal Electrocardiogram.

Uninterrupted monitoring of fetal cardiac health is essential for the timely diagnosis of congenital diseases. The maternal Electrocardiogram (mECG), which has the most significant impact, always tampers with the signals collected from the pregnant woman's abdomen. So, an efficient nonlinear filtering network based on artificial neural network (ANN) is required to eliminate the maternal part from the abdominal Electrocardiogram (aECG) that is traveled from the thoracic of the mother to the abdomen following nonlinear dynamics. In this work, we have presented an adaptive noise canceler (ANC) using 3-layer perceptron architecture where the inputs are expanded by the functional link expansion using the second-order Volterra series, and the weights are updated using backpropagation. The adaptive filter approximates the nonlinear mapping between the thoracic Electrocardiogram (tECG) and the maternal component present in the aECG. Here the thoracic signal is the reference signal, and the abdominal signal is the desired signal to the adaptive filter. The proposed methodology uses the advantages of both multilayer perceptron (MLP) as well as functional link neural network (FLNN) in mapping the nonlinearity and effectively determining the fetal Electrocardiogram (fECG) from the aECG. For the detailed analysis, we have used the real Daisy database, the Non-invasive Fetal ECG database, and the fetal ECG synthetic database from Physionet. The results show that the nonlinear functional link MLP using the Volterra series gives a high-level performance compared to other classical adaptive filtering techniques, as all the evaluation metrics are above 90%.

A Simplified Volterra Equalizer Based on System Characteristics for Direct Modulation Laser (DML)-Based Intensity Modulation and Direct Detection (IM/DD) Transmission Systems

The nonlinear Volterra equalizer has been proved to be able to solve the problem of nonlinear distortion, but it has high computational complexity and is difficult to implement. In this paper, a simplified second-order Volterra nonlinear equalizer designed for intensity modulation/direct detection systems based on direct modulated laser is proposed and demonstrated, taking into account the characteristics of the system. It has been proved that the received signal of direct modulation laser/direct detection system can be expressed in Volterra series form, but its form is too complex, and the device parameters should also be considered. We re-derived it and obtained a more concise form. At the same time, we proposed a method to simplify the second-order Volterra nonlinear equalizer without relying on device parameters. The performance of the proposed Volterra nonlinear equalizer is evaluated experimentally on a 56 Gb/s 4-ary pulse amplitude modulation link implemented by using a 1.55 µm direct modulation laser. The results show that, compared with the traditional Volterra nonlinear equalizer, the receiver sensitivity of the equalizer is only reduced by 0.2 dB at most, but the complexity can be reduced by 50%; compared with diagonally pruned Volterra nonlinear equalizers, the complexity of the equalizer is the same, but the reception sensitivity can be improved by 0.5 dB.

Identification of the parametric array loudspeaker system using differential Volterra filter

The self-demodulation property of a parametric array loudspeaker (PAL) system can peel off audible sound modulated on ultrasonic wave and deliver it with high directivity, but also introduces nonlinear distortion. The PAL system can be modeled by the Volterra series expansion in any acoustic field region. However, measurements of Volterra kernel using simple frequency sweep excitation signals may not provide adequate information about the nonlinear system, and the use of wideband signals such as speech signals makes convergence difficult. Based on the Westervelt model, we propose a PAL system identification method utilizing the differential Volterra kernel. The results demonstrate superior fitting performance for real swept frequency output data and speech signal output data compared to the existing one-dimensional Volterra filter (ODVF). Additionally, the proposed method exhibits fewer parameters and faster convergence speed than the traditional volterra kernel.

Residual Tensor Train: A Quantum-Inspired Approach for Learning Multiple Multilinear Correlations

States of quantum many-body systems are defined in a high-dimensional Hilbert space, where rich and complex interactions among subsystems can be modelled. In machine learning, complex multiple multilinear correlations may also exist within input features. In this paper, we present a quantum-inspired multilinear model, named Residual Tensor Train (ResTT), to capture the multiple multilinear correlations of features, from low to high orders, within a single model. ResTT is able to build a robust decision boundary in a high-dimensional space for solving fitting and classification tasks. In particular, we prove that the fully-connected layer and the Volterra series can be taken as special cases of ResTT. Furthermore, we derive the rule for weight initialization that stabilizes the training of ResTT based on a mean-field analysis. We prove that such a rule is much more relaxed than that of TT, which means ResTT can easily address the vanishing and exploding gradient problem that exists in the existing TT models. Numerical experiments demonstrate that ResTT outperforms the state-of-the-art tensor network and benchmark deep learning models on MNIST and Fashion-MNIST datasets. Moreover, ResTT achieves better performance than other statistical methods on two practical examples with limited data which are known to have complex feature interactions.