Related papers: Computational aspects of the Volterra Signature
Modern approaches for learning from non-Markovian time series, such as recurrent neural networks, neural controlled differential equations or transformers, typically rely on implicit memory mechanisms that can be difficult to interpret or…
This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths.…
Volterra analysis and its variants have long been prominent among methods for modeling multi-input non-linear systems. The product of Volterra analysis, the Volterra kernels, are particularly suited to quantifying intra- and inter-input…
We propose the numerical methods for solution of the weakly regular linear and nonlinear evolutionary (Volterra) integral equation of the first kind. The kernels of such equations have jump discontinuities along the continuous curves…
Modeling nonlinear systems with Volterra series is challenging because the number of kernel coefficients grows exponentially with the model order. This work introduces Bayesian Tensor Network Volterra kernel machines (BTN-V), extending the…
Volterra series are especially useful for nonlinear system identification, also thanks to their capability to approximate a broad range of input-output maps. However, their identification from a finite set of data is hard, due to the curse…
We make an attempt to understanding convolutional neural network by exploring the relationship between (deep) convolutional neural networks and Volterra convolutions. We propose a novel approach to explain and study the overall…
The Volterra series is a powerful tool in modelling a broad range of nonlinear dynamic systems. However, due to its nonparametric nature, the number of parameters in the series increases rapidly with memory length and series order, with the…
We extend the new approach introduced in arXiv:1912.02064v2 [math.PR] and arXiv:2102.10119v1 [math.PR] for dealing with stochastic Volterra equations using the ideas of Rough Path theory and prove global existence and uniqueness results.…
A universal kernel is constructed whose sections approximate any causal and time-invariant filter in the fading memory category with inputs and outputs in a finite-dimensional Euclidean space. This kernel is built using the reservoir…
Volterra series representation is a powerful mathematical model for nonlinear circuits. However, the difficulties in determining higher-order Volterra kernels limited its broader applications. In this work, a systematic approach that…
There have been increasing interests on the Volterra series identification with the kernel-based regularization method. The major difficulties are on the kernel design and efficiency of the corresponding implementation. In this paper, we…
This paper studies existence and uniqueness of solutions to generalized Volterra integral equations. Since our proof for existence and uniqueness does not make use of Banach fixed point theorem unlike the previous papers focused on this…
The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretisation, the basic problem can be represented as a matrix-vector…
This work defines and studies one-dimensional convolution kernels that preserve nonnegativity. When the past dynamics of a process is integrated with a convolution kernel like in Stochastic Volterra Equations or in the jump intensity of…
We propose a deep structure encoder using the recently introduced Volterra Neural Networks (VNNs) to seek a latent representation of multi-modal data whose features are jointly captured by a union of subspaces. The so-called…
Higher-order learning is fundamentally rooted in exploiting compositional features. It clearly hinges on enriching the representation by more elaborate interactions of the data which, in turn, tends to increase the model complexity of…
This article introduces two Tensor Network-based iterative algorithms for the identification of high-order discrete-time nonlinear multiple-input multiple-output (MIMO) Volterra systems. The system identification problem is rewritten in…
The importance of inference in Machine Learning (ML) has led to an explosive number of different proposals in ML, and particularly in Deep Learning. In an attempt to reduce the complexity of Convolutional Neural Networks, we propose a…
Tensor algebras give rise to one of the most powerful measures of similarity for sequences of arbitrary length called the signature kernel accompanied with attractive theoretical guarantees from stochastic analysis. Previous algorithms to…