# Joint Signal Estimation and Nonlinear Topology Identification from Noisy Data with Missing Entries

31 Oct 2022-

TL;DR: In this article , a non convex optimization problem is formed with lasso regularization and solved via block coordinate descent (BCD) for joint signal estimation and topology identification with a nonlinear model, under the modelling assumption that signals are generated by a sparse VAR model in a latent space and then transformed by a set of invertible, componentwise nonlinearities.

Abstract: Topology identification from multiple time series has been proved to be useful for system identification, anomaly detection, denoising, and data completion. Vector autoregressive (VAR) methods have proved well in identifying directed topology from complex networks. The task of inferring topology in the presence of noise and missing observations has been studied for linear models. As a first approach to joint signal estimation and topology identification with a nonlinear model, this paper proposes a method to do so under the modelling assumption that signals are generated by a sparse VAR model in a latent space and then transformed by a set of invertible, component-wise nonlinearities. A non convex optimization problem is formed with lasso regularisation and solved via block coordinate descent (BCD). Initial experiments conducted on synthetic data sets show the identifying capability of the proposed method.

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TL;DR: It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.

Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p-penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.

4,339 citations

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25 Apr 2018TL;DR: The main goal of this paper is to outline overarching advances, and develop a principled framework to capture nonlinearities through kernels, which are judiciously chosen from a preselected dictionary to optimally fit the data.

Abstract: Identifying graph topologies as well as processes evolving over graphs emerge in various applications involving gene-regulatory, brain, power, and social networks, to name a few. Key graph-aware learning tasks include regression, classification, subspace clustering, anomaly identification, interpolation, extrapolation, and dimensionality reduction. Scalable approaches to deal with such high-dimensional tasks experience a paradigm shift to address the unique modeling and computational challenges associated with data-driven sciences. Albeit simple and tractable, linear time-invariant models are limited since they are incapable of handling generally evolving topologies, as well as nonlinear and dynamic dependencies between nodal processes. To this end, the main goal of this paper is to outline overarching advances, and develop a principled framework to capture nonlinearities through kernels, which are judiciously chosen from a preselected dictionary to optimally fit the data. The framework encompasses and leverages (non) linear counterparts of partial correlation and partial Granger causality, as well as (non)linear structural equations and vector autoregressions, along with attributes such as low rank, sparsity, and smoothness to capture even directional dependencies with abrupt change points, as well as time-evolving processes over possibly time-evolving topologies. The overarching approach inherits the versatility and generality of kernel-based methods, and lends itself to batch and computationally affordable online learning algorithms, which include novel Kalman filters over graphs. Real data experiments highlight the impact of the nonlinear and dynamic models on consumer and financial networks, as well as gene-regulatory and functional connectivity brain networks, where connectivity patterns revealed exhibit discernible differences relative to existing approaches.

187 citations

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TL;DR: This paper proposed a class of nonlinear methods by applying structured multilayer perceptrons (MLP) or recurrent neural networks (RNNs) combined with sparsityinducing penalties on the weights.

Abstract: While most classical approaches to Granger causality detection assume linear dynamics, many interactions in applied domains, like neuroscience and genomics, are inherently nonlinear. In these cases, using linear models may lead to inconsistent estimation of Granger causal interactions. We propose a class of nonlinear methods by applying structured multilayer perceptrons (MLPs) or recurrent neural networks (RNNs) combined with sparsity-inducing penalties on the weights. By encouraging specific sets of weights to be zero---in particular through the use of convex group-lasso penalties---we can extract the Granger causal structure. To further contrast with traditional approaches, our framework naturally enables us to efficiently capture long-range dependencies between series either via our RNNs or through an automatic lag selection in the MLP. We show that our neural Granger causality methods outperform state-of-the-art nonlinear Granger causality methods on the DREAM3 challenge data. This data consists of nonlinear gene expression and regulation time courses with only a limited number of time points. The successes we show in this challenging dataset provide a powerful example of how deep learning can be useful in cases that go beyond prediction on large datasets. We likewise illustrate our methods in detecting nonlinear interactions in a human motion capture dataset.

84 citations

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TL;DR: In this article, a batch solver is proposed that iterates between inferring directed graphs that best fit the sequence of observations, and estimating the network processes at reduced computational complexity by leveraging tools related to Kalman smoothing.

Abstract: A task of major practical importance in network science is inferring the graph structure from noisy observations at a subset of nodes. Available methods for topology inference typically assume that the process over the network is observed at all nodes. However, application-specific constraints may prevent acquiring network-wide observations. Alleviating the limited flexibility of existing approaches, this work advocates structural models for graph processes and develops novel algorithms for joint inference of the network topology and processes from partial nodal observations. Structural equation models (SEMs) and structural vector autoregressive models (SVARMs) have well documented merits in identifying even directed topologies of complex graphs; while SEMs capture contemporaneous causal dependencies among nodes, SVARMs further account for time-lagged influences. A batch solver is proposed that iterates between inferring directed graphs that “best” fit the sequence of observations, and estimating the network processes at reduced computational complexity by leveraging tools related to Kalman smoothing. To further accommodate delay-sensitive applications, an online joint inference approach is put forth that even tracks time-evolving topologies. Furthermore, we specify novel conditions for identifying the network topology given partial observations. We prove that the required number of observations for unique identification reduces significantly when the network structure is sparse. Numerical tests with synthetic as well as real datasets corroborate the effectiveness of the proposed approach.

49 citations

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TL;DR: The merits of kernel-based methods are extended here to the task of learning the effective brain connectivity, and an efficient regularized estimator is put forth to leverage the edge sparsity inherent to real-world complex networks.

Abstract: Structural equation models (SEMs) and vector autoregressive models (VARMs) are two broad families of approaches that have been shown useful in effective brain connectivity studies. While VARMs postulate that a given region of interest in the brain is directionally connected to another one by virtue of time-lagged influences, SEMs assert that directed dependencies arise due to instantaneous effects, and may even be adopted when nodal measurements are not necessarily multivariate time series. To unify these complementary perspectives, linear structural vector autoregressive models (SVARMs) that leverage both instantaneous and time-lagged nodal data have recently been put forth. Albeit simple and tractable, linear SVARMs are quite limited since they are incapable of modeling nonlinear dependencies between neuronal time series. To this end, the overarching goal of the present paper is to considerably broaden the span of linear SVARMs by capturing nonlinearities through kernels, which have recently emerged as a powerful nonlinear modeling framework in canonical machine learning tasks, e.g., regression, classification, and dimensionality reduction. The merits of kernel-based methods are extended here to the task of learning the effective brain connectivity, and an efficient regularized estimator is put forth to leverage the edge sparsity inherent to real-world complex networks. Judicious kernel choice from a preselected dictionary of kernels is also addressed using a data-driven approach. Numerical tests on ECoG data captured through a study on epileptic seizures demonstrate that it is possible to unveil previously unknown directed links between brain regions of interest.

41 citations