Online Tensor Decomposition with optimized Stochastic Gradient Descent: an Application in Structural Damage Identification

Abstract

Structural Health Monitoring (SHM) provides an economic approach which aims to enhance understanding the behavior of structures by continuously collects data through multiple networked sensors attached to the structure. This data is then utilized to gain insight into the health of a structure and make timely and economic decisions about its maintenance. The generated SHM sensing data is non-stationary and exists in a correlated multi-way form which makes the batch/off-line learning and standard two-way matrix analysis unable to capture all of these correlations and relationships. In this sense, the online tensor data analysis has become an essential tool for capturing underlying structures in higher-order datasets stored in a tensor X ∈ R I1ו••×I N . The CANDECOMP/PARAFAC (CP) decomposition has been extensively studied and applied to approximate X by N loading matrices A(1), . . . ,A(N) where N represents the order of the tensor. We propose a novel algorithm, FP-CPD, to parallelize the CANDECOMP/PARAFAC (CP) decomposition of a tensor X ∈ R I1ו••×I N . Our approach is based on stochastic gradient descent (SGD) algorithm which allows us to parallelize the learning process and it is very useful in online setting since it updates X t+1 in one single step. Our SGD algorithm is augmented with Nesterov's Accelerated Gradient (NAG) and perturbation methods to accelerate and guarantee convergence. The

Online Rank-Revealing Block-Term Tensor Decomposition

“…Online CPD was also extensively studied after the pioneering work of [13]; see, e.g., [34]- [36]. The basic idea (proposed in [13]) is, given a new frontal slice, to find the new row of the C factor, use this to find the updated subspace spanned by modes 1 and 2 and finally split it (one way or another) into its A and B factors.…”

“…Applications of OTF abound. They include unveiling the topology of evolving networks [70], spatio-temporal prediction or image in-painting [41], multiple-input multiple-output (MIMO) wireless communications [13], [71], brain imaging [72], monitoring heart-related features from wearable sensors for multi-lead electro-cardiography (ECG) [73], anomaly detection in networks and topic modeling [16], structural health monitoring (in an internet of things (IoT) context) [36], online cartography (spectrum map reconstruction in cognitive radio networks) [14], detection of anomalies in the process of 3D printing [74], data traffic monitoring in networks [10], [16], cardiac MRI [10], stream data compression (e.g., in power distribution systems [75] or in video [76]), and online completion [10], [77], [78], among others.…”

Online Nonnegative and Sparse Canonical Polyadic Decomposition of Fluorescence Tensors

“…However, in this work we are interested in an other practical situation, called online CPD where the data tensor grows with time and its decomposition have to be updated regularly [31]. We now precise the concept of online CPD in the particular context of fluorescence data and environmental sciences but it also applies in other scientific fields [1,12]. First, we assume that the data tensor is a 3-way array that gathers on its last mode a collection of EEMs and that new collections of EEMs called sub-tensors are recorded regularly as illustrated in Figure 2.…”