A literature survey of matrix methods for data science

Stoll, Martin

doi:10.1002/gamm.202000013

“…As we use a linear algebraic formulation of the considered multilayer graphs in terms of their supraadjacency matrices, well-studied methods from numerical linear algebra can be put to new use as it is currently the case in many data-driven applications [49]. In particular, Krylov subspace methods for the approximation of matrix functions provide very mature computational means to evaluate matrix function-based centrality measures even for large-scale problems [31,35,47].…”

Section: Introductionmentioning

confidence: 99%

Fast computation of matrix function-based centrality measures for layer-coupled multiplex networks

Bergermann,

Stoll

2021

Preprint

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Centrality measures identify the most important nodes in a complex network. In recent years, multilayer networks have emerged as a flexible tool to create increasingly realistic models of complex systems. In this paper, we generalize matrix function-based centrality and communicability measures to the case of layer-coupled multiplex networks. We use the supra-adjacency matrix as the network representation, which has already been used to generalize eigenvector centrality to temporal and multiplex networks. With this representation, the definition of single-layer matrix function-based centrality measures in terms of walks on the networks carries over naturally to the multilayer case. Several aggregation techniques allow the ranking of nodes, layers, as well as node-layer pairs in terms of their importance in the network. We present efficient and scalable numerical methods based on Krylov subspace techniques and Gauss quadrature rules, which provide a high accuracy in only a few iterations and which scale linearly in the network size under the assumption of sparsity in the supra-adjacency matrix. Finally, we present extensive numerical studies for both directed and undirected as well as weighted and unweighted multiplex networks. While we focus on social and transportation applications the networks' size ranges between 89 and 2.28 • 10 6 nodes and between 3 and 37 layers.

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“…Graph-based approaches have become a standard tool in many learning tasks (cf. [19][20][21][22][23][24] and the references mentioned therein). The matrix representation of the graph via its Laplacian [25] leads to studying the network using matrix properties.…”

Section: Introductionmentioning

confidence: 99%

An Empirical Study of Graph-Based Approaches for Semi-supervised Time Series Classification

Alfke¹,

Gondos²,

Peroche³

et al. 2022

Front. Appl. Math. Stat.

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Time series data play an important role in many applications and their analysis reveals crucial information for understanding the underlying processes. Among the many time series learning tasks of great importance, we here focus on semi-supervised learning based on a graph representation of the data. Two main aspects are studied in this paper. Namely, suitable distance measures to evaluate the similarities between different time series, and the choice of learning method to make predictions based on a given number of pre-labeled data points. However, the relationship between the two aspects has never been studied systematically in the context of graph-based learning. We describe four different distance measures, including (Soft) DTW and MPDist, a distance measure based on the Matrix Profile, as well as four successful semi-supervised learning methods, including the recently introduced graph Allen–Cahn method and Graph Convolutional Neural Network method. We provide results for the novel combination of these distance measures with both the Allen-Cahn method and the GCN algorithm for binary semi-supervised learning tasks for various time-series data sets. In our findings we compare the chosen graph-based methods using all distance measures and observe that the results vary strongly with respect to the accuracy. We then observe that no clear best combination to employ in all cases is found. Our study provides a reproducible framework for future work in the direction of semi-supervised learning for time series with a focus on graph representations.

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“…Graph-based approaches have become a standard tool in many learning tasks (cf. [45,41,34,10,38,14] and the references mentioned therein). The matrix representation of the graph via its Laplacian [23] leads to studying the network using matrix properties.…”

Section: Introductionmentioning

confidence: 99%

An Empirical Study of Graph-Based Approaches for Semi-Supervised Time Series Classification

Alfke¹,

Gondos²,

Peroche³

et al. 2021

Preprint

Self Cite

0

View full text Add to dashboard Cite

Time series data play an important role in many applications and their analysis reveals crucial information for understanding the underlying processes. Among the many time series learning tasks of great importance, we here focus on semi-supervised learning which benefits of a graph representation of the data. Two main aspects are involved in this task: A suitable distance measure to evaluate the similarities between time series, and a learning method to make predictions based on these distances. However, the relationship between the two aspects has never been studied systematically. We describe four different distance measures, including (Soft) DTW and Matrix Profile, as well as four successful semi-supervised learning methods, including the graph Allen-Cahn method and a Graph Convolutional Neural Network. We then compare the performance of the algorithms on standard data sets. Our findings show that all measures and methods vary strongly in accuracy between data sets and that there is no clear best combination to employ in all cases.

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A literature survey of matrix methods for data science

Cited by 17 publications

References 241 publications

Fast computation of matrix function-based centrality measures for layer-coupled multiplex networks

Fast computation of matrix function-based centrality measures for layer-coupled multiplex networks

An Empirical Study of Graph-Based Approaches for Semi-supervised Time Series Classification

An Empirical Study of Graph-Based Approaches for Semi-Supervised Time Series Classification

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