A Randomized Greedy Algorithm for Near-Optimal Sensor Scheduling in Large-Scale Sensor Networks

We study the problem of scheduling sensors in a resource-constrained linear dynamical system, where the objective is to select a small subset of sensors from a large network to perform the state estimation task. We formulate this problem as the maximization of a monotone set function under a matroid constraint. We propose a randomized greedy algorithm that is significantly faster than state-of-the-art methods. By introducing the notion of curvature which quantifies how close a function is to being submodular, we analyze the performance of the proposed algorithm and find a bound on the expected mean square error (MSE) of the estimator that uses the selected sensors in terms of the optimal MSE. Moreover, we derive a probabilistic bound on the curvature for the scenario where{\color{black}{ the measurements are i.i.d. random vectors with bounded $\ell_2$ norm.}} Simulation results demonstrate efficacy of the randomized greedy algorithm in a comparison with greedy and semidefinite programming relaxation methods

Percentage of variance of each LF that is explained (<i>R</i>²) by mouse group (column 2), week (column 3), and by both mouse group and week (based on a linear model comprising interaction terms, column 4).

Percentage of variance of each LF that is explained (R2) by mouse group (column 2), week (column 3), and by both mouse group and week (based on a linear model comprising interaction terms, column 4).</p

Overview of the multiomic data analysed in this article.

A) Schematic representation of the experiment. WT, mdx, mdx++ and mdx+- mice were monitored for 30 weeks. Blood samples were drawn every 6 weeks, allowing measurement of RNA, lipids and metabolites. At week 30, mice were sacrificed and muscle samples drawn, allowing measurement of RNA expression in muscle. B) Weekly mean weight (in grams) of the mouse groups, measured from week 5 to week 30. C-F) Heatmaps showing the distribution of blood RNA (panel C), muscle RNA (panel D), lipids (panel E) and metabolites (panel F) across samples.</p

S4 Fig -

Comparison of the percentage of variance explained in the 4 omic views versus number of LFs for MOFA models fitted including the top 1000 (left) or 2500 (center) genes by variance in blood and muscle RNA seq, or all expressed genes (11243 in muscle and 10349 in blood; right). (PDF)</p

MOFA factor 3: Top 10 molecules by factor weight in each omic view.

A) Factor 3 weights for the top 10 genes, lipids and metabolites by factor weight in muscle RNA, blood RNA, lipids and metabolites. B-E) Heatmaps showing the distribution of the top 10 molecules by factor weight in muscle RNA (panel B), blood RNA (panel C), lipids (panel D) and metabolites (panel E) across samples.</p

Graphical representation of MOFA’s matrix decomposition of the data in each view into the product of a view-specific matrix of factor loadings and a matrix of shared latent factors.

Graphical representation of MOFA’s matrix decomposition of the data in each view into the product of a view-specific matrix of factor loadings and a matrix of shared latent factors.</p

Summary of the multi-omics factor analysis.

A) Data overview showing the number of samples (n) and molecules (d) available in each omic view. A grey bar indicates that the sample is missing in the given omic view. B) Cumulative proportion of variance explained for MOFA fits with increasing number of factors. C) Cumulative proportion of variance explained by omic view for MOFA fits with increasing number of factors. D) Total percentage of variance explained (R2) by omic view (top), and percentage of variance explained by each latent factor in the different omic views (bottom) for the selected MOFA model. E) Comparison of the distribution of the factor loadings across different omic views.</p

Results of the hypothesis tests on difference in mean of each latent factor across mouse groups and between weeks.

Adjusted p-values are obtained using Benjamini-Hochberg’s method. (XLSX)</p

Lipid data (see the “Data availability” section for the RNA-seq and metabolite data).

Lipid data (see the “Data availability” section for the RNA-seq and metabolite data).</p

Latent space representation of the 8 MOFA factors.

A) Beeswarm plots representing the distribution of each latent factor across the 4 mouse groups. B) Beeswarm plots representing the distribution of each latent factor by week. C) Latent space representations of factors 1 and 5. The colour of dots denotes mouse group, and the shape denotes the week of sampling. D) Latent space representations of factors 1 and 5. The colour of dots denotes week of sampling, and the shape denotes mouse group.</p