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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.03768 |
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| _version_ | 1866908425635495936 |
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| author | Rayas, Anirudh Cheng, Jiajun Anguluri, Rajasekhar Deka, Deepjyoti Dasarathy, Gautam |
| author_facet | Rayas, Anirudh Cheng, Jiajun Anguluri, Rajasekhar Deka, Deepjyoti Dasarathy, Gautam |
| contents | Complex networked systems driven by latent inputs are common in fields like neuroscience, finance, and engineering. A key inference problem here is to learn edge connectivity from node outputs (potentials). We focus on systems governed by steady-state linear conservation laws: $X_t = {L^{\ast}}Y_{t}$, where $X_t, Y_t \in \mathbb{R}^p$ denote inputs and potentials, respectively, and the sparsity pattern of the $p \times p$ Laplacian $L^{\ast}$ encodes the edge structure. Assuming $X_t$ to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of $L^{\ast}$ from temporally correlated samples of $Y_t$ via an $\ell_1$-regularized Whittle's maximum likelihood estimator (MLE). The regularization is particularly useful for learning large-scale networks in the high-dimensional setting where the network size $p$ significantly exceeds the number of samples $n$.
We show that the MLE problem is strictly convex, admitting a unique solution. Under a novel mutual incoherence condition and certain sufficient conditions on $(n, p, d)$, we show that the ML estimate recovers the sparsity pattern of $L^\ast$ with high probability, where $d$ is the maximum degree of the graph underlying $L^{\ast}$. We provide recovery guarantees for $L^\ast$ in element-wise maximum, Frobenius, and operator norms. Finally, we complement our theoretical results with several simulation studies on synthetic and benchmark datasets, including engineered systems (power and water networks), and real-world datasets from neural systems (such as the human brain). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning Networks from Wide-Sense Stationary Stochastic Processes Rayas, Anirudh Cheng, Jiajun Anguluri, Rajasekhar Deka, Deepjyoti Dasarathy, Gautam Machine Learning Signal Processing Complex networked systems driven by latent inputs are common in fields like neuroscience, finance, and engineering. A key inference problem here is to learn edge connectivity from node outputs (potentials). We focus on systems governed by steady-state linear conservation laws: $X_t = {L^{\ast}}Y_{t}$, where $X_t, Y_t \in \mathbb{R}^p$ denote inputs and potentials, respectively, and the sparsity pattern of the $p \times p$ Laplacian $L^{\ast}$ encodes the edge structure. Assuming $X_t$ to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of $L^{\ast}$ from temporally correlated samples of $Y_t$ via an $\ell_1$-regularized Whittle's maximum likelihood estimator (MLE). The regularization is particularly useful for learning large-scale networks in the high-dimensional setting where the network size $p$ significantly exceeds the number of samples $n$. We show that the MLE problem is strictly convex, admitting a unique solution. Under a novel mutual incoherence condition and certain sufficient conditions on $(n, p, d)$, we show that the ML estimate recovers the sparsity pattern of $L^\ast$ with high probability, where $d$ is the maximum degree of the graph underlying $L^{\ast}$. We provide recovery guarantees for $L^\ast$ in element-wise maximum, Frobenius, and operator norms. Finally, we complement our theoretical results with several simulation studies on synthetic and benchmark datasets, including engineered systems (power and water networks), and real-world datasets from neural systems (such as the human brain). |
| title | Learning Networks from Wide-Sense Stationary Stochastic Processes |
| topic | Machine Learning Signal Processing |
| url | https://arxiv.org/abs/2412.03768 |