This approach provides greater flexibility than fixed-coefficient models and is widely used for longitudinal and functional data analysis.
Can we also add VCVS here as an extension of VC?
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adaptinfer.github.io adaptinfer.github.io
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Structured Regularization for Graphical and Network Models
This body of review largely ignored the work from:
M. Kolar, H. Liu and E. P. Xing, Graph Estimation From Multi-attribute Data , Journal of Machine Learning Research, 15:1713-1750, 2014.
M. Kolar, H. Liu and E. P. Xing, Markov Network Estimation From Multi-attribute Data, The 30th International Conference on Machine Learningy (ICML 2013).
M. Kolar and E. P. Xing, On Time Varying Undirected Graphs, Proceedings of the 14th International Conference on Artifical Intelligence and Statistics (AISTAT 2011).
M. Kolar and E. P. Xing, Estimating Networks With Jumps, Electronic Journal of Statistics Vol. 6 (2012) 2069-2106. (arXiv:1012.3795, communicated 17 Dec 2010.)
M. Kolar, L. Song, A. Ahmed, and E. P. Xing, Estimating Time-Varying Networks , Annals of Applied Statistics, Vol. 4, No. 1, 94 - 123, 2010.
M. Kolar, L. Song and E. P. Xing, Sparsistent Learning of Varying-coefficient Models with Structural Changes, Proceeding of the 23rd Neural Information Processing Systems, (NIPS 2009). [appendix available soon]
L. Song, M. Kolar and E. P. Xing, Time-Varying Dynamic Bayesian Networks, Proceeding of the 23rd Neural Information Processing Systems, (NIPS 2009). [appendix available soon]
A. Ahmed and E. P. Xing, TESLA: Recovering Time-Varying Networks of Dependencies in Social and Biological Studies, Proc. Natl. Acad. Sci. USA, vol. 106, no. 29, 11878-11883
A. Parikh, R. Curtis, I. Kuhn, S. Becker, M. Bissell, E. P. Xing and W. Wu, Network Analysis of Breast Cancer Progression and Reversal Using a Tree-evolving Network Algorithm, PLoS Computational Biology, Volume 10, Issue 7, e1003713, 2014.
A. Parikh, W. Wu (equal contributing 1st author), R. Curtis and E. P. Xing, Reverse Engineering Tree-Evolving Gene Networks Underlying Developing Biological Lineages, the Nineteenth International Conference on Intelligence Systems for Molecular Biology (ISMB 2011). Bioinformatics, Volume 27, Issue 13Pp. i196-i204
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Classical Remedies: Grouped and Distance-Based Models
There is one more method used distanced reweighed Parzan kernel density estimator, not sure which category it bellows to:
L. Song, M. Kolar and E. P. Xing, KELLER: Estimating Time-Evolving Interactions Between Genes, The Seventeenth International Conference on Intelligence Systems for Molecular Biology (ISMB 2009). Bioinformatics 2009 25(12):i128-i136. [software]
L. Song, M. Kolar and E. P. Xing, Time-Varying Dynamic Bayesian Networks, Proceeding of the 23rd Neural Information Processing Systems, (NIPS 2009).
It is also discussed in equation 7-8 in M. Kolar, L. Song, A. Ahmed, and E. P. Xing, Estimating Time-Varying Networks , Annals of Applied Statistics, Vol. 4, No. 1, 94 - 123, 2010.
which analyzed the smooth and jumpy conditions of VCVS estimators.
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A more flexible alternative assumes that observations with similar contexts should have similar parameters. This is encoded as a regularization penalty that discourages large differences in θiθi\theta_i for nearby cicic_i:
please cite Ahmed and Xing PNAS 2009 on TV-network.
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Statistical Methods with Varying Coefficient Models[7]
Please add reference to varying coefficient varying structured model:
Sparsistent Learning of Varying-coefficient Models with Structural Changes
Mladen Kolar, Le Song, Eric P. Xing
Advances in Neural Information Processing Systems 22 (NIPS 2009)
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For a new unit with context ccc, we write a unified empirical objective: ˆθ(c)∈argminθ∈Θ∑(i,j)∈S(c)ℓ(hθ(xij),yij)context-dependent support+R(θ;c)context-structured regularization,(★)(★)θ^(c)∈argminθ∈Θ∑(i,j)∈S(c)ℓ(hθ(xij),yij)⏟context-dependent support+R(θ;c)⏟context-structured regularization, \widehat{\theta}(c)\in\arg\min_{\theta\in\Theta}\; \underbrace{\sum_{(i,j)\in S(c)} \ell\!\big(h_\theta(x_{ij}),y_{ij}\big)}_{\text{context-dependent support}} \;+\; \underbrace{\mathcal{R}(\theta;\,c)}_{\text{context-structured regularization}}, \tag{★} where ℓℓ\ell is a proper loss (e.g., squared, logistic), S(c)⊆{1,…,n}×NS(c)⊆{1,…,n}×NS(c)\subseteq\{1,\dots,n\}\times\mathbb{N} is a support set selected for context ccc, and R(θ;c)R(θ;c)\mathcal{R}(\theta;c) encodes how parameters are allowed to vary with context (smoothness, sparsity, low-rank, hierarchy, etc.).
I believe this particular form was used in Mladen, Le, Xing 2029, and then systematically studied in Estimating time-varying networks, AOAO 2010 Mladen Kolar, Le Song, Amr Ahmed, Eric P Xing
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We formalize this by assuming each observation xixix_i is drawn from a distribution governed by parameters θiθi\theta_i: xi∼P(x;θi).
The notion of x_i ~ P( , \theta_i), was probably first brought up in the sequence of papers from Xing's group in their work on estimating time varying networks, in a couple of papers such as Ahmed and Xing (PNAS 09, on total variation estimator), Le (2009 on kernel reweighting estimator), Mladen, Le, xing (Nips 2009 on the VCVS model), and later with the same sample specific models in various papers. Please explicitly refer to the original of this notion.
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