--- _id: '1092' abstract: - lang: eng text: 'A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations can be represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. We show that there is an analogous polytope for conditional independence relations coming from a regular Gaussian model, and it can be defined using multiinformation or relative entropy. For directed acyclic graphical models we give a construction of this polytope as a Minkowski sum of matroid polytopes. Finally, we apply this geometric insight to construct a new ordering-based search algorithm for causal inference via directed acyclic graphical models. ' author: - first_name: Fatemeh full_name: Mohammadi, Fatemeh id: 2C29581E-F248-11E8-B48F-1D18A9856A87 last_name: Mohammadi - first_name: Caroline full_name: Uhler, Caroline id: 49ADD78E-F248-11E8-B48F-1D18A9856A87 last_name: Uhler orcid: 0000-0002-7008-0216 - first_name: Charles full_name: Wang, Charles last_name: Wang - first_name: Josephine full_name: Yu, Josephine last_name: Yu citation: ama: Mohammadi F, Uhler C, Wang C, Yu J. Generalized permutohedra from probabilistic graphical models. SIAM Journal on Discrete Mathematics. 2018;32(1):64-93. doi:10.1137/16M107894X apa: Mohammadi, F., Uhler, C., Wang, C., & Yu, J. (2018). Generalized permutohedra from probabilistic graphical models. SIAM Journal on Discrete Mathematics. SIAM. https://doi.org/10.1137/16M107894X chicago: Mohammadi, Fatemeh, Caroline Uhler, Charles Wang, and Josephine Yu. “Generalized Permutohedra from Probabilistic Graphical Models.” SIAM Journal on Discrete Mathematics. SIAM, 2018. https://doi.org/10.1137/16M107894X. ieee: F. Mohammadi, C. Uhler, C. Wang, and J. Yu, “Generalized permutohedra from probabilistic graphical models,” SIAM Journal on Discrete Mathematics, vol. 32, no. 1. SIAM, pp. 64–93, 2018. ista: Mohammadi F, Uhler C, Wang C, Yu J. 2018. Generalized permutohedra from probabilistic graphical models. SIAM Journal on Discrete Mathematics. 32(1), 64–93. mla: Mohammadi, Fatemeh, et al. “Generalized Permutohedra from Probabilistic Graphical Models.” SIAM Journal on Discrete Mathematics, vol. 32, no. 1, SIAM, 2018, pp. 64–93, doi:10.1137/16M107894X. short: F. Mohammadi, C. Uhler, C. Wang, J. Yu, SIAM Journal on Discrete Mathematics 32 (2018) 64–93. date_created: 2018-12-11T11:50:06Z date_published: 2018-01-01T00:00:00Z date_updated: 2021-01-12T06:48:13Z day: '01' doi: 10.1137/16M107894X extern: '1' intvolume: ' 32' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1606.01814 month: '01' oa: 1 oa_version: Preprint page: 64-93 publication: SIAM Journal on Discrete Mathematics publication_status: published publisher: SIAM publist_id: '6284' quality_controlled: '1' status: public title: Generalized permutohedra from probabilistic graphical models type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 32 year: '2018' ... --- _id: '1547' abstract: - lang: eng text: Let G be a graph on the vertex set V(G) = {x1,…,xn} with the edge set E(G), and let R = K[x1,…, xn] be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials xixj with {xi,xj} ∈ E(G), and the vertex cover ideal IG generated by monomials ∏xi∈Cxi for all minimal vertex covers C of G. A minimal vertex cover of G is a subset C ⊂ V(G) such that each edge has at least one vertex in C and no proper subset of C has the same property. Indeed, the vertex cover ideal of G is the Alexander dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we consider the lattice of vertex covers LG and we explicitly describe the minimal free resolution of the ideal associated to LG which is exactly the vertex cover ideal of G. Then we compute depth, projective dimension, regularity and extremal Betti numbers of R/I(G) in terms of the associated lattice. author: - first_name: Fatemeh full_name: Mohammadi, Fatemeh id: 2C29581E-F248-11E8-B48F-1D18A9856A87 last_name: Mohammadi - first_name: Somayeh full_name: Moradi, Somayeh last_name: Moradi citation: ama: Mohammadi F, Moradi S. Resolution of unmixed bipartite graphs. Bulletin of the Korean Mathematical Society. 2015;52(3):977-986. doi:10.4134/BKMS.2015.52.3.977 apa: Mohammadi, F., & Moradi, S. (2015). Resolution of unmixed bipartite graphs. Bulletin of the Korean Mathematical Society. Korean Mathematical Society. https://doi.org/10.4134/BKMS.2015.52.3.977 chicago: Mohammadi, Fatemeh, and Somayeh Moradi. “Resolution of Unmixed Bipartite Graphs.” Bulletin of the Korean Mathematical Society. Korean Mathematical Society, 2015. https://doi.org/10.4134/BKMS.2015.52.3.977. ieee: F. Mohammadi and S. Moradi, “Resolution of unmixed bipartite graphs,” Bulletin of the Korean Mathematical Society, vol. 52, no. 3. Korean Mathematical Society, pp. 977–986, 2015. ista: Mohammadi F, Moradi S. 2015. Resolution of unmixed bipartite graphs. Bulletin of the Korean Mathematical Society. 52(3), 977–986. mla: Mohammadi, Fatemeh, and Somayeh Moradi. “Resolution of Unmixed Bipartite Graphs.” Bulletin of the Korean Mathematical Society, vol. 52, no. 3, Korean Mathematical Society, 2015, pp. 977–86, doi:10.4134/BKMS.2015.52.3.977. short: F. Mohammadi, S. Moradi, Bulletin of the Korean Mathematical Society 52 (2015) 977–986. date_created: 2018-12-11T11:52:39Z date_published: 2015-05-31T00:00:00Z date_updated: 2021-01-12T06:51:31Z day: '31' department: - _id: CaUh doi: 10.4134/BKMS.2015.52.3.977 intvolume: ' 52' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/0901.3015 month: '05' oa: 1 oa_version: Preprint page: 977 - 986 publication: Bulletin of the Korean Mathematical Society publication_identifier: eissn: - 2234-3016 publication_status: published publisher: Korean Mathematical Society publist_id: '5624' quality_controlled: '1' scopus_import: 1 status: public title: Resolution of unmixed bipartite graphs type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 52 year: '2015' ...