---
_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'
...