Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866917671256195072 |
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| author | Hollering, Benjamin Johnson, Joseph Portakal, Irem Solus, Liam |
| author_facet | Hollering, Benjamin Johnson, Joseph Portakal, Irem Solus, Liam |
| contents | Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gröbner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gröbner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and propose directions for future work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_01834 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing Hollering, Benjamin Johnson, Joseph Portakal, Irem Solus, Liam Statistics Theory Commutative Algebra Algebraic Geometry Combinatorics 62R01, 13P10, 62A09, 13P25 Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gröbner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gröbner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and propose directions for future work. |
| title | Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing |
| topic | Statistics Theory Commutative Algebra Algebraic Geometry Combinatorics 62R01, 13P10, 62A09, 13P25 |
| url | https://arxiv.org/abs/2209.01834 |