TY - GEN
T1 - Learning Ising models from one or multiple samples
AU - Dagan, Yuval
AU - Daskalakis, Constantinos
AU - Dikkala, Nishanth
AU - Kandiros, Anthimos Vardis
N1 - Publisher Copyright: © 2021 ACM.
PY - 2021/6/15
Y1 - 2021/6/15
N2 - There have been two main lines of work on estimating Ising models: (1) estimating them from multiple independent samples under minimal assumptions about the model's interaction matrix ; and (2) estimating them from one sample in restrictive settings. We propose a unified framework that smoothly interpolates between these two settings, enabling significantly richer estimation guarantees from one, a few, or many samples. Our main theorem provides guarantees for one-sample estimation, quantifying the estimation error in terms of the metric entropy of a family of interaction matrices. As corollaries of our main theorem, we derive bounds when the model's interaction matrix is a (sparse) linear combination of known matrices, or it belongs to a finite set, or to a high-dimensional manifold. In fact, our main result handles multiple independent samples by viewing them as one sample from a larger model, and can be used to derive estimation bounds that are qualitatively similar to those obtained in the afore-described multiple-sample literature. Our technical approach benefits from sparsifying a model's interaction network, conditioning on subsets of variables that make the dependencies in the resulting conditional distribution sufficiently weak. We use this sparsification technique to prove strong concentration and anti-concentration results for the Ising model, which we believe have applications beyond the scope of this paper.
AB - There have been two main lines of work on estimating Ising models: (1) estimating them from multiple independent samples under minimal assumptions about the model's interaction matrix ; and (2) estimating them from one sample in restrictive settings. We propose a unified framework that smoothly interpolates between these two settings, enabling significantly richer estimation guarantees from one, a few, or many samples. Our main theorem provides guarantees for one-sample estimation, quantifying the estimation error in terms of the metric entropy of a family of interaction matrices. As corollaries of our main theorem, we derive bounds when the model's interaction matrix is a (sparse) linear combination of known matrices, or it belongs to a finite set, or to a high-dimensional manifold. In fact, our main result handles multiple independent samples by viewing them as one sample from a larger model, and can be used to derive estimation bounds that are qualitatively similar to those obtained in the afore-described multiple-sample literature. Our technical approach benefits from sparsifying a model's interaction network, conditioning on subsets of variables that make the dependencies in the resulting conditional distribution sufficiently weak. We use this sparsification technique to prove strong concentration and anti-concentration results for the Ising model, which we believe have applications beyond the scope of this paper.
KW - Concentration Inequalities
KW - Dependent Data
KW - Ising model
KW - Low temperature
KW - Maximum Likelihood Estimation
KW - Single-Sample Estimation
UR - http://www.scopus.com/inward/record.url?scp=85108171059&partnerID=8YFLogxK
U2 - https://doi.org/10.1145/3406325.3451074
DO - https://doi.org/10.1145/3406325.3451074
M3 - منشور من مؤتمر
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 161
EP - 168
BT - STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
A2 - Khuller, Samir
A2 - Williams, Virginia Vassilevska
T2 - 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Y2 - 21 June 2021 through 25 June 2021
ER -