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Abstract:
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We establish a new framework for learning a directed acyclic graph (DAG) when data are generated from a Gaussian, linear structural equation model. Our framework consists of two parts: (1) introducing a permutation matrix as a new parameter within a regularized Gaussian log-likelihood to estimate variable ordering; and (2) given the ordering, estimating the DAG structure through sparse Cholesky factor of the inverse covariance matrix. For a permutation matrix estimation, we propose a relaxation technique that avoids the hard combinatorial problem of order estimation. Given ordering, a sparse Cholesky factor is estimated utilizing a cyclic coordinatewise descent algorithm by decoupling row-wise. Our framework recovers DAGs without a need for an expensive verification of the acyclicity constraint or enumeration of possible parent sets. We establish the convergence of the algorithm in each step under mild conditions. On the statistical side, we show the consistency of the Cholesky factor estimator, assuming the order of variables is known. Through several simulated and macro-economic datasets, we study the scope and performance of the proposed methodology.
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