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Joint approximate diagonalization is one of well-known methods for solving independent component analysis and blind source separation. It calculates an orthonormal separating matrix which diagonalizes many cumulant matrices of given observed signals as accurately as possible. It has been known that such diagonalization can be carried out efficiently by the Jacobi method, where the optimization for each pair of signals is repeated until the convergence of the whole separating matrix. The Jacobi method decides whether the optimization is actually applied to a given pair by a convergence decision condition. Generally, a fixed threshold is used as the condition. Though a sufficiently small threshold is desirable for the accuracy of results, the speed of convergence is quite slow if the threshold is too small. In this paper, we propose a new decision condition with an adaptive threshold for joint approximate diagonalization. The condition is theoretically derived by a model selection approach to a simple generative model of cumulants in the similar way as in Akaike information criterion. In consequence, the adaptive threshold is given as the current average of all the cumulants. Only if the expected reduction of the cumulants on each pair is larger than the adaptive threshold, the pair is actually optimized. Numerical results verify that the method can choose a suitable threshold for artificial data and image separation.