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A regression with multi-dimensional responses is quite common nowadays in the so-called big data era. In such regression, to relieve the curse of dimension due to high-dimension of responses, the dimension reduction of predictors is essential in analysis. Sufficient dimension reduction provides effective tools for the reduction, but there are few sufficient dimension reduction methodologies for multivariate regression. To fill this gap, we newly propose two fused slice-based inverse regressiondoi:10.29220/csam.2021.28.3.267 fatcat:taf6ty4ln5defjjd47uwu4e4d4