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We propose a novel Multi-Manifold Matrix Decomposition for Co-clustering (M3DC) algorithm that considers the geometric structures of both the sample manifold and the feature manifold simultaneously. Specifically, multiple candidate manifolds are constructed separately to take local invariance into account. Then, we employ multi-manifold learning to approximate the optimal intrinsic manifold, which better reflects the local geometrical structure, by linearly combining these candidate manifolds. In M3DC, the candidate manifolds are obtained using various manifold-based dimensionality reduction methods. These methods are based on different rationales and use different metrics for data distances. Experimental results on several real data sets demonstrate the effectiveness of our proposed M3DC. HighlightsWe consider the geometric structures of both sample and feature manifolds.To reduces the complexity, we use two low-dimensional intermediate matrices.We employ multi-manifold learning to approximate the intrinsic manifold.The intrinsic manifold is constructed by linearly combining multiple manifolds.The candidate manifolds are constructed using six dimensionality reduction methods.