Identifying accurate correspondences information among different shapes is of great importance in shape analysis such as shape registration, segmentation and retrieval. This paper aims to develop a paradigm to address the challenging issues posed by shape structural variation and symmetry ambiguity. Specifically, the proposed research developed a novel shape signature based on local diffusion map on 3D surface, which is used to identify the shape correspondence through graph matching process. The developed shape signature, named local diffusion map signature (LDMS), is obtained by projecting heat diffusion distribution on 3D surface into 2D images along the surface normal direction with orientation determined by gradients of heat diffusion field. The local diffusion map signature is able to capture the concise geometric essence that is deformation-insensitive and symmetry-aware. Experimental results on 3D shape correspondence demonstrate the superior performance of our proposed method over other state-of-the-art techniques in identifying correspondences for non-rigid shapes with symmetry ambiguity.