  While impressive progress has recently been made in image-oriented facial
attribute translation, shape-oriented 3D facial attribute translation remains
an unsolved issue. This is primarily limited by the lack of 3D generative
models and ineffective usage of 3D facial data. We propose a learning framework
for 3D facial attribute translation to relieve these limitations. Firstly, we
customize a novel geometric map for 3D shape representation and embed it in an
end-to-end generative adversarial network. The geometric map represents 3D
shapes symmetrically on a square image grid, while preserving the neighboring
relationship of 3D vertices in a local least-square sense. This enables
effective learning for the latent representation of data with different
attributes. Secondly, we employ a unified and unpaired learning framework for
multi-domain attribute translation. It not only makes effective usage of data
correlation from multiple domains, but also mitigates the constraint for hardly
accessible paired data. Finally, we propose a hierarchical architecture for the
discriminator to guarantee robust results against both global and local
artifacts. We conduct extensive experiments to demonstrate the advantage of the
proposed framework over the state-of-the-art in generating high-fidelity facial
shapes. Given an input 3D facial shape, the proposed framework is able to
synthesize novel shapes of different attributes, which covers some downstream
applications, such as expression transfer, gender translation, and aging. Code
at https://github.com/NaughtyZZ/3D_facial_shape_attribute_translation_ssgmap.
