It is a challenging problem to accurately estimate gazes from low-resolution eye images that do not provide fine and detailed features for eyes. Existing methods attempt to establish the mapping between the visual appearance space to the gaze space. Different from the direct regression approach, the reconstruction-based approach represents appearance and gaze via local linear reconstruction in their own spaces. A common treatment is to use the same local reconstruction in the two spaces, i.e., the reconstruction weights in the appearance space are transferred to the gaze space for gaze reconstruction. However, this questionable treatment is taken for granted but has never been justified, leading to significant errors in gaze estimation. This paper is focused on the study of this fundamental issue. It shows that the distance metric in the appearance space needs to be adjusted, before the same reconstruction can be used. A novel method is proposed to learn the metric, such that the affinity structure of the appearance space under this new metric is as close as possible to the affinity structure of the gaze space under the normal Euclidean metric. Furthermore, the local affinity structure invariance is utilized to further regularize the solution to the reconstruction weights, so as to obtain a more robust and accurate solution. Effectiveness of the proposed method is validated and demonstrated through extensive experiments on different subjects.