The purpose of this study is to develop an appearance-based method for estimating gaze directions from low resolution images. The problem of estimating directions using low resolution images is that the position of an eye region cannot be determined accurately. In this work, we introduce two key ideas to cope with the problem: incorporating training images of eye regions with artificially added positioning errors, and separating the factor of gaze variation from that of positioning error based on -mode SVD (Singular Value Decomposition). We show that estimation of gaze direction in this framework is formulated as a bilinear problem that is then solved by alternatively minimizing a bilinear cost function with respect to gaze direction and position of the eye region. In this paper, we describe the details of our proposed method and show experimental results that demonstrate the merits of our method.