Psychophysical spaces of facial expressions have been built by measurement of discrimination threshold hyperellipsoids in the facial expression image space. These spaces revealed various novel attributions therefore suggested new applications in representation and recognition of expressions. An remaining open problem is to determine the effective dimensions of the psychophysical expression spaces. In this paper, we approach the problem from both local and global viewpoints. The local effective dimensions are investigated by preservation of local squared sensitivities or the trace of the Riemannian metric tensors, which provide lower bounds for the global dimension. Spatial distribution of Riemannian metric tensor is then analyzed to find upper bounds for the effective dimension. In particular, we discover a high uncertainty of the Riemannian metric tensor in dimensions higher than nine such that 2D cross-sections of the discrimination hyperellipsoids, the 2D submetrics, the sectional curvatures and principal axes of the hyperellipsoids have discontinuous and random fluctuations between neighboring points. As a result, the lower and upper bounds of effective dimensions of the expression space are obtained.