Hyperspectral image visualization reduces color bands to three, but prevailing linear methods fail to address data characteristics, and nonlinear embeddings are computationally demanding. Qualitative evaluation of an embedding is also lacking. We propose Faithful Stochastic Proximity Embedding (FSPE), a scalable, nonlinear dimensionality reduction method. FSPE considers the nonlinear characteristics of spectral signatures, yet it avoids the costly computation of geodesic distances that are often required by other nonlinear methods. Furthermore, we introduce a point-wise metric that measures the quality of hyperspectral image visualization at each pixel. FSPE outperforms the state-of-art methods by at least 12% on average, and up to 25% in the proposed qualitative measure. An implementation on Graphics Processing Units (GPUs) is two magnitudes faster than the baseline. Our method opens the path to high-fidelity, real-time analysis of hyperspectral images.