Due to the wide existence of mixed pixels, the derivation of constituent components (endmembers) and their fractional proportions (abundances) at the subpixel scale has been given a lot of attention. The entire process is often referred to as mixed-pixel decomposition or spectral unmixing. Although various algorithms have been proposed to solve this problem, two potential issues still need to be further investigated. First, assuming the endmembers are known, the abundance estimation is commonly performed by employing a least-squares error criterion, which, however, makes the estimation sensitive to noise and outliers. Second, the mathematical intractability of the abundance non-negative constraint results in computationally expensive numerical approaches. In this paper, we propose an unsupervised decomposition method based on the classic maximum entropy principle, termed the gradient descent maximum entropy (GDME), aiming at robust and effective estimates. We address the importance of the maximum entropy principle for mixed-pixel decomposition from a geometric point of view and demonstrate that when the given data present strong noise or when the endmember signatures are close to each other, the proposed method has the potential of providing more accurate estimates than the popular least-squares methods (e.g., fully constrained least squares). We apply the proposed GDME to the subject of unmixing multispectral and hyperspectral data. The experimental results obtained from both simulated and real images show the effectiveness of the proposed method.