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Multiplex Immunohistochemistry (mIHC) is a cost-effective and accessible method for in situ labeling of multiple protein biomarkers in a tissue sample. By assigning a different stain to each biomarker, it allows the visualization of different types of cells within the tumor vicinity for downstream analysis. However, to detect different types of stains in a given mIHC image is a challenging problem, especially when the number of stains is high. Previous deep-learning-based methods mostly assume full supervision; yet the annotation can be costly. In this paper, we propose a novel unsupervised stain decomposition method to detect different stains simultaneously. Our method does not require any supervision, except for color samples of different stains. A main technical challenge is that the problem is underdetermined and can have multiple solutions. To conquer this issue, we propose a novel inversion regulation technique, which eliminates most undesirable solutions. On a 7-plexed IHC images dataset, the proposed method achieves high quality stain decomposition results without human annotation.