In this paper, we address the segmentation of tumor subtypes in whole slide images (WSI) by utilizing incomplete label proportions. Specifically, we utilize `partial' label proportions, which give the proportions among tumor subtypes but do not give the proportion between tumor and non-tumor. Partial label proportions are recorded as the standard diagnostic information by pathologists, and we, therefore, want to use them for realizing the segmentation model that can classify each WSI patch into one of the tumor subtypes or non-tumor. We call this problem ``learning from partial label proportions (LPLP)'' and formulate the problem as a weakly supervised learning problem. Then, we propose an efficient algorithm for this challenging problem by decomposing it into two weakly supervised learning subproblems: multiple instance learning (MIL) and learning from label proportions (LLP). These subproblems are optimized efficiently in the end-to-end manner. The effectiveness of our algorithm is demonstrated through experiments conducted on two WSI datasets.
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