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Agreed, item response theory (IRT) would be a principled approach to use. I recently released a Python package called DeepIRTools that could be helpful here. It uses a deep learning approach to fit IRT models and provides a method for determining the latent dimensionality (i.e., how many "components" to retain). To get the dimension-reduced data (called embeddings in deep learning, factor scores in IRT, and components in PCA), you would just call model.scores(data).
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