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Consider a classification problem where we have 2D data and there are 3 classes. We train a model on the data, and obtain a parameter w. Now, lets consider a target scenario, where we know that the data distributions have shifted. I know that the shifting is such that the class means (i.e. mean of all data in each class) have shifted in different directions in the feature space from mu_i to mu_i' for i=1,2,3. How can I leverage this information to adapt my input, parameter or ouput(which is probabilistic classification) for the target scenario? Is there pertaining literature on such class specific shifting ?

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Considering the shift in the class means, it means that there is a shift in the distribution of the features. We have no information if there is a shift in the distribution of the labels (classes). Since you know the new class-means, then for any new unlabeled instances you could perform a semi-supervised clustering method using for example k-means and getting as initial centroids the new class-means.

About the parameter w that you have obtained from training a model using the old data, probably you could modify it in a way that you are taking into account the shift. For being more precise, we need to know how you have obtained this parameter, which means what kind of learning algorithm you have used.

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