# Structural Correspondence Learning for Domain Adaptation - how is the augmented data formed?

Structural Correspondence Learning (SCL) is a method for dealing with domain adaptation (different data distributions in the training and testing sets). It was proposed by Blitzer et al. but I am having difficulty understanding exactly how the data is represented in this technique.

SCL deals with Domain Adaptation by choosing m 'pivot features' which are consistent across both domains, then for each one, creating a binary classification task, using the remaining features to predict whether or not a the pivot in question will appear in the instance.

The weight vectors which were learned in this task form W,a mxn matrix , where n is the number of original features. Each element corresponds to the strength of correlation between one original feature with one pivot.

Singular value decomposition is applied to W, factorising it into UDVT. θ is then formed by taking the first h elements of UT.

θ is added to the original feature set; from the paper:

'For training instance t, the augmented feature vector will contain all the original features xt plus the new shared features θxt.

Now, the part which I don't understand:

As far as I can tell, θ has dimensions h x n. The original features have dimensions m x n.

So how can we get θxt for each of the original m instances in the original dataset when θ is hxn?

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