Affine transformation on $X$ is a transformation of the following form

$$Y = wX + b$$

In general, $w, X, Y$ and $b$ tensors.

We generally call tensor $X$ as an input to affine transformation or the tensor which we want to transform. We call $w, b$ as weight and bias tensors respectively. We call $Y$ as output tensor after transformation. Every layer of multi layer percetron contains an affine transformation.

Suppose I have two types of inputs, say $X_1, X_2$. Now, I want to apply affine transformation on them using other.

Consider the following

#1: Combining using individual affine transformations

$$Y_1 = w_1X_1 + b_1$$ $$Y_2 = w_2X_2 + b_2$$ $$Y_1Y_2 = w_1w_2X_1X_2 + w_1b_2X_1 +w_2b_1X_2 + b_1b_2$$

#2 multiplying them and applying affine

$$Y = wX_1X_2 + b$$

#3 concatenating them and applying affine

$$Y = w (X_1, X_2) + b$$

Does anyone of the above eligible to call affine transformation in terms of $X$ and $Y$ (not $XY$)? If not, is it true that there is nothing like affine transformation on two inputs taken together?


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