The Better of Two Weight Matrices

A classifier's first layer is a single weight matrix $W$ applied to two-dimensional embeddings. Two points from different classes arrive:

$c = (3, 1) \quad \text{(class A)}, \qquad d = (1, 2) \quad \text{(class B)}$

A later layer can only separate the classes with a straight line, so what matters is how far apart this layer leaves the two points. Two candidate weight matrices are on the table:

Candidate	Weight matrix
$W_A$	$\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$
$W_B$	$\begin{bmatrix} 1 & -2 \\ 4 & 1 \end{bmatrix}$

For each candidate, transform both points and measure the straight-line distance between the two outputs. Then report the size of the gap between those two separations:

$\Big\lvert\; \lVert W_B c - W_B d \rVert \;-\; \lVert W_A c - W_A d \rVert \;\Big\rvert$

Round your answer to two decimal places.

The Better of Two Weight Matrices

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