How Alike Are Two Applicants?

A bank's loan data has an awkward shape. Plotted by two scaled features, the approved applicants sit in a ring, with denied applicants both inside it and all around it. No straight line can separate a ring from what surrounds it, so a plain linear SVM is out.

A kernel SVM gets a curved boundary by asking a different question. It doesn't ask "which side of a line is this point on?" It asks "how similar is this point to each training point?" For that it needs similarity as a number, and the most common choice is the RBF (Gaussian) kernel:

• $\lVert a - b \rVert^2$ is the squared straight-line distance between the two points: square the difference in every feature, then add those up.
• $\sigma$ (sigma) is the kernel's width, in the same units as the features. It sets how quickly similarity fades as points move apart.
• Identical points score exactly $1$. The score falls toward $0$ as the points get further apart, and it is never negative.

Task: write rbf_similarity(points_a, points_b, sigma) that returns the kernel matrix. This is a list with one row per point in points_a, where entry [i][j] is $K$(points_a[i], points_b[j]), rounded to 4 decimal places.

• Every point is a list of numbers, and all points have the same number of features (one, two, three or more).
• points_a and points_b can have different lengths, so the matrix need not be square. They can also be the same list.
• sigma is greater than 0.

Each row of this matrix shows how one new applicant looks to the model: a list of "how close am I to each stored applicant?" A kernel SVM decides from those closeness scores instead of from a line, and that is how its boundary can bend into a circle.