Fuzzy c-means

Soft memberships instead of hard labels

k-means assigns each point to exactly one cluster. Fuzzy c-means gives every point a membership in every cluster, with memberships that sum to 1. Points between two groups get split membership rather than a forced choice.

Implement fuzzy_c_means(X, c, m=2.0, max_iter=100, tol=1e-5). Return (centers, U), where centers has shape (c, d) and U has shape (n, c).

1. Initialize centers with farthest-first picks: row 0, then repeatedly the row farthest from the chosen centers.
2. Membership update: u(i, k) = 1 / Σ over j of (d(i, k) / d(i, j))^(2 / (m − 1)), where d is Euclidean distance.
3. If a point sits exactly on a center (distance 0), its membership is 1 for that center and 0 for the others.
4. Center update: v(k) = Σ over i of u(i, k)^m · x(i) divided by Σ over i of u(i, k)^m.
5. Stop when the largest change in U is below tol, or after max_iter rounds.

Constraints

• m must be greater than 1, otherwise raise ValueError.
• 1 ≤ c ≤ n, otherwise raise ValueError.
• Each row of U sums to 1 and is nonnegative.
• Do not modify X.

Hints