Live Coding: Vectorized K-Means

Live Coding Vectorized K-Means (Pure Numpy) evaluates an ML engineer's mastery over unsupervised Expectation-Maximization (EM) iterations and vectorized NumPy broadcasting; the algorithm alternates between: 1) E-step (Cluster Assignment): computing the Euclidean distance matrix $D \in \mathbb{R}^{N \times K}$ between $N$ data points and $K$ centroids using the expanded matrix identity $|x - c|^2 = |x|^2 + |c|^2 - 2 x c^T$ without Python loops, assigning each sample to its nearest centroid $r_{ik} = \arg\min_j D_{ij}$; 2) M-step (Centroid Re-computation): recalculating centroid coordinates as the arithmetic mean of assigned points $\mu_k = \frac{1}{|C_k|} \sum_{i \in C_k} x_i$; 3) Convergence Check: stopping when centroid shifts fall below tolerance $\epsilon$.

Product quantization codebook learning in vector DBs, user cohort segmentation, and unsupervised anomaly clustering.

Nested Python loops over 100k samples take minutes; fully vectorized matrix broadcasting accelerates execution by 100x and avoids memory bottlenecks.

Write the complete zero-loop vectorized distance matrix computation in Pure Numpy using Euclidean expansions?

Explain K-Means++ probabilistic initialization $P(x) \propto D(x)^2$ and how it mathematically prevents poor local minima?

How should K-Means handle empty clusters (clusters with zero assigned points) gracefully during M-step updates?

Prove that K-Means iterations monotonically decrease the Within-Cluster Sum of Squares (WCSS) objective function?

Explain how Mini-Batch K-Means samples stochastic batches to scale clustering across millions of records without OOM?