11.7 ML clustering

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §11.6. Rewritten in original wording — no KmeansCluster.py / Perceptron.py / Pandas listing dumps. Continues 11.6b. Next: 11.7b.

Grouping data by shared features reveals structure, outliers, and (for nets) useful training signals. Labeled groups ⇒ supervised learning; no labels ⇒ unsupervised. Clustering is computationally hard in general (NP-hard): expect heuristics, not guaranteed global optima in poly-time.

$k$-means and linear classifiers

Pick $k$ clusters (centroids = means that minimize squared distance to members). Initialize, then iterate assign → update until centroids barely move; sklearn’s KMeans Loss is within-cluster sum of squares (inertia). Supervised linear / perceptron models score $w^{T}x+b$ and step weights with a learning rate. Walk assign, centroid update, linear score, and the weight step — pick each matching equation among lookalikes.

Particle-mass demo

Eighteen PDG-style masses (MeV/$c^2$; $\nu$ treated as $0$ for the toy): $0$, $0.511$, $105.65$, $105.658$, $134.98$, $139.57$, $139.57$, $547.862$, $497.611$, $493.677$, $938.272$, $939.565$, $1115.683$, $1180.37$, $1197.449$, $1314.86$, $1321.71$, $1672.45$.

With $k=3$ you typically see a low-mass clump (leptons / light mesons), a medium band (heavier mesons ± nucleons depending on init), and a baryon / heavy clump. Exact membership can shift with initialization — another reason to try several seeds.

Pandas for tables

Hard-coding arrays fails at scale. Pandas reads whitespace / CSV tables into dataframes (columns = named series). Drop unused name columns, feed the mass column into $k$-means. (Sandbox labs keep a fixed Python list; use Pandas locally.)

Perceptron / SGD boundaries

Fire if $\phi(\sum w_j x_j)>\theta$. Sklearn’s Perceptron / SGDClassifier scale features (zero mean, unit variance), split train/test, and report accuracy. Boundaries are hyperplanes in feature space. A decaying $\eta(t)=1/(\alpha(t_0+t))$ shrinks steps late in training (picked above).

SGD clustering / classification

Same particle table, supervised on (mass, type): shuffle each epoch to avoid cycles, descend the Loss stochastically. Regions resemble the perceptron’s but need not match exactly — randomness and loss details matter.

Takeaway

Method	Labels?	Idea
$k$-means	No	Nearest centroid; update means
Perceptron	Yes	Linear threshold; weight updates
SGD classifier	Yes	Stochastic Loss descent; hyperplanes

Next: pure-Python assign / update / standardize / $\eta(t)$ locks.

Quiz

ML clustering

1 / 4

Unsupervised clustering means…

Pick an answer to see if you are right.

Next: 11.7b Clustering labs.