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Clustering groups observations according to a specified notion of similarity or shared structure. Unlike supervised learning, it need not begin with known class labels. The representation, distance and objective determine what counts as a group: nearby numerical values, a connected dense region and membership in a probabilistic component are different criteria.

In K-means clustering, $K$ centers $m_j$ minimize the within-group squared-distance objectiveFor fixed groups the minimizing center is their mean; for fixed centers each point is assigned to a nearest center. These elementary minimizations explain the alternating updates. Stuart P. Lloyd's original quantization paper develops the corresponding nearest-representative and mean conditions for scalar signal reconstruction.^[ref]

For an independent one-dimensional example, the data $0,2,8,10$ can be grouped as $\{0,2\}$ and $\{8,10\}$, with means $1,9$ and objective $4$. A single center at $5$ gives objective $68$. Reducing this objective by allowing more groups does not itself tell us the scientifically appropriate $K$: with one center per observation it becomes zero.

Clustering describes the chosen representation, not an automatic discovery of natural kinds. Changing units can change distances; elongated or nested groups can challenge a squared-distance criterion; different initial centers can produce different fitted solutions. Stability, the scientific question and knowledge of how observations were collected help determine whether a partition is informative.

Hinrich Schütze is a researcher in natural language processing and computational linguistics whose work includes learning representations of language from text. His LMU profile identifies his research and teaching at its Center for Information and Language Processing.^[ref]

In his 1998 paper Automatic Word Sense Discrimination by Hinrich Schütze, occurrences of an ambiguous word are represented by their surrounding context and grouped using clustering. The groups can distinguish uses without requiring a human to supply a sense label for every training example. The paper investigates alternative representations and evaluates how well the resulting groups separate uses.^[ref]

For an illustrative example, occurrences of “bank” near “loan” and “account” differ from occurrences near “river” and “flood”. A context vector can record such differences, and clustering can group similar vectors. This is a problem of discovering useful structure from data; the algorithm's groups need not coincide perfectly with the entries of a human dictionary. The example connects his work to unsupervised learning and to the broader use of word embeddings.

Roberto Bruno is a physicist who studies the solar wind as a natural laboratory for turbulence. INAF's 2018 institutional interview documents his work on nonlinear processes in space plasmas and the American Geophysical Union's recognition of that work.^[ref]

His 2024 first-authored study examined proton and alpha particle populations measured by Solar Orbiter. Instead of characterizing all particles by one bulk value, the team used clustering to separate core and beam populations within the velocity distribution. That distinction changed the interpretation of temperature and its directional dependence, because two populations moving relative to one another can appear broader when combined.^[ref]

An illustrative calculation shows the issue. Two equally populated, individually narrow groups moving at $u-d$ and $u+d$ have combined mean $u$, but their relative drift contributes $d^2$ to the combined variance. Treating that whole width as random thermal motion would hide the organized relative motion. Measurements of a full velocity distribution can therefore reveal physics that one average speed cannot.