Unsupervised Learning and Representation

Introduction and Context

Unsupervised learning addresses situations in which observations are available but reliable target labels are absent. Rather than learning a mapping from inputs to known outputs, the learner seeks structure in the data itself: groups of similar observations, lower-dimensional representations, recurring co-occurrences, unusual cases, or latent factors that explain observed variation.

This distinction is important in modern machine learning. A retailer may have millions of customers but no authoritative labels describing their needs. A cybersecurity team may possess network logs but only a small number of confirmed attacks. A biologist may measure thousands of gene-expression variables without knowing which biological mechanisms are active. Unsupervised methods can provide hypotheses, summaries and operational signals, but they do not automatically discover objective truth. Their results depend strongly on representation, distance, scaling, assumptions and evaluation choices.

This chapter develops five connected families of methods:

1. clustering for discovering groups;
2. dimensionality reduction for compression and visualization;
3. manifold learning for nonlinear geometric structure;
4. association analysis for co-occurrence and rule discovery; and
5. anomaly detection for identifying observations that differ from a reference population.

A central theme is representation. An algorithm does not operate on an abstract object called “meaning”; it operates on a numerical representation and a similarity or probability model. Consequently, representation learning and evaluation must be treated as first-class design decisions.

Learning Objectives

By the end of this chapter, a student should be able to:

• distinguish clustering, dimensionality reduction, manifold learning, association analysis and anomaly detection;
• formulate and interpret common unsupervised-learning objectives;
• select distance measures, scaling procedures and algorithms appropriate to data type;
• use PCA, k-means, hierarchical clustering, density-based methods and association rules responsibly;
• explain the assumptions and limitations of latent representations;
• evaluate results using internal, external and stability-based evidence;
• design an end-to-end unsupervised workflow for segmentation or discovery; and
• identify ethical, governance and operational risks in deploying unsupervised models.

1. A General View of Unsupervised Learning

Let a dataset be represented by a matrix X ∈ ℝⁿˣᵖ, where n is the number of observations and p is the number of measured features. Observation i is the vector xᵢ. In supervised learning, a target y is supplied. In unsupervised learning, the task is to infer a useful structure z, a representation h, a density p(x), or a set of relations from X alone.

The phrase “structure” has several meanings. A cluster is a group whose members are more similar to one another under a chosen criterion than to members of other groups. A latent representation is a lower-dimensional or hidden description that preserves selected information. A density model describes regions of high and low probability. An association rule describes frequent co-occurrence, not necessarily causation.

Representation, similarity and inductive bias

Three choices control most unsupervised outcomes:

• Representation: Which variables, transformations and encodings describe each observation?
• Similarity: What does it mean for two observations to be close?
• Inductive bias: What type of structure is the algorithm encouraged to find?

For numerical vectors, Euclidean distance is

where xᵢₖ is feature k for observation i. Euclidean distance is meaningful when features are commensurable and differences are appropriately measured. If income is recorded in dollars and age in years, the income variable may dominate unless scaling is applied. For standardized variables, each feature is transformed as

where μₖ and σₖ are the training-sample mean and standard deviation of feature k. This transformation is appropriate when relative deviations from the mean are meaningful and extreme outliers do not make σₖ misleading. Robust alternatives use the median and interquartile range.

For binary or categorical data, Hamming, Jaccard, Gower or model-based similarities may be preferable. Applying Euclidean distance mechanically to arbitrary categories can create artificial relationships.

2. Clustering

Clustering partitions or organizes observations into groups. It is used for customer segmentation, image organization, document exploration, biological subtype discovery and operational monitoring. Clusters are not inherent facts waiting to be revealed; they are interpretations produced under a particular representation and objective.

2.1 k-means clustering

Given a selected number K of clusters, k-means seeks centroids μ₁,…,μK that minimize within-cluster squared Euclidean distance:

Here ||·|| is the Euclidean norm. The objective assumes numerical features, meaningful averages and roughly compact, convex groups. The standard algorithm alternates two steps:

1. assign each observation to its nearest centroid;
2. replace each centroid with the mean of its assigned observations.

The process stops when assignments or the objective change negligibly. Because initialization affects the result, multiple random starts are recommended. k-means is sensitive to scale, outliers and K. It tends to produce spherical clusters of similar scale even when the data do not have that geometry.

Illustrative calculation. Suppose one-dimensional observations are 2, 3, 8 and 9, and K = 2. Assigning {2, 3} to centroid 2.5 and {8, 9} to centroid 8.5 gives

J = (2 − 2.5)² + (3 − 2.5)² + (8 − 8.5)² + (9 − 8.5)² = 1.

This is a compact partition, but it is not evidence that two natural populations exist. Alternative K values and stability should also be examined.

2.2 k-medoids and hierarchical clustering

k-medoids is similar to k-means but chooses actual observations as representatives, called medoids. Because medoids are less affected by extreme values and can use arbitrary dissimilarities, the method is useful when means are not meaningful.

Hierarchical clustering produces a nested structure rather than requiring one final K immediately. In agglomerative clustering, every observation begins as a separate cluster and the closest clusters are repeatedly merged. Linkage determines distance between clusters:

• single linkage: minimum pairwise distance;
• complete linkage: maximum pairwise distance;
• average linkage: average pairwise distance; and
• Ward linkage: merge that produces the smallest increase in within-cluster squared error, generally for Euclidean numerical data.

The output is a dendrogram. Cutting it at a chosen height produces a partition. Single linkage can detect elongated chains but is vulnerable to bridging through noise; complete linkage favors compact groups.

2.3 Density-based clustering

Density-based spatial clustering methods identify dense regions separated by sparse regions. DBSCAN uses a radius ε and a minimum number of points minPts. A point is a core point if at least minPts observations lie within its ε-neighborhood. Density-reachable points are assigned to the same cluster, while isolated observations may be labeled noise.

DBSCAN can discover nonconvex shapes and explicitly identify noise. It does not require K, but ε and minPts can be difficult to select, especially when cluster densities differ. HDBSCAN extends the density perspective by constructing a hierarchy and extracting relatively stable density-based clusters.

Gaussian mixture models (GMMs) provide a probabilistic alternative. A K-component mixture models

where πⱼ ≥ 0, ∑πⱼ = 1, μⱼ is a component mean, Σⱼ is a covariance matrix, and 𝓝 denotes a multivariate Gaussian density. The expectation-maximization algorithm alternates between estimating component-membership probabilities and updating parameters. GMMs supply soft assignments, but their Gaussian and covariance assumptions must be checked.

2.4 Choosing and interpreting clusters

The silhouette coefficient for observation i compares its average distance a(i) to its own cluster with the smallest average distance b(i) to another cluster:

It lies between −1 and 1 when distances are nonnegative. Larger values indicate better separation relative to cohesion, but the measure favors some geometries and should not be used alone.

The Calinski–Harabasz index compares between-cluster dispersion with within-cluster dispersion; the Davies–Bouldin index summarizes similarity between each cluster and its most similar competitor, with lower values preferred. These are internal metrics: they evaluate structure using the same data and assumptions used to create it.

If domain labels exist after the analysis, adjusted Rand index or normalized mutual information can provide external comparisons. However, agreement with an existing label is not automatically the scientific objective. Stability under resampling, perturbation, alternative preprocessing and alternative algorithms is often more informative than one favorable score.

3. Dimensionality Reduction and Latent Representation

High-dimensional data are difficult to visualize, store, model and interpret. Dimensionality reduction maps x ∈ ℝᵖ to a representation z ∈ ℝᵈ where d < p. The goal may be compression, noise reduction, visualization, computational efficiency or discovery of latent factors. No reduction preserves all relationships simultaneously; every method preserves selected properties.

3.1 Principal component analysis

PCA finds orthogonal directions of maximum variance. After centering the data, let S be the sample covariance matrix. Its eigenvectors v₁,…,vₚ satisfy

where λⱼ is the variance explained by component j. Components are ordered λ₁ ≥ λ₂ ≥ … ≥ 0. The score of observation i on component j is zᵢⱼ = (xᵢ − x̄)ᵀvⱼ, where x̄ is the feature mean vector.

The proportion of variance explained by the first d components is

This ratio is dimensionless. PCA is equivalent to finding the rank-d linear reconstruction with minimum squared reconstruction error under standard centering assumptions. It is not a causal model, and a high-variance direction is not necessarily the most predictive or meaningful direction.

PCA should be performed inside a training pipeline when it precedes a downstream predictive model, particularly if preprocessing parameters could leak information from a test set. Standardization is often appropriate when units differ, but not when absolute scale carries the intended meaning.

3.2 Nonnegative matrix factorization and autoencoders

Nonnegative matrix factorization approximates a nonnegative matrix X by WH, with W and H constrained to be nonnegative. This can produce additive, parts-based representations in text, images and biological data. The solution is generally nonunique and depends on rank, regularization and initialization.

An autoencoder consists of an encoder h = fθ(x) and decoder x̂ = gϕ(h). It minimizes a reconstruction loss such as

for continuous data under a squared-error objective. For binary features, cross-entropy may be more appropriate. Bottlenecks, sparsity, denoising objectives or variational constraints can encourage useful representations, but an autoencoder can also learn nuisance variation or memorize training data.

3.3 t-SNE and UMAP

t-SNE constructs a low-dimensional map that attempts to preserve local neighborhood relationships. It is valuable for exploratory visualization but distances between far-separated groups, cluster sizes and apparent gaps can be misleading. Perplexity, initialization and random seed influence results.

UMAP constructs a graph-based representation intended to preserve local and some global structure. It is often faster and more scalable than t-SNE, but its interpretation remains parameter-dependent. Neither method should be treated as proof that visible islands are real clusters. A visualization should be checked against original variables, alternative settings and quantitative stability.

4. Manifold Learning

The manifold hypothesis proposes that high-dimensional observations may lie near a lower-dimensional, smoothly varying structure. For example, images of one object may vary primarily with pose, illumination and viewpoint even though each image has thousands of pixels.

Isomap preserves approximate geodesic distances: distances measured along a neighborhood graph rather than through the ambient space. Locally Linear Embedding reconstructs each point from nearby points and seeks low-dimensional coordinates preserving those reconstruction weights. Laplacian eigenmaps preserve local graph relationships through a graph Laplacian.

Manifold methods depend on neighborhood construction. Too few neighbors can fragment the graph; too many can connect distinct regions and destroy local geometry. Noise, disconnected components, nonuniform sampling and boundaries can distort the result. Manifold learning is therefore best understood as a hypothesis about geometry, not an automatic discovery of intrinsic truth.

5. Association Analysis

Association analysis discovers items or events that occur together. In market-basket data, a transaction T is a set of items drawn from an item universe I. An association rule has the form A → B, where A and B are disjoint itemsets.

Support measures prevalence:

In a finite dataset, this is the proportion of transactions containing every item in A ∪ B.

Confidence measures conditional frequency:

provided P(A) > 0. Confidence can be high simply because B is common. Therefore, lift is often more informative:

Lift greater than 1 indicates positive association relative to independence; lift equal to 1 is consistent with independence under the measure; lift below 1 indicates negative association. None of these measures establishes causation.

Apriori exploits the downward-closure property: if an itemset is infrequent, every larger itemset containing it is also infrequent. This prunes candidate search. FP-Growth compresses transactions into an FP-tree and avoids generating many explicit candidates. In large-scale applications, minimum support must balance computational feasibility with business relevance. Statistical significance, multiple testing and temporal confounding should also be considered.

6. Anomaly Detection

An anomaly is an observation that is unusual relative to a reference distribution, operating regime or population. The meaning of unusual is context-dependent: a rare transaction may be fraud, a sensor reading may be a fault, or a legitimate holiday peak may merely be uncommon.

6.1 Statistical and distance-based methods

If a variable is modeled as Gaussian with mean μ and standard deviation σ, a standardized score is

Large absolute values may indicate unusual observations, but the threshold depends on distributional assumptions and the cost of errors. In multivariate settings, the squared Mahalanobis distance is

where Σ must be invertible or suitably regularized. This distance accounts for correlation and reduces to squared standardized distance in the one-dimensional case. It is unreliable when covariance is poorly estimated, dimensions are high relative to sample size, or the population is multimodal.

6.2 Isolation Forest, LOF and one-class learning

Isolation Forest randomly partitions features and tends to isolate anomalies in fewer splits because they are rare or different. It is computationally useful for tabular data but its score depends on subsampling and feature representation.

Local Outlier Factor compares an observation’s local density with those of its neighbors. It can detect local anomalies that global distance methods miss, although neighborhood size is consequential.

One-class support vector machines estimate a boundary around typical data. Kernel choice, ν and scaling affect the boundary. These methods are generally novelty detectors when trained mostly on normal data; contamination by many anomalies can distort the reference.

6.3 Evaluation

Anomaly labels are often scarce and delayed. Precision, recall and precision–recall curves are useful when positive cases are rare. If a threshold produces TP true positives, FP false positives, FN false negatives and TN true negatives, then

and

Threshold selection should reflect the relative cost of investigation, missed incidents and customer harm. Time-based validation is essential for streaming systems; random splits can allow future information to influence the past.

7. A Professional Workflow for Unsupervised Projects

A defensible project commonly follows these stages:

1. Define the decision or discovery question. “Find clusters” is insufficient; specify how the result will be used.
2. Audit data provenance and quality. Examine missingness, duplicates, leakage, changing definitions and sampling bias.
3. Choose the unit of analysis. A customer, transaction, session and customer-month can produce different structures.
4. Engineer and scale representations. Use domain-informed transformations, text or image embeddings where justified, and fit preprocessing only on permitted data.
5. Establish baselines. Compare simple summaries, random structure and meaningful domain rules.
6. Fit several plausible methods. Vary K, distance, hyperparameters and random seeds.
7. Evaluate quantitatively and qualitatively. Use internal metrics, stability, external evidence, visualization and expert review.
8. Profile and label discovered groups. Describe distributions, not merely centroids; identify actionable differences and uncertainty.
9. Test operational consequences. Estimate workload, drift, false alarms, fairness and intervention effects.
10. Monitor and govern. Track data drift, cluster instability, threshold performance, access controls and documentation.

8. Detailed Illustrative Case: Retail Customer Segmentation

This is a fictional illustrative case.

Background and problem

A subscription retailer wants to tailor retention programs. Its data contain twelve months of customer-level purchase frequency, average order value, discount usage, return rate, tenure and customer-service contacts. Marketing requests “four customer segments,” but no verified segment labels exist.

Analysis and approach

The team first defines the unit as one active customer over the previous twelve months and excludes information recorded after the planned campaign date. Monetary variables are log-transformed because of strong right skew. Features are standardized using training-period parameters. Missing service-contact counts are distinguished from unknown values rather than automatically replaced by zero.

The team compares k-means, k-medoids and a Gaussian mixture model for K from 2 through 6. Silhouette values alone favor K = 2, but K = 4 yields more stable profiles under bootstrap resampling and supports distinct operational actions. A two-dimensional PCA plot is used for communication, not as the clustering input alone. The fourth group is small and has high returns and service contacts; the team investigates whether it represents a data-quality artifact before naming it a “high-risk” segment.

Outcome

The resulting profiles are described as behavioral patterns: frequent full-price purchasers, discount-sensitive occasional purchasers, new low-history customers and high-return customers. Marketing receives profile distributions and uncertainty intervals rather than deterministic claims about individual identity. A controlled campaign tests whether segment-specific interventions improve retention without increasing discounts unnecessarily.

Learning points

The project demonstrates that segmentation is not complete when an algorithm returns labels. The unit of analysis, leakage controls, scaling, algorithm comparison, stability, interpretation and intervention testing determine whether the result is useful. It also illustrates that labels such as “valuable” or “risky” can embed managerial judgments and should be documented.

9. Challenges, Ethics and Governance

Unsupervised systems can reproduce historical inequality even without protected attributes. Proxy variables, unequal measurement quality and different population densities may cause one group to be labeled anomalous more often. A “high-risk” cluster can become a self-fulfilling label if it triggers reduced service or surveillance.

Important safeguards include documenting features and exclusions, testing subgroup stability, examining disparate false-positive rates, limiting access to sensitive representations, and providing human review for consequential decisions. Organizations should distinguish exploratory use from automated decision-making and preserve an audit trail of preprocessing, seeds, model versions and threshold changes.

Privacy risks also arise because embeddings can encode sensitive information or permit linkage attacks. Data minimization, retention limits, encryption and purpose limitation remain relevant even when the model is unsupervised. Association rules can reveal private co-occurrences, and anomaly detection can expose legitimate but unusual behavior. Governance must therefore address both model output and downstream use.

10. Contemporary Developments and Best Practices

Representation learning increasingly uses pretrained language, vision and multimodal encoders. These embeddings can make semantic similarity practical, but they inherit training-data bias, may be difficult to interpret and can encode information inappropriate for a particular decision. Before clustering embeddings, practitioners should examine normalization, cosine versus Euclidean similarity, duplicate content and domain shift.

Self-supervised learning creates supervisory signals from the data itself, such as masked-token prediction or contrastive agreement between augmented views. It bridges unsupervised representation learning and later supervised tasks. Generative models provide another route to latent variables, but latent dimensions are not automatically identifiable or semantically disentangled without additional assumptions.

Best practice is therefore not to seek a universally best unsupervised algorithm. It is to align the representation and objective with the decision, quantify uncertainty and stability, compare plausible alternatives, involve domain experts, and validate downstream value.

Chapter Summary

Unsupervised learning discovers structure without supplied target labels. Clustering organizes observations according to a distance, density or probabilistic model; k-means is efficient but favors compact groups, while hierarchical, density-based and mixture methods provide different geometries and uncertainty representations. Dimensionality reduction compresses or visualizes data, with PCA preserving linear variance and manifold methods emphasizing neighborhood geometry. Association analysis measures item co-occurrence through support, confidence and lift, none of which proves causation. Anomaly detection identifies observations unusual relative to a reference population and requires careful thresholding and cost-sensitive evaluation.

Across all methods, representation, scaling, similarity, assumptions and validation determine the result. Internal scores are insufficient: stability, external evidence, domain interpretation and operational outcomes matter. Ethical practice requires attention to privacy, proxy discrimination, subgroup performance, human oversight and documentation.

Key Terms

Representation: Numerical or structured description supplied to a learning algorithm.

Unsupervised learning: Learning patterns from data without target labels.

Distance measure: Function quantifying dissimilarity between observations.

Clustering: Grouping observations according to a chosen similarity criterion.

Centroid: Mean vector representing a cluster in k-means.

Medoid: Actual observation selected as a cluster representative.

Density-based clustering: Clustering based on dense regions separated by sparse regions.

Gaussian mixture model: Probabilistic model representing data as a mixture of Gaussian components.

Silhouette coefficient: Internal measure comparing within-cluster cohesion with nearest-cluster separation.

Dimensionality reduction: Mapping data into fewer dimensions while preserving selected properties.

Principal component analysis: Linear method finding orthogonal maximum-variance directions.

Explained variance: Fraction of total variance represented by selected principal components.

Manifold learning: Learning lower-dimensional nonlinear geometric structure.

Association rule: Statement of the form A → B describing item co-occurrence.

Support: Proportion of records containing an itemset.

Confidence: Conditional frequency of the consequent given the antecedent.

Lift: Association strength relative to independence.

Anomaly detection: Identification of observations unusual relative to a reference population.

Novelty detection: Detecting deviations from a model trained primarily on normal data.

Representation drift: Change in the distribution or meaning of model inputs or embeddings over time.

Review Questions

Short Answer Questions

1. Why can feature scaling change the result of k-means clustering?
2. State the difference between support, confidence and lift.
3. What does PCA optimize under its standard linear formulation?
4. Why can t-SNE visualizations not by themselves establish the existence of clusters?
5. Distinguish anomaly detection from novelty detection.

Long Answer Questions

1. Compare k-means, hierarchical clustering, DBSCAN and Gaussian mixture models in terms of assumptions, strengths and limitations.
2. Explain PCA mathematically and discuss when standardization is and is not appropriate.
3. Describe a complete workflow for developing a customer-segmentation system, including validation and governance.
4. Explain why internal clustering metrics should be supplemented by stability and domain-based evaluation.
5. Discuss ethical and privacy risks in embedding-based segmentation and anomaly detection.

Analytical / Application Questions

1. A dataset contains age in years, annual income in dollars and a binary membership indicator. Recommend a representation and similarity strategy before clustering, and justify each preprocessing decision.
2. An association rule has support 0.08, confidence 0.40 and consequent prevalence 0.20. Compute its lift and interpret the result. What conclusion cannot be drawn from the lift?
3. An anomaly detector produces 18 true positives, 12 false positives and 30 false negatives. Compute precision and recall, and recommend what additional information is needed before selecting a threshold.

Multiple Choice Questions with Answers

Question 1

Which objective is minimized by standard k-means clustering?

A. Within-cluster sum of squared distances to centroids

B. Between-cluster sum of squared distances to medoids

C. Total entropy of categorical variables

D. Average pairwise covariance between features

Correct Answer: A. Within-cluster sum of squared distances to centroids

Question 2

What does a lift value greater than 1 for an association rule indicate?

A. The antecedent occurs in every transaction

B. The consequent is independent of the antecedent

C. The consequent occurs with the antecedent more often than expected under independence

D. The rule proves that the antecedent causes the consequent

Correct Answer: C. The consequent occurs with the antecedent more often than expected under independence

Question 3

In PCA, the eigenvalue associated with a principal component represents what quantity?

A. The number of observations assigned to a cluster

B. The variance explained by that component

C. The probability that an observation is anomalous

D. The confidence of an association rule

Correct Answer: B. The variance explained by that component

Question 4

Which clustering method can explicitly label sparse observations as noise without requiring the number of clusters in advance?

A. k-means

B. DBSCAN

C. PCA

D. Apriori

Correct Answer: B. DBSCAN

Question 5

If TP = 18 and FP = 12, what is anomaly-detection precision?

A. 0.40

B. 0.50

C. 0.60

D. 0.67

Correct Answer: C. 0.60

Question 6

Why is Euclidean distance often inappropriate for unscaled features measured in very different units?

A. It cannot be computed for numerical data

B. The feature with the largest numerical scale may dominate the distance

C. It always produces negative distances

D. It requires target labels

Correct Answer: B. The feature with the largest numerical scale may dominate the distance

Question 7

Which statement about t-SNE is most accurate?

A. It guarantees that displayed clusters are statistically real

B. It preserves all global distances exactly

C. It is primarily an exploratory visualization method emphasizing local neighborhoods

D. It requires a predefined number of clusters

Correct Answer: C. It is primarily an exploratory visualization method emphasizing local neighborhoods

Question 8

What is the main purpose of bootstrap stability analysis for clustering?

A. To increase the number of features automatically

B. To examine whether discovered assignments persist under data perturbations

C. To convert an unsupervised task into a supervised task

D. To guarantee causal interpretation of clusters

Correct Answer: B. To examine whether discovered assignments persist under data perturbations

Question 9

In the Mahalanobis distance, what role does the covariance matrix play?

A. It converts categorical variables into labels

B. It accounts for feature scale and correlation

C. It determines the number of clusters

D. It measures association-rule confidence

Correct Answer: B. It accounts for feature scale and correlation

Question 10

Which statement is a valid governance principle for unsupervised models used in consequential decisions?

A. Unsupervised models need no fairness testing because they use no labels

B. Embeddings should always be publicly released for transparency

C. Human review, subgroup analysis and documentation should accompany consequential use

D. A high silhouette score proves that deployment is safe

Correct Answer: C. Human review, subgroup analysis and documentation should accompany consequential use