A Survey on Archetypal Analysis

Aleix Alcacer; Irene Epifanio; Sebastian Mair; Morten Mørup

stat.MEApr 16, 2025

A Survey on Archetypal Analysis

Authors: Aleix Alcacer, Irene Epifanio, Sebastian Mair, Morten Mørup

Organizations: Jaume I University and ValgrAI · Linköping University · Technical University of Denmark

Abstract

Archetypal analysis (AA) was originally proposed in 1994 by Adele Cutler and Leo Breiman as a computational procedure for extracting distinct aspects, so-called archetypes, from observations, with each observational record approximated as a mixture (i.e., convex combination) of these archetypes. AA thereby provides straightforward, interpretable, and explainable representations for feature extraction and dimensionality reduction, facilitating the understanding of the structure of high-dimensional data and enabling wide applications across the sciences. However, AA also faces challenges, particularly as the associated optimization problem is nonconvex. This is the first survey that provides researchers and data mining practitioners with an overview of the methodologies and opportunities that AA offers, surveying the many applications of AA across disparate fields of science, as well as best practices for modeling data with AA and its limitations. The survey concludes by explaining crucial future research directions concerning AA.

Figures & tables

Fig. 1: AA computed on the subset of the MNIST handwritten digit dataset showing solely the digit 9 with three archetypes. The left part depicts the mixing weights of the nines whereas the right part depicts the actual handwritten digits.

Fig. 2: Citation overview of paper [ 1 ] in Google Scholar and SCOPUS until 5th November 2025.

Fig. 3: An example of AA with

K=3

archetypes on two-dimensional toy data. After initializing the archetypes (purple squares), the optimization procedure pushes the archetypes (green squares) to lie on the boundary of the convex hull of data (see Theorem 1 ). The overall objective is to minimize the sum of projection errors (i.e., the sum of all dashed lines).

Fig. 4: An example of AA in two dimensions for various numbers (

K=2,3,4

) of archetypes

\{\mathbf{a}_{1},\ldots,\mathbf{a}_{K}\}

. The archetypes are always located on the boundary of the convex hull of data.

Fig. 5: AA with

K=3

archetypes compared to a nonnegative matrix factorization (NMF) with

K=2

components, a principal component analysis (PCA) with

K=2

components (first is solid, second dashed),

k

-means clustering with

K=3

clusters, and

k

-maxoids clustering with

K=3

clusters.

Algorithm	Restrictions on $\mathbf{C}$ and $\mathbf{A}$		Restrictions on $\mathbf{S}$
PCA	$\mathbf{C}$		$\mathbf{S}$
NMF [ 6 ]		$\mathbf{A}\geq 0$	$\mathbf{S}\geq 0$
Convex NMF [ 17 ]	$\mathbf{C}\geq 0$	$\lVert\mathbf{c}_{k}\rVert_{1}=1$	$\mathbf{S}\geq 0$
Affine Coding [ 18 ]		$0\leq a_{km}\leq 1$		$\lVert\mathbf{s}_{n}\rVert_{1}=1$
Convex Coding [ 18 ]		$0\leq a_{km}\leq 1$	$\mathbf{S}\geq 0$	$\lVert\mathbf{s}_{n}\rVert_{1}=1$
Conic Coding [ 18 ]		$0\leq a_{km}\leq 1$	$\mathbf{S}\geq 0$

TABLE I: An overview of related methods and their restrictions.

Fig. 6: Only

M+1

archetypes

\mathbf{A}^{\prime}

are needed to define a data point. Hence, in the case of

K>M+1

, the weights

\mathbf{s}_{n}

are not consistent. For the highlighted data point in the middle of the dataset, two different subsets

\mathbf{A}^{\prime}

(blue shaded area) of

M+1

archetypes can be used to define the point. Thus, the representation

\mathbf{s}_{n}

is not consistent. Here,

M=2

and

K=4

Method	Archetypes	Location	Representation space	Purpose / when to use
AA	Mixtures of observations	Boundary of the convex hull	Original data space	Standard choice for interpretable extremal profiles
Relaxed AA	Relaxed mixtures of observations	May extend outside the convex hull	Original data space	When extreme profiles may not be well represented by the observed convex hull
ADA	Actual observations	Boundary or interior of the convex hull	Original data space	When archetypes should correspond to tangible, observed cases
Kernel AA	Mixtures of mapped observations	Boundary of the convex hull in feature space	Nonlinear kernel space	When nonlinear structure or pairwise similarities are important
AAnet / DeepAA	Latent archetypal representations	Latent archetypal space	Learned nonlinear space	When the nonlinear representation should be learned jointly with the archetypal structure
Sparse AA	Sparse mixtures	May extend outside the convex hull	Original data space	For sparse representations in hyperspectral unmixing

TABLE II: Taxonomy of representative extensions of archetypal analysis.

Optimization algorithm	Abbreviation	Computational Cost $\mathcal{O}_{\mathbf{S}}$	Computational Cost $\mathcal{O}_{\mathbf{C}}$
Nonnegative Least Squares [ 81 , 1 ]	NNLS	$\mathcal{O}_{\mathbf{S}}(NK^{3})$	$\mathcal{O}_{\mathbf{C}}(K\|\mathcal{A}\|^{3})$
Active Set [ 70 ]	AS	$\mathcal{O}_{\mathbf{S}}(KM+\|\mathcal{A}\|^{2})$	$\mathcal{O}_{\mathbf{C}}(KM+\|\mathcal{A}\|^{2})$
Principal Convex Hull Analysis [ 8 ]	PCHA	$\mathcal{O}_{\mathbf{S}}(NMK)$	$\mathcal{O}_{\mathbf{C}}(NMK)$
Frank–Wolfe [ 84 ]	FW	$\mathcal{O}_{\mathbf{S}}(NMK)$	$\mathcal{O}_{\mathbf{C}}(NMK)$
Softmax + Adam [ 56 , 66 ]	SA	$\mathcal{O}_{\mathbf{S}}(NMK)$	$\mathcal{O}_{\mathbf{C}}(NMK)$
Block Coordinate Descent [ 78 ]	BCD	$\mathcal{O}_{\mathbf{S}}(NMK)$	$\mathcal{O}_{\mathbf{C}}(NMK)$

TABLE III: Comparison of the principal optimization algorithms used in AA. Costs are per update and exclude the

\mathcal{O}(NMK)

shared by every update for forming

\mathbf{A}=\mathbf{C}\mathbf{X}

and

\mathbf{S}\mathbf{C}\mathbf{X}

, which for the gradient-based procedures is already the dominant term.

Fig. 7: A comparison of the following initialization techniques of AA for

K=3

: Uniform, FurthestFirst, FurthestSum, and AA++.

Name	Language	Description	Models	Optimization algorithms	Sparsity	Nonlinearity	Robustness	Weighted
LEAST-SQUARES AA
archetypes [ 111 ]	R	Package for AA	AA	NNLS [ 1 ]	✗	✗	✓	✓
adamethods [ 115 , 100 ]	R	Package for archetypoid analysis	ADA	NNLS [ 1 ] , ASA [ 19 ]	✗	✗	✓	✗
archetypes [ 46 ]	Python	Package for AA and visualization tools	AA, ADA, BiAA	PCHA [ 8 ] , NNLS [ 1 ] , PAA [ 3 ]	✗	✓	✗	✓
SPAMS [ 70 ]	R, Matlab, Python	Sparse modeling package including AA	AA	AS [ 70 ]	✗	✗	✗	✗
ParetoTI [ 110 ]	R	Package for AA and Pareto task inference	AA	PCHA [ 8 ]	✗	✗	✗	✗

TABLE IV: Summary of the main available packages and tools for archetypal analysis.

Fig. 8: Left panel: Simplex plot representing the scores, where each observation is a point, and the archetypes are positioned at the colored vertices. The location of each point indicates its relative contribution from the archetypes. Right panel: Stacked bar chart of the scores, where each bar represents an observation and the colored segments correspond to the contributions of each archetype.

Fig. 9: Bubble map with the main areas of AA applications.

Fig. 10: Analysis of data on Finches 6 6 6 Data taken from https://datadryad.org/dataset/doi:10.5061/dryad.g6g3h in which two features are measured based on beak length and depth for scandens (blue dots) and fortis (black dots) finches illustrated for a

K=2

K=3

, and

K=4

AA model. When inspecting the variance explained we observe substantial improvements up until

K=3

, and reproducibility as measured using

\text{NMI}(\mathbf{S},\mathbf{S}^{\prime})

and

\text{sim}(\mathbf{A},\mathbf{A}^{\prime})

we observe that a

K=3

model is also reliably recovering

\mathbf{S}

and

\mathbf{A}

. Whereas the

K=2

AA model mainly discriminates between the two types of finches, the

K=3

characterizes in particular variability in the fortis finch class whereas the

K=4

further subdivides the scandens finch class.

Fig. 11: Analysis of data on mixture designed NMR experiment containing systematically different fractions of propanol, butanol and pentanol 8 8 8 Data taken from https://ucphchemometrics.com/datasets/ . When inspecting the variance explained we observe substantial improvements up until

K=3

and reproducibility as measured using

\text{NMI}(\mathbf{S},\mathbf{S}^{\prime})

and

\text{sim}(\mathbf{A},\mathbf{A}^{\prime})

we observe that a

K=3

reliably recovers

\mathbf{S}

and

\mathbf{A}

. This three component model well corresponds to the three compounds whereas the estimated concentration fractions (given at the bottom) well correspond to the true concentration levels of the three compounds in each sample.

Fig. 12: Analysis of a remote sensing hyperspectral image data 10 10 10 Data taken from https://www.ehu.eus/ccwintco/index.php?title=Hyperspectral_Remote_Sensing_Scenes#Indian_Pines in which an image of a landscape in Indiana, US is measured across 200 wavelengths (removing bands pertaining to water absorption) in a $145\times 145$ image. The image is turned into a matrix of $200$ wavelength features by $145^{2}$ pixel observations. When inspecting the variance explained and reproducibility as measured using $\text{NMI}(\mathbf{S},\mathbf{S}^{\prime})$ and $\text{sim}(\mathbf{A},\mathbf{A}^{\prime})$ we observe that $K=6$ is the largest number of components that can be extracted while attaining reliable solutions. Inspecting the learned AA representation for $K=6$ we observe that the archetypes provide distinct spectral profiles with their convex combinations defining different regions of the image in which these profiles are present.

Fig. 13: On the left, we analyze a two-dimensional dataset where points follow a linear relationship, except for an outlier shown in red. The outlier appears clearly separated from the rest in the representation for $K$ = 3 AA model, unlike the original space or PCA projection. The table shows AUC results (first row) and the rank of outlierness of the outlier (second row) for the outlier detection methods by [ 131 ] . As there are 11 points, the highest possible rank is 11, which corresponds to the lowest degree of outlierness. Projection in the archetypal space improves distance based outlier detection techniques such as $k$ -NN. AA + $k$ NN is the only method that detects the outlier. On the right, we reproduce an example by [ 132 ] where functional AA projections and $k$ -NN are used to detect the anomaly in the landscape image.

Fig. 14: Analysis of data of congressional voting records 12 12 12 Data taken from https://archive.ics.uci.edu/dataset/105/congressional+voting+records considering 16 bills with votes missing by some congress members on some bills. The Kernel AA procedure is here used considering the Jaccard similarity between congress members in terms of bills they both voted for ignoring bills either had as missing value, i.e.,

\mathbf{K}^{\text{Jac.}}_{ij}=\frac{\sum_{m:(x_{im}\in\{0,1\},x_{jm}\in\{0,1\})}x_{im}x_{jm}}{\sum_{m^{\prime}:(x_{im^{\prime}}\in\{0,1\},x_{jm^{\prime}}\in\{0,1\})}1-(1-x_{im^{\prime}})(1-x_{jm^{\prime}})}

. The kernel has been double centered, i.e.,

\mathbf{TK}^{\text{Jac.}}\mathbf{T}^{\mkern-1.5mu\mathsf{T}}

prior to analyses, corresponding to removing the mean. Black dots correspond to Republicans, whereas blue dots correspond to Democrats. The learned models are illustrated for a

K=2

K=3

, and

K=4

AA model. When inspecting the variance explained and reproducibility as measured using

\text{NMI}(\mathbf{S},\mathbf{S}^{\prime})

and

\text{sim}(\mathbf{A},\mathbf{A}^{\prime})

we observe that up to

K=4

the AA models can reliably be inferred whereas the variance explained steadily improves as the number of components is increased. Inspecting the model for

K=2

K=3

, and

K=4

we observe that the models well separate Democrats (blue dots) from Republicans (black dots).

Irene Epifanio received her MS degree in mathematics and her PhD degree in statistics from València University, Spain. She is currently a full professor of Statistics at Jaume I University, Spain, and Senior Research Fellow at valgrAI. Her research interests include statistical learning, functional data analysis, computer vision and equality. She was the recipient of various awards in research, teaching, scientific dissemination and social commitment, including the Margarita Salas Prize of Talent Woman Spain.

Table 20

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