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$k$-POD: A Method for $k$-Means Clustering of Missing Data
Jocelyn T. Chi, Eric C. Chi, Richard G. Baraniuk
TL;DR
Missing data is common, but clustering methods often require complete data, and deletion or imputation can entail information loss or substantial costs. k-POD adapts k-means to partially observed data by iteratively filling missing entries from cluster centroids and reclustering. The paper reports accurate clustering across missingness mechanisms and at high missingness, while retaining limitations shared with k-means.
Problem
Clustering methods typically require complete data despite missing data being common, and deletion or imputation can be costly approaches to incomplete datasets.
Source: §1 · pp. 3–4Method
k-POD uses an MM algorithm to cluster partially observed data, updating missing entries from cluster centroids and reclustering without assumptions on missingness patterns.
Source: §1 · pp. 4–5, §2 · p. 6Results
Across missingness patterns, k-POD produces accurate clusterings even at large overall missingness and within reasonable time.
Source: §1 · p. 5Takeaways & Limitations
k-POD offers a simple, fast, tuning-free alternative that works without external information or assumptions on the missingness mechanism.
Source: §1 · p. 5, §6 · p. 23As a method built around k-means, k-POD retains difficulty with overlapping clusters or widely varying within-cluster scatter, where mixture-model extensions may be required.
Source: §6 · p. 22
Abstract
from arXiv · showhide
The $k$-means algorithm is often used in clustering applications but its usage requires a complete data matrix. Missing data, however, is common in many applications. Mainstream approaches to clustering missing data reduce the missing data problem to a complete data formulation through either deletion or imputation but these solutions may incur significant costs. Our $k$-POD method presents a simple extension of $k$-means clustering for missing data that works even when the missingness mechanism is unknown, when external information is unavailable, and when there is significant missingness in the data.
1. INTRODUCTION
Clustering partially observed data is challenging because common conversion strategies—deletion and imputation—can discard information, require costly modeling, or yield poor clusterings. The paper introduces k-POD, which clusters using observed data and reports accurate, efficient results across missingness patterns, including high overall missingness.
1. INTRODUCTION
- Practitioners commonly convert incomplete data through deletion or imputation, despite missing data being common and clustering methods typically requiring complete data.The paper notes examples in astronomy and social surveys.Source: §1 · p. 3
- Deletion removes variables with missing entries, which may be inadequate when many variables are affected or their missingness matters to clustering.Imputation fills missing entries but may require modeling joint distributions and can be laborious, uncertain, or harmful when assumptions are wrong.Source: §1 · pp. 3–4
- k-POD extends k-means to partially observed data with an MM algorithm, retaining observed information without assumptions about missingness patterns.It seeks clusterings in accord with the observed data.Source: §1 · p. 4
- k-POD combines a matrix-completion formulation with MM descent, avoiding the setup and computational expense typically required by probabilistic imputation methods.The authors describe this as a way to produce clusterings that agree with observed data.Source: §1 · p. 5
- Across missingness patterns, k-POD produces accurate clusterings even at large overall missingness and within reasonable time, while alternatives may fail or take prohibitively long.The experiments also show that imputation can require impractical preprocessing time on some modestly sized datasets.Source: §1 · p. 5
- The paper distinguishes its goal of extending simple, efficient k-means from more sophisticated mixture models that may identify more complicated cluster structures with missing data.The precise relationship between k-POD and mixture-model EM algorithms is beyond the paper’s scope.Source: §1 · p. 6
2. PREVIEW
The preview explains k-POD through iterative centroid-based completion: use cluster centroids to fill missing entries, then recluster. This procedure is formulated as an MM algorithm for a missing-data k-means objective.
2. PREVIEW
- Given cluster memberships, k-POD fills missing entries with the corresponding centroid values, reclusters the completed matrix, and updates those fills as assignments and centroids change.This cycle repeats as the clustering estimates change.Source: §2 · p. 6
- The preview identifies this iterative completion-and-reclustering procedure as an MM algorithm for minimizing a missing-data version of the k-means objective.The method assumes each observation is a noisy instantiation of a cluster centroid; its MM framework ensures subsequent missing-value estimates improve the objective.Source: §2 · pp. 6–7
- The paper next develops the mathematical formulations for k-means and an MM algorithm for its missing-data version.Source: §2 · p. 7
3. THE k-MEANS PROBLEM
The section expresses k-means as choosing cluster memberships and centroids to minimize squared residuals, then recasts the missing-data problem as minimizing residuals over observed entries. The membership matrix encodes a partition in which each observation belongs to exactly one cluster.
3. THE k-MEANS PROBLEM
- The section sets up the data matrix Y as n observations by p features and represents cluster centroids with the rows of B.Source: §3 · p. 7
- The k-means problem seeks a partition and centroid values minimizing the sum of squared residuals.The problem is NP-hard and is typically addressed with greedy alternating minimization, including Lloyd’s method.Source: §3 · p. 7
- A binary membership matrix A encodes cluster assignments, with each observation assigned to one and only one partition.The constraint A1 = 1 expresses this one-cluster-per-observation requirement.Source: §3 · p. 8
- For missing data, the proposed objective minimizes the squared discrepancy between Y and AB over observed entries Ω.The formulation uses a matrix-completion loss, while the paper emphasizes that its primary concern is clustering rather than imputation.Source: §3 · p. 8
4. MAJORIZATION-MINIMIZATION ALGORITHM
The algorithm majorizes the observed-entry clustering loss with a simpler surrogate and applies k-means to successive completed matrices. The resulting MM iterates decrease the missing-data objective even when k-means only inexactly minimizes the surrogate.
4. MAJORIZATION-MINIMIZATION ALGORITHM
- The MM principle replaces a hard optimization with simpler surrogate minimizations, using a surrogate that touches the objective at the current iterate and dominates it elsewhere.Source: §4 · pp. 9–10
- MM iterates decrease the objective even when the surrogate is not exactly minimized, as occurs when k-means performs the update.The paper states that k-POD drives the missing-data objective downhill.Source: §4 · pp. 9–10
- The MM surrogate combines observed-entry residuals with a penalty for changing the model on unobserved entries, and majorizes the missing-data objective at the current iterate.Its completed matrix uses observed values from Y and missing values from the current model.Source: §4 · p. 9
- Each k-POD iteration fills unobserved entries from the current cluster model AB, then applies k-means to update assignments and centroids.The algorithm initializes with inexpensive column-wise means followed by k-means; the experiments use k-means++ starting points.Source: §4 · pp. 10–11
- Unlike probabilistic imputation approaches, k-POD requires no assumptions on missingness patterns and targets clustering agreement with observed data rather than imputation quality.Source: §4 · p. 11
5. NUMERICAL EXPERIMENTS
The experiments compare k-POD with imputation and deletion approaches across missingness mechanisms, datasets, and missingness levels. k-POD often produces the most accurate or comparable clusterings and, in several comparisons, does so with less computation or completes scenarios where some imputation methods do not.
5.1 Mechanisms of Missingness
- The study covers MCAR, MAR, and NMAR, distinguished by whether missingness depends on the data, observed values, or unobserved values.The experiments do not simulate MAR for the simulated datasets because restricting missing entries to some columns makes the target overall missingness levels impractical.Source: §5.1 · p. 12, §5.4 · p. 14
5.2 Clustering Methods
- The experiments compare imputation-based k-means methods and deletion with k-POD, assuming the missingness mechanism is unknown.The imputation comparisons use Amelia, mice, and mi; deletion removes variables with missing entries.Source: §5.2 · p. 13
5.3 Data
- Experiments use a perturbed wine dataset and simulated mixtures with 10 or 25 clusters, with 500 observations on 100 variables in the simulated data.The wine dataset has 178 observations, 14 variables, and three clusters; the simulated datasets are generated under conditions where k-means performs well without missing data.Source: §5.3 · p. 14
5.4 Experimental Setup
- Each method–mechanism–missingness combination has 100 trials; simulated-data scenarios are terminated after 55 hours, and incomplete methods have no aggregate Rand scores.Rand scores compare each clustering with true class labels; higher scores indicate greater agreement.Source: §5.4 · pp. 15–16
5.5 Results
- On the wine data, k-POD is most accurate from 45% MCAR and 12% MAR missingness, while other approaches sometimes fail to complete as missingness increases.At low missingness all methods perform well; Amelia and mice fail to complete some imputation scenarios at higher missingness.Source: §5.5 · pp. 16–17, Table 2 · p. 17
- On the wine data, k-POD is essentially as fast as Amelia, faster than mice in MCAR and NMAR scenarios, and faster than mi in all scenarios.mi is the slowest in all scenarios, while deletion is very expeditious in MAR scenarios; mi alone completes all experiments regardless of mechanism or missingness percentage, but requires more time, particularly when missingness occurs in all variables.Source: §5.5 · pp. 17–18, Table 3 · p. 18
- For 10 simulated clusters, k-POD is most accurate at 75% MCAR and 50% NMAR missingness, and is essentially equivalent to mice elsewhere.Amelia and mi did not produce results within 55 hours; Table 5 timing results show mice requires significantly more computation than k-POD for essentially the same results.Source: §5.5 · p. 18, Table 4 · p. 19, Table 5 · p. 20
- With 25 simulated clusters, k-POD is most accurate in all but one scenario, while mice results remain similar and generally require significantly more computation.Amelia and mi did not produce results within 55 hours.Source: §5.5 · p. 18, Table 6 · p. 20, Table 7 · p. 21
6. DISCUSSION
k-POD extends k-means to missing data by minimizing an observed-value loss through a simple majorization step that can be minimized with k-means. The discussion presents it as a fast alternative while noting that it retains k-means’ limitations.
6. DISCUSSION
- k-POD formulates k-means as an optimal rank-k decomposition and minimizes residual sum of squares over observed values using a majorization that can be inexactly minimized with k-means.This provides a missing-data clustering loss using the observed data alone.Source: §6 · p. 22
- The numerical results show k-POD is accurate and fast, particularly with higher missingness, because its update copies centroid estimates into missing entries and then runs k-means.Each k-means iteration requires O(knp) computations; reasonable imputation setup and computation may become prohibitively expensive on larger data.Source: §6 · p. 22
- As a k-means-based meta-algorithm, k-POD retains k-means’ weakness with overlapping clusters or drastically varying within-cluster scatter, where mixture-model extensions may be needed.For complete data, EM clustering can produce superior results in these cases.Source: §6 · p. 22
- k-POD builds on the often-preferred k-means algorithm to offer a simple, quick alternative that works with unknown missingness mechanisms, no external information, and substantial missingness.The R package kpodclustr is available on CRAN.Source: §6 · pp. 22–23