The paper "Stop using the elbow criterion for k-means" argues against the common practice of using the elbow method to determine the optimal number of clusters (k) in k-means clustering. The authors demonstrate that the elbow method is unreliable, often identifying spurious elbows or missing genuine ones. They show this through theoretical analysis and empirical examples across various datasets and distance metrics, revealing how the within-cluster sum of squares (WCSS) curve, on which the elbow method relies, can behave unexpectedly. The paper advocates for abandoning the elbow method entirely in favor of more robust and theoretically grounded alternatives like the gap statistic, silhouette analysis, or information criteria, which offer statistically sound approaches to k selection.
The arXiv preprint "Stop using the elbow criterion for k-means" argues vehemently against the common practice of employing the elbow method for determining the optimal number of clusters (k) in k-means clustering. The authors meticulously demonstrate that the elbow method, which relies on identifying a "kink" or "elbow" in the plot of within-cluster sum of squares (WCSS) against the number of clusters, is fundamentally flawed and often leads to inaccurate and misleading results. They highlight the subjective nature of visually identifying this "elbow," making the method prone to interpreter bias and lacking reproducibility. Different observers might identify different optimal k values based on the same WCSS plot, rendering the method unreliable for scientific rigor.
The paper underscores that the WCSS metric inherently decreases monotonically with increasing k. This means that adding more clusters will always reduce the WCSS, albeit at a diminishing rate. The elbow, representing the point of diminishing returns, is thus not a definitive indicator of an inherently optimal clustering structure within the data but rather a natural consequence of the algorithm's behavior. Furthermore, the paper illustrates how the elbow, even if discernible, can occur at an incorrect k, particularly in datasets exhibiting complex cluster shapes or varying cluster densities. The authors provide numerous simulated and real-world examples where the elbow method fails to identify the true number of clusters, sometimes dramatically overestimating or underestimating the optimal k.
As a compelling alternative to the elbow method, the authors advocate for the use of gap statistics. The gap statistic compares the within-cluster dispersion of the observed data to the expected dispersion under a null reference distribution representing a dataset with no discernible clustering structure. By calculating the gap statistic for different k values and identifying the k for which the gap is maximized, one obtains a more statistically principled and robust estimate of the optimal cluster number. This approach avoids the subjective interpretation inherent in the elbow method and provides a quantifiable measure for comparing different clustering solutions. The authors emphasize that the gap statistic, while computationally more intensive than the elbow method, offers a significantly more reliable and objective way to determine k, leading to more accurate and insightful clustering results. They conclude by strongly recommending abandoning the elbow method in favor of more robust alternatives like the gap statistic, promoting a more rigorous and statistically sound approach to k-means clustering analysis.
Summary of Comments ( 13 )
https://news.ycombinator.com/item?id=43450550
HN users discuss the problems with the elbow method for determining the optimal number of clusters in k-means, agreeing it's often unreliable and subjective. Several commenters suggest superior alternatives, such as the silhouette coefficient, gap statistic, and information criteria like AIC/BIC. Some highlight the importance of considering the practical context and the "business need" when choosing the number of clusters, rather than relying solely on statistical methods. Others point out that k-means itself may not be the best clustering algorithm for all datasets, recommending DBSCAN and hierarchical clustering as potentially better suited for certain situations, particularly those with non-spherical clusters. A few users mention the difficulty in visualizing high-dimensional data and interpreting the results of these metrics, emphasizing the iterative nature of cluster analysis.
The Hacker News post titled "Stop using the elbow criterion for k-means" (https://news.ycombinator.com/item?id=43450550) discusses the linked arXiv paper which argues against using the elbow method for determining the optimal number of clusters in k-means clustering. The comments section is relatively active, featuring a variety of perspectives on the topic.
Several commenters agree with the premise of the article. They point out that the elbow method is often subjective and unreliable, leading to arbitrary choices for the number of clusters. Some users share anecdotal experiences of the elbow method failing to produce meaningful results or being difficult to interpret. One commenter suggests the gap statistic as a more robust alternative.
A recurring theme in the comments is the inherent difficulty of choosing the "right" number of clusters, especially in high-dimensional spaces. Some users argue that the optimal number of clusters is often dependent on the specific application and downstream analysis, rather than being an intrinsic property of the data. They suggest that domain knowledge and interpretability should play a significant role in the decision-making process.
One commenter points out that the elbow method is particularly problematic when the clusters are not well-separated or when the data has a complex underlying structure. They suggest using visualization techniques, like dimensionality reduction, to gain a better understanding of the data before attempting to cluster it.
Another comment thread discusses the limitations of k-means clustering itself, regardless of the method used to choose k. Users highlight the algorithm's sensitivity to initial conditions and its assumption of spherical clusters. They propose alternative clustering methods, such as DBSCAN and hierarchical clustering, which may be more suitable for certain types of data.
A few commenters defend the elbow method, arguing that it can be a useful starting point for exploratory data analysis. They acknowledge its limitations but suggest that it can provide a rough estimate of the number of clusters, which can be refined using other techniques.
Finally, some commenters discuss the practical implications of choosing the wrong number of clusters. They highlight the potential for misleading results and incorrect conclusions, emphasizing the importance of careful consideration and validation. One commenter suggests using metrics like silhouette score or Calinski-Harabasz index to assess the quality of the clustering.
Overall, the comments section reflects a general consensus that the elbow method is not a reliable technique for determining the optimal number of clusters in k-means. Commenters offer various alternative approaches, emphasize the importance of domain knowledge and data visualization, and discuss the broader challenges of clustering high-dimensional data.