Poisson Subspace Clustering: Focusing on the Essentials in Count Data

Collin Leiber; Kai Puolamäki; Heikki Mannila

cs.LGAug 24, 2026

Poisson Subspace Clustering: Focusing on the Essentials in Count Data

Authors: Collin Leiber, Kai Puolamäki, Heikki Mannila

Organizations: Aalto University, Espoo, Finland. · University of Helsinki, Helsinki, Finland.

Abstract

Count data represented as a matrix of non-negative integer values, such as contingency tables, are prevalent across diverse domains. When clustering such data sets, specific methods are required, as generic algorithms often fail to consider their unique distributional properties, leading to unreliable outputs. An effective strategy is to use well-established statistical models such as the Poisson and negative binomial distributions. We present 3CPO, a clustering algorithm based on statistically solid modeling of count data. In addition to the cluster labels, it identifies a subset of relevant columns, enhancing the interpretability of the results. We propose a simple iterative algorithm that maximizes the posterior probability to find good clustering solutions and discuss its properties. Extensive experiments demonstrate its ability to define high-quality clusters within associated subspaces for various data domains, ranging from gene expressions and texts to economics. Our findings suggest that 3CPO is a robust solution for clustering count data in a statistically sound and interpretable manner. Our code is available at https://github.com/collinleiber/3CPO.

Figures & tables

Panel of Figure 2 — Figure 2: Loss of 3CPO (blue line) and number of columns in $C_{1}$ (green line) after each iteration. The dashes below indicate changes of $C_{1}$ (red), $C_{0}$ (blue) and $C_{-}$ (gray).

Data set	3CPO	PoissonL	PoissonC	SKM	KM	STD+KM	MM+KM	RF+KM	RCA+KM
Synth	$95.5\pm 0.0$	$33.3\pm 0.1$	$29.9\pm 0.0$	$42.7\pm 0.0$	$0.2\pm 0.0$	$42.1\pm 0.0$	$42.9\pm 0.1$	$42.5\pm 0.0$	$0.0\pm 0.0$
Wholesales	$30.1\pm 0.0$	$31.3\pm 0.0$	$32.8\pm 0.0$	$26.5\pm 0.0$	$-3.1\pm 0.0$	$28.2\pm 0.0$	$27.9\pm 0.2$	$21.3\pm 0.0$	$47.0\pm 0.3$
SportA	$32.3\pm 0.5$	$32.5\pm 0.2$	$21.9\pm 1.5$	$29.5\pm 0.0$	$21.3\pm 0.0$	$31.9\pm 0.0$	$25.9\pm 0.1$	$7.5\pm 0.0$	$0.1\pm 0.0$
Optdigits	$66.7\pm 2.2$	$67.0\pm 1.8$	$45.0\pm 3.1$	$67.5\pm 0.1$	$67.1\pm 0.2$	$67.6\pm 0.1$	$67.3\pm 0.3$	$66.5\pm 0.2$	$0.0\pm 0.0$
BBCSports	$90.1\pm 2.9$	$51.6\pm 8.1$	$27.7\pm 4.0$	$10.4\pm 1.7$	$0.3\pm 0.2$	$9.5\pm 0.9$	$6.9\pm 3.6$	$9.8\pm 1.7$	$3.7\pm 2.6$
BBCNews	$89.9\pm 0.9$	$89.2\pm 0.8$	$63.4\pm 6.7$	$31.5\pm 4.0$	$6.0\pm 0.3$	$24.6\pm 1.6$	$16.4\pm 1.5$	$25.6\pm 1.8$	$15.4\pm 12.0$

Table 1: ARI results (in %) of traditional clustering (top) and co-clustering algorithms (bottom). Entries correspond to the mean of ten executions

\pm

the standard deviation. The best result for each data set and group of algorithms (traditional/co-clustering) is marked in bold , the runner-up is underlined , and the third place is dashed-underlined .

Data set	3CPO	TF-IDF+KM	BM25+KM	TF-IDF+SKM	BM25+SKM
BBCSports	$90.1\pm 2.9$	$71.0\pm 11.4$	$43.7\pm 17.0$	$76.5\pm 7.1$	$81.1\pm 6.8$
BBCNews	$89.9\pm 0.9$	$84.2\pm 2.8$	$87.7\pm 6.8$	$86.4\pm 0.8$	$90.7\pm 0.3$
WebKB	$31.5\pm 3.2$	$18.8\pm 1.0$	$14.6\pm 1.3$	$22.4\pm 1.1$	$29.0\pm 1.9$
Reuters	$67.8\pm 1.6$	$19.3\pm 0.2$	$34.8\pm 5.0$	$34.0\pm 0.2$	$55.8\pm 1.2$
20NewsG	$23.0\pm 1.3$	$5.6\pm 0.6$	$5.5\pm 0.5$	$8.2\pm 0.4$	$21.1\pm 1.2$

Table 2: ARI results (in %) regarding text data sets pre-processed by TF-IDF and BM25. Entries correspond to the mean of ten executions

\pm

the standard deviation. The best performance is highlighted in bold .

Data set	$\text{ACC}^{*}$	$\text{NMI}^{*}$	$\text{ARI}^{*}$	$\|O\|$	Data set	$\text{ACC}^{*}$	$\text{NMI}^{*}$	$\text{ARI}^{*}$	$\|O\|$
Synth	$91.9\pm 0.0$	$81.7\pm 0.0$	$85.4\pm 0.0$	$336\pm 0$	WebKB	$56.8\pm 3.7$	$39.3\pm 1.1$	$33.1\pm 3.0$	$431\pm 38$
Wholesales	$86.2\pm 0.2$	$47.4\pm 0.5$	$52.2\pm 0.7$	$144\pm 1$	Reuters	$85.3\pm 7.3$	$75.3\pm 7.0$	$73.9\pm 8.2$	$427\pm 275$
SportA	$85.3\pm 0.1$	$37.2\pm 0.2$	$49.4\pm 0.2$	$341\pm 4$	20NewsG	$39.8\pm 2.6$	$42.7\pm 1.1$	$25.4\pm 1.3$	$978\pm 49$
Optdigits	$82.1\pm 2.0$	$79.2\pm 1.0$	$72.6\pm 2.4$	$443\pm 3$	MouseAtlas	$72.4\pm 7.7$	$75.8\pm 3.5$	$64.5\pm 8.2$	$1326\pm 84$
BBCSports	$98.4\pm 1.1$	$95.5\pm 2.0$	$95.9\pm 2.7$	$56\pm 4$	GeneExp	$100.0\pm 0.0$	$100.0\pm 0.0$	$100.0\pm 0.0$	$14\pm 2$
BBCNews	$97.9\pm 0.4$	$93.3\pm 0.8$	$94.9\pm 0.9$	$137\pm 11$	HDendritic	$87.7\pm 7.3$	$81.5\pm 3.7$	$82.6\pm 5.5$	$22\pm 2$

Table 3: Clustering results (in %) and the number of outliers

|O|

when using the outlier detection feature. An asterisk

*

indicates that only non-outliers are used for the evaluation. Entries correspond to the mean of ten executions

\pm

the standard deviation. Improvements compared to results without outlier detection are marked in bold .

	Synth	Wholesales	SportA	Optdigits	BBCSports	BBCNews	WebKB	Reuters	20NewsG	MouseAtlas	GeneExp	HDendritic
$m$	$6$	$6$	$55$	$64$	$2000$	$2000$	$2000$	$2000$	$2000$	$15006$	$20531$	$26593$
$\|C_{1}\|$	$2\pm 0$	$5\pm 0$	$41\pm 1$	$56\pm 1$	$799\pm 8$	$1379\pm 6$	$1315\pm 18$	$1322\pm 8$	$1454\pm 15$	$14566\pm 59$	$6445\pm 3$	$16673\pm 406$

Table 4: Total (

m

) and cluster-relevant columns (

|C_{1}|

) as identified by 3CPO. Entries for

|C_{1}|

correspond to the mean of ten executions

\pm

the standard deviation.

Panel of Figure 3 — Figure 3: Histograms showing the amount of zeros per column in each column partition.

Data set	Partition + Cluster	Most important words w.r.t. $\lambda_{j\|l}^{C}-\lambda_{j\|0}^{C}$ or $\lambda_{j\|l}^{C}$ (stemmed)	Ground Truth Fit
BBCSports	$C_{1}\qquad 1~(\Rightarrow l=1)$	‘half’, ‘six’, ‘their’, ‘franc’, ‘nation’, ‘the’, ‘rugbi’, ‘ireland’, ‘wale’, ‘england’	Rugby
BBCSports	$C_{1}\qquad 2~(\Rightarrow l=2)$	‘year’, ‘indoor’, ‘world’, ‘race’, ‘athlet’, ‘her’, ‘olymp’, ‘she’, ‘in’, ‘the’	Athletics
BBCSports	$C_{1}\qquad 3~(\Rightarrow l=3)$	‘four’, ‘day’, ‘by’, ‘run’, ‘his’, ‘test’, ‘wicket’, ‘over’, ‘off’, ‘ball’	Cricket
BBCSports	$C_{1}\qquad 4~(\Rightarrow l=4)$	‘the’, ‘australian’, ‘she’, ‘her’, ‘final’, ‘in’, ‘seed’, ‘roddick’, ‘set’, ‘open’	Tennis
BBCSports	$C_{1}\qquad 5~(\Rightarrow l=5)$	‘unit’, ‘leagu’, ‘have’, ‘chelsea’, ‘club’, ‘that’, ‘he’, ‘is’, ‘we’, ‘it’	Football
BBCSports	$C_{0}\qquad~~(\Rightarrow l=0)$	‘as’, ‘at’, ‘with’, ‘but’, ‘was’, ‘on’, ‘for’, ‘of’, ‘and’, ‘to’	–

Table 5: The ten most important words (based on the largest difference

\lambda_{j|l}^{C}-\lambda_{j|0}^{C}

for

j\in C_{1}

and based on the largest

\lambda_{j|l}^{C}

for

j\in C_{0}\cup C_{-}

) within the BBCSports and BBCNews data sets per column partition and cluster as identified by 3CPO.

Panel of Figure 4 — Figure 4: Robustness experiments investigating the impact of columns containing uniformly distributed noise on the clustering performance (measured by ARI). The upper plots show traditional clustering algorithms and the lower plots show co-clustering algorithms. In (a) we increase the amount noise columns with values within $[0,15]$ and in (b) we have a single noise column with values within $[0,u]$ ( $u$ is given on the x-axis).

Data set	Metric	3CPO	$C_{-}=\emptyset$	$C_{0}=\emptyset$	$b({\bf X}_{\cdot j})=0$	$b({\bf X}_{\cdot j})=\text{BIC}({\bf X}_{\cdot j})$
Synth	ARI	$95.5\pm 0.0$	$33.3\pm 0.1$	$95.5\pm 0.0$	$95.5\pm 0.0$	$95.5\pm 0.0$
	$\|C_{1}\|$	$2.0\pm 0.0$	$5.8\pm 0.6$	$4.0\pm 0.0$	$4.0\pm 0.0$	$2.0\pm 0.0$
Wholesales	ARI	$30.1\pm 0.0$	$31.3\pm 0.0$	$30.1\pm 0.0$	$30.1\pm 0.0$	$30.1\pm 0.0$
	$\|C_{1}\|$	$5.0\pm 0.0$	$6.0\pm 0.0$	$5.0\pm 0.0$	$5.0\pm 0.0$	$5.0\pm 0.0$
SportA	ARI	$32.3\pm 0.5$	$32.3\pm 0.1$	$32.6\pm 0.2$	$32.6\pm 0.2$	$32.5\pm 0.3$
	$\|C_{1}\|$	$41.1\pm 1.4$	$42.5\pm 1.1$	$52.0\pm 0.0$	$52.0\pm 0.0$	$41.3\pm 0.9$

Table 6: ARI results (in %) and the final number of columns included in

C_{1}

of various ablation studies. Entries correspond to the mean of ten executions

\pm

the standard deviation. Colors indicate whether the average performance of 3CPO lies above (green), within (yellow) or below (red) the standard deviation band, where above is better for ARI and below is better for

|C_{1}|

Panel of Figure 5 — Figure 5: Loss of 3CPO (incl. the penalty term $v(K)$ – blue line) and number of columns in $C_{1}$ (green line) for varying $K$ . Each entry corresponds to the best loss after $20$ runs. The red line marks the $K$ for which the minimum loss occurs.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Data set	$n$	$m$	$K$	Range	Sparsity	Imbalance	Domain
Synth	$1000$	$6$	$3$	$[0,125]$	$1\%$	$1.00$	Tabular (synthetic)
Wholesales	$440$	$6$	$2$	$[3,112151]$	$0\%$	$0.48$	Tabular (economics)
SportA	$1000$	$55$	$2$	$[0,897]$	$31\%$	$0.57$	Tabular (text characteristics)
Optdigits	$5620$	$64$	$10$	$[0,16]$	$49\%$	$0.97$	Image (digits)
BBCSports	$737$	$2000^{\dagger}$	$5$	$[0,145]$	$92\%$	$0.38$	Text (sport articles)
BBCNews	$2225$	$2000^{\dagger}$	$5$	$[0,245]$	$92\%$	$0.76$	Text (news articles)

Appendix

Table 7: Data set characteristics: the number of rows

n

, columns

m

and clusters

K

as well as the data range, the sparsity (ratio of zeros) and the imbalance (ratio between the smallest and largest ground truth cluster, i.e.,

\text{min}_{k\in[K]}|C_{k}|/\text{max}_{k\in[K]}|C_{k}|

). Furthermore, the table includes information about the data domain. A dagger

\dagger

indicates that the original number of features has been modified.

Data set	Metric	3CPO	PoissonL	PoissonC	SKM	KM	STD+KM	MM+KM	RF+KM	RCA+KM
Synth	ACC	$98.5\pm 0.0$	$58.0\pm 0.0$	$57.2\pm 0.0$	$61.3\pm 0.1$	$36.6\pm 0.2$	$61.0\pm 0.0$	$61.1\pm 0.0$	$61.0\pm 0.0$	$35.8\pm 0.0$
	NMI	$92.5\pm 0.0$	$42.4\pm 0.1$	$39.9\pm 0.0$	$49.0\pm 0.1$	$0.3\pm 0.0$	$49.1\pm 0.0$	$49.5\pm 0.1$	$48.3\pm 0.0$	$0.2\pm 0.0$
	ARI	$95.5\pm 0.0$	$33.3\pm 0.1$	$29.9\pm 0.0$	$42.7\pm 0.0$	$0.2\pm 0.0$	$42.1\pm 0.0$	$42.9\pm 0.1$	$42.5\pm 0.0$	$0.0\pm 0.0$
Wholesales	ACC	$77.5\pm 0.0$	$78.2\pm 0.0$	$78.9\pm 0.0$	$75.9\pm 0.0$	$59.3\pm 0.0$	$76.8\pm 0.0$	$76.6\pm 0.1$	$73.2\pm 0.0$	$84.4\pm 0.1$
	NMI	$31.9\pm 0.0$	$23.1\pm 0.0$	$24.0\pm 0.0$	$19.5\pm 0.0$	$0.9\pm 0.0$	$20.0\pm 0.0$	$20.3\pm 0.2$	$17.2\pm 0.0$	$36.1\pm 0.2$
	ARI	$30.1\pm 0.0$	$31.3\pm 0.0$	$32.8\pm 0.0$	$26.5\pm 0.0$	$-3.1\pm 0.0$	$28.2\pm 0.0$	$27.9\pm 0.2$	$21.3\pm 0.0$	$47.0\pm 0.3$

Appendix

Table 8: Clustering results (in %) of traditional clustering algorithms. Entries correspond to the mean of ten executions

\pm

the standard deviation. The best result for each data set and evaluation metric is marked in bold , the runner-up is underlined , and the third place is dashed-underlined .

Data set	Metric	3CPO	CROINFO	CCMod	CCSMod	ELBM	SELBM	TauCC
Synth	ACC	$98.5\pm 0.0$	$57.9\pm 0.1$	$60.0\pm 0.2$	$45.8\pm 14.5$	$45.4\pm 7.7$	$42.5\pm 0.2$	$59.8\pm 0.0$
	NMI	$92.5\pm 0.0$	$42.0\pm 0.2$	$34.0\pm 0.4$	$20.0\pm 23.6$	$14.7\pm 14.1$	$7.3\pm 0.1$	$33.7\pm 0.5$
	ARI	$95.5\pm 0.0$	$32.8\pm 0.3$	$25.2\pm 0.1$	$17.8\pm 20.8$	$9.2\pm 8.6$	$3.5\pm 1.6$	$25.1\pm 0.6$
Wholesales	ACC	$77.5\pm 0.0$	$79.2\pm 0.2$	$78.4\pm 0.0$	$68.8\pm 3.1$	$64.7\pm 4.3$	$65.5\pm 0.1$	$75.7\pm 0.0$
	NMI	$31.9\pm 0.0$	$25.1\pm 0.5$	$25.1\pm 0.0$	$2.3\pm 6.7$	$1.6\pm 3.9$	$0.3\pm 0.1$	$21.8\pm 0.0$
	ARI	$30.1\pm 0.0$	$33.6\pm 0.5$	$32.0\pm 0.0$	$3.2\pm 9.1$	$2.0\pm 5.0$	$1.9\pm 0.6$	$27.6\pm 0.0$

Appendix

Table 9: Clustering results (in %) of co-clustering algorithms. Entries correspond to the mean of ten executions

\pm

the standard deviation. The best result for each data set and evaluation metric is marked in bold , the runner-up is underlined , and the third place is dashed-underlined .

Data set	Metric	3CPO	TF-IDF+KM	BM25+KM	TF-IDF+SKM	BM25+SKM
BBCSports	ACC	$96.1\pm 1.2$	$86.9\pm 6.3$	$64.1\pm 8.0$	$89.2\pm 5.0$	$92.4\pm 2.9$
	NMI	$89.4\pm 2.0$	$74.7\pm 6.3$	$58.9\pm 10.5$	$77.8\pm 4.7$	$82.0\pm 4.6$
	ARI	$90.1\pm 2.9$	$71.0\pm 11.4$	$43.7\pm 17.0$	$76.5\pm 7.1$	$81.1\pm 6.8$
BBCNews	ACC	$95.7\pm 0.4$	$93.2\pm 1.2$	$94.0\pm 5.3$	$94.1\pm 0.4$	$96.0\pm 0.2$
	NMI	$87.2\pm 0.8$	$81.6\pm 1.8$	$85.9\pm 3.9$	$83.3\pm 0.8$	$87.9\pm 0.4$
	ARI	$89.9\pm 0.9$	$84.2\pm 2.8$	$87.7\pm 6.8$	$86.4\pm 0.8$	$90.7\pm 0.3$

Appendix

Table 10: Clustering results (in %) regarding text data sets pre-processed by TF-IDF and BM25. Entries correspond to the mean of ten executions

\pm

the standard deviation. The best performance is highlighted in bold .

Data set	Metric	3CPO	$C_{-}=\emptyset$	$C_{0}=\emptyset$	$b({\bf X}_{\cdot j})=0$	$b({\bf X}_{\cdot j})=\text{BIC}({\bf X}_{\cdot j})$
Synth	ACC	$98.5\pm 0.0$	$58.0\pm 0.0$	$98.5\pm 0.0$	$98.5\pm 0.0$	$98.5\pm 0.0$
	NMI	$92.5\pm 0.0$	$42.4\pm 0.1$	$92.5\pm 0.0$	$92.5\pm 0.0$	$92.5\pm 0.0$
	ARI	$95.5\pm 0.0$	$33.3\pm 0.1$	$95.5\pm 0.0$	$95.5\pm 0.0$	$95.5\pm 0.0$
	$\|C_{1}\|$	$2.0\pm 0.0$	$5.8\pm 0.6$	$4.0\pm 0.0$	$4.0\pm 0.0$	$2.0\pm 0.0$
Wholesales	ACC	$77.5\pm 0.0$	$78.2\pm 0.0$	$77.5\pm 0.0$	$77.5\pm 0.0$	$77.5\pm 0.0$
	NMI	$31.9\pm 0.0$	$23.1\pm 0.0$	$31.9\pm 0.0$	$31.9\pm 0.0$	$31.9\pm 0.0$

Appendix

Table 11: Clustering results (in %) and the final number of columns included in

C_{1}

of various ablation studies. Entries correspond to the mean of ten executions

\pm

|C_{1}|

Data set	Metric	Poisson- $k$ -Means++	random labels	random centers	$k$ -Means++	$k$ -Means (full)	RF+ $k$ -Means++	RF+ $k$ -Means (full)
Synth	ACC	$98.5\pm 0.0$	$98.5\pm 0.0$	$98.5\pm 0.0$	$98.5\pm 0.0$	$98.5\pm 0.0$	$98.5\pm 0.0$	$60.0\pm 0.0$
	NMI	$92.5\pm 0.0$	$92.5\pm 0.0$	$92.5\pm 0.0$	$92.5\pm 0.0$	$92.5\pm 0.0$	$92.5\pm 0.0$	$44.6\pm 0.0$
	ARI	$95.5\pm 0.0$	$95.5\pm 0.0$	$95.5\pm 0.0$	$95.5\pm 0.0$	$95.5\pm 0.0$	$95.5\pm 0.0$	$37.1\pm 0.0$
	$\|C_{1}\|$	$2.0\pm 0.0$	$2.0\pm 0.0$	$2.0\pm 0.0$	$2.0\pm 0.0$	$2.0\pm 0.0$	$2.0\pm 0.0$	$3.0\pm 0.0$
Wholesales	ACC	$77.5\pm 0.0$	$77.5\pm 0.0$	$77.5\pm 0.0$	$77.7\pm 0.3$	$77.5\pm 0.0$	$77.5\pm 0.0$	$77.5\pm 0.0$
	NMI	$31.9\pm 0.0$	$31.9\pm 0.0$	$31.9\pm 0.0$	$29.2\pm 3.9$	$31.9\pm 0.0$	$31.9\pm 0.0$	$31.9\pm 0.0$

Appendix

Table 12: Clustering results (in %) and the final number of columns included in

C_{1}

for different initialization strategies for

R_{k}

. Poisson-

k

-Means++ corresponds to the strategy of 3CPO, proposed in Sect. 3.5 . Representation corresponds to Tab. 1 .

Panel of Figure 6 — Figure 6: Runtime experiments investigating the impact of the number of rows $n$ and columns $m$ . Both experiments build on the Synth data set, where the number of clusters $K=3$ . The experiments are run on an Apple M3 Pro with $16$ GB memory.

Panel of Figure 7 — Figure 7: Visual examples of eight inliers and eight outliers identified by 3CPO in the Optdigits data set.

Panel of Figure 8 — Figure 8: Loss (inertia), Silhouette score, and BIC score of the $k$ -Means-based comparison algorithms for an increasing number of clusters $K$ . The colored dots indicate the suggested number of clusters for each algorithm and evaluation method.

Data set	Metric	3CPO	TauCC	SKM			KM			STD+KM			MM+KM			RF+KM			RCA+KM
Data set	Metric	3CPO	(mean)	Elbow	SIL	BIC	Elbow	SIL	BIC	Elbow	SIL	BIC	Elbow	SIL	BIC	Elbow	SIL	BIC	Elbow	SIL	BIC
Synth	$K$	$3$	$3.0$	$8$	$2$	$30$	$7$	$2$	$29$	$7$	$2$	$29$	$7$	$2$	$30$	$8$	$2$	$30$	$8$	$2$	$29$
( $K_{gt}=3$ )	ARI	$95.5$	$25.1$	$53.1$	$-0.1$	$25.2$	$16.7$	$0.4$	$10.2$	$60.4$	$0.0$	$31.7$	$63.3$	$-0.1$	$22.0$	$50.6$	$0.0$	$23.1$	$21.7$	$0.0$	$17.7$
	Purity	$98.5$	$59.8$	$86.0$	$34.1$	$95.1$	$64.8$	$36.7$	$88.3$	$89.0$	$34.2$	$94.6$	$88.1$	$34.1$	$94.3$	$85.3$	$34.1$	$95.3$	$57.4$	$34.3$	$83.1$
Wholesales	$K$	$30$	$3.0$	$8$	$2$	$29$	$8$	$2$	$30$	$8$	$2$	$30$	$9$	$2$	$30$	$8$	$2$	$30$	$7$	$4$	$30$
( $K_{gt}=2$ )	ARI	$3.9$	$27.6$	$12.7$	$26.5$	$5.1$	$23.2$	$-3.1$	$7.3$	$13.4$	$28.2$	$5.6$	$13.3$	$27.8$	$7.9$	$8.8$	$21.3$	$2.8$	$15.3$	$18.8$	$3.9$

Appendix

Table 13: The estimated number of clusters

K

and the corresponding ARI and Purity values obtained by different

K

-estimation strategies. TauCC intrinsically estimates

K

, 3CPO uses the method described in Sect. 4.6 , and the

k

-Means-based approaches use the elbow method, Silhouette score (SIL) and BIC score. The result that best matches the ground truth is highlighted in bold .

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Apr 18, 2026

Neighbor Embedding for High-Dimensional Sparse Poisson Data

Across many scientific fields, measurements often represent the number of times an event occurs. For example, a document can be represented by word occurrence counts, neural activity by spike counts per time window, or online communication by daily email counts. These measurements yield high-dimensional count data that often approximate a Poisson distribution, frequently with low rates that produce substantial sparsity and complicate downstream analysis. A useful approach is to embed the data into a low-dimensional space that preserves meaningful structure, commonly termed dimensionality reduction. Yet existing dimensionality reduction methods, including both linear (e.g., PCA) and nonlinear approaches (e.g., t-SNE), often assume continuous Euclidean geometry, thereby misaligning with the discrete, sparse nature of low-rate count data. Here, we propose p-SNE (Poisson Stochastic Neighbor Embedding), a nonlinear neighbor embedding method designed around the Poisson structure of count data, using KL divergence between Poisson distributions to measure pairwise dissimilarity and Hellinger distance to optimize the embedding. We test p-SNE on synthetic Poisson data and demonstrate its ability to recover meaningful structure in real-world count datasets, including weekday patterns in email communication, research area clusters in OpenReview papers, and temporal drift and stimulus gradients in neural spike recordings.

Noga Mudrik, Adam S. Charles

Department of Biomedical Engineering Center for Imaging Science Kavli Neuroscience Discovery Institute Johns Hopkins University Baltimore, MD 21218, USA

Manifold Learning Dimensionality Reduction

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Jun 12, 2026

Cluster LOCO: Feature Importance For Interpreting Clusters

Clustering is widely used for exploratory analysis and scientific discovery, driving insights from market segmentation to biological data analysis, but its outputs can be difficult to interpret, audit, and reproduce as modern datasets become increasingly large and complex. Reliable use of clustering requires understanding which features drive the discovered structure, yet feature-level explanations for clustering remain scarce compared with methods in supervised learning. Furthermore, existing clustering feature importance scores are often tied to specific algorithms and data assumptions. To address these challenges, we propose Cluster LOCO (Leave-One-Covariate-Out), a family of model-agnostic feature importance scores for clustering. Cluster LOCO is built on feature occlusion and clustering generalizability, defined as whether cluster labels learned on one subset of the data can be accurately predicted on held-out samples. For any chosen clustering algorithm, Cluster LOCO quantifies a feature's importance by measuring how much its removal degrades generalizability. We first introduce Cluster LOCO-Split, which relies on data splitting, and then extend it to Cluster LOCO-MP, a minipatch ensemble-based version designed for large-scale data. Across synthetic simulations and an application to cell-type discovery in single-cell transcriptomics, we show that Cluster LOCO more reliably recovers informative features than existing clustering feature importance methods.

Claire M. He, Genevera I. Allen

Department of Statistics · Columbia University · New York, NY 10027, USA

Clustering

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Apr 24, 2026

Associativity-Peakiness Metric for Contingency Tables

For the use case of comparing the performance of clustering algorithms whose output is a contingency table, a single performance metric for contingency tables is needed. Such a metric is vital for comparative performance analysis of clustering algorithms. A survey of publicly available literature did not show the presence of such a metric. Metrics do exist for vector pairs of truth values and predicted values, which are an alternative form of output of clustering algorithms. However, the metrics for vector pairs do not reveal the presence of detailed features that are apparent in contingency tables. This paper presents the Associativity Peakiness (AP) metric, which characterizes aspects of clustering algorithm performance that are critical for predicting a clustering algorithm's performance when deployed. The AP metric is analogous to measures of quality for confusion matrices that are outputs of supervised learning algorithms. This paper presents results from simulations in which 500 contingency tables were generated for multiple test scenarios. The results show that for the use case of evaluating clustering algorithms, the AP metric characterizes performance of contingency tables with higher dynamic range than publicly available metrics, and that it is computationally more efficient than comparable publicly available metrics.

Naomi E. Zirkind, William J. Diehl

Army Research Directorate, DEVCOM Army Research Laboratory, 2800 Powder Mill Road, Adelphi, 20783, MD, USA.

Clustering

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Abstract

Clustering

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Neighbor Embedding for High-Dimensional Sparse Poisson Data

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