Selective Inference for Deep Clustering in Latent Spaces

Summary

This paper develops a selective inference (SI) framework for testing feature-wise mean differences between clusters discovered by sequential deep clustering: a fixed, pretrained piecewise-affine encoder (ReLU MLP) maps high-dimensional data to a latent space where k-means is run. Because cluster assignments depend on the same data used for testing (double dipping), naive p-values are invalid. The authors characterize the selection event as a truncation region on the SI search line by combining linear constraints from the encoder's activation pattern (via Auto-Conditioning) with quadratic constraints from Lloyd's algorithm assignment history, then stitch intervals with matching clustering output using parametric programming to avoid over-conditioning. Theorems 3.1-3.2 and the validity guarantee are explicit restatements of Chen and Witten (2023) and Chen and Gao (2025); the contribution is computing the truncation region for the encoder-plus-clustering pipeline. In synthetic experiments (10,000 trials, alpha=0.05), the method controls Type I error where naive unadjusted tests become severely anticonservative (the plotted error axis extends to 0.5), and achieves higher power than Bonferroni and single-interval conditioning. Applications to SEQC neuroblastoma and 10x PBMC RNA-seq are descriptive only, without multiplicity control across genes, cluster pairs, or K. Main limitations: encoders must be pretrained on independent data, Gaussian errors with known covariance (robustness to estimated variance and non-Gaussian noise is empirical only), thin baselines (no data-splitting comparator), and unreported computational cost.