Independent Component Discovery in Temporal Count Data

TL;DR

The paper tackles interpretable latent decomposition of multivariate temporal count data by developing a Poisson log-normal independent component analysis (PLN-ICA) framework with regime-switching dynamics. It establishes identifiability of the log-intensity mixing up to permutation and scaling under mild conditions and provides a structured amortized variational inference procedure to learn the model efficiently. The ARPLN-ICA approach yields interpretable latent components and regime-driven dynamics, with finite-sample simulations showing partial recovery of the mixing and a microbiome case study demonstrating regime-aligned perturbations and biologically meaningful taxon co-variation. The work further demonstrates competitive forecasting performance and contributes a scalable PyTorch implementation, facilitating representation learning and perturbation analysis in temporal count data. Overall, PLN-ICA offers a principled, interpretable framework for disentangling complex temporal count trajectories across domains such as microbiomics and beyond.

Abstract

Advances in data collection are producing growing volumes of temporal count observations, making adapted modeling increasingly necessary. In this work, we introduce a generative framework for independent component analysis of temporal count data, combining regime-adaptive dynamics with Poisson log-normal emissions. The model identifies disentangled components with regime-dependent contributions, enabling representation learning and perturbations analysis. Notably, we establish the identifiability of the model, supporting principled interpretation. To learn the parameters, we propose an efficient amortized variational inference procedure. Experiments on simulated data evaluate recovery of the mixing function and latent sources across diverse settings, while an in vivo longitudinal gut microbiome study reveals microbial co-variation patterns and regime shifts consistent with clinical perturbations.

Figures (19)

• Figure 1: ARPLN-ICA graphical dependencies representation. MC indicates the discrete Markov chain on the latent switching labels modeling regimes, AR the Gaussian auto-regressive prior on the sources conditional to the latent regimes, and PLN the Poisson Log-Normal linear mixing emission of counts at time $t$.
• Figure 2: Generative distribution quality and mixing recovery in simulation scenarios with ARPLN-ICA, comparing Auto-Regressive (AR) and Mean-Field (MF) inference of the mixing in ARPLN-ICA, and linear ICA mixing estimation algorithms. (Left) Sliced Wasserstein evaluates the distance to the baseline distribution with the fitted model (mean with 95% CI; lower is better). (Right) Cosine similarity evaluates the absolute scalar product between the columns of the baseline mixing and the fitted mixing (1: recovery, 0: orthogonality).
• Figure 3: Mixing components recovery with varying training size $n$ using the Auto-Regressive variational approximation in the moderate coherence scenario. Recovery is assessed via cosine similarity per component over $10$ training with different initializations.
• Figure 4: Average probability of being in latent state $0$ for each component, computed on test samples across LOO CV. The probability is computed from empirical inhomogeneous derivation in Appendix \ref{['app:empirical_switching']}. Evaluation per test mouse is provided in Figure \ref{['fig:MDSINE_perturbation_components_all_mice']}.
• Figure 5: Independent components inferred by ARPLN-ICA on the gnotobiotic mice dataset. (Left) Taxa loading per component (column) of the medoid mixing across LOO CV, with standard deviation to the medoid across folds. (Right) Latent components evolution through time for test mice, aligned with the medoid mixing across LOO, with standard deviation. The fourth component displays a sharp reaction to the perturbation, consistent with Figure \ref{['fig:MDSINE_perturbation_components']}, with taxa loadings separating opportunistic pathogens from healthy and commensal microbial species. See Appendix \ref{['app:stability_and_medoid']} for the computation of the medoid mixing and stability of components.

Theorems & Definitions (15)

• Proposition 3.1
• Corollary 3.2
• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• proof
• Corollary 1.4
• proof
• Lemma 1.5
• proof