The proof of infeasibility condition (Theorem 3.2) is provided in Section B. Explanations on conditions derived in Theorem 3.2 are included in Section C. The proof of properties of the proposed model (r)LogSpecT (Proposition 3.4 The truncated Hausdorff distance based proof details of Theorem 4.1 and Corollary 4.4 are Details of L-ADMM and its convergence analysis are in Section F. Additional experiments and discussions on synthetic data are included in Section G. ( m 1) Again, from Farkas' lemma, this implies that the following linear system does not have a solution: Example 3.1 we know δ = 2|h Since the constraint set S is a cone, it follows that for all γ > 0, γ S = S . Opt(C, α) = α Opt(C, 1), which completes the proof. The proof will be conducted by constructing a feasible solution for rLogSpecT. Since the LogSpecT is a convex problem and Slater's condition holds, the KKT conditions We show that it is feasible for rLogSpecT. R, its epigraph is defined as epi f: = {( x, y) | y f ( x) }. Before presenting the proof, we first introduce the following lemma.

First, note that we shall use the same MDPs defined in Appendix D.1 as follows

This appendix presents the technical details of efficiently implementing Algorithm 2. B.1 Computing Intermediate Quantities We argue that in the setting of neural networks, Algorithm 2 can obtain the intermediate quantities ζ Algorithm 3 gives a subroutine for computing the necessary scalars used in the efficient squared norm function of the embedding layer.Algorithm 3 Computing the Nonzero V alues of n In the former case, it is straightforward to see that we incur a compute (resp. F .1 Effect of Batch Size on Fully-Connected Layers Figure 4 presents numerical results for the same set of experiments as in Subsection 5.1 but for different batch sizes |B | instead of the output dimension q . Similar to Subsection 5.1, the results in Figure 4 are more favorable towards Adjoint compared to GhostClip.

Personalized PageRank (PPR) is a fundamental tool in unsupervised learning of graph representations such as node ranking, labeling, and graph embedding.

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.