We develop a scalable second-order algorithm for a recently proposed $\ell_1$-regularized pseudolikelihood-based partial correlation network estimation framework. While the latter method admits statistical guarantees and is inherently scalable compared to likelihood-based methods such as graphical lasso, the currently available implementations rely only on first-order information and require thousands of iterations to obtain reliable estimates even on high-performance supercomputers. In this paper, we further investigate the inherent scalability of the framework and propose locally and globally convergent semismooth Newton methods. Despite the nonsmoothness of the problem, these second-order algorithms converge at a locally quadratic rate, and require only a few tens of iterations in practice. Each iteration reduces to solving linear systems of small dimensions or linear complementary problems of smaller dimensions, making the computation also suitable for less powerful computing environments. Experiments on both simulated and real-world genomic datasets demonstrate the superior convergence behavior and computational efficiency of the proposed algorithm, which position our method as a promising tool for massive-scale network analysis sought for in, e.g., modern multi-omics research.

Autoencoders have become an indispensable tool for generative modeling and representation learning in high dimensions. Imposing structural constraints such as conditional independence in order to capture invariance of latent variables to nuisance information has been attempted through adding *ad hoc* penalties to the loss function mostly in the variational autoencoder (VAE) context, often based on heuristics. This paper demonstrates that Wasserstein autoencoders (WAEs) are highly flexible in embracing such structural constraints. Well-known extensions of VAEs for this purpose are gracefully handled within the framework of WAEs. In particular, given a conditional independence structure of the generative model (decoder), corresponding encoder structure and penalties are derived from the functional constraints that define the WAE. These structural uses of WAEs, termed StrWAEs (“stairways”), open up a principled way of penalizing autoencoders to impose structural constraints. Utilizing these advantages, we present handful of results on semi-supervised classification, conditional generation, and invariant representation tasks.

The implicit stochastic gradient descent (ISGD), a proximal version of SGD, is gaining interest in the literature due to its stability over (explicit) SGD. In this paper, we conduct an in-depth analysis of the two modes of ISGD for smooth convex functions, namely proximal Robbins-Monro (proxRM) and proximal Poylak-Ruppert (proxPR) procedures, for their use in statistical inference on model parameters. Specifically, we derive non-asymptotic point estimation error bounds of both proxRM and proxPR iterates and their limiting distributions, and propose on-line estimators of their asymptotic covariance matrices that require only a single run of ISGD. The latter estimators are used to construct valid confidence intervals for the model parameters. Our analysis is free of the generalized linear model assumption that has limited the preceding analyses, and employs feasible procedures. Our on-line covariance matrix estimators appear to be the first of this kind in the ISGD literature.

Many of real-world data, e.g., the VGGFace2 dataset, which is a collection of multiple portraits of individuals, come with nested structures due to grouped observation. The Ornstein auto-encoder (OAE) is an emerging framework for representation learning from nested data, based on an optimal transport distance between random processes. An attractive feature of OAE is its ability to generate new variations nested within an observational unit, whether or not the unit is known to the model. A previously proposed algorithm for OAE, termed the random-intercept OAE (RIOAE), showed an impressive performance in learning nested representations, yet lacks theoretical justification. In this work, we show that RIOAE minimizes a loose upper bound of the employed optimal transport distance. After identifying several issues with RIOAE, we present the product-space OAE (PSOAE) that minimizes a tighter upper bound of the distance and achieves orthogonality in the representation space. PSOAE alleviates the instability of RIOAE and provides more flexible representation of nested data. We demonstrate the high performance of PSOAE in the three key tasks of generative models: exemplar generation, style transfer, and new concept generation.