This paper describes a simple technique to analyze Generative Adversarial Networks (GANs) and create interpretable controls for image synthesis, such as change of viewpoint, aging, lighting, and time of day. We identify important latent directions based on Principal Component Analysis (PCA) applied either in latent space or feature space. Then, we show that a large number of interpretable controls can be defined by layer-wise perturbation along the principal directions. Moreover, we show that BigGAN can be controlled with layer-wise inputs in a StyleGAN-like manner. We show results on different GANs trained on various datasets, and demonstrate good qualitative matches to edit directions found through earlier supervised approaches.

Suppose certain data points are overly contaminated, then the existing principal component analysis (PCA) methods are frequently incapable of filtering out and eliminating the excessively polluted ones, which potentially lead to the functional degeneration of the corresponding models. To tackle the issue, we propose a general framework namely robust weight learning with adaptive neighbors (RWL-AN), via which adaptive weight vector is automatically obtained with both robustness and sparse neighbors. More significantly, the degree of the sparsity is steerable such that only exact k well-fitting samples with least reconstruction errors are activated during the optimization, while the residual samples, i.e., the extreme noised ones are eliminated for the global robustness. Additionally, the framework is further applied to PCA problem to demonstrate the superiority and effectiveness of the proposed RWL-AN model.

Missing Not At Random (MNAR) values where the probability of having missing data may depend on the missing value itself, are notoriously difficult to account for in analyses, although very frequent in the data. One solution to handle MNAR data is to specify a model for the missing data mechanism, which makes inference or imputation tasks more complex. Furthermore, this implies a strong \textit{a priori} on the parametric form of the distribution. However, some works have obtained guarantees on the estimation of parameters in the presence of MNAR data, without specifying the distribution of missing data \citep{mohan2018estimation, tang2003analysis}. This is very useful in practice, but is limited to simple cases such as few self-masked MNAR variables in data generated according to linear regression models.

We present a federated, asynchronous, and (\varepsilon, \delta) -differentially private algorithm for \PCA in the memory-limited setting.

We consider streaming principal component analysis when the stochastic data-generating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja's algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments.

Fair Principal Component Analysis (PCA) is a problem setting where we aim to perform PCA while making the resulting representation fair in that the projected distributions, conditional on the sensitive attributes, match one another. However, existing approaches to fair PCA have two main problems: theoretically, there has been no statistical foundation of fair PCA in terms of learnability; practically, limited memory prevents us from using existing approaches, as they explicitly rely on full access to the entire data. On the theoretical side, we rigorously formulate fair PCA using a new notion called probably approximately fair and optimal (PAFO) learnability. On the practical side, motivated by recent advances in streaming algorithms for addressing memory limitation, we propose a new setting called fair streaming PCA along with a memory-efficient algorithm, fair noisy power method (FNPM). We then provide its statistical guarantee in terms of PAFO-learnability, which is the first of its kind in fair PCA literature. We verify our algorithm in the CelebA dataset without any pre-processing; while the existing approaches are inapplicable due to memory limitations, by turning it into a streaming setting, we show that our algorithm performs fair PCA efficiently and effectively.

We study efficient distributed algorithms for the fundamental problem of principal component analysis and leading eigenvector computation on the sphere, when the data are randomly distributed among a set of computational nodes. We propose a new quantized variant of Riemannian gradient descent to solve this problem, and prove that the algorithm converges with high probability under a set of necessary spherical-convexity properties. We give bounds on the number of bits transmitted by the algorithm under common initialization schemes, and investigate the dependency on the problem dimension in each case.

In this paper, we propose to adopt the diffusion approximation tools to study the dynamics of Oja's iteration which is an online stochastic gradient descent method for the principal component analysis. Oja's iteration maintains a running estimate of the true principal component from streaming data and enjoys less temporal and spatial complexities. We show that the Oja's iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere. We characterize the Oja's iteration in three phases using diffusion approximation and weak convergence tools. Our three-phase analysis further provides a finite-sample error bound for the running estimate, which matches the minimax information lower bound for principal component analysis under the additional assumption of bounded samples.

The performance of the standard Online Robust Principal Component Analysis (OR-PCA) technique depends on the optimum tuning of the explicit regularizers and this tuning is dataset sensitive. We aim to remove the dependency on these tuning parameters by using implicit regularization. We propose to use the implicit regularization effect of various modified gradient descents to make OR-PCA tuning free. Our method incorporates three different versions of modified gradient descent that separately but naturally encourage sparsity and low-rank structures in the data. The proposed method performs comparable or better than the tuned OR-PCA for both simulated and real-world datasets. Tuning-free ORPCA makes it more scalable for large datasets since we do not require dataset-dependent parameter tuning.

Dimension reduction is often the first step in statistical modeling or prediction of multivariate spatial data. However, most existing dimension reduction techniques do not account for the spatial correlation between observations and do not take the downstream modeling task into consideration when finding the lower-dimensional representation. We formalize the closeness of approximation to the original data and the utility of lower-dimensional scores for downstream modeling as two complementary, sometimes conflicting, metrics for dimension reduction. We illustrate how existing methodologies fall into this framework and propose a flexible dimension reduction algorithm that achieves the optimal trade-off. We derive a computationally simple form for our algorithm and illustrate its performance through simulation studies, as well as two applications in air pollution modeling and spatial transcriptomics.