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Generative adversarial networks (GANs) have emerged as a powerful unsupervised method to model the statistical patterns of real-world data sets, such as natural images. These networks are trained to map random inputs in their latent space to new samples representative of the learned data. However, the structure of the latent space is hard to intuit due to its high dimensionality and the non-linearity of the generator, which limits the usefulness of the models. Understanding the latent space requires a way to identify input codes for existing real-world images (inversion), and a way to identify directions with known image transformations (interpretability). Here, we use a geometric framework to address both issues simultaneously. We develop an architecture-agnostic method to compute the Riemannian metric of the image manifold created by GANs. The eigen-decomposition of the metric isolates axes that account for different levels of image variability. An empirical analysis of several pretrained GANs shows that image variation around each position is concentrated along surprisingly few major axes (the space is highly anisotropic) and the directions that create this large variation are similar at different positions in the space (the space is homogeneous). We show that many of the top eigenvectors correspond to interpretable transforms in the image space, with a substantial part of eigenspace corresponding to minor transforms which could be compressed out. This geometric understanding unifies key previous results related to GAN interpretability. We show that the use of this metric allows for more efficient optimization in the latent space (e.g. GAN inversion) and facilitates unsupervised discovery of interpretable axes. Our results illustrate that defining the geometry of the GAN image manifold can serve as a general framework for understanding GANs.

This study presents PRISM, a probabilistic simplex component analysis approach to identifying the vertices of a data-circumscribing simplex from data. The problem has a rich variety of applications, the most notable being hyperspectral unmixing in remote sensing and non-negative matrix factorization in machine learning. PRISM uses a simple probabilistic model, namely, uniform simplex data distribution and additive Gaussian noise, and it carries out inference by maximum likelihood. The inference model is sound in the sense that the vertices are provably identifiable under some assumptions, and it suggests that PRISM can be effective in combating noise when the number of data points is large. PRISM has strong, but hidden, relationships with simplex volume minimization, a powerful geometric approach for the same problem. We study these fundamental aspects, and we also consider algorithmic schemes based on importance sampling and variational inference. In particular, the variational inference scheme is shown to resemble a matrix factorization problem with a special regularizer, which draws an interesting connection to the matrix factorization approach. Numerical results are provided to demonstrate the potential of PRISM.

Image virtual try-on replaces the clothes on a person image with a desired in-shop clothes image. It is challenging because the person and the in-shop clothes are unpaired. Existing methods formulate virtual try-on as either in-painting or cycle consistency. Both of these two formulations encourage the generation networks to reconstruct the input image in a self-supervised manner. However, existing methods do not differentiate clothing and non-clothing regions. A straight-forward generation impedes virtual try-on quality because of the heavily coupled image contents. In this paper, we propose a Disentangled Cycle-consistency Try-On Network (DCTON). The DCTON is able to produce highly-realistic try-on images by disentangling important components of virtual try-on including clothes warping, skin synthesis, and image composition. To this end, DCTON can be naturally trained in a self-supervised manner following cycle consistency learning. Extensive experiments on challenging benchmarks show that DCTON outperforms state-of-the-art approaches favorably.

Many-query problems, arising from uncertainty quantification, Bayesian inversion, Bayesian optimal experimental design, and optimization under uncertainty-require numerous evaluations of a parameter-to-output map. These evaluations become prohibitive if this parametric map is high-dimensional and involves expensive solution of partial differential equations (PDEs). To tackle this challenge, we propose to construct surrogates for high-dimensional PDE-governed parametric maps in the form of projected neural networks that parsimoniously capture the geometry and intrinsic low-dimensionality of these maps. Specifically, we compute Jacobians of these PDE-based maps, and project the high-dimensional parameters onto a low-dimensional derivative-informed active subspace; we also project the possibly high-dimensional outputs onto their principal subspace. This exploits the fact that many high-dimensional PDE-governed parametric maps can be well-approximated in low-dimensional parameter and output subspace. We use the projection basis vectors in the active subspace as well as the principal output subspace to construct the weights for the first and last layers of the neural network, respectively. This frees us to train the weights in only the low-dimensional layers of the neural network. The architecture of the resulting neural network captures to first order, the low-dimensional structure and geometry of the parametric map. We demonstrate that the proposed projected neural network achieves greater generalization accuracy than a full neural network, especially in the limited training data regime afforded by expensive PDE-based parametric maps. Moreover, we show that the number of degrees of freedom of the inner layers of the projected network is independent of the parameter and output dimensions, and high accuracy can be achieved with weight dimension independent of the discretization dimension.

We consider the estimation of the transition matrix of a hidden Markovian process by using information geometry with respect to transition matrices. In this paper, only the histogram of $k$-memory data is used for the estimation. To establish our method, we focus on a partial observation model with the Markovian process and we propose an efficient estimator whose asymptotic estimation error is given as the inverse of projective Fisher information of transition matrices. This estimator is applied to the estimation of the transition matrix of the hidden Markovian process. In this application, we carefully discuss the equivalence problem for hidden Markovian process on the tangent space.

Flexible estimation of multiple conditional quantiles is of interest in numerous applications, such as studying the effect of pregnancy-related factors on very low or high birth weight. We propose a Bayesian non-parametric method to simultaneously estimate non-crossing, non-linear quantile curves. We expand the conditional distribution function of the response in I-spline basis functions where the covariate-dependent coefficients are modeled using neural networks. By leveraging the approximation power of splines and neural networks, our model can approximate any continuous quantile function. Compared to existing models, our model estimates all rather than a finite subset of quantiles, scales well to high dimensions, and accounts for estimation uncertainty. While the model is arbitrarily flexible, interpretable marginal quantile effects are estimated using accumulative local effect plots and variable importance measures. A simulation study shows that our model can better recover quantiles of the response distribution when the data is sparse, and illustrative applications providing new insights on analyses of birth weight and tropical cyclone intensity are presented.

We study object recognition under the constraint that each object class is only represented by very few observations. In such cases, naive supervised learning would lead to severe over-fitting in deep neural networks due to limited training data. We tackle this problem by creating much more training data through label propagation from the few labeled examples to a vast collection of unannotated images. Our main insight is that such a label propagation scheme can be highly effective when the similarity metric used for propagation is learned and transferred from other related domains with lots of data. We test our approach on semi-supervised learning, transfer learning and few-shot recognition, where we learn our similarity metric using various supervised/unsupervised pretraining methods, and transfer it to unlabeled data across different data distributions. By taking advantage of unlabeled data in this way, we achieve significant improvements on all three tasks. Notably, our approach outperforms current state-of-the-art techniques by an absolute $20\%$ for semi-supervised learning on CIFAR10, $10\%$ for transfer learning from ImageNet to CIFAR10, and $6\%$ for few-shot recognition on mini-ImageNet, when labeled examples are limited.

An important problem in geostatistics is to build models of the subsurface of the Earth given physical measurements at sparse spatial locations. Typically, this is done using spatial interpolation methods or by reproducing patterns from a reference image. However, these algorithms fail to produce realistic patterns and do not exhibit the wide range of uncertainty inherent in the prediction of geology. In this paper, we show how semantic inpainting with Generative Adversarial Networks can be used to generate varied realizations of geology which honor physical measurements while matching the expected geological patterns. In contrast to other algorithms, our method scales well with the number of data points and mimics a distribution of patterns as opposed to a single pattern or image. The generated conditional samples are state of the art.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

Large margin nearest neighbor (LMNN) is a metric learner which optimizes the performance of the popular $k$NN classifier. However, its resulting metric relies on pre-selected target neighbors. In this paper, we address the feasibility of LMNN's optimization constraints regarding these target points, and introduce a mathematical measure to evaluate the size of the feasible region of the optimization problem. We enhance the optimization framework of LMNN by a weighting scheme which prefers data triplets which yield a larger feasible region. This increases the chances to obtain a good metric as the solution of LMNN's problem. We evaluate the performance of the resulting feasibility-based LMNN algorithm using synthetic and real datasets. The empirical results show an improved accuracy for different types of datasets in comparison to regular LMNN.