Improving Relational Regularized Autoencoders with Spherical Sliced Fused Gromov Wasserstein

Relational regularized autoencoder (RAE) is a framework to learn the distribution of data by minimizing a reconstruction loss together with a relational regularization on the latent space. A recent attempt to reduce the inner discrepancy between the prior and aggregated posterior distributions is to incorporate sliced fused Gromov-Wasserstein (SFG) between these distributions. That approach has a weakness since it treats every slicing direction similarly, meanwhile several directions are not useful for the discriminative task. To improve the discrepancy and consequently the relational regularization, we propose a new relational discrepancy, named spherical sliced fused Gromov Wasserstein (SSFG), that can find an important area of projections characterized by a von Mises-Fisher distribution. Then, we introduce two variants of SSFG to improve its performance. The first variant, named mixture spherical sliced fused Gromov Wasserstein (MSSFG), replaces the vMF distribution by a mixture of von Mises-Fisher distributions to capture multiple important areas of directions that are far from each other. The second variant, named power spherical sliced fused Gromov Wasserstein (PSSFG), replaces the vMF distribution by a power spherical distribution to improve the sampling time in high dimension settings. We then apply the new discrepancies to the RAE framework to achieve its new variants. Finally, we conduct extensive experiments to show that the new proposed autoencoders have favorable performance in learning latent manifold structure, image generation, and reconstruction.

Vec2Face: Unveil Human Faces from their Blackbox Features in Face Recognition

Unveiling face images of a subject given his/her high-level representations extracted from a blackbox Face Recognition engine is extremely challenging. It is because the limitations of accessible information from that engine including its structure and uninterpretable extracted features. This paper presents a novel generative structure with Bijective Metric Learning, namely Bijective Generative Adversarial Networks in a Distillation framework (DiBiGAN), for synthesizing faces of an identity given that person's features. In order to effectively address this problem, this work firstly introduces a bijective metric so that the distance measurement and metric learning process can be directly adopted in image domain for an image reconstruction task. Secondly, a distillation process is introduced to maximize the information exploited from the blackbox face recognition engine. Then a Feature-Conditional Generator Structure with Exponential Weighting Strategy is presented for a more robust generator that can synthesize realistic faces with ID preservation. Results on several benchmarking datasets including CelebA, LFW, AgeDB, CFP-FP against matching engines have demonstrated the effectiveness of DiBiGAN on both image realism and ID preservation properties.

Distributional Sliced-Wasserstein and Applications to Generative Modeling

Sliced-Wasserstein distance (SWD) and its variation, Max Sliced-Wasserstein distance (Max-SWD), have been widely used in the recent years due to their fast computation and scalability when the probability measures lie in very high dimension. However, these distances still have their weakness, SWD requires a lot of projection samples because it uses the uniform distribution to sample projecting directions, Max-SWD uses only one projection, causing it to lose a large amount of information. In this paper, we propose a novel distance that finds optimal penalized probability measure over the slices, which is named Distributional Sliced-Wasserstein distance (DSWD). We show that the DSWD is a generalization of both SWD and Max-SWD, and the proposed distance could be found by searching for the push-forward measure over a set of measures satisfying some certain constraints. Moreover, similar to SWD, we can extend Generalized Sliced-Wasserstein distance (GSWD) to Distributional Generalized Sliced-Wasserstein distance (DGSWD). Finally, we carry out extensive experiments to demonstrate the favorable generative modeling performances of our distances over the previous sliced-based distances in large-scale real datasets.

On Unbalanced Optimal Transport: An Analysis of Sinkhorn Algorithm

We provide a computational complexity analysis for the Sinkhorn algorithm that solves the entropic regularized Unbalanced Optimal Transport (UOT) problem between two measures of possibly different masses with at most $n$ components. We show that the complexity of the Sinkhorn algorithm for finding an $\varepsilon$-approximate solution to the UOT problem is of order $\widetilde{\mathcal{O}}(n^2/ \varepsilon)$, which is near-linear time. To the best of our knowledge, this complexity is better than the complexity of the Sinkhorn algorithm for solving the Optimal Transport (OT) problem, which is of order $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$. Our proof technique is based on the geometric convergence of the Sinkhorn updates to the optimal dual solution of the entropic regularized UOT problem and some properties of the primal solution. It is also different from the proof for the complexity of the Sinkhorn algorithm for approximating the OT problem since the UOT solution does not have to meet the marginal constraints.

On Efficient Multilevel Clustering via Wasserstein Distances

We propose a novel approach to the problem of multilevel clustering, which aims to simultaneously partition data in each group and discover grouping patterns among groups in a potentially large hierarchically structured corpus of data. Our method involves a joint optimization formulation over several spaces of discrete probability measures, which are endowed with Wasserstein distance metrics. We propose several variants of this problem, which admit fast optimization algorithms, by exploiting the connection to the problem of finding Wasserstein barycenters. Consistency properties are established for the estimates of both local and global clusters. Finally, the experimental results with both synthetic and real data are presented to demonstrate the flexibility and scalability of the proposed approach.

Prediction, Consistency, Curvature: Representation Learning for Locally-Linear Control

Many real-world sequential decision-making problems can be formulated as optimal control with high-dimensional observations and unknown dynamics. A promising approach is to embed the high-dimensional observations into a lower-dimensional latent representation space, estimate the latent dynamics model, then utilize this model for control in the latent space. An important open question is how to learn a representation that is amenable to existing control algorithms? In this paper, we focus on learning representations for locally-linear control algorithms, such as iterative LQR (iLQR). By formulating and analyzing the representation learning problem from an optimal control perspective, we establish three underlying principles that the learned representation should comprise: 1) accurate prediction in the observation space, 2) consistency between latent and observation space dynamics, and 3) low curvature in the latent space transitions. These principles naturally correspond to a loss function that consists of three terms: prediction, consistency, and curvature (PCC). Crucially, to make PCC tractable, we derive an amortized variational bound for the PCC loss function. Extensive experiments on benchmark domains demonstrate that the new variational-PCC learning algorithm benefits from significantly more stable and reproducible training, and leads to superior control performance. Further ablation studies give support to the importance of all three PCC components for learning a good latent space for control.

SpectralLeader: Online Spectral Learning for Single Topic Models

We study the problem of learning a latent variable model from a stream of data. Latent variable models are popular in practice because they can explain observed data in terms of unobserved concepts. These models have been traditionally studied in the offline setting. In the online setting, on the other hand, the online EM is arguably the most popular algorithm for learning latent variable models. Although the online EM is computationally efficient, it typically converges to a local optimum. In this work, we develop a new online learning algorithm for latent variable models, which we call SpectralLeader. SpectralLeader always converges to the global optimum, and we derive a sublinear upper bound on its $n$-step regret in the bag-of-words model. In both synthetic and real-world experiments, we show that SpectralLeader performs similarly to or better than the online EM with tuned hyper-parameters.

Robust Locally-Linear Controllable Embedding

Embed-to-control (E2C) is a model for solving high-dimensional optimal control problems by combining variational auto-encoders with locally-optimal controllers. However, the E2C model suffers from two major drawbacks: 1) its objective function does not correspond to the likelihood of the data sequence and 2) the variational encoder used for embedding typically has large variational approximation error, especially when there is noise in the system dynamics. In this paper, we present a new model for learning robust locally-linear controllable embedding (RCE). Our model directly estimates the predictive conditional density of the future observation given the current one, while introducing the bottleneck between the current and future observations. Although the bottleneck provides a natural embedding candidate for control, our RCE model introduces additional specific structures in the generative graphical model so that the model dynamics can be robustly linearized. We also propose a principled variational approximation of the embedding posterior that takes the future observation into account, and thus, makes the variational approximation more robust against the noise. Experimental results show that RCE outperforms the E2C model, and does so significantly when the underlying dynamics is noisy.

Graphical Model Sketch

Structured high-cardinality data arises in many domains, and poses a major challenge for both modeling and inference. Graphical models are a popular approach to modeling structured data but they are unsuitable for high-cardinality variables. The count-min (CM) sketch is a popular approach to estimating probabilities in high-cardinality data but it does not scale well beyond a few variables. In this work, we bring together the ideas of graphical models and count sketches; and propose and analyze several approaches to estimating probabilities in structured high-cardinality streams of data. The key idea of our approximations is to use the structure of a graphical model and approximately estimate its factors by "sketches", which hash high-cardinality variables using random projections. Our approximations are computationally efficient and their space complexity is independent of the cardinality of variables. Our error bounds are multiplicative and significantly improve upon those of the CM sketch, a state-of-the-art approach to estimating probabilities in streams. We evaluate our approximations on synthetic and real-world problems, and report an order of magnitude improvements over the CM sketch.

Boosted Markov Networks for Activity Recognition

We explore a framework called boosted Markov networks to combine the learning capacity of boosting and the rich modeling semantics of Markov networks and applying the framework for video-based activity recognition. Importantly, we extend the framework to incorporate hidden variables. We show how the framework can be applied for both model learning and feature selection. We demonstrate that boosted Markov networks with hidden variables perform comparably with the standard maximum likelihood estimation. However, our framework is able to learn sparse models, and therefore can provide computational savings when the learned models are used for classification.