Furthermore, we show that the WAE objective is related to other statistical quantities such as thef-divergence and in particular, upper bounded by the Wasserstein distance, which then allows us to tap into existing efficient(regularized)optimaltransportsolvers.

To close this gap, in this paper, we investigate the statistical properties of WAE.

The introduction of Variational Autoencoders (VAE) has been marked as a breakthrough in the history of representation learning models. Besides having several accolades of its own, VAE has successfully flagged off a series of inventions in the form of its immediate successors. Wasserstein Autoencoder (WAE), being an heir to that realm carries with it all of the goodness and heightened generative promises, matching even the generative adversarial networks (GANs). Needless to say, recent years have witnessed a remarkable resurgence in statistical analyses of the GANs. Similar examinations for Autoencoders however, despite their diverse applicability and notable empirical performance, remain largely absent. To close this gap, in this paper, we investigate the statistical properties of WAE. Firstly, we provide statistical guarantees that WAE achieves the target distribution in the latent space, utilizing the Vapnik-Chervonenkis (VC) theory. The main result, consequently ensures the regeneration of the input distribution, harnessing the potential offered by Optimal Transport of measures under the Wasserstein metric. This study, in turn, hints at the class of distributions WAE can reconstruct after suffering a compression in the form of a latent law.

The introduction of Variational Autoencoders (VAE) has been marked as a breakthrough in the history of representation learning models. Besides having several accolades of its own, VAE has successfully flagged off a series of inventions in the form of its immediate successors. Wasserstein Autoencoder (WAE), being an heir to that realm carries with it all of the goodness and heightened generative promises, matching even the generative adversarial networks (GANs). Needless to say, recent years have witnessed a remarkable resurgence in statistical analyses of the GANs. Similar examinations for Autoencoders however, despite their diverse applicability and notable empirical performance, remain largely absent.

Variational Autoencoders (VAEs) have been a pioneering force in the realm of deep generative models. Amongst its legions of progenies, Wasserstein Autoencoders (WAEs) stand out in particular due to the dual offering of heightened generative quality and a strong theoretical backbone. WAEs consist of an encoding and a decoding network forming a bottleneck with the prime objective of generating new samples resembling the ones it was catered to. In the process, they aim to achieve a target latent representation of the encoded data. Our work is an attempt to offer a theoretical understanding of the machinery behind WAEs. From a statistical viewpoint, we pose the problem as concurrent density estimation tasks based on neural network-induced transformations. This allows us to establish deterministic upper bounds on the realized errors WAEs commit. We also analyze the propagation of these stochastic errors in the presence of adversaries. As a result, both the large sample properties of the reconstructed distribution and the resilience of WAE models are explored.

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In this paper, we develop the Laplacian pyramid-like autoencoder (LPAE) by adding the Laplacian pyramid (LP) concept widely used to analyze images in Signal Processing. LPAE decomposes an image into the approximation image and the detail image in the encoder part and then tries to reconstruct the original image in the decoder part using the two components. We use LPAE for experiments on classifications and super-resolution areas. Using the detail image and the smaller-sized approximation image as inputs of a classification network, our LPAE makes the model lighter. Moreover, we show that the performance of the connected classification networks has remained substantially high. In a super-resolution area, we show that the decoder part gets a high-quality reconstruction image by setting to resemble the structure of LP. Consequently, LPAE improves the original results by combining the decoder part of the autoencoder and the super-resolution network.

A number of recent works have employed decision trees for the construction of explainable partitions that aim to minimize the $k$-means cost function. These works, however, largely ignore metrics related to the depths of the leaves in the resulting tree, which is perhaps surprising considering how the explainability of a decision tree depends on these depths. To fill this gap in the literature, we propose an efficient algorithm that takes into account these metrics. In experiments on 16 datasets, our algorithm yields better results than decision-tree clustering algorithms such as the ones presented in \cite{dasgupta2020explainable}, \cite{frost2020exkmc}, \cite{laber2021price} and \cite{DBLP:conf/icml/MakarychevS21}, typically achieving lower or equivalent costs with considerably shallower trees. We also show, through a simple adaptation of existing techniques, that the problem of building explainable partitions induced by binary trees for the $k$-means cost function does not admit an $(1+\epsilon)$-approximation in polynomial time unless $P=NP$, which justifies the quest for approximation algorithms and/or heuristics.

Variational Autoencoders (VAEs) have experienced recent success as data-generating models by using simple architectures that do not require significant fine-tuning of hyperparameters. However, VAEs are known to suffer from over-regularization which can lead to failure to escape local maxima. This phenomenon, known as posterior collapse, prevents learning a meaningful latent encoding of the data. Recent methods have mitigated this issue by deterministically moment-matching an aggregated posterior distribution to an aggregate prior. However, abandoning a probabilistic framework (and thus relying on point estimates) can both lead to a discontinuous latent space and generate unrealistic samples. Here we present Hierarchical Empirical Bayes Autoencoder (HEBAE), a computationally stable framework for probabilistic generative models. Our key contributions are two-fold. First, we make gains by placing a hierarchical prior over the encoding distribution, enabling us to adaptively balance the trade-off between minimizing the reconstruction loss function and avoiding over-regularization. Second, we show that assuming a general dependency structure between variables in the latent space produces better convergence onto the mean-field assumption for improved posterior inference. Overall, HEBAE is more robust to a wide-range of hyperparameter initializations than an analogous VAE. Using data from MNIST and CelebA, we illustrate the ability of HEBAE to generate higher quality samples based on FID score than existing autoencoder-based approaches.