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Register a new userAutoencoders and Probabilistic Inference with Missing Data: An Exact Solution for The Factor Analysis Case
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Latent variable models can be used to probabilistically fill-in missing data entries. The variational autoencoder architecture (Kingma and Welling, 2014; Rezende et al., 2014) includes a recognition or encoder network that infers the latent variables given the data variables. However, it is not clear how to handle missing data variables in this network. The factor analysis (FA) model is a basic autoencoder, using linear encoder and decoder networks. We show how to calculate exactly the latent posterior distribution for the factor analysis (FA) model in the presence of missing data, and note that this solution implies that a different encoder network is required for each pattern of missingness. We also discuss various approximations to the exact solution. Experiments compare the effectiveness of various approaches to filling in the missing data.
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In many machine learning tasks, learning a good representation of the data can be the key to building a well-performant solution. This is because most learning algorithms operate with the features in order to find models for the data. For instance, classification performance can improve if the data is mapped to a space where classes are easily separated, and regression can be facilitated by finding a manifold of data in the feature space. As a general rule, features are transformed by means of statistical methods such as principal component analysis, or manifold learning techniques such as Isomap or locally linear embedding. From a plethora of representation learning methods, one of the most versatile tools is the autoencoder. In this paper we aim to demonstrate how to influence its learned representations to achieve the desired learning behavior. To this end, we present a series of learning tasks: data embedding for visualization, image denoising, semantic hashing, detection of abnormal behaviors and instance generation. We model them from the representation learning perspective, following the state of the art methodologies in each field. A solution is proposed for each task employing autoencoders as the only learning method. The theoretical developments are put into practice using a selection of datasets for the different problems and implementing each solution, followed by a discussion of the results in each case study and a brief explanation of other six learning applications. We also explore the current challenges and approaches to explainability in the context of autoencoders. All of this helps conclude that, thanks to alterations in their structure as well as their objective function, autoencoders may be the core of a possible solution to many problems which can be modeled as a transformation of the feature space.
In high-stakes applications of data-driven decision making like healthcare, it is of paramount importance to learn a policy that maximizes the reward while avoiding potentially dangerous actions when there is uncertainty. There are two main challenges usually associated with this problem. Firstly, learning through online exploration is not possible due to the critical nature of such applications. Therefore, we need to resort to observational datasets with no counterfactuals. Secondly, such datasets are usually imperfect, additionally cursed with missing values in the attributes of features. In this paper, we consider the problem of constructing personalized policies using logged data when there are missing values in the attributes of features in both training and test data. The goal is to recommend an action (treatment) when $Xt$, a degraded version of $Xb$ with missing values, is observed. We consider three strategies for dealing with missingness. In particular, we introduce the textit{conservative strategy} where the policy is designed to safely handle the uncertainty due to missingness. In order to implement this strategy we need to estimate posterior distribution $p(Xb|Xt)$, we use variational autoencoder to achieve this. In particular, our method is based on partial variational autoencoders (PVAE) which are designed to capture the underlying structure of features with missing values.
A remarkable recent discovery in machine learning has been that deep neural networks can achieve impressive performance (in terms of both lower training error and higher generalization capacity) in the regime where they are massively over-parameterized. Consequently, over the past year, the community has devoted growing interest in analyzing optimization and generalization properties of over-parameterized networks, and several breakthrough works have led to important theoretical progress. However, the majority of existing work only applies to supervised learning scenarios and hence are limited to settings such as classification and regression. In contrast, the role of over-parameterization in the unsupervised setting has gained far less attention. In this paper, we study the gradient dynamics of two-layer over-parameterized autoencoders with ReLU activation. We make very few assumptions about the given training dataset (other than mild non-degeneracy conditions). Starting from a randomly initialized autoencoder network, we rigorously prove the linear convergence of gradient descent in two learning regimes, namely: (i) the weakly-trained regime where only the encoder is trained, and (ii) the jointly-trained regime where both the encoder and the decoder are trained. Our results indicate the considerable benefits of joint training over weak training for finding global optima, achieving a dramatic decrease in the required level of over-parameterization. We also analyze the case of weight-tied autoencoders (which is a commonly used architectural choice in practical settings) and prove that in the over-parameterized setting, training such networks from randomly initialized points leads to certain unexpected degeneracies.
Real-world clinical time series data sets exhibit a high prevalence of missing values. Hence, there is an increasing interest in missing data imputation. Traditional statistical approaches impose constraints on the data-generating process and decouple imputation from prediction. Recent works propose recurrent neural network based approaches for missing data imputation and prediction with time series data. However, they generate deterministic outputs and neglect the inherent uncertainty. In this work, we introduce a unified Bayesian recurrent framework for simultaneous imputation and prediction on time series data sets. We evaluate our approach on two real-world mortality prediction tasks using the MIMIC-III and PhysioNet benchmark datasets. We demonstrate strong performance gains over state-of-the-art (SOTA) methods, and provide strategies to use the resulting probability distributions to better assess reliability of the imputations and predictions.
This paper studies how to construct confidence regions for principal component analysis (PCA) in high dimension, a problem that has been vastly under-explored. While computing measures of uncertainty for nonlinear/nonconvex estimators is in general difficult in high dimension, the challenge is further compounded by the prevalent presence of missing data and heteroskedastic noise. We propose a suite of solutions to perform valid inference on the principal subspace based on two estimators: a vanilla SVD-based approach, and a more refined iterative scheme called $textsf{HeteroPCA}$ (Zhang et al., 2018). We develop non-asymptotic distributional guarantees for both estimators, and demonstrate how these can be invoked to compute both confidence regions for the principal subspace and entrywise confidence intervals for the spiked covariance matrix. Particularly worth highlighting is the inference procedure built on top of $textsf{HeteroPCA}$, which is not only valid but also statistically efficient for broader scenarios (e.g., it covers a wider range of missing rates and signal-to-noise ratios). Our solutions are fully data-driven and adaptive to heteroskedastic random noise, without requiring prior knowledge about the noise levels and noise distributions.
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