Training neural networks with binary weights and activations is a challenging problem due to the lack of gradients and difficulty of optimization over discrete weights. Many successful experimental results have been achieved with empirical straight-through (ST) approaches, proposing a variety of ad-hoc rules for propagating gradients through non-differentiable activations and updating discrete weights. At the same time, ST methods can be truly derived as estimators in the stochastic binary network (SBN) model with Bernoulli weights. We advance these derivations to a more complete and systematic study. We analyze properties, estimation accuracy, obtain different forms of correct ST estimators for activations and weights, explain existing empirical approaches and their shortcomings, explain how latent weights arise from the mirror descent method when optimizing over probabilities. This allows to reintroduce ST methods, long known empirically, as sound approximations, apply them with clarity and develop further improvements.
We formulate a general framework for building structural causal models (SCMs) with deep learning components. The proposed approach employs normalising flows and variational inference to enable tractable inference of exogenous noise variables - a crucial step for counterfactual inference that is missing from existing deep causal learning methods. Our framework is validated on a synthetic dataset built on MNIST as well as on a real-world medical dataset of brain MRI scans. Our experimental results indicate that we can successfully train deep SCMs that are capable of all three levels of Pearl's ladder of causation: association, intervention, and counterfactuals, giving rise to a powerful new approach for answering causal questions in imaging applications and beyond. The code for all our experiments is available at https://github.com/biomedia-mira/deepscm.
Mobility-on-demand (MoD) systems represent a rapidly developing mode of transportation wherein travel requests are dynamically handled by a coordinated fleet of vehicles. Crucially, the efficiency of an MoD system highly depends on how well supply and demand distributions are aligned in spatio-temporal space (i.e., to satisfy user demand, cars have to be available in the correct place and at the desired time). To do so, we argue that predictive models should aim to explicitly disentangle between temporal} and spatial variability in the evolution of urban mobility demand. However, current approaches typically ignore this distinction by either treating both sources of variability jointly, or completely ignoring their presence in the first place. In this paper, we propose recurrent flow networks (RFN), where we explore the inclusion of (i) latent random variables in the hidden state of recurrent neural networks to model temporal variability, and (ii) normalizing flows to model the spatial distribution of mobility demand. We demonstrate how predictive models explicitly disentangling between spatial and temporal variability exhibit several desirable properties, and empirically show how this enables the generation of distributions matching potentially complex urban topologies.
We consider the problem of lossy image compression with deep latent variable models. State-of-the-art methods build on hierarchical variational autoencoders (VAEs) and learn inference networks to predict a compressible latent representation of each data point. Drawing on the variational inference perspective on compression, we identify three approximation gaps which limit performance in the conventional approach: an amortization gap, a discretization gap, and a marginalization gap. We propose remedies for each of these three limitations based on ideas related to iterative inference, stochastic annealing for discrete optimization, and bits-back coding, resulting in the first application of bits-back coding to lossy compression. In our experiments, which include extensive baseline comparisons and ablation studies, we achieve new state-of-the-art performance on lossy image compression using an established VAE architecture, by changing only the inference method.
Probabilistic Latent Variable Models (LVMs) provide an alternative to self-supervised learning approaches for linguistic representation learning from speech. LVMs admit an intuitive probabilistic interpretation where the latent structure shapes the information extracted from the signal. Even though LVMs have recently seen a renewed interest due to the introduction of Variational Autoencoders (VAEs), their use for speech representation learning remains largely unexplored. In this work, we propose Convolutional Deep Markov Model (ConvDMM), a Gaussian state-space model with non-linear emission and transition functions modelled by deep neural networks. This unsupervised model is trained using black box variational inference. A deep convolutional neural network is used as an inference network for structured variational approximation. When trained on a large scale speech dataset (LibriSpeech), ConvDMM produces features that significantly outperform multiple self-supervised feature extracting methods on linear phone classification and recognition on the Wall Street Journal dataset. Furthermore, we found that ConvDMM complements self-supervised methods like Wav2Vec and PASE, improving on the results achieved with any of the methods alone. Lastly, we find that ConvDMM features enable learning better phone recognizers than any other features in an extreme low-resource regime with few labeled training examples.
We introduce a stochastic variational inference procedure for training scalable Gaussian process (GP) models whose per-iteration complexity is independent of both the number of training points, $n$, and the number basis functions used in the kernel approximation, $m$. Our central contributions include an unbiased stochastic estimator of the evidence lower bound (ELBO) for a Gaussian likelihood, as well as a stochastic estimator that lower bounds the ELBO for several other likelihoods such as Laplace and logistic. Independence of the stochastic optimization update complexity on $n$ and $m$ enables inference on huge datasets using large capacity GP models. We demonstrate accurate inference on large classification and regression datasets using GPs and relevance vector machines with up to $m = 10^7$ basis functions.
Modeling the temporal behavior of data is of primordial importance in many scientific and engineering fields. Baseline methods assume that both the dynamic and observation equations follow linear-Gaussian models. However, there are many real-world processes that cannot be characterized by a single linear behavior. Alternatively, it is possible to consider a piecewise-linear model which, combined with a switching mechanism, is well suited when several modes of behavior are needed. Nevertheless, switching dynamical systems are intractable because of their computational complexity increases exponentially with time. In this paper, we propose a variational approximation of piecewise linear dynamical systems. We provide full details of the derivation of two variational expectation-maximization algorithms, a filter and a smoother. We show that the model parameters can be split into two sets, static and dynamic parameters, and that the former parameters can be estimated off-line together with the number of linear modes, or the number of states of the switching variable. We apply the proposed method to a visual tracking problem, namely head-pose tracking, and we thoroughly compare our algorithm with several state of the art trackers.
Submodular functions have been studied extensively in machine learning and data mining. In particular, the optimization of submodular functions over the integer lattice (integer submodular functions) has recently attracted much interest, because this domain relates naturally to many practical problem settings, such as multilabel graph cut, budget allocation and revenue maximization with discrete assignments. In contrast, the use of these functions for probabilistic modeling has received surprisingly little attention so far. In this work, we firstly propose the Generalized Multilinear Extension, a continuous DR-submodular extension for integer submodular functions. We study central properties of this extension and formulate a new probabilistic model which is defined through integer submodular functions. Then, we introduce a block-coordinate ascent algorithm to perform approximate inference for those class of models. Finally, we demonstrate its effectiveness and viability on several real-world social connection graph datasets with integer submodular objectives.
We propose VarFA, a variational inference factor analysis framework that extends existing factor analysis models for educational data mining to efficiently output uncertainty estimation in the model's estimated factors. Such uncertainty information is useful, for example, for an adaptive testing scenario, where additional tests can be administered if the model is not quite certain about a students' skill level estimation. Traditional Bayesian inference methods that produce such uncertainty information are computationally expensive and do not scale to large data sets. VarFA utilizes variational inference which makes it possible to efficiently perform Bayesian inference even on very large data sets. We use the sparse factor analysis model as a case study and demonstrate the efficacy of VarFA on both synthetic and real data sets. VarFA is also very general and can be applied to a wide array of factor analysis models.
Deep neural networks have revolutionized medical image analysis and disease diagnosis. Despite their impressive performance, it is difficult to generate well-calibrated probabilistic outputs for such networks, which makes them uninterpretable black boxes. Bayesian neural networks provide a principled approach for modelling uncertainty and increasing patient safety, but they have a large computational overhead and provide limited improvement in calibration. In this work, by taking skin lesion classification as an example task, we show that by shifting Bayesian inference to the functional space we can craft meaningful priors that give better calibrated uncertainty estimates at a much lower computational cost.
Gaussian processes offer an attractive framework for predictive modeling from longitudinal data, i.e., irregularly sampled, sparse observations from a set of individuals over time. However, such methods have two key shortcomings: (i) They rely on ad hoc heuristics or expensive trial and error to choose the effective kernels, and (ii) They fail to handle multilevel correlation structure in the data. We introduce Longitudinal deep kernel Gaussian process regression (L-DKGPR), which to the best of our knowledge, is the only method to overcome these limitations by fully automating the discovery of complex multilevel correlation structure from longitudinal data. Specifically, L-DKGPR eliminates the need for ad hoc heuristics or trial and error using a novel adaptation of deep kernel learning that combines the expressive power of deep neural networks with the flexibility of non-parametric kernel methods. L-DKGPR effectively learns the multilevel correlation with a novel addictive kernel that simultaneously accommodates both time-varying and the time-invariant effects. We derive an efficient algorithm to train L-DKGPR using latent space inducing points and variational inference. Results of extensive experiments on several benchmark data sets demonstrate that L-DKGPR significantly outperforms the state-of-the-art longitudinal data analysis (LDA) methods.
Recent studies of cancer somatic mutation profiles seek to identify mutations for targeted therapy in personalized medicine. Analysis of profiles, however, is not trivial, as each profile is heterogeneous and there are multiple confounding factors that influence the cause-and-effect relationships between cancer genes such as cancer (sub)type, biological processes, total number of mutations, and non-linear mutation interactions. Moreover, cancer is biologically redundant, i.e., distinct mutations can result in the alteration of similar biological processes, so it is important to identify all possible combinatorial sets of mutations for effective patient treatment. To model this phenomena, we propose the correlated zero-inflated negative binomial process to infer the inherent structure of somatic mutation profiles through latent representations. This stochastic process takes into account different, yet correlated, co-occurring mutations using profile-specific negative binomial dispersion parameters that are mixed with a correlated beta-Bernoulli process and a probability parameter to model profile heterogeneity. These model parameters are inferred by iterative optimization via amortized and stochastic variational inference using the Pan Cancer dataset from The Cancer Genomic Archive (TCGA). By examining the the latent space, we identify biologically relevant correlations between somatic mutations.
Inspired by recent developments in learning smoothed densities with empirical Bayes, we study variational autoencoders with a decoder that is tailored for the random variable $Y=X+N(0,\sigma^2 I_d)$. A notion of smoothed variational inference emerges where the smoothing is implicitly enforced by the noise model of the decoder; "implicit", since during training the encoder only sees clean samples. This is the concept of imaginary noise model, where the noise model dictates the functional form of the variational lower bound $\mathcal{L}(\sigma)$, but the noisy data are never seen during learning. The model is named $\sigma$-VAE. We prove that all $\sigma$-VAEs are equivalent to each other via a simple $\beta$-VAE expansion: $\mathcal{L}(\sigma_2) \equiv \mathcal{L}(\sigma_1,\beta)$, where $\beta=\sigma_2^2/\sigma_1^2$. We prove a similar result for the Laplace distribution in exponential families. Empirically, we report an intriguing power law $\mathcal{D}_{\rm KL} \sim \sigma^{-\nu}$ for the learned models and we study the inference in the $\sigma$-VAE for unseen noisy data. The experiments were performed on MNIST, where we show that quite remarkably the model can make reasonable inferences on extremely noisy samples even though it has not seen any during training. The vanilla VAE completely breaks down in this regime. We finish with a hypothesis (the XYZ hypothesis) on the findings here.
The coronavirus disease 2019 (COVID-19) global pandemic has led many countries to impose unprecedented lockdown measures in order to slow down the outbreak. Questions on whether governments have acted promptly enough, and whether lockdown measures can be lifted soon have since been central in public discourse. Data-driven models that predict COVID-19 fatalities under different lockdown policy scenarios are essential for addressing these questions and informing governments on future policy directions. To this end, this paper develops a Bayesian model for predicting the effects of COVID-19 lockdown policies in a global context -- we treat each country as a distinct data point, and exploit variations of policies across countries to learn country-specific policy effects. Our model utilizes a two-layer Gaussian process (GP) prior -- the lower layer uses a compartmental SEIR (Susceptible, Exposed, Infected, Recovered) model as a prior mean function with "country-and-policy-specific" parameters that capture fatality curves under "counterfactual" policies within each country, whereas the upper layer is shared across all countries, and learns lower-layer SEIR parameters as a function of a country's features and its policy indicators. Our model combines the solid mechanistic foundations of SEIR models (Bayesian priors) with the flexible data-driven modeling and gradient-based optimization routines of machine learning (Bayesian posteriors) -- i.e., the entire model is trained end-to-end via stochastic variational inference. We compare the projections of COVID-19 fatalities by our model with other models listed by the Center for Disease Control (CDC), and provide scenario analyses for various lockdown and reopening strategies highlighting their impact on COVID-19 fatalities.
We propose a framework called HyperVAE for encoding distributions of distributions. When a target distribution is modeled by a VAE, its neural network parameters \theta is drawn from a distribution p(\theta) which is modeled by a hyper-level VAE. We propose a variational inference using Gaussian mixture models to implicitly encode the parameters \theta into a low dimensional Gaussian distribution. Given a target distribution, we predict the posterior distribution of the latent code, then use a matrix-network decoder to generate a posterior distribution q(\theta). HyperVAE can encode the parameters \theta in full in contrast to common hyper-networks practices, which generate only the scale and bias vectors as target-network parameters. Thus HyperVAE preserves much more information about the model for each task in the latent space. We discuss HyperVAE using the minimum description length (MDL) principle and show that it helps HyperVAE to generalize. We evaluate HyperVAE in density estimation tasks, outlier detection and discovery of novel design classes, demonstrating its efficacy.
Variational Bayesian inference is an important machine-learning tool that finds application from statistics to robotics. The goal is to find an approximate probability density function (PDF) from a chosen family that is in some sense 'closest' to the full Bayesian posterior. Closeness is typically defined through the selection of an appropriate loss functional such as the Kullback-Leibler (KL) divergence. In this paper, we explore a new formulation of variational inference by exploiting the fact that (most) PDFs are members of a Bayesian Hilbert space under careful definitions of vector addition, scalar multiplication and an inner product. We show that, under the right conditions, variational inference based on KL divergence can amount to iterative projection, in the Euclidean sense, of the Bayesian posterior onto a subspace corresponding to the selected approximation family. We work through the details of this general framework for the specific case of the Gaussian approximation family and show the equivalence to another Gaussian variational inference approach. We furthermore discuss the implications for systems that exhibit sparsity, which is handled naturally in Bayesian space, and give an example of a high-dimensional robotic state estimation problem that can be handled as a result. We provide some preliminary examples of how the approach could be applied to non-Gaussian inference and discuss the limitations of the approach in detail to encourage follow-on work along these lines.
The ability to recognize objects despite there being differences in appearance, known as Core Object Recognition, forms a critical part of human perception. While it is understood that the brain accomplishes Core Object Recognition through feedforward, hierarchical computations through the visual stream, the underlying algorithms that allow for invariant representations to form downstream is still not well understood. (DiCarlo et al., 2012) Various computational perceptual models have been built to attempt and tackle the object identification task in an artificial perceptual setting. Artificial Neural Networks, computational graphs consisting of weighted edges and mathematical operations at vertices, are loosely inspired by neural networks in the brain and have proven effective at various visual perceptual tasks, including object characterization and identification. (Pinto et al., 2008) (DiCarlo et al., 2012) For many data analysis tasks, learning representations where each dimension is statistically independent and thus disentangled from the others is useful. If the underlying generative factors of the data are also statistically independent, Bayesian inference of latent variables can form disentangled representations. This thesis constitutes a research project exploring a generalization of the Variational Autoencoder (VAE), $\beta$-VAE, that aims to learn disentangled representations using variational inference. $\beta$-VAE incorporates the hyperparameter $\beta$, and enforces conditional independence of its bottleneck neurons, which is in general not compatible with the statistical independence of latent variables. This text examines this architecture, and provides analytical and numerical arguments, with the goal of demonstrating that this incompatibility leads to a non-monotonic inference performance in $\beta$-VAE with a finite optimal $\beta$.
This paper describes a statistically-principled semi-supervised method of automatic chord estimation (ACE) that can make effective use of music signals regardless of the availability of chord annotations. The typical approach to ACE is to train a deep classification model (neural chord estimator) in a supervised manner by using only annotated music signals. In this discriminative approach, prior knowledge about chord label sequences (model output) has scarcely been taken into account. In contrast, we propose a unified generative and discriminative approach in the framework of amortized variational inference. More specifically, we formulate a deep generative model that represents the generative process of chroma vectors (observed variables) from discrete labels and continuous features (latent variables), which are assumed to follow a Markov model favoring self-transitions and a standard Gaussian distribution, respectively. Given chroma vectors as observed data, the posterior distributions of the latent labels and features are computed approximately by using deep classification and recognition models, respectively. These three models form a variational autoencoder and can be trained jointly in a semi-supervised manner. The experimental results show that the regularization of the classification model based on the Markov prior of chord labels and the generative model of chroma vectors improved the performance of ACE even under the supervised condition. The semi-supervised learning using additional non-annotated data can further improve the performance.
Upcoming experiments such as Hydrogen Epoch of Reionization Array (HERA) and Square Kilometre Array (SKA) are intended to measure the 21cm signal over a wide range of redshifts, representing an incredible opportunity in advancing our understanding about the nature of cosmic Reionization. At the same time these kind of experiments will present new challenges in processing the extensive amount of data generated, calling for the development of automated methods capable of precisely estimating physical parameters and their uncertainties. In this paper we employ Variational Inference, and in particular Bayesian Neural Networks, as an alternative to MCMC in 21 cm observations to report credible estimations for cosmological and astrophysical parameters and assess the correlations among them.
Functional Distributional Semantics provides a linguistically interpretable framework for distributional semantics, by representing the meaning of a word as a function (a binary classifier), instead of a vector. However, the large number of latent variables means that inference is computationally expensive, and training a model is therefore slow to converge. In this paper, I introduce the Pixie Autoencoder, which augments the generative model of Functional Distributional Semantics with a graph-convolutional neural network to perform amortised variational inference. This allows the model to be trained more effectively, achieving better results on two tasks (semantic similarity in context and semantic composition), and outperforming BERT, a large pre-trained language model.