Duration modelling has become an important research problem once more with the rise of non-attention neural text-to-speech systems. The current approaches largely fall back to relying on previous statistical parametric speech synthesis technology for duration prediction, which poorly models the expressiveness and variability in speech. In this paper, we propose two alternate approaches to improve duration modelling. First, we propose a duration model conditioned on phrasing that improves the predicted durations and provides better modelling of pauses. We show that the duration model conditioned on phrasing improves the naturalness of speech over our baseline duration model. Second, we also propose a multi-speaker duration model called Cauliflow, that uses normalising flows to predict durations that better match the complex target duration distribution. Cauliflow performs on par with our other proposed duration model in terms of naturalness, whilst providing variable durations for the same prompt and variable levels of expressiveness. Lastly, we propose to condition Cauliflow on parameters that provide an intuitive control of the pacing and pausing in the synthesised speech in a novel way.
Non-parallel voice conversion (VC) is typically achieved using lossy representations of the source speech. However, ensuring only speaker identity information is dropped whilst all other information from the source speech is retained is a large challenge. This is particularly challenging in the scenario where at inference-time we have no knowledge of the text being read, i.e., text-free VC. To mitigate this, we investigate information-preserving VC approaches. Normalising flows have gained attention for text-to-speech synthesis, however have been under-explored for VC. Flows utilize invertible functions to learn the likelihood of the data, thus provide a lossless encoding of speech. We investigate normalising flows for VC in both text-conditioned and text-free scenarios. Furthermore, for text-free VC we compare pre-trained and jointly-learnt priors. Flow-based VC evaluations show no degradation between text-free and text-conditioned VC, resulting in improvements over the state-of-the-art. Also, joint-training of the prior is found to negatively impact text-free VC quality.
Recently a machine learning approach to Monte-Carlo simulations called Neural Markov Chain Monte-Carlo (NMCMC) is gaining traction. In its most popular form it uses neural networks to construct normalizing flows which are then trained to approximate the desired target distribution. In this contribution we present new gradient estimator for Stochastic Gradient Descent algorithm (and the corresponding \texttt{PyTorch} implementation) and show that it leads to better training results for $\phi^4$ model. For this model our estimator achieves the same precision in approximately half of the time needed in standard approach and ultimately provides better estimates of the free energy. We attribute this effect to the lower variance of the new estimator. In contrary to the standard learning algorithm our approach does not require estimation of the action gradient with respect to the fields, thus has potential of further speeding up the training for models with more complicated actions.
Statistical models are central to machine learning with broad applicability across a range of downstream tasks. The models are controlled by free parameters that are typically estimated from data by maximum-likelihood estimation or approximations thereof. However, when faced with real-world data sets many of the models run into a critical issue: they are formulated in terms of fully-observed data, whereas in practice the data sets are plagued with missing data. The theory of statistical model estimation from incomplete data is conceptually similar to the estimation of latent-variable models, where powerful tools such as variational inference (VI) exist. However, in contrast to standard latent-variable models, parameter estimation with incomplete data often requires estimating exponentially-many conditional distributions of the missing variables, hence making standard VI methods intractable. We address this gap by introducing variational Gibbs inference (VGI), a new general-purpose method to estimate the parameters of statistical models from incomplete data. We validate VGI on a set of synthetic and real-world estimation tasks, estimating important machine learning models such as variational autoencoders and normalising flows from incomplete data. The proposed method, whilst general-purpose, achieves competitive or better performance than existing model-specific estimation methods.
Normalizing flows are flexible, parameterized distributions that can be used to approximate expectations from intractable distributions via importance sampling. However, current flow-based approaches are limited on challenging targets where they either suffer from mode seeking behaviour or high variance in the training loss, or rely on samples from the target distribution, which may not be available. To address these challenges, we combine flows with annealed importance sampling (AIS), while using the $\alpha$-divergence as our objective, in a novel training procedure, FAB (Flow AIS Bootstrap). Thereby, the flow and AIS improve each other in a bootstrapping manner. We demonstrate that FAB can be used to produce accurate approximations to complex target distributions, including Boltzmann distributions, in problems where previous flow-based methods fail.
Normalising flows offer a flexible way of modelling continuous probability distributions. We consider expressiveness, fast inversion and exact Jacobian determinant as three desirable properties a normalising flow should possess. However, few flow models have been able to strike a good balance among all these properties. Realising that the integral of a convex sum of sinusoidal functions squared leads to a bijective residual transformation, we propose Sinusoidal Flow, a new type of normalising flows that inherits the expressive power and triangular Jacobian from fully autoregressive flows while guaranteed by Banach fixed-point theorem to remain fast invertible and thereby obviate the need for sequential inversion typically required in fully autoregressive flows. Experiments show that our Sinusoidal Flow is not only able to model complex distributions, but can also be reliably inverted to generate realistic-looking samples even with many layers of transformations stacked.
We are interested in the challenging problem of modelling densities on Riemannian manifolds with a known symmetry group using normalising flows. This has many potential applications in physical sciences such as molecular dynamics and quantum simulations. In this work we combine ideas from implicit neural layers and optimal transport theory to propose a generalisation of existing work on exponential map flows, Implicit Riemannian Concave Potential Maps, IRCPMs. IRCPMs have some nice properties such as simplicity of incorporating symmetries and are less expensive than ODE-flows. We provide an initial theoretical analysis of its properties and layout sufficient conditions for stable optimisation. Finally, we illustrate the properties of IRCPMs with density estimation experiments on tori and spheres.
One major drawback of state-of-the-art artificial intelligence is its lack of explainability. One approach to solve the problem is taking causality into account. Causal mechanisms can be described by structural causal models. In this work, we propose a method for estimating bivariate structural causal models using a combination of normalising flows applied to density estimation and variational Gaussian process regression for post-nonlinear models. It facilitates causal discovery, i.e. distinguishing cause and effect, by either the independence of cause and residual or a likelihood ratio test. Our method which estimates post-nonlinear models can better explain a variety of real-world cause-effect pairs than a simple additive noise model. Though it remains difficult to exploit this benefit regarding all pairs from the T\"ubingen benchmark database, we demonstrate that combining the additive noise model approach with our method significantly enhances causal discovery.
We propose Parallelised Diffeomorphic Sampling-based Motion Planning (PDMP). PDMP is a novel parallelised framework that uses bijective and differentiable mappings, or diffeomorphisms, to transform sampling distributions of sampling-based motion planners, in a manner akin to normalising flows. Unlike normalising flow models which use invertible neural network structures to represent these diffeomorphisms, we develop them from gradient information of desired costs, and encode desirable behaviour, such as obstacle avoidance. These transformed sampling distributions can then be used for sampling-based motion planning. A particular example is when we wish to imbue the sampling distribution with knowledge of the environment geometry, such that drawn samples are less prone to be in collisions. To this end, we propose to learn a continuous occupancy representation from environment occupancy data, such that gradients of the representation defines a valid diffeomorphism and is amenable to fast parallel evaluation. We use this to "morph" the sampling distribution to draw far fewer collision-prone samples. PDMP is able to leverage gradient information of costs, to inject specifications, in a manner similar to optimisation-based motion planning methods, but relies on drawing from a sampling distribution, retaining the tendency to find more global solutions, thereby bridging the gap between trajectory optimisation and sampling-based planning methods.
The ability to generate high-fidelity synthetic data is crucial when available (real) data is limited or where privacy and data protection standards allow only for limited use of the given data, e.g., in medical and financial data-sets. Current state-of-the-art methods for synthetic data generation are based on generative models, such as Generative Adversarial Networks (GANs). Even though GANs have achieved remarkable results in synthetic data generation, they are often challenging to interpret.Furthermore, GAN-based methods can suffer when used with mixed real and categorical variables.Moreover, loss function (discriminator loss) design itself is problem specific, i.e., the generative model may not be useful for tasks it was not explicitly trained for. In this paper, we propose to use a probabilistic model as a synthetic data generator. Learning the probabilistic model for the data is equivalent to estimating the density of the data. Based on the copula theory, we divide the density estimation task into two parts, i.e., estimating univariate marginals and estimating the multivariate copula density over the univariate marginals. We use normalising flows to learn both the copula density and univariate marginals. We benchmark our method on both simulated and real data-sets in terms of density estimation as well as the ability to generate high-fidelity synthetic data
As data volumes continue to grow, the labelling process increasingly becomes a bottleneck, creating demand for methods that leverage information from unlabelled data. Impressive results have been achieved in semi-supervised learning (SSL) for image classification, nearing fully supervised performance, with only a fraction of the data labelled. In this work, we propose a probabilistically principled general approach to SSL that considers the distribution over label predictions, for labels of different complexity, from "one-hot" vectors to binary vectors and images. Our method regularises an underlying supervised model, using a normalising flow that learns the posterior distribution over predictions for labelled data, to serve as a prior over the predictions on unlabelled data. We demonstrate the general applicability of this approach on a range of computer vision tasks with varying output complexity: classification, attribute prediction and image-to-image translation.
Normalising flows (NFs) for discrete data are challenging because parameterising bijective transformations of discrete variables requires predicting discrete/integer parameters. Having a neural network architecture predict discrete parameters takes a non-differentiable activation function (eg, the step function) which precludes gradient-based learning. To circumvent this non-differentiability, previous work has employed biased proxy gradients, such as the straight-through estimator. We present an unbiased alternative where rather than deterministically parameterising one transformation, we predict a distribution over latent transformations. With stochastic transformations, the marginal likelihood of the data is differentiable and gradient-based learning is possible via score function estimation. To test the viability of discrete-data NFs we investigate performance on binary MNIST. We observe great challenges with both deterministic proxy gradients and unbiased score function estimation. Whereas the former often fails to learn even a shallow transformation, the variance of the latter could not be sufficiently controlled to admit deeper NFs.
Normalizing flows (NFs) have become a prominent method for deep generative models that allow for an analytic probability density estimation and efficient synthesis. However, a flow-based network is considered to be inefficient in parameter complexity because of reduced expressiveness of bijective mapping, which renders the models unfeasibly expensive in terms of parameters. We present an alternative parameterization scheme called NanoFlow, which uses a single neural density estimator to model multiple transformation stages. Hence, we propose an efficient parameter decomposition method and the concept of flow indication embedding, which are key missing components that enable density estimation from a single neural network. Experiments performed on audio and image models confirm that our method provides a new parameter-efficient solution for scalable NFs with significant sublinear parameter complexity.
Normalising flows are tractable probabilistic models that leverage the power of deep learning to describe a wide parametric family of distributions, all while remaining trainable using maximum likelihood. We discuss how these methods can be further improved based on insights from robust (in particular, resistant) statistics. Specifically, we propose to endow flow-based models with fat-tailed latent distributions such as multivariate Student's $t$, as a simple drop-in replacement for the Gaussian distribution used by conventional normalising flows. While robustness brings many advantages, this paper explores two of them: 1) We describe how using fatter-tailed base distributions can give benefits similar to gradient clipping, but without compromising the asymptotic consistency of the method. 2) We also discuss how robust ideas lead to models with reduced generalisation gap and improved held-out data likelihood. Experiments on several different datasets confirm the efficacy of the proposed approach in both regards.
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.
Normalizing flows are deep generative models that allow efficient likelihood calculation and sampling. The core requirement for this advantage is that they are constructed using functions that can be efficiently inverted and for which the determinant of the function's Jacobian can be efficiently computed. Researchers have introduced various such flow operations, but few of these allow rich interactions among variables without incurring significant computational costs. In this paper, we introduce Woodbury transformations, which achieve efficient invertibility via the Woodbury matrix identity and efficient determinant calculation via Sylvester's determinant identity. In contrast with other operations used in state-of-the-art normalizing flows, Woodbury transformations enable (1) high-dimensional interactions, (2) efficient sampling, and (3) efficient likelihood evaluation. Other similar operations, such as 1x1 convolutions, emerging convolutions, or periodic convolutions allow at most two of these three advantages. In our experiments on multiple image datasets, we find that Woodbury transformations allow learning of higher-likelihood models than other flow architectures while still enjoying their efficiency advantages.
Generative flows are promising tractable models for density modeling that define probabilistic distributions with invertible transformations. However, tractability imposes architectural constraints on generative flows, making them less expressive than other types of generative models. In this work, we study a previously overlooked constraint that all the intermediate representations must have the same dimensionality with the original data due to invertibility, limiting the width of the network. We tackle this constraint by augmenting the data with some extra dimensions and jointly learning a generative flow for augmented data as well as the distribution of augmented dimensions under a variational inference framework. Our approach, VFlow, is a generalization of generative flows and therefore always performs better. Combining with existing generative flows, VFlow achieves a new state-of-the-art 2.98 bits per dimension on the CIFAR-10 dataset and is more compact than previous models to reach similar modeling quality.
We present an algorithm for learning an approximate action-value soft Q-function in the relative entropy regularised reinforcement learning setting, for which an optimal improved policy can be recovered in closed form. We use recent advances in normalising flows for parametrising the policy together with a learned value-function; and show how this combination can be used to implicitly represent Q-values of an arbitrary policy in continuous action space. Using simple temporal difference learning on the Q-values then leads to a unified objective for policy and value learning. We show how this approach considerably simplifies standard Actor-Critic off-policy algorithms, removing the need for a policy optimisation step. We perform experiments on a range of established reinforcement learning benchmarks, demonstrating that our approach allows for complex, multimodal policy distributions in continuous action spaces, while keeping the process of sampling from the policy both fast and exact.
Variational inference has had great success in scaling approximate Bayesian inference to big data by exploiting mini-batch training. To date, however, this strategy has been most applicable to models of independent data. We propose an extension to state space models of time series data based on a novel generative model for latent temporal states: the neural moving average model. This permits a subsequence to be sampled without drawing from the entire distribution, enabling training iterations to use mini-batches of the time series at low computational cost. We illustrate our method on autoregressive, Lotka-Volterra, FitzHugh-Nagumo and stochastic volatility models, achieving accurate parameter estimation in a short time.
We show that normalising flows become pathological when used to model targets whose supports have complicated topologies. In this scenario, we prove that a flow must become arbitrarily numerically noninvertible in order to approximate the target closely. This result has implications for all flow-based models, and especially Residual Flows (ResFlows), which explicitly control the Lipschitz constant of the bijection used. To address this, we propose Continuously Indexed Flows (CIFs), which replace the single bijection used by normalising flows with a continuously indexed family of bijections, and which can intuitively "clean up" mass that would otherwise be misplaced by a single bijection. We show theoretically that CIFs are not subject to the same topological limitations as normalising flows, and obtain better empirical performance on a variety of models and benchmarks.