The Riemannian Geometry of Deep Generative Models

Deep generative models learn a mapping from a low dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate the Riemannian geometry of these generated manifolds. First, we develop efficient algorithms for computing geodesic curves, which provide an intrinsic notion of distance between points on the manifold. Second, we develop an algorithm for parallel translation of a tangent vector along a path on the manifold. We show how parallel translation can be used to generate analogies, i.e., to transport a change in one data point into a semantically similar change of another data point. Our experiments on real image data show that the manifolds learned by deep generative models, while nonlinear, are surprisingly close to zero curvature. The practical implication is that linear paths in the latent space closely approximate geodesics on the generated manifold. However, further investigation into this phenomenon is warranted, to identify if there are other architectures or datasets where curvature plays a more prominent role. We believe that exploring the Riemannian geometry of deep generative models, using the tools developed in this paper, will be an important step in understanding the high-dimensional, nonlinear spaces these models learn.Comment: 9 page

Hierarchical Graphical Models for Multigroup Shape Analysis using Expectation Maximization with Sampling in Kendall's Shape Space

This paper proposes a novel framework for multi-group shape analysis relying on a hierarchical graphical statistical model on shapes within a population.The framework represents individual shapes as point setsmodulo translation, rotation, and scale, following the notion in Kendall shape space.While individual shapes are derived from their group shape model, each group shape model is derived from a single population shape model. The hierarchical model follows the natural organization of population data and the top level in the hierarchy provides a common frame of reference for multigroup shape analysis, e.g. classification and hypothesis testing. Unlike typical shape-modeling approaches, the proposed model is a generative model that defines a joint distribution of object-boundary data and the shape-model variables. Furthermore, it naturally enforces optimal correspondences during the process of model fitting and thereby subsumes the so-called correspondence problem. The proposed inference scheme employs an expectation maximization (EM) algorithm that treats the individual and group shape variables as hidden random variables and integrates them out before estimating the parameters (population mean and variance and the group variances). The underpinning of the EM algorithm is the sampling of pointsets, in Kendall shape space, from their posterior distribution, for which we exploit a highly-efficient scheme based on Hamiltonian Monte Carlo simulation. Experiments in this paper use the fitted hierarchical model to perform (1) hypothesis testing for comparison between pairs of groups using permutation testing and (2) classification for image retrieval. The paper validates the proposed framework on simulated data and demonstrates results on real data.Comment: 9 pages, 7 figures, International Conference on Machine Learning 201

Optimal energy management strategy for a fuel cell hybrid electric vehicle

The Energy Management Strategy (EMS) has a huge effect on the performance of any hybrid vehicle because it determines the operating point of almost every component associated with the powertrain. This means that its optimisation is an incredibly complex task which must consider a number of objectives including the fuel consumption, drive-ability, component degradation and straight-line performance. The EMS is of particular importance for Fuel Cell Hybrid Electric Vehicles (FCHEVs), not only to minimise the fuel consumption, but also to reduce the electrical stress on the fuel cell and maximise its useful lifetime. This is because the durability and cost of the fuel cell stack is one of the major obstacles preventing FCHEVs from being competitive with conventional vehicles. In this work, a novel EMS is developed, specifcally for Fuel Cell Hybrid Electric Vehicles (FCHEVs), which considers not only the fuel consumption, but also the degradation of the fuel cell in order to optimise the overall running cost of the vehicle. This work is believed to be the first of its kind to quantify effect of decisions made by the EMS on the fuel cell degradation, inclusive of multiple causes of voltage degradation. The performance of this new strategy is compared in simulation to a recent strategy from the literature designed solely to optimise the fuel consumption. It is found that the inclusion of the degradation metrics results in a 20% increase in fuel cell lifetime for only a 3.7% increase in the fuel consumption, meaning that the overall running cost is reduced by 9%. In addition to direct implementation on board a vehicle, this technique for optimising the degradation alongside the fuel consumption also allows alternative vehicle designs to be compared in an unbiased way. In order to demonstrate this, the novel optimisation technique is subsequently used to compare alternative system designs in order to identify the optimal economic sizing of the fuel cell and battery pack. It is found that the overall running cost can be minimised by using the smallest possible fuel cell stack that will satisfy the average power requirement of the duty cycle, and by using an oversized battery pack to maximise the fuel cell effciency and minimise the transient loading on the stack. This research was undertaken at Loughborough University as part of the Doctoral Training Centre (DTC) in Hydrogen, Fuel Cells and Their Applications in collaboration with the University of Birmingham and Nottingham University and with sponsorship from HORIBA-MIRA (Nuneaton, UK). A Microcab H4 test vehicle has been made available for use in testing for this research which was previously used for approximately 2 years at the University of Birmingham. The Microcab H4 is a small campus based vehicle designed for passenger transport and mail delivery at low speeds as seen on a university campus. It has a top speed of approximately 30mph, and is fitted with a 1.2kW fuel cell and a 2kWh battery pack

Feature Gradient Flow for Interpreting Deep Neural Networks in Head and Neck Cancer Prediction

This paper introduces feature gradient flow, a new technique for interpreting deep learning models in terms of features that are understandable to humans. The gradient flow of a model locally defines nonlinear coordinates in the input data space representing the information the model is using to make its decisions. Our idea is to measure the agreement of interpretable features with the gradient flow of a model. To then evaluate the importance of a particular feature to the model, we compare that feature's gradient flow measure versus that of a baseline noise feature. We then develop a technique for training neural networks to be more interpretable by adding a regularization term to the loss function that encourages the model gradients to align with those of chosen interpretable features. We test our method in a convolutional neural network prediction of distant metastasis of head and neck cancer from a computed tomography dataset from the Cancer Imaging Archive

The Lyman Continuum Escape Survey: Ionizing Radiation from [O III]-Strong Sources at a Redshift of 3.1

We present results from the LymAn Continuum Escape Survey (LACES), a Hubble Space Telescope (HST) program designed to characterize the ionizing radiation emerging from a sample of Lyman alpha emitting galaxies at redshift $z\simeq 3.1$. As many show intense [O III] emission characteristic of $z>6.5$ star-forming galaxies, they may represent valuable low redshift analogs of galaxies in the reionization era. Using HST Wide Field Camera 3 / UVIS $F336W$ to image Lyman continuum emission, we investigate the escape fraction of ionizing photons in this sample. For 61 sources, of which 77% are spectroscopically confirmed and 53 have measures of [O III] emission, we detect Lyman continuum leakage in 20%, a rate significantly higher than is seen in individual continuum-selected Lyman break galaxies. We estimate there is a 98% probability that $\leq 2$ of our detections could be affected by foreground contamination. Fitting multi-band spectral energy distributions (SEDs) to take account of the varying stellar populations, dust extinctions and metallicities, we derive individual Lyman continuum escape fractions corrected for foreground intergalactic absorption. We find escape fractions of 15 to 60% for individual objects, and infer an average 20% escape fraction by fitting composite SEDs for our detected samples. Surprisingly however, even a deep stack of those sources with no individual $F336W$ detections provides a stringent upper limit on the average escape fraction of less than 0.5%. We examine various correlations with source properties and discuss the implications in the context of the popular picture that cosmic reionization is driven by such compact, low metallicity star-forming galaxies.Comment: 26 pages, 16 figures, accepted for publication in Ap

Hierarchical Geodesic Models in Diffeomorphisms

Hierarchical linear models (HLMs) are a standard approach for analyzing data where individuals are measured repeatedly over time. However, such models are only applicable to longitudinal studies of Euclidean data. This paper develops the theory of hierarchical geodesic models (HGMs), which generalize HLMs to the manifold setting. Our proposed model quantifies longitudinal trends in shapes as a hierarchy of geodesics in the group of diffeomorphisms. First, individual-level geodesics represent the trajectory of shape changes within individuals. Second, a group-level geodesic represents the average trajectory of shape changes for the population. Our proposed HGM is applicable to longitudinal data from unbalanced designs, i.e., varying numbers of timepoints for individuals, which is typical in medical studies. We derive the solution of HGMs on diffeomorphisms to estimate individual-level geodesics, the group geodesic, and the residual diffeomorphisms. We also propose an efficient parallel algorithm that easily scales to solve HGMs on a large collection of 3D images of several individuals. Finally, we present an effective model selection procedure based on cross validation. We demonstrate the effectiveness of HGMs for longitudinal analysis of synthetically generated shapes and 3D MRI brain scans

Characterizing Distances of Networks on the Tensor Manifold

At the core of understanding dynamical systems is the ability to maintain and control the systems behavior that includes notions of robustness, heterogeneity, or regime-shift detection. Recently, to explore such functional properties, a convenient representation has been to model such dynamical systems as a weighted graph consisting of a finite, but very large number of interacting agents. This said, there exists very limited relevant statistical theory that is able cope with real-life data, i.e., how does perform analysis and/or statistics over a family of networks as opposed to a specific network or network-to-network variation. Here, we are interested in the analysis of network families whereby each network represents a point on an underlying statistical manifold. To do so, we explore the Riemannian structure of the tensor manifold developed by Pennec previously applied to Diffusion Tensor Imaging (DTI) towards the problem of network analysis. In particular, while this note focuses on Pennec definition of geodesics amongst a family of networks, we show how it lays the foundation for future work for developing measures of network robustness for regime-shift detection. We conclude with experiments highlighting the proposed distance on synthetic networks and an application towards biological (stem-cell) systems.Comment: This paper is accepted at 8th International Conference on Complex Networks 201