Bayesian deep learning all too often underfits so that the Bayesian prediction is less accurate than a simple point estimate. Uncertainty quantification then comes at the cost of accuracy. For linearized models, the null space of the generalized Gauss-Newton matrix corresponds to parameters that preserve the training predictions of the point estimate. We propose to build Bayesian approximations in this null space, thereby guaranteeing that the Bayesian predictive does not underfit. We suggest a matrix-free algorithm for projecting onto this null space, which scales linearly with the number of parameters and quadratically with the number of output dimensions. We further propose an approximation that only scales linearly with parameters to make the method applicable to generative models. An extensive empirical evaluation shows that the approach scales to large models, including vision transformers with 28 million parameters. Code is available at: \url{https://github.com/h-roy/projected-bayes}

Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches ignore the underlying geometric constraints or fail to provide meaningful metrics in the latent space. To address these limitations, we propose to learn Riemannian latent representations of such geometric data. To do so, we estimate the pullback metric induced by a Wrapped Gaussian Process Latent Variable Model, which explicitly accounts for the data geometry. This enables us to define geometry-aware notions of distance and shortest paths in the latent space, while ensuring that our model only assigns probability mass to the data manifold. This generalizes previous work and allows us to handle complex tasks in various domains, including robot motion synthesis and analysis of brain connectomes.

Deep latent variable models learn condensed representations of data that, hopefully, reflect the inner workings of the studied phenomena. Unfortunately, these latent representations are not statistically identifiable, meaning they cannot be uniquely determined. Domain experts, therefore, need to tread carefully when interpreting these. Current solutions limit the lack of identifiability through additional constraints on the latent variable model, e.g. by requiring labeled training data, or by restricting the expressivity of the model. We change the goal: instead of identifying the latent variables, we identify relationships between them such as meaningful distances, angles, and volumes. We prove this is feasible under very mild model conditions and without additional labeled data. We empirically demonstrate that our theory results in more reliable latent distances, offering a principled path forward in extracting trustworthy conclusions from deep latent variable models.

Accurately modeling and predicting complex dynamical systems, particularly those involving force exchange and dissipation, is crucial for applications ranging from fluid dynamics to robotics, but presents significant challenges due to the intricate interplay of geometric constraints and energy transfer. This paper introduces Geometric Contact Flows (GFC), a novel framework leveraging Riemannian and Contact geometry as inductive biases to learn such systems. GCF constructs a latent contact Hamiltonian model encoding desirable properties like stability or energy conservation. An ensemble of contactomorphisms then adapts this model to the target dynamics while preserving these properties. This ensemble allows for uncertainty-aware geodesics that attract the system’s behavior toward the data support, enabling robust generalization and adaptation to unseen scenarios. Experiments on learning dynamics for physical systems and for controlling robots on interaction tasks demonstrate the effectiveness of our approach.

Variational mean field approximations tend to struggle with contemporary overparametrized deep neural networks. Where a Bayesian treatment is usually associated with high-quality predictions and uncertainties, the practical reality has been the opposite, with unstable training, poor predictive power, and subpar calibration. Building upon recent work on reparametrizations of neural networks, we propose a simple variational family that considers two independent linear subspaces of the parameter space. These represent functional changes inside and outside the support of training data. This allows us to build a fully-correlated approximate posterior reflecting the overparametrization that tunes easy-to-interpret hyperparameters. We develop scalable numerical routines that maximize the associated evidence lower bound (ELBO) and sample from the approximate posterior. Empirically, we observe state-of-the-art performance across tasks, models, and datasets compared to a wide array of baseline methods. Our results show that approximate Bayesian inference applied to deep neural networks is far from a lost cause when constructing inference mechanisms that reflect the geometry of reparametrizations.

Neural models learn representations of high-dimensional data on low-dimensional manifolds. Multiple factors, including stochasticities in the training process, model architectures, and additional inductive biases, may induce different representations, even when learning the same task on the same data. However, it has recently been shown that when a latent structure is shared between distinct latent spaces, relative distances between representations can be preserved, up to distortions. Building on this idea, we demonstrate that exploiting the differential-geometric structure of latent spaces of neural models, it is possible to capture *precisely* the transformations between representational spaces trained on similar data distributions. Specifically, we assume that distinct neural models parametrize approximately the same underlying manifold, and introduce a representation based on the *pullback metric* that captures the intrinsic structure of the latent space, while scaling efficiently to large models. We validate experimentally our method on model stitching and retrieval tasks, covering autoencoders and vision foundation discriminative models, across diverse architectures, datasets, pretraining schemes and modalities. Code is available at the following [link](https://github.com/marc0git/RelativeGeodesics).

Stability guarantees are crucial when ensuring that a fully autonomous robot does not take undesirable or potentially harmful actions. Unfortunately, global stability guarantees are hard to provide in dynamical systems learned from data, especially when the learned dynamics are governed by neural networks. We propose a novel methodology to learn \emph{neural contractive dynamical systems}, where our neural architecture ensures contraction, and hence, global stability. To efficiently scale the method to high-dimensional dynamical systems, we develop a variant of the variational autoencoder that learns dynamics in a low-dimensional latent representation space while retaining contractive stability after decoding. We further extend our approach to learning contractive systems on the Lie group of rotations to account for full-pose end-effector dynamic motions. The result is the first highly flexible learning architecture that provides contractive stability guarantees with capability to perform obstacle avoidance. Empirically, we demonstrate that our approach encodes the desired dynamics more accurately than the current state-of-the-art, which provides less strong stability guarantees.

Generative models have shown strong generation ability while efficient likelihood estimation is less explored. Energy-based models (EBMs) define a flexible energy function to parameterize unnormalized densities efficiently but are notorious for being difficult to train. Adversarial EBMs introduce a generator to form a minimax training game to avoid expensive MCMC sampling used in traditional EBMs, but a noticeable gap between adversarial EBMs and other strong generative models still exists. Inspired by diffusion-based models, we embedded EBMs into each denoising step to split a long-generated process into several smaller steps. Besides, we employ a symmetric Jeffrey divergence and introduce a variational posterior distribution for the generator's training to address the main challenges that exist in adversarial EBMs. Our experiments show significant improvement in generation compared to existing adversarial EBMs, while also providing a useful energy function for efficient density estimation.

Bessel functions are critical in scientific computing for applications such as machine learning, protein structure modeling, and robotics. However, currently, available routines lack precision or fail for certain input ranges, such as when the order v is large, and GPU-specific implementations are limited. We address the precision limitations of current numerical implementations while dramatically improving the runtime. We propose two novel algorithms for computing the logarithm of modified Bessel functions of the first and second kinds by computing intermediate values on a logarithmic scale. Our algorithms are robust and never have issues with underflows or overflows while having relative errors on the order of machine precision, even for inputs where existing libraries fail. In C++/CUDA, our algorithms have median and maximum speedups of 45x and 6150x for GPU and 17x and 3403x for CPU, respectively, over the ranges of inputs and third-party libraries tested. Compared to SciPy, the algorithms have median and maximum speedups of 77x and 300x for GPU and 35x and 98x for CPU, respectively, over the ranges of inputs tested.

Optimizing discrete black-box functions is key in several domains, e.g. protein engineering and drug design. Due to the lack of gradient information and the need for sample efficiency, Bayesian optimization is an ideal candidate for these tasks. Several methods for high-dimensional continuous and categorical Bayesian optimization have been proposed recently. However, our survey of the field reveals highly heterogeneous experimental set-ups across methods and technical barriers for the replicability and application of published algorithms to real-world tasks. To address these issues, we develop a unified framework to test a vast array of high-dimensional Bayesian optimization methods and a collection of standardized black-box functions representing real-world application domains in chemistry and biology. These two components of the benchmark are each supported by flexible, scalable, and easily extendable software libraries (poli and poli-baselines), allowing practitioners to readily incorporate new optimization objectives or discrete optimizers. Project website: https://machinelearninglifescience.github.io/hdbo_benchmark.

Tuning scientific and probabilistic machine learning models - for example, partial differential equations, Gaussian processes, or Bayesian neural networks - often relies on evaluating functions of matrices whose size grows with the data set or the number of parameters. While the state-of-the-art for _evaluating_ these quantities is almost always based on Lanczos and Arnoldi iterations, the present work is the first to explain how to _differentiate_ these workhorses of numerical linear algebra efficiently. To get there, we derive previously unknown adjoint systems for Lanczos and Arnoldi iterations, implement them in JAX, and show that the resulting code can compete with Diffrax when it comes to differentiating PDEs, GPyTorch for selecting Gaussian process models and beats standard factorisation methods for calibrating Bayesian neural networks. All this is achieved without any problem-specific code optimisation. Find the code at [link redacted] and install the library with *pip install [redacted]*.

Current uncertainty quantification is memory and compute expensive, which hinders practical uptake. To counter, we develop Sketched Lanczos Uncertainty (SLU): an architecture-agnostic uncertainty score that can be applied to pre-trained neural networks with minimal overhead. Importantly, the memory use of SLU only grows logarithmically with the number of model parameters. We combine Lanczos' algorithm with dimensionality reduction techniques to compute a sketch of the leading eigenvectors of a matrix. Applying this novel algorithm to the Fisher information matrix yields a cheap and reliable uncertainty score. Empirically, SLU yields well-calibrated uncertainties, reliably detects out-of-distribution examples, and consistently outperforms existing methods in the low-memory regime.

Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a crucial limitation: they fail to maintain invariance under reparameterization, i.e. BNNs assign different posterior densities to different parametrizations of identical functions. This creates a fundamental flaw in the application of Bayesian principles as it breaks the correspondence between uncertainty over the parameters with uncertainty over the parametrized function. In this paper, we investigate this issue in the context of the increasingly popular linearized Laplace approximation. Specifically, it has been observed that linearized predictives alleviate the common underfitting problems of the Laplace approximation. We develop a new geometric view of reparametrizations from which we explain the success of linearization. Moreover, we demonstrate that these reparameterization invariance properties can be extended to the original neural network predictive using a Riemannian diffusion process giving a straightforward algorithm for approximate posterior sampling, which empirically improves posterior fit.

We present a method to approximate Gaussian process regression models to large datasets by considering only a subset of the data. Our approach is novel in that the size of the subset is selected on the fly during exact inference with little computational overhead. From an empirical observation that the log-marginal likelihood often exhibits a linear trend once a sufficient subset of a dataset has been observed, we conclude that many large datasets contain redundant information that only slightly affects the posterior. Based on this, we provide probabilistic bounds on the full model evidence that can identify such subsets. Remarkably, these bounds are largely composed of terms that appear in intermediate steps of the standard Cholesky decomposition, allowing us to modify the algorithm to adaptively stop the decomposition once enough data have been observed.

We present WineSensed, a large multimodal wine dataset for studying the relations between visual perception, language, and flavor. The dataset encompasses 897k images of wine labels and 824k reviews of wines curated from the Vivino platform. It has over 350k unique vintages, annotated with year, region, rating, alcohol percentage, price, and grape composition. We obtained fine-grained flavor annotations on a subset by conducting a wine-tasting experiment with 256 participants who were asked to rank wines based on their similarity in flavor, resulting in more than 5k pairwise flavor distances. We propose a low-dimensional concept embedding algorithm that combines human experience with automatic machine similarity kernels. We demonstrate that this shared concept embedding space improves upon separate embedding spaces for coarse flavor classification (alcohol percentage, country, grape, price, rating) and representing human perception of flavor.

Bayesian neural networks often approximate the weight-posterior with a Gaussian distribution. However, practical posteriors are often, even locally, highly non-Gaussian, and empirical performance deteriorates. We propose a simple parametric approximate posterior that adapts to the shape of the true posterior through a Riemannian metric that is determined by the log-posterior gradient. We develop a Riemannian Laplace approximation where samples naturally fall into weight-regions with low negative log-posterior. We show that these samples can be drawn by solving a system of ordinary differential equations, which can be done efficiently by leveraging the structure of the Riemannian metric and automatic differentiation. Empirically, we demonstrate that our approach consistently improves over the conventional Laplace approximation across tasks. We further show that, unlike the conventional Laplace approximation, our method is not overly sensitive to the choice of prior, which alleviates a practical pitfall of current approaches.

Masked pre-training removes random input dimensions and learns a model that can predict the missing values. Empirical results indicate that this intuitive form of self-supervised learning yields models that generalize very well to new domains. A theoretical understanding is, however, lacking. This paper shows that masked pre-training with a suitable cumulative scoring function corresponds to maximizing the model's marginal likelihood, which is de facto the Bayesian model selection measure of generalization. Beyond shedding light on the success of masked pre-training, this insight also suggests that Bayesian models can be trained with appropriately designed self-supervision. Empirically, we confirm the developed theory and explore the main learning principles of masked pre-training in large language models.

We propose a Bayesian encoder for metric learning. Rather than relying on neural amortization as done in prior works, we learn a distribution over the network weights with the Laplace Approximation. We first prove that the contrastive loss is a negative log-likelihood on the spherical space. We propose three methods that ensure a positive definite covariance matrix. Lastly, we present a novel decomposition of the Generalized Gauss-Newton approximation. Empirically, we show that our Laplacian Metric Learner (LAM) yields well-calibrated uncertainties, reliably detects out-of-distribution examples, and has state-of-the-art predictive performance.

Latent space geometry has shown itself to provide a rich and rigorous framework for interacting with the latent variables of deep generative models. The existing theory, however, relies on the decoder being a Gaussian distribution as its simple reparametrization allows us to interpret the generating process as a random projection of a deterministic manifold. Consequently, this approach breaks down when applied to decoders that are not as easily reparametrized. We here propose to use the Fisher-Rao metric associated with the space of decoder distributions as a reference metric, which we pull back to the latent space. We show that we can achieve meaningful latent geometries for a wide range of decoder distributions for which the previous theory was not applicable, opening the door to ’black box’ latent geometries.

We present simple methods for out-of-distribution detection using a trained generative model. These techniques, based on classical statistical tests, are model-agnostic in the sense that they can be applied to any differentiable generative model. The idea is to combine a classical parametric test (Rao’s score test) with the recently introduced typicality test. These two test statistics are both theoretically well-founded and exploit different sources of information based on the likelihood for the typicality test and its gradient for the score test. We show that combining them using Fisher’s method overall leads to a more accurate out-of-distribution test. We also discuss the benefits of casting out-of-distribution detection as a statistical testing problem, noting in particular that false positive rate control can be valuable for practical out-of-distribution detection. Despite their simplicity and generality, these methods can be competitive with model-specific out-of-distribution detection algorithms without any assumptions on the out-distribution.

Deep generative models can automatically create content of diverse types. However, there are no guarantees that such content will satisfy the criteria necessary to present it to end-users and be functional, e.g. the generated levels could be unsolvable or incoherent. In this paper we study this problem from a geometric perspective, and provide a method for reliable interpolation and random walks in the latent spaces of Categorical VAEs based on Riemannian geometry. We test our method with “Super Mario Bros” and “The Legend of Zelda” levels, and against simpler baselines inspired by current practice. Results show that the geometry we propose is better able to interpolate and sample, reliably staying closer to parts of the latent space that decode to playable content.

Established methods for unsupervised representation learning such as variational autoencoders produce none or poorly calibrated uncertainty estimates making it difficult to evaluate if learned representations are stable and reliable. In this work, we present a Bayesian autoencoder for unsupervised representation learning, which is trained using a novel variational lower-bound of the autoencoder evidence. This is maximized using Monte Carlo EM with a variational distribution that takes the shape of a Laplace approximation. We develop a new Hessian approximation that scales linearly with data size allowing us to model high-dimensional data. Empirically, we show that our Laplacian autoencoder estimates well-calibrated uncertainties in both latent and output space. We demonstrate that this results in improved performance across a multitude of downstream tasks.

Decoders built on Gaussian processes (GPs) are enticing due to the marginalisation over the non-linear function space. Such models (also known as GP-LVMs) are often expensive and notoriously difficult to train in practice, but can be scaled using variational inference and inducing points. In this paper, we revisit active set approximations. We develop a new stochastic estimate of the log-marginal likelihood based on recently discovered links to cross-validation, and we propose a computationally efficient approximation thereof. We demonstrate that the resulting stochastic active sets (SAS) approximation significantly improves the robustness of GP decoder training, while reducing computational cost. The SAS-GP obtains more structure in the latent space, scales to many datapoints, and learns better representations than variational autoencoders, which is rarely the case for GP decoders.

Spatial Transformer Networks (STNs) estimate image transformations that can improve downstream tasks by ‘zooming in’ on relevant regions in an image. However, STNs are hard to train and sensitive to mis-predictions of transformations. To circumvent these limitations, we propose a probabilistic extension that estimates a stochastic transformation rather than a deterministic one. Marginalizing transformations allows us to consider each image at multiple poses, which makes the localization task easier and the training more robust. As an additional benefit, the stochastic transformations act as a localized, learned data augmentation that improves the downstream tasks. We show across standard imaging benchmarks and on a challenging real-world dataset that these two properties lead to improved classification performance, robustness and model calibration. We further demonstrate that the approach generalizes to non-visual domains by improving model performance on time-series data.

A common assumption in generative models is that the generator immerses the latent space into a Euclidean ambient space. Instead, we consider the ambient space to be a Riemannian manifold, which allows for encoding domain knowledge through the associated Riemannian metric. Shortest paths can then be defined accordingly in the latent space to both follow the learned manifold and respect the ambient geometry. Through careful design of the ambient metric we can ensure that shortest paths are well-behaved even for deterministic generators that otherwise would exhibit a misleading bias. Experimentally we show that our approach improves the interpretability and the functionality of learned representations both using stochastic and deterministic generators.

Uncertainty quantification in image retrieval is crucial for downstream decisions, yet it remains a challenging and largely unexplored problem. Current methods for estimating uncertainties are poorly calibrated, computationally expensive, or based on heuristics. We present a new method that views image embeddings as stochastic features rather than deterministic features. Our two main contributions are (1) a likelihood that matches the triplet constraint and that evaluates the probability of an anchor being closer to a positive than a negative; and (2) a prior over the feature space that justifies the conventional l2normalization. To ensure computational efficiency, we derive a variational approximation of the posterior, called the Bayesian triplet loss, that produces state-of-the-art uncertainty estimates and matches the predictive performance of current state-of-the-art methods.

Deep generative models have been demonstrated as state-of-the-art density estimators. Yet, recent work has found that they often assign a higher likelihood to data from outside the training distribution. This seemingly paradoxical behavior has caused concerns over the quality of the attained density estimates. In the context of hierarchical variational autoencoders, we provide evidence to explain this behavior by out-of-distribution data having in-distribution low-level features. We argue that this is both expected and desirable behavior. With this insight in hand, we develop a fast, scalable and fully unsupervised likelihood-ratio score for OOD detection that requires data to be in-distribution across all feature-levels. We benchmark the method on a vast set of data and model combinations and achieve state-of-the-art results on out-of-distribution detection.

We present a probabilistic model where the latent variable respects both the distances and the topology of the modeled data. The model leverages the Riemannian geometry of the generated manifold to endow the latent space with a well-defined stochastic distance measure, which is modeled locally as Nakagami distributions. These stochastic distances are sought to be as similar as possible to observed distances along a neighborhood graph through a censoring process. The model is inferred by variational inference based on observations of pairwise distances. We demonstrate how the new model can encode invariances in the learned manifolds.

Energy-based models (EBMs) provide an elegant framework for density estimation, but they are notoriously difficult to train. Recent work has established links to generative adversarial networks, where the EBM is trained through a minimax game with a variational value function. We propose a bidirectional bound on the EBM log-likelihood, such that we maximize a lower bound and minimize an upper bound when solving the minimax game. We link one bound to a gradient penalty that stabilizes training, thereby provide grounding for best engineering practice. To evaluate the bounds we develop a new and efficient estimator of the Jacobi-determinant of the EBM generator. We demonstrate that these developments stabilize training and yield high-quality density estimation and sample generation.

Principal component analysis (PCA) and Kernel principal component analysis (KPCA) are fundamental methods in machine learning for dimensionality reduction. The former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional reproducing Kernel Hilbert-space (RKHS). In this paper, we present a geometric framework for computing the principal linear subspaces in both (finite and infinite) situations as well as for the robust PCA case, that amounts to computing the intrinsic average on the space of all subspaces: the Grassmann manifold. Points on this manifold are defined as the subspaces spanned by K-tuples of observations. The intrinsic Grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a Gaussian distribution. We show similar results in the RKHS case and provide an efficient algorithm for computing the projection onto the this average subspace. The result is a method akin to KPCA which is substantially faster. Further, we present a novel online version of the KPCA using our geometric framework. Competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets.

Lifelong place recognition is an essential and challenging task in computer vision with vast applications in robust localization and efficient large-scale 3D reconstruction. Progress is currently hindered by a lack of large, diverse, publicly available datasets. We contribute with Mapillary Street-Level Sequences (SLS), a large dataset for urban and suburban place recognition from image sequences. It contains more than 1.6 million images curated from the Mapillary collaborative mapping platform. The dataset is orders of magnitude larger than current data sources, and is designed to reflect the diversities of true lifelong learning. It features images from 30 major cities across six continents, hundreds of distinct cameras, and substantially different viewpoints and capture times, spanning all seasons over a nine year period. All images are geo-located with GPS and compass, and feature high-level attributes such as road type. We propose a set of benchmark tasks designed to push state-of-the-art performance and provide baseline studies. We show that current state-of-the-art methods still have a long way to go, and that the lack of diversity in existing datasets have prevented generalization to new environments. The dataset and benchmarks are available for academic research.

Variational Autoencoders (VAEs) represent the given data in a low-dimensional latent space, which is generally assumed to be Euclidean. This assumption naturally leads to the common choice of a standard Gaussian prior over continuous latent variables. Recent work has, however, shown that this prior has a detrimental effect on model capacity, leading to subpar performance. We propose that the Euclidean assumption lies at the heart of this failure mode. To counter this, we assume a Riemannian structure over the latent space, which constitutes a more principled geometric view of the latent codes, and replace the standard Gaussian prior with a Riemannian Brownian motion prior. We propose an efficient inference scheme that does not rely on the unknown normalizing factor of this prior. Finally, we demonstrate that this prior significantly increases model capacity using only one additional scalar parameter.

We propose a fast, simple and robust algorithm for computing shortest paths and distances on Riemannian manifolds learned from data. This amounts to solving a system of ordinary differential equations (ODEs) subject to boundary conditions. Here standard solvers perform poorly because they require well-behaved Jacobians of the ODE, and usually, manifolds learned from data imply unstable and ill-conditioned Jacobians. Instead, we propose a fixed-point iteration scheme for solving the ODE that avoids Jacobians. This enhances the stability of the solver, while reduces the computational cost. In experiments involving both Riemannian metric learning and deep generative models we demonstrate significant improvements in speed and stability over both general-purpose state-of-the-art solvers as well as over specialized solvers.

Latent variable models (LVMs) learn probabilistic models of data manifolds lying in an ambient Euclidean space. In a number of applications, a priori known spatial constraints can shrink the ambient space into a considerably smaller manifold. Additionally, in these applications the Euclidean geometry might induce a suboptimal similarity measure, which could be improved by choosing a different metric. Euclidean models ignore such information and assign probability mass to data points that can never appear as data, and vastly different likelihoods to points that are similar under the desired metric.We propose the wrapped Gaussian process latent variable model (WGPLVM), that extends Gaussian process latent variable models to take values strictly on a given Riemannian manifold, making the model blind to impossible data points. This allows non-linear, probabilistic inference of low-dimensional Riemannian submanifolds from data. Our evaluation on diverse datasets show that we improve performance on several tasks, including encoding, visualization and uncertainty quantification.

Disentangled representation learning finds compact, independent and easy-to-interpret factors of the data. Learning such has been shown to require an inductive bias, which we explicitly encode in a generative model of images. Specifically, we propose a model with two latent spaces: one that represents spatial transformations of the input data, and another that represents the transformed data. We find that the latter naturally captures the intrinsic appearance of the data. To realize the generative model, we propose a Variationally Inferred Transformational Autoencoder (VITAE) that incorporates a spatial ransformer into a variational autoencoder. We show how to perform inference in the model efficiently by carefully designing the encoders and restricting the transformation class to be diffeomorphic. Empirically, our model separates the visual style from digit type on MNIST, separates shape and pose in images of human bodies and facial features from facial shape on CelebA.

We propose and investigate new complementary methodologies for estimating predictive variance networks in regression neural networks. We derive a locally aware mini-batching scheme that results in sparse robust gradients, and we show how to make unbiased weight updates to a variance network. Further, we formulate a heuristic for robustly fitting both the mean and variance networks post hoc. Finally, we take inspiration from posterior Gaussian processes and propose a network architecture with similar extrapolation properties to Gaussian processes. The proposed methodologies are complementary, and improve upon baseline methods individually. Experimentally, we investigate the impact of predictive uncertainty on multiple datasets and tasks ranging from regression, active learning and generative modeling. Experiments consistently show significant improvements in predictive uncertainty estimation over state-of-the-art methods across tasks and datasets.

Spatial Transformer layers allow neural networks, at least in principle, to be invariant to large spatial transformations in image data. The model has, however, seen limited uptake as most practical implementations support only transformations that are too restricted, e.g. affine or homographic maps, and/or destructive maps, such as thin plate splines. We investigate the use of flexible diffeomorphic image transformations within such networks and demonstrate that significant performance gains can be attained over currently-used models. The learned transformations are found to be both simple and intuitive, thereby providing insights into individual problem domains. With the proposed framework, a standard convolutional neural network matches state-of-the-art results on face verification with only two extra lines of simple TensorFlow code.

A well-known problem in directional statistics - the study of data distributed on the unit sphere - is that current models disregard the curvature of the underlying sample space. This ensures computationally efficiency, but can influence results. To investigate this, we develop efficient inference techniques for data distributed by the curvature-aware spherical normal distribution. We derive closed-form expressions for the normalization constant when the distribution is isotropic, and a fast and accurate approximation for the anisotropic case on the two-sphere. We further develop approximate posterior inference techniques for the mean and concentration of the distribution, and propose a fast sampling algorithm for simulating the distribution. Combined, this provides the tools needed for practical inference on the unit sphere in a manner that respects the curvature of the underlying sample space.

Principal Component Analysis (PCA) is a fundamental method for estimating a linear subspace approximation to high-dimensional data. Many algorithms exist in literature to achieve a statistically robust version of PCA called RPCA. In this paper, we present a geometric framework for computing the principal linear subspaces in both situations that amounts to computing the intrinsic average on the space of all subspaces (the Grassmann manifold). Points on this manifold are defined as the subspaces spanned by K-tuples of observations. We show that the intrinsic Grassmann average of these subspaces coincide with the principal components of the observations when they are drawn from a Gaussian distribution. Similar results are also shown to hold for the RPCA. Further, we propose an efficient online algorithm to do subspace averaging which is of linear complexity in terms of number of samples and has a linear convergence rate. When the data has outliers, our proposed online robust subspace averaging algorithm shows significant performance (accuracy and computation time) gain over a recently published RPCA methods with publicly accessible code. We have demonstrated competitive performance of our proposed online subspace algorithm method on one synthetic and two real data sets. Experimental results depicting stability of our proposed method are also presented. Furthermore, on two real outlier corrupted datasets, we present comparison experiments showing lower reconstruction error using our online RPCA algorithm. In terms of reconstruction error and time required, both our algorithms outperform the competition.

We propose novel finite-dimensional spaces of well-behaved transformations. The latter are obtained by (fast and highly-accurate) integration of continuous piecewise-affine velocity fields. The proposed method is simple yet highly expressive, effortlessly handles optional constraints (e.g., volume preservation and/or boundary conditions), and supports convenient modeling choices such as smoothing priors and coarse-to-fine analysis. Importantly, the proposed approach, partly due to its rapid likelihood evaluations and partly due to its other properties, facilitates tractable inference over rich transformation spaces, including using Markov-Chain Monte-Carlo methods. Its applications include, but are not limited to: monotonic regression (more generally, optimization over monotonic functions); modeling cumulative distribution functions or histograms; time-warping; image warping; image registration; real-time diffeomorphic image editing; data augmentation for image classifiers. Our GPU-based code is publicly available.

Data augmentation is a key element in training high-dimensional models. In this approach, one synthesizes new observations by applying pre-specified transformations to the original training data; e.g. new images are formed by rotating old ones. Current augmentation schemes, however, rely on manual specification of the applied transformations, making data augmentation an implicit form of feature engineering. With an eye towards true end-to-end learning, we suggest learning the applied transformations on a per-class basis. Particularly, we align image pairs within each class under the assumption that the spatial transformation between images belongs to a large class of diffeomorphisms. We then learn a class-specific probabilistic generative models of the transformations in a Riemannian submanifold of the Lie group of diffeomorphisms. We demonstrate significant performance improvements in training deep neural nets over manually-specified augmentation schemes. Our code and augmented datasets are available online.

Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geodesic metric spaces after all. Here, however, we present evidence that large intervals of bandwidths exist where geodesic exponential kernels have high probability of being positive definite over finite datasets, while still having significant predictive power. From this we formulate conjectures on the probability of a positive definite kernel matrix for a finite random sample, depending on the geometry of the data space and the spread of the sample.

The multivariate normal density is a monotonic function of the distance to the mean, and its ellipsoidal shape is due to the underlying Euclidean metric. We suggest to replace this metric with a locally adaptive, smoothly changing (Riemannian) metric that favors regions of high local density. The resulting locally adaptive normal distribution (LAND) is a generalization of the normal distribution to the "manifold" setting, where data is assumed to lie near a potentially low-dimensional manifold embedded in R^D. The LAND is parametric, depending only on a mean and a covariance, and is the maximum entropy distribution under the given metric. The underlying metric is, however, non-parametric. We develop a maximum likelihood algorithm to infer the distribution parameters that relies on a combination of gradient descent and Monte Carlo integration. We further extend the LAND to mixture models, and provide the corresponding EM algorithm. We demonstrate the efficiency of the LAND to fit non-trivial probability distributions over both synthetic data, and EEG measurements of human sleep.

Euclidean statistics are often generalized to Riemannian manifolds by replacing straight-line interpolations with geodesic ones. While these Riemannian models are familiar-looking, they are restricted by the inflexibility of geodesics, and they rely on constructions which are optimal only in Euclidean domains. We consider extensions of Principal Component Analysis (PCA) to Riemannian manifolds. Classic Riemannian approaches seek a geodesic curve passing through the mean that optimizes a criteria of interest. The requirements that the solution both is geodesic and must pass through the mean tend to imply that the methods only work well when the manifold is mostly flat within the support of the generating distribution. We argue that instead of generalizing linear Euclidean models, it is more fruitful to generalize non-linear Euclidean models. Specifically, we extend the classic Principal Curves from Hastie & Stuetzle to data residing on a complete Riemannian manifold. We show that for elliptical distributions in the tangent of spaces of constant curvature, the standard principal geodesic is a principal curve. The proposed model is simple to compute and avoids many of the pitfalls of traditional geodesic approaches. We empirically demonstrate the effectiveness of the Riemannian principal curves on several manifolds and datasets.

In large datasets, manual data verification is impossible, and we must expect the number of outliers to increase with data size. While principal component analysis (PCA) can reduce data size, and scalable solutions exist, it is well-known that outliers can arbitrarily corrupt the results. Unfortunately, state-of-the-art approaches for robust PCA are not scalable. We note that in a zero-mean dataset, each observation spans a one-dimensional subspace, giving a point on the Grassmann manifold. We show that the average subspace corresponds to the leading principal component for Gaussian data. We provide a simple algorithm for computing this Grassmann Average ( GA), and show that the subspace estimate is less sensitive to outliers than PCA for general distributions. Because averages can be efficiently computed, we immediately gain scalability. We exploit robust averaging to formulate the Robust Grassmann Average (RGA) as a form of robust PCA. The resulting Trimmed Grassmann Average ( TGA) is appropriate for computer vision because it is robust to pixel outliers. The algorithm has linear computational complexity and minimal memory requirements. We demonstrate TGA for background modeling, video restoration, and shadow removal. We show scalability by performing robust PCA on the entire Star Wars IV movie; a task beyond any current method. Source code is available online.

We consider kernel methods on general geodesic metric spaces and provide both negative and positive results. First we show that the common Gaussian kernel can only be generalized to a positive definite kernel on a geodesic metric space if the space is flat. As a result, for data on a Riemannian manifold, the geodesic Gaussian kernel is only positive definite if the Riemannian manifold is Euclidean. This implies that any attempt to design geodesic Gaussian kernels on curved Riemannian manifolds is futile. However, we show that for spaces with conditionally negative definite distances the geodesic Laplacian kernel can be generalized while retaining positive definiteness. This implies that geodesic Laplacian kernels can be generalized to some curved spaces, including spheres and hyperbolic spaces. Our theoretical results are verified empirically.

We propose novel finite-dimensional spaces of Rn→ Rntransformations, n ∈ {1, 2, 3}, derived from (continuously-defined) parametric stationary velocity fields. Particularly, we obtain these transformations, which are diffeomorphisms, by fast and highly-accurate integration of continuous piecewise-affine velocity fields, we also provide an exact solution for n = 1. The simple-yet-highly-expressive proposed representation handles optional constraints (e.g., volume preservation) easily and supports convenient modeling choices and rapid likelihood evaluations (facilitating tractable inference over latent transformations). Its applications include, but are not limited to: unconstrained optimization over monotonic functions, modeling cumulative distribution functions or histograms, time warping, image registration, landmark-based warping, real-time diffeomorphic image editing. Our code is available at https://github.com/freifeld/cpabDiffeo.

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtually all computations. The probabilistic formulation permits marginalising the uncertainty of the numerical solution such that statistics are less sensitive to inaccuracies. This leads to new Riemannian algorithms for mean value computations and principal geodesic analysis. Marginalisation also means results can be less precise than point estimates, enabling a noticeable speed-up over the state of the art. Our approach is an argument for a wider point that uncertainty caused by numerical calculations should be tracked throughout the pipeline of machine learning algorithms.

We consider the intersection of two research fields: transfer learning and statistics on manifolds. In particular, we consider, for manifold-valued data, transfer learning of tangent-space models such as Gaussians distributions, PCA, regression, or classifiers. Though one would hope to simply use ordinary Rn-transfer learning ideas, the manifold structure prevents it. We overcome this by basing our method on inner-product-preserving parallel transport, a well-known tool widely used in other problems of statistics on manifolds in computer vision. At first, this straight-forward idea seems to suffer from an obvious shortcoming: Transporting large datasets is prohibitively expensive, hindering scalability. Fortunately, with our approach, we never transport data. Rather, we show how the statistical models themselves can be transported, and prove that for the tangent-space models above, the transport "commutes" with learning. Consequently, our compact framework, ap- plicable to a large class of manifolds, is not restricted by the size of either the training or test sets. We demonstrate the approach by transferring PCA and logistic-regression models of real-world data involving 3D shapes and image descriptors.

As the collection of large datasets becomes increasingly automated, the occurrence of outliers will increase -- "big data" implies "big outliers". While principal component analysis (PCA) is often used to reduce the size of data, and scalable solutions exist, it is well-known that outliers can arbitrarily corrupt the results. Unfortunately, state-of-the-art approaches for robust PCA do not scale beyond small-to-medium sized datasets. To address this, we introduce the Grassmann Average (GA), which expresses dimensionality reduction as an average of the subspaces spanned by the data. Because averages can be efficiently computed, we immediately gain scalability. GA is inherently more robust than PCA, but we show that they coincide for Gaussian data. We exploit that averages can be made robust to formulate the Robust Grassmann Average (RGA) as a form of robust PCA. Robustness can be with respect to vectors (subspaces) or elements of vectors, we focus on the latter and use a trimmed average. The resulting Trimmed Grassmann Average (TGA) is particularly appropriate for computer vision because it is robust to pixel outliers. The algorithm has low computational complexity and minimal memory requirements, making it scalable to "big noisy data." We demonstrate TGA for background modeling, video restoration, and shadow removal. We show scalability by performing robust PCA on the entire Star Wars IV movie.

Multi-metric learning techniques learn local metric tensors in different parts of a feature space. With such an approach, even simple classifiers can be competitive with the state-of-the-art because the distance measure locally adapts to the structure of the data. The learned distance measure is, however, non-metric, which has prevented multi-metric learning from generalizing to tasks such as dimensionality reduction and regression in a principled way. We prove that, with appropriate changes, multi-metric learning corresponds to learning the structure of a Riemannian manifold. We then show that this structure gives us a principled way to perform dimensionality reduction and regression according to the learned metrics. Algorithmically, we provide the first practical algorithm for computing geodesics according to the learned metrics, as well as algorithms for computing exponential and logarithmic maps on the Riemannian manifold. Together, these tools let many Euclidean algorithms take advantage of multi-metric learning. We illustrate the approach on regression and dimensionality reduction tasks that involve predicting measurements of the human body from shape data.

The mean is often the most important statistic of a dataset as it provides a single point that summarizes the entire set. While the mean is readily defined and computed in Euclidean spaces, no commonly accepted solutions are currently available in more complicated spaces, such as spaces of tree-structured data. In this paper we study the notion of means, both generally in Gromov's CAT(0)-spaces (metric spaces of non-positive curvature), but also specifically in the space of tree-like shapes. We prove local existence and uniqueness of means in such spaces and discuss three different algorithms for computing means. We make an experimental evaluation of the three algorithms through experiments on three different sets of data with tree-like structure: a synthetic dataset, a leaf morphology dataset from images, and a set of human airway subtrees from medical CT scans. This experimental study provides great insight into the behavior of the different methods and how they relate to each other. More importantly, it also provides mathematically well-founded, tractable and robust “average trees”. This statistic is of utmost importance due to the ever-presence of tree-like structures in human anatomy, e.g., airways and vascularization systems.