Geodesic Slice Sampler for Multimodal Distributions with Strong Curvature

Traditional Markov Chain Monte Carlo sampling methods often struggle with sharp curvatures, intricate geometries, and multimodal distributions. Slice sampling can resolve local exploration inefficiency issues and Riemannian geometries help with sharp curvatures. Recent extensions enable slice sampling on Riemannian manifolds, but they are restricted to cases where geodesics are available in closed form. We propose a method that generalizes Hit-and-Run slice sampling to more general geometries tailored to the target distribution, by approximating geodesics as solutions to differential equations. Our approach enables exploration of regions with strong curvature and rapid transitions between modes in multimodal distributions. We demonstrate the advantages of the approach over challenging sampling problems.

Density Ratio Estimation with Conditional Probability Paths

Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.

Preferential Normalizing Flows

Eliciting a high-dimensional probability distribution from an expert via noisy judgments is notoriously challenging, yet useful for many applications, such as prior elicitation and reward modeling. We introduce a method for eliciting the expert's belief density as a normalizing flow based solely on preferential questions such as comparing or ranking alternatives. This allows eliciting in principle arbitrarily flexible densities, but flow estimation is susceptible to the challenge of collapsing or diverging probability mass that makes it difficult in practice. We tackle this problem by introducing a novel functional prior for the flow, motivated by a decision-theoretic argument, and show empirically that the belief density can be inferred as the function-space maximum a posteriori estimate. We demonstrate our method by eliciting multivariate belief densities of simulated experts, including the prior belief of a general-purpose large language model over a real-world dataset.

Stochastic variance-reduced Gaussian variational inference on the Bures-Wasserstein manifold

Optimization in the Bures-Wasserstein space has been gaining popularity in the machine learning community since it draws connections between variational inference and Wasserstein gradient flows. The variational inference objective function of Kullback-Leibler divergence can be written as the sum of the negative entropy and the potential energy, making forward-backward Euler the method of choice. Notably, the backward step admits a closed-form solution in this case, facilitating the practicality of the scheme. However, the forward step is no longer exact since the Bures-Wasserstein gradient of the potential energy involves "intractable" expectations. Recent approaches propose using the Monte Carlo method -- in practice a single-sample estimator -- to approximate these terms, resulting in high variance and poor performance. We propose a novel variance-reduced estimator based on the principle of control variates. We theoretically show that this estimator has a smaller variance than the Monte-Carlo estimator in scenarios of interest. We also prove that variance reduction helps improve the optimization bounds of the current analysis. We demonstrate that the proposed estimator gains order-of-magnitude improvements over the previous Bures-Wasserstein methods.

Non-geodesically-convex optimization in the Wasserstein space

We study a class of optimization problems in the Wasserstein space (the space of probability measures) where the objective function is \emph{nonconvex} along generalized geodesics. When the regularization term is the negative entropy, the optimization problem becomes a sampling problem where it minimizes the Kullback-Leibler divergence between a probability measure (optimization variable) and a target probability measure whose logarithmic probability density is a nonconvex function. We derive multiple convergence insights for a novel {\em semi Forward-Backward Euler scheme} under several nonconvex (and possibly nonsmooth) regimes. Notably, the semi Forward-Backward Euler is just a slight modification of the Forward-Backward Euler whose convergence is -- to our knowledge -- still unknown in our very general non-geodesically-convex setting.

Riemannian Laplace Approximation with the Fisher Metric

The Laplace's method approximates a target density with a Gaussian distribution at its mode. It is computationally efficient and asymptotically exact for Bayesian inference due to the Bernstein-von Mises theorem, but for complex targets and finite-data posteriors it is often too crude an approximation. A recent generalization of the Laplace Approximation transforms the Gaussian approximation according to a chosen Riemannian geometry providing a richer approximation family, while still retaining computational efficiency. However, as shown here, its properties heavily depend on the chosen metric, indeed the metric adopted in previous work results in approximations that are overly narrow as well as being biased even at the limit of infinite data. We correct this shortcoming by developing the approximation family further, deriving two alternative variants that are exact at the limit of infinite data, extending the theoretical analysis of the method, and demonstrating practical improvements in a range of experiments.

Warped geometric information on the optimisation of Euclidean functions

We consider the fundamental task of optimizing a real-valued function defined in a potentially high-dimensional Euclidean space, such as the loss function in many machine-learning tasks or the logarithm of the probability distribution in statistical inference. We use the warped Riemannian geometry notions to redefine the optimisation problem of a function on Euclidean space to a Riemannian manifold with a warped metric, and then find the function's optimum along this manifold. The warped metric chosen for the search domain induces a computational friendly metric-tensor for which optimal search directions associate with geodesic curves on the manifold becomes easier to compute. Performing optimization along geodesics is known to be generally infeasible, yet we show that in this specific manifold we can analytically derive Taylor approximations up to third-order. In general these approximations to the geodesic curve will not lie on the manifold, however we construct suitable retraction maps to pull them back onto the manifold. Therefore, we can efficiently optimize along the approximate geodesic curves. We cover the related theory, describe a practical optimization algorithm and empirically evaluate it on a collection of challenging optimisation benchmarks. Our proposed algorithm, using third-order approximation of geodesics, outperforms standard Euclidean gradient-based counterparts in term of number of iterations until convergence and an alternative method for Hessian-based optimisation routines.

Scalable Stochastic Gradient Riemannian Langevin Dynamics in Non-Diagonal Metrics

Bayesian neural network inference is often carried out using stochastic gradient sampling methods. For best performance the methods should use a Riemannian metric that improves posterior exploration by accounting for the local curvature, but the existing methods resort to simple diagonal metrics to remain computationally efficient. This loses some of the gains. We propose two non-diagonal metrics that can be used in stochastic samplers to improve convergence and exploration but that have only a minor computational overhead over diagonal metrics. We show that for neural networks with complex posteriors, caused e.g. by use of sparsity-inducing priors, using these metrics provides clear improvements. For some other choices the posterior is sufficiently easy also for the simpler metrics.

Estimating the Contamination Factor's Distribution in Unsupervised Anomaly Detection

Anomaly detection methods identify examples that do not follow the expected behaviour, typically in an unsupervised fashion, by assigning real-valued anomaly scores to the examples based on various heuristics. These scores need to be transformed into actual predictions by thresholding, so that the proportion of examples marked as anomalies equals the expected proportion of anomalies, called contamination factor. Unfortunately, there are no good methods for estimating the contamination factor itself. We address this need from a Bayesian perspective, introducing a method for estimating the posterior distribution of the contamination factor of a given unlabeled dataset. We leverage on outputs of several anomaly detectors as a representation that already captures the basic notion of anomalousness and estimate the contamination using a specific mixture formulation. Empirically on 22 datasets, we show that the estimated distribution is well-calibrated and that setting the threshold using the posterior mean improves the anomaly detectors' performance over several alternative methods. All code is publicly available for full reproducibility.

Neural network for determining an asteroid mineral composition from reflectance spectra

Chemical and mineral compositions of asteroids reflect the formation and history of our Solar System. This knowledge is also important for planetary defence and in-space resource utilisation. We aim to develop a fast and robust neural-network-based method for deriving the mineral modal and chemical compositions of silicate materials from their visible and near-infrared spectra. The method should be able to process raw spectra without significant pre-processing. We designed a convolutional neural network with two hidden layers for the analysis of the spectra, and trained it using labelled reflectance spectra. For the training, we used a dataset that consisted of reflectance spectra of real silicate samples stored in the RELAB and C-Tape databases, namely olivine, orthopyroxene, clinopyroxene, their mixtures, and olivine-pyroxene-rich meteorites. We used the model on two datasets. First, we evaluated the model reliability on a test dataset where we compared the model classification with known compositional reference values. The individual classification results are mostly within 10 percentage-point intervals around the correct values. Second, we classified the reflectance spectra of S-complex (Q-type and V-type, also including A-type) asteroids with known Bus-DeMeo taxonomy classes. The predicted mineral chemical composition of S-type and Q-type asteroids agree with the chemical composition of ordinary chondrites. The modal abundances of V-type and A-type asteroids show a dominant contribution of orthopyroxene and olivine, respectively. Additionally, our predictions of the mineral modal composition of S-type and Q-type asteroids show an apparent depletion of olivine related to the attenuation of its diagnostic absorptions with space weathering. This trend is consistent with previous results of the slower pyroxene response to space weathering relative to olivine.