This paper investigates methods for estimating uncertainty in semantic segmentation predictions derived from satellite imagery. Estimating uncertainty for segmentation presents unique challenges compared to standard image classification, requiring scalable methods producing per-pixel estimates. While most research on this topic has focused on scene understanding or medical imaging, this work benchmarks existing methods specifically for remote sensing and Earth observation applications. Our evaluation focuses on the practical utility of uncertainty measures, testing their ability to identify prediction errors and noise-corrupted input image regions. Experiments are conducted on two remote sensing datasets, PASTIS and ForTy, selected for their differences in scale, geographic coverage, and label confidence. We perform an extensive evaluation featuring several models, such as Stochastic Segmentation Networks and ensembles, in combination with a number of neural architectures and uncertainty metrics. We make a number of practical recommendations based on our findings.

Mass transport problems arise in many areas of machine learning whereby one wants to compute a map transporting one distribution to another. Generative modeling techniques like Generative Adversarial Networks (GANs) and Denoising Diffusion Models (DDMs) have been successfully adapted to solve such transport problems, resulting in CycleGAN and Bridge Matching respectively. However, these methods do not approximate Optimal Transport (OT) maps, which are known to have desirable properties. Existing techniques approximating OT maps for high-dimensional data-rich problems, such as DDM-based Rectified Flow and Schr\"odinger Bridge procedures, require fully training a DDM-type model at each iteration, or use mini-batch techniques which can introduce significant errors. We propose a novel algorithm to compute the Schr\"odinger Bridge, a dynamic entropy-regularised version of OT, that eliminates the need to train multiple DDM-like models. This algorithm corresponds to a discretisation of a flow of path measures, which we call the Schr\"odinger Bridge Flow, whose only stationary point is the Schr\"odinger Bridge. We demonstrate the performance of our algorithm on a variety of unpaired data translation tasks.

Neural Posterior Estimation methods for simulation-based inference can be ill-suited for dealing with posterior distributions obtained by conditioning on multiple observations, as they tend to require a large number of simulator calls to learn accurate approximations. In contrast, Neural Likelihood Estimation methods can handle multiple observations at inference time after learning from individual observations, but they rely on standard inference methods, such as MCMC or variational inference, which come with certain performance drawbacks. We introduce a new method based on conditional score modeling that enjoys the benefits of both approaches. We model the scores of the (diffused) posterior distributions induced by individual observations, and introduce a way of combining the learned scores to approximately sample from the target posterior distribution. Our approach is sample-efficient, can naturally aggregate multiple observations at inference time, and avoids the drawbacks of standard inference methods.

Training large-scale mixture of experts models efficiently on modern hardware requires assigning datapoints in a batch to different experts, each with a limited capacity. Recently proposed assignment procedures lack a probabilistic interpretation and use biased estimators for training. As an alternative, we propose two unbiased estimators based on principled stochastic assignment procedures: one that skips datapoints which exceed expert capacity, and one that samples perfectly balanced assignments using an extension of the Gumbel-Matching distribution [29]. Both estimators are unbiased, as they correct for the used sampling procedure. On a toy experiment, we find the `skip'-estimator is more effective than the balanced sampling one, and both are more robust in solving the task than biased alternatives.

Efficient low-variance gradient estimation enabled by the reparameterization trick (RT) has been essential to the success of variational autoencoders. Doubly-reparameterized gradients (DReGs) improve on the RT for multi-sample variational bounds by applying reparameterization a second time for an additional reduction in variance. Here, we develop two generalizations of the DReGs estimator and show that they can be used to train conditional and hierarchical VAEs on image modelling tasks more effectively. First, we extend the estimator to hierarchical models with several stochastic layers by showing how to treat additional score function terms due to the hierarchical variational posterior. We then generalize DReGs to score functions of arbitrary distributions instead of just those of the sampling distribution, which makes the estimator applicable to the parameters of the prior in addition to those of the posterior.

Training models with discrete latent variables is challenging due to the high variance of unbiased gradient estimators. While low-variance reparameterization gradients of a continuous relaxation can provide an effective solution, a continuous relaxation is not always available or tractable. Dong et al. (2020) and Yin et al. (2020) introduced a performant estimator that does not rely on continuous relaxations; however, it is limited to binary random variables. We introduce a novel derivation of their estimator based on importance sampling and statistical couplings, which we extend to the categorical setting. Motivated by the construction of a stick-breaking coupling, we introduce gradient estimators based on reparameterizing categorical variables as sequences of binary variables and Rao-Blackwellization. In systematic experiments, we show that our proposed categorical gradient estimators provide state-of-the-art performance, whereas even with additional Rao-Blackwellization, previous estimators (Yin et al., 2019) underperform a simpler REINFORCE with a leave-one-out-baseline estimator (Kool et al., 2019).

Training models with discrete latent variables is challenging due to the difficulty of estimating the gradients accurately. Much of the recent progress has been achieved by taking advantage of continuous relaxations of the system, which are not always available or even possible. The Augment-REINFORCE-Merge (ARM) estimator provides an alternative that, instead of relaxation, uses continuous augmentation. Applying antithetic sampling over the augmenting variables yields a relatively low-variance and unbiased estimator applicable to any model with binary latent variables. However, while antithetic sampling reduces variance, the augmentation process increases variance. We show that ARM can be improved by analytically integrating out the randomness introduced by the augmentation process, guaranteeing substantial variance reduction. Our estimator, \emph{DisARM}, is simple to implement and has the same computational cost as ARM. We evaluate DisARM on several generative modeling benchmarks and show that it consistently outperforms ARM and a strong independent sample baseline in terms of both variance and log-likelihood. Furthermore, we propose a local version of DisARM designed for optimizing the multi-sample variational bound, and show that it outperforms VIMCO, the current state-of-the-art method.

Lipschitz constants of neural networks have been explored in various contexts in deep learning, such as provable adversarial robustness, estimating Wasserstein distance, stabilising training of GANs, and formulating invertible neural networks. Such works have focused on bounding the Lipschitz constant of fully connected or convolutional networks, composed of linear maps and pointwise non-linearities. In this paper, we investigate the Lipschitz constant of self-attention, a non-linear neural network module widely used in sequence modelling. We prove that the standard dot-product self-attention is not Lipschitz, and propose an alternative L2 self-attention that is Lipschitz. We derive an upper bound on the Lipschitz constant of L2 self-attention and provide empirical evidence for its asymptotic tightness. To demonstrate the practical relevance of the theory, we formulate invertible self-attention and use it in a Transformer-based architecture for a character-level language modelling task.

Neural probabilistic language models (NPLMs) have been shown to be competitive with and occasionally superior to the widely-used n-gram language models. The main drawback of NPLMs is their extremely long training and testing times. Morin and Bengio have proposed a hierarchical language model built around a binary tree of words that was two orders of magnitude faster than the non-hierarchical language model it was based on. However, it performed considerably worse than its non-hierarchical counterpart in spite of using a word tree created using expert knowledge. We introduce a fast hierarchical language model along with a simple feature-based algorithm for automatic construction of word trees from the data.

User preferences for items can be inferred from either explicit feedback, such as item ratings, or implicit feedback, such as rental histories. Research in collaborative filtering has concentrated on explicit feedback, resulting in the development of accurate and scalable models. However, since explicit feedback is often difficult to collect it is important to develop effective models that take advantage of the more widely available implicit feedback. We introduce a probabilistic approach to collaborative filtering with implicit feedback based on modelling the user's item selection process. In the interests of scalability, we restrict our attention to tree-structured distributions over items and develop a principled and efficient algorithm for learning item trees from data. We also identify a problem with a widely used protocol for evaluating implicit feedback models and propose a way of addressing it using a small quantity of explicit feedback data.