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We study the numerical approximation of a coupled hyperbolic-parabolic system by a family of discontinuous Galerkin space-time finite element methods. The model is rewritten as a first-order evolutionary problem that is treated by the unified abstract solution theory of R.\ Picard. To preserve the mathematical structure of the evolutionary equation on the fully discrete level, suitable generalizations of the distribution gradient and divergence operators on broken polynomial spaces on which the discontinuous Galerkin approach is built on are defined. Well-posedness of the fully discrete problem and error estimates for the discontinuous Galerkin approximation in space and time are proved.

Stochastic filtering is a vibrant area of research in both control theory and statistics, with broad applications in many scientific fields. Despite its extensive historical development, there still lacks an effective method for joint parameter-state estimation in SDEs. The state-of-the-art particle filtering methods suffer from either sample degeneracy or information loss, with both issues stemming from the dynamics of the particles generated to represent system parameters. This paper provides a novel and effective approach for joint parameter-state estimation in SDEs via Rao-Blackwellization and modularization. Our method operates in two layers: the first layer estimates the system states using a bootstrap particle filter, and the second layer marginalizes out system parameters explicitly. This strategy circumvents the need to generate particles representing system parameters, thereby mitigating their associated problems of sample degeneracy and information loss. Moreover, our method employs a modularization approach when integrating out the parameters, which significantly reduces the computational complexity. All these designs ensure the superior performance of our method. Finally, a numerical example is presented to illustrate that our method outperforms existing approaches by a large margin.

The study of time-inhomogeneous Markov jump processes is a traditional topic within probability theory that has recently attracted substantial attention in various applications. However, their flexibility also incurs a substantial mathematical burden which is usually circumvented by using well-known generic distributional approximations or simulations. This article provides a novel approximation method that tailors the dynamics of a time-homogeneous Markov jump process to meet those of its time-inhomogeneous counterpart on an increasingly fine Poisson grid. Strong convergence of the processes in terms of the Skorokhod $J_1$ metric is established, and convergence rates are provided. Under traditional regularity assumptions, distributional convergence is established for unconditional proxies, to the same limit. Special attention is devoted to the case where the target process has one absorbing state and the remaining ones transient, for which the absorption times also converge. Some applications are outlined, such as univariate hazard-rate density estimation, ruin probabilities, and multivariate phase-type density evaluation.

In this article, we study the parameterized complexity of the Set Cover problem restricted to d-semi-ladder-free hypergraphs, a class defined by Fabianski et al. [Proceedings of STACS 2019]. We observe that two algorithms introduced by Langerman and Morin [Discrete \& Computational Geometry 2005] in the context of geometric covering problems can be adapted to this setting, yielding simple FPT and kernelization algorithms for Set Cover in d-semi-ladder-free hypergraphs. We complement our algorithmic results with a compression lower bound for the problem, that proves the tightness of our kernelization under standard complexity-theoretic assumptions.

Exploring the semantic context in scene images is essential for indoor scene recognition. However, due to the diverse intra-class spatial layouts and the coexisting inter-class objects, modeling contextual relationships to adapt various image characteristics is a great challenge. Existing contextual modeling methods for indoor scene recognition exhibit two limitations: 1) During training, space-independent information, such as color, may hinder optimizing the network's capacity to represent the spatial context. 2) These methods often overlook the differences in coexisting objects across different scenes, suppressing scene recognition performance. To address these limitations, we propose SpaCoNet, which simultaneously models the Spatial relation and Co-occurrence of objects based on semantic segmentation. Firstly, the semantic spatial relation module (SSRM) is designed to explore the spatial relation among objects within a scene. With the help of semantic segmentation, this module decouples the spatial information from the image, effectively avoiding the influence of irrelevant features. Secondly, both spatial context features from the SSRM and deep features from the Image Feature Extraction Module are used to distinguish the coexisting object across different scenes. Finally, utilizing the discriminative features mentioned above, we employ the self-attention mechanism to explore the long-range co-occurrence among objects, and further generate a semantic-guided feature representation for indoor scene recognition. Experimental results on three widely used scene datasets demonstrate the effectiveness and generality of the proposed method. The code will be made publicly available after the blind review process is completed.

Supervised learning problems with side information in the form of a network arise frequently in applications in genomics, proteomics and neuroscience. For example, in genetic applications, the network side information can accurately capture background biological information on the intricate relations among the relevant genes. In this paper, we initiate a study of Bayes optimal learning in high-dimensional linear regression with network side information. To this end, we first introduce a simple generative model (called the Reg-Graph model) which posits a joint distribution for the supervised data and the observed network through a common set of latent parameters. Next, we introduce an iterative algorithm based on Approximate Message Passing (AMP) which is provably Bayes optimal under very general conditions. In addition, we characterize the limiting mutual information between the latent signal and the data observed, and thus precisely quantify the statistical impact of the network side information. Finally, supporting numerical experiments suggest that the introduced algorithm has excellent performance in finite samples.

We propose InCA, a lightweight method for transfer learning that cross-attends to any activation layer of a pre-trained model. During training, InCA uses a single forward pass to extract multiple activations, which are passed to external cross-attention adapters, trained anew and combined or selected for downstream tasks. We show that, even when selecting a single top-scoring adapter, InCA achieves performance comparable to full fine-tuning, at a cost comparable to fine-tuning just the last layer. For example, with a cross-attention probe 1.3% the size of a pre-trained ViT-L/16 model, we achieve performance within 0.2% of the full fine-tuning paragon at a computational training cost of 51% of the baseline, on average across 11 downstream classification. Unlike other forms of efficient adaptation, InCA does not require backpropagating through the pre-trained model, thus leaving its execution unaltered at both training and inference. The versatility of InCA is best illustrated in fine-grained tasks, which may require accessing information absent in the last layer but accessible in intermediate layer activations. Since the backbone is fixed, InCA allows parallel ensembling as well as parallel execution of multiple tasks. InCA achieves state-of-the-art performance in the ImageNet-to-Sketch multi-task benchmark.

We systematically study a wide variety of generative models spanning semantically-diverse image datasets to understand and improve the feature extractors and metrics used to evaluate them. Using best practices in psychophysics, we measure human perception of image realism for generated samples by conducting the largest experiment evaluating generative models to date, and find that no existing metric strongly correlates with human evaluations. Comparing to 17 modern metrics for evaluating the overall performance, fidelity, diversity, rarity, and memorization of generative models, we find that the state-of-the-art perceptual realism of diffusion models as judged by humans is not reflected in commonly reported metrics such as FID. This discrepancy is not explained by diversity in generated samples, though one cause is over-reliance on Inception-V3. We address these flaws through a study of alternative self-supervised feature extractors, find that the semantic information encoded by individual networks strongly depends on their training procedure, and show that DINOv2-ViT-L/14 allows for much richer evaluation of generative models. Next, we investigate data memorization, and find that generative models do memorize training examples on simple, smaller datasets like CIFAR10, but not necessarily on more complex datasets like ImageNet. However, our experiments show that current metrics do not properly detect memorization: none in the literature is able to separate memorization from other phenomena such as underfitting or mode shrinkage. To facilitate further development of generative models and their evaluation we release all generated image datasets, human evaluation data, and a modular library to compute 17 common metrics for 9 different encoders at //github.com/layer6ai-labs/dgm-eval.

Many of the tools available for robot learning were designed for Euclidean data. However, many applications in robotics involve manifold-valued data. A common example is orientation; this can be represented as a 3-by-3 rotation matrix or a quaternion, the spaces of which are non-Euclidean manifolds. In robot learning, manifold-valued data are often handled by relating the manifold to a suitable Euclidean space, either by embedding the manifold or by projecting the data onto one or several tangent spaces. These approaches can result in poor predictive accuracy, and convoluted algorithms. In this paper, we propose an "intrinsic" approach to regression that works directly within the manifold. It involves taking a suitable probability distribution on the manifold, letting its parameter be a function of a predictor variable, such as time, then estimating that function non-parametrically via a "local likelihood" method that incorporates a kernel. We name the method kernelised likelihood estimation. The approach is conceptually simple, and generally applicable to different manifolds. We implement it with three different types of manifold-valued data that commonly appear in robotics applications. The results of these experiments show better predictive accuracy than projection-based algorithms.

We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.