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Local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. For many classic global properties, like checking the acyclicity of the network, the optimal size of the certificates depends on the size of the network, $n$. In this paper, we focus on properties for which the size of the certificates does not depend on $n$ but on other parameters. We focus on three such important properties and prove tight bounds for all of them. Namely, we prove that the optimal certification size is: $\Theta(\log k)$ for $k$-colorability (and even exactly $\lceil \log k \rceil$ bits in the anonymous model while previous works had only proved a $2$-bit lower bound); $(1/2)\log t+o(\log t)$ for dominating sets at distance $t$ (an unexpected and tighter-than-usual bound) ; and $\Theta(\log \Delta)$ for perfect matching in graphs of maximum degree $\Delta$ (the first non-trivial bound parameterized by $\Delta$). We also prove some surprising upper bounds, for example, certifying the existence of a perfect matching in a planar graph can be done with only two bits. In addition, we explore various specific cases for these properties, in particular improving our understanding of the trade-off between locality of the verification and certificate size.

Advances in large language models (LLMs) have driven an explosion of interest about their societal impacts. Much of the discourse around how they will impact social equity has been cautionary or negative, focusing on questions like "how might LLMs be biased and how would we mitigate those biases?" This is a vital discussion: the ways in which AI generally, and LLMs specifically, can entrench biases have been well-documented. But equally vital, and much less discussed, is the more opportunity-focused counterpoint: "what promising applications do LLMs enable that could promote equity?" If LLMs are to enable a more equitable world, it is not enough just to play defense against their biases and failure modes. We must also go on offense, applying them positively to equity-enhancing use cases to increase opportunities for underserved groups and reduce societal discrimination. There are many choices which determine the impact of AI, and a fundamental choice very early in the pipeline is the problems we choose to apply it to. If we focus only later in the pipeline -- making LLMs marginally more fair as they facilitate use cases which intrinsically entrench power -- we will miss an important opportunity to guide them to equitable impacts. Here, we highlight the emerging potential of LLMs to promote equity by presenting four newly possible, promising research directions, while keeping risks and cautionary points in clear view.

Microring resonators (MRRs) are promising devices for time-delay photonic reservoir computing, but the impact of the different physical effects taking place in the MRRs on the reservoir computing performance is yet to be fully understood. We numerically analyze the impact of linear losses as well as thermo-optic and free-carrier effects relaxation times on the prediction error of the time-series task NARMA-10. We demonstrate the existence of three regions, defined by the input power and the frequency detuning between the optical source and the microring resonance, that reveal the cavity transition from linear to nonlinear regimes. One of these regions offers very low error in time-series prediction under relatively low input power and number of nodes while the other regions either lack nonlinearity or become unstable. This study provides insight into the design of the MRR and the optimization of its physical properties for improving the prediction performance of time-delay reservoir computing.

Polarization information of the light can provide rich cues for computer vision and scene understanding tasks, such as the type of material, pose, and shape of the objects. With the advent of new and cheap polarimetric sensors, this imaging modality is becoming accessible to a wider public for solving problems such as pose estimation, 3D reconstruction, underwater navigation, and depth estimation. However, we observe several limitations regarding the usage of this sensorial modality, as well as a lack of standards and publicly available tools to analyze polarization images. Furthermore, although polarization camera manufacturers usually provide acquisition tools to interface with their cameras, they rarely include processing algorithms that make use of the polarization information. In this paper, we review recent advances in applications that involve polarization imaging, including a comprehensive survey of recent advances on polarization for vision and robotics perception tasks. We also introduce a complete software toolkit that provides common standards to communicate with and process information from most of the existing micro-grid polarization cameras on the market. The toolkit also implements several image processing algorithms for this modality, and it is publicly available on GitHub: //github.com/vibot-lab/Pola4all_JEI_2023.

Forecast reconciliation is the post-forecasting process aimed to revise a set of incoherent base forecasts into coherent forecasts in line with given data structures. Most of the point and probabilistic regression-based forecast reconciliation results ground on the so called "structural representation" and on the related unconstrained generalized least squares reconciliation formula. However, the structural representation naturally applies to genuine hierarchical/grouped time series, where the top- and bottom-level variables are uniquely identified. When a general linearly constrained multiple time series is considered, the forecast reconciliation is naturally expressed according to a projection approach. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, so far it is not completely understood if and how a structural-like reconciliation formula may be derived for a general linearly constrained multiple time series. Such an expression would permit to extend reconciliation definitions, theorems and results in a straightforward manner. In this paper, we show that for general linearly constrained multiple time series it is possible to express the reconciliation formula according to a "structural-like" approach that keeps distinct free and constrained, instead of bottom and upper (aggregated), variables, establish the probabilistic forecast reconciliation framework, and apply these findings to obtain fully reconciled point and probabilistic forecasts for the aggregates of the Australian GDP from income and expenditure sides, and for the European Area GDP disaggregated by income, expenditure and output sides and by 19 countries.

This work studies nonparametric Bayesian estimation of the intensity function of an inhomogeneous Poisson point process in the important case where the intensity depends on covariates, based on the observation of a single realisation of the point pattern over a large area. It is shown how the presence of covariates allows to borrow information from far away locations in the observation window, enabling consistent inference in the growing domain asymptotics. In particular, optimal posterior contraction rates under both global and point-wise loss functions are derived. The rates in global loss are obtained under conditions on the prior distribution resembling those in the well established theory of Bayesian nonparametrics, here combined with concentration inequalities for functionals of stationary processes to control certain random covariate-dependent loss functions appearing in the analysis. The local rates are derived with an ad-hoc study that builds on recent advances in the theory of P\'olya tree priors, extended to the present multivariate setting with a novel construction that makes use of the random geometry induced by the covariates.

Deep neural networks (DNNs) often fail silently with over-confident predictions on out-of-distribution (OOD) samples, posing risks in real-world deployments. Existing techniques predominantly emphasize either the feature representation space or the gradient norms computed with respect to DNN parameters, yet they overlook the intricate gradient distribution and the topology of classification regions. To address this gap, we introduce GRadient-aware Out-Of-Distribution detection in interpolated manifolds (GROOD), a novel framework that relies on the discriminative power of gradient space to distinguish between in-distribution (ID) and OOD samples. To build this space, GROOD relies on class prototypes together with a prototype that specifically captures OOD characteristics. Uniquely, our approach incorporates a targeted mix-up operation at an early intermediate layer of the DNN to refine the separation of gradient spaces between ID and OOD samples. We quantify OOD detection efficacy using the distance to the nearest neighbor gradients derived from the training set, yielding a robust OOD score. Experimental evaluations substantiate that the introduction of targeted input mix-upamplifies the separation between ID and OOD in the gradient space, yielding impressive results across diverse datasets. Notably, when benchmarked against ImageNet-1k, GROOD surpasses the established robustness of state-of-the-art baselines. Through this work, we establish the utility of leveraging gradient spaces and class prototypes for enhanced OOD detection for DNN in image classification.

Partitioned neural network functions are used to approximate the solution of partial differential equations. The problem domain is partitioned into non-overlapping subdomains and the partitioned neural network functions are defined on the given non-overlapping subdomains. Each neural network function then approximates the solution in each subdomain. To obtain the convergent neural network solution, certain continuity conditions on the partitioned neural network functions across the subdomain interface need to be included in the loss function, that is used to train the parameters in the neural network functions. In our work, by introducing suitable interface values, the loss function is reformulated into a sum of localized loss functions and each localized loss function is used to train the corresponding local neural network parameters. In addition, to accelerate the neural network solution convergence, the localized loss function is enriched with an augmented Lagrangian term, where the interface condition and the boundary condition are enforced as constraints on the local solutions by using Lagrange multipliers. The local neural network parameters and Lagrange multipliers are then found by optimizing the localized loss function. To take the advantage of the localized loss function for the parallel computation, an iterative algorithm is also proposed. For the proposed algorithms, their training performance and convergence are numerically studied for various test examples.

In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.

Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.