Rank and PIT histograms are established tools to assess the calibration of probabilistic forecasts. They not only check whether an ensemble forecast is calibrated, but they also reveal what systematic biases (if any) are present in the forecasts. Several extensions of rank histograms have been proposed to evaluate the calibration of probabilistic forecasts for multivariate outcomes. These extensions introduce a so-called pre-rank function that condenses the multivariate forecasts and observations into univariate objects, from which a standard rank histogram can be produced. Existing pre-rank functions typically aim to preserve as much information as possible when condensing the multivariate forecasts and observations into univariate objects. Although this is sensible when conducting statistical tests for multivariate calibration, it can hinder the interpretation of the resulting histograms. In this paper, we demonstrate that there are few restrictions on the choice of pre-rank function, meaning forecasters can choose a pre-rank function depending on what information they want to extract from their forecasts. We introduce the concept of simple pre-rank functions, and provide examples that can be used to assess the location, scale, and dependence structure of multivariate probabilistic forecasts, as well as pre-rank functions tailored to the evaluation of probabilistic spatial field forecasts. The simple pre-rank functions that we introduce are easy to interpret, easy to implement, and they deliberately provide complementary information, meaning several pre-rank functions can be employed to achieve a more complete understanding of multivariate forecast performance. We then discuss how e-values can be employed to formally test for multivariate calibration over time. This is demonstrated in an application to wind speed forecasting using the EUPPBench post-processing benchmark data set.
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We provide a new sequent calculus that enjoys syntactic cut-elimination and strongly terminating backward proof search for the intuitionistic Strong L\"ob logic $\sf{iSL}$, an intuitionistic modal logic with a provability interpretation. A novel measure on sequents is used to prove both the termination of the naive backward proof search strategy, and the admissibility of cut in a syntactic and direct way, leading to a straightforward cut-elimination procedure. All proofs have been formalised in the interactive theorem prover Coq.
Deep neural networks have shown remarkable performance when trained on independent and identically distributed data from a fixed set of classes. However, in real-world scenarios, it can be desirable to train models on a continuous stream of data where multiple classification tasks are presented sequentially. This scenario, known as Continual Learning (CL) poses challenges to standard learning algorithms which struggle to maintain knowledge of old tasks while learning new ones. This stability-plasticity dilemma remains central to CL and multiple metrics have been proposed to adequately measure stability and plasticity separately. However, none considers the increasing difficulty of the classification task, which inherently results in performance loss for any model. In that sense, we analyze some limitations of current metrics and identify the presence of setup-induced forgetting. Therefore, we propose new metrics that account for the task's increasing difficulty. Through experiments on benchmark datasets, we demonstrate that our proposed metrics can provide new insights into the stability-plasticity trade-off achieved by models in the continual learning environment.
We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes.
Textual geographic information is indispensable and heavily relied upon in practical applications. The absence of clear distribution poses challenges in effectively harnessing geographic information, thereby driving our quest for exploration. We contend that geographic information is influenced by human behavior, cognition, expression, and thought processes, and given our intuitive understanding of natural systems, we hypothesize its conformity to the Gamma distribution. Through rigorous experiments on a diverse range of 24 datasets encompassing different languages and types, we have substantiated this hypothesis, unearthing the underlying regularities governing the dimensions of quantity, length, and distance in geographic information. Furthermore, theoretical analyses and comparisons with Gaussian distributions and Zipf's law have refuted the contingency of these laws. Significantly, we have estimated the upper bounds of human utilization of geographic information, pointing towards the existence of uncharted territories. Also, we provide guidance in geographic information extraction. Hope we peer its true countenance uncovering the veil of geographic information.
Neural networks have gained much interest because of their effectiveness in many applications. However, their mathematical properties are generally not well understood. If there is some underlying geometric structure inherent to the data or to the function to approximate, it is often desirable to take this into account in the design of the neural network. In this work, we start with a non-autonomous ODE and build neural networks using a suitable, structure-preserving, numerical time-discretisation. The structure of the neural network is then inferred from the properties of the ODE vector field. Besides injecting more structure into the network architectures, this modelling procedure allows a better theoretical understanding of their behaviour. We present two universal approximation results and demonstrate how to impose some particular properties on the neural networks. A particular focus is on 1-Lipschitz architectures including layers that are not 1-Lipschitz. These networks are expressive and robust against adversarial attacks, as shown for the CIFAR-10 and CIFAR-100 datasets.
Text-to-SQL is a task that converts a natural language question into a structured query language (SQL) to retrieve information from a database. Large language models (LLMs) work well in natural language generation tasks, but they are not specifically pre-trained to understand the syntax and semantics of SQL commands. In this paper, we propose an LLM-based framework for Text-to-SQL which retrieves helpful demonstration examples to prompt LLMs. However, questions with different database schemes can vary widely, even if the intentions behind them are similar and the corresponding SQL queries exhibit similarities. Consequently, it becomes crucial to identify the appropriate SQL demonstrations that align with our requirements. We design a de-semanticization mechanism that extracts question skeletons, allowing us to retrieve similar examples based on their structural similarity. We also model the relationships between question tokens and database schema items (i.e., tables and columns) to filter out scheme-related information. Our framework adapts the range of the database schema in prompts to balance length and valuable information. A fallback mechanism allows for a more detailed schema to be provided if the generated SQL query fails. Ours outperforms state-of-the-art models and demonstrates strong generalization ability on three cross-domain Text-to-SQL benchmarks.
We extend several relative perturbation bounds to Hermitian matrices that are possibly singular, and also develop a general class of relative bounds for Hermitian matrices. As a result, corresponding relative bounds for singular values of rank-deficient $m\times n$ matrices are also obtained using the Jordan-Wielandt matrices. We also present that the main relative bound derived would be invariant with respect to congruence transformation under certain conditions, and compare its sharpness with the Weyl's absolute perturbation bound.
Recently, several algorithms have been proposed for decomposing reactive synthesis specifications into independent and simpler sub-specifications. Being inspired by one of the approaches, developed by Antonio Iannopollo (2018), who designed the so-called (DC) algorithm, we present here our solution that takes his ideas further and provides mathematical formalisation of the strategy behind DC. We rigorously define the main notions involved in the algorithm, explain the technique, and demonstrate its application on examples. The core technique of DC is based on the detection of independent variables in linear temporal logic formulae by exploiting the power and efficiency of a model checker. Although the DC algorithm is sound, it is not complete, as its author already pointed out. In this paper, we provide a counterexample that shows this fact and propose relevant changes to adapt the original DC strategy to ensure its correctness. The modification of DC and the detailed proof of its soundness and completeness are the main contributions of this work.
We describe an algorithm that allows one to find dense packing configurations of a number of congruent disks in arbitrary domains in two or more dimensions. We have applied it to a large class of two dimensional domains such as rectangles, ellipses, crosses, multiply connected domains and even to the cardioid. For many of the cases that we have studied no previous result was available. The fundamental idea in our approach is the introduction of "image" disks, which allows one to work with a fixed container, thus lifting the limitations of the packing algorithms of \cite{Nurmela97,Amore21,Amore23}. We believe that the extension of our algorithm to three (or higher) dimensional containers (not considered here) can be done straightforwardly.
Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.
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