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In this paper, we propose two algorithms for a hybrid construction of all $n\times n$ MDS and involutory MDS matrices over a finite field $\mathbb{F}_{p^m}$, respectively. The proposed algorithms effectively narrow down the search space to identify $(n-1) \times (n-1)$ MDS matrices, facilitating the generation of all $n \times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. To the best of our knowledge, existing literature lacks methods for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. In our approach, we introduce a representative matrix form for generating all $n\times n$ MDS and involutory MDS matrices over $\mathbb{F}_{p^m}$. The determination of these representative MDS matrices involves searching through all $(n-1)\times (n-1)$ MDS matrices over $\mathbb{F}_{p^m}$. Our contributions extend to proving that the count of all $3\times 3$ MDS matrices over $\mathbb{F}_{2^m}$ is precisely $(2^m-1)^5(2^m-2)(2^m-3)(2^{2m}-9\cdot 2^m+21)$. Furthermore, we explicitly provide the count of all $4\times 4$ MDS and involutory MDS matrices over $\mathbb{F}_{2^m}$ for $m=2, 3, 4$.

This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region and (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data. We study the properties of these three main discretization methods both theoretically and experimentally, and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and derivatives as well as the integrated Gaussian kernels and derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing.

We present experimental results on the single file motion of a group of robots interacting with each other through position sensors. We successfully replicate the fundamental diagram typical of these systems, with a transition from free flow to congested traffic as the density of the system increases. In the latter scenario we also observe the characteristic stop-and-go waves. The unique advantages of this novel system, such as experimental stability and repeatability, allow for extended experimental runs, facilitating a comprehensive statistical analysis of the global dynamics. Above a certain density, we observe a divergence of the average jam duration and the average number of robots involved in it. This discovery enables us to precisely identify another transition: from congested intermittent flow (for intermediate densities) to a totally congested scenario for high densities. Beyond this finding, the present work demonstrates the suitability of robot swarms to model complex behaviors in many particle systems.

Convolutional Neural Networks (CNNs) are nowadays the model of choice in Computer Vision, thanks to their ability to automatize the feature extraction process in visual tasks. However, the knowledge acquired during training is fully subsymbolic, and hence difficult to understand and explain to end users. In this paper, we propose a new technique called HOLMES (HOLonym-MEronym based Semantic inspection) that decomposes a label into a set of related concepts, and provides component-level explanations for an image classification model. Specifically, HOLMES leverages ontologies, web scraping and transfer learning to automatically construct meronym (parts)-based detectors for a given holonym (class). Then, it produces heatmaps at the meronym level and finally, by probing the holonym CNN with occluded images, it highlights the importance of each part on the classification output. Compared to state-of-the-art saliency methods, HOLMES takes a step further and provides information about both where and what the holonym CNN is looking at, without relying on densely annotated datasets and without forcing concepts to be associated to single computational units. Extensive experimental evaluation on different categories of objects (animals, tools and vehicles) shows the feasibility of our approach. On average, HOLMES explanations include at least two meronyms, and the ablation of a single meronym roughly halves the holonym model confidence. The resulting heatmaps were quantitatively evaluated using the deletion/insertion/preservation curves. All metrics were comparable to those achieved by GradCAM, while offering the advantage of further decomposing the heatmap in human-understandable concepts, thus highlighting both the relevance of meronyms to object classification, as well as HOLMES ability to capture it. The code is available at //github.com/FrancesC0de/HOLMES.

Authorship Verification (AV) is the process of analyzing a set of documents to determine whether they were written by a specific author. This problem often arises in forensic scenarios, e.g., in cases where the documents in question constitute evidence for a crime. Existing state-of-the-art AV methods use computational solutions that are not supported by a plausible scientific explanation for their functioning and that are often difficult for analysts to interpret. To address this, we propose a method relying on calculating a quantity we call $\lambda_G$ (LambdaG): the ratio between the likelihood of a document given a model of the Grammar for the candidate author and the likelihood of the same document given a model of the Grammar for a reference population. These Grammar Models are estimated using $n$-gram language models that are trained solely on grammatical features. Despite not needing large amounts of data for training, LambdaG still outperforms other established AV methods with higher computational complexity, including a fine-tuned Siamese Transformer network. Our empirical evaluation based on four baseline methods applied to twelve datasets shows that LambdaG leads to better results in terms of both accuracy and AUC in eleven cases and in all twelve cases if considering only topic-agnostic methods. The algorithm is also highly robust to important variations in the genre of the reference population in many cross-genre comparisons. In addition to these properties, we demonstrate how LambdaG is easier to interpret than the current state-of-the-art. We argue that the advantage of LambdaG over other methods is due to fact that it is compatible with Cognitive Linguistic theories of language processing.

The extrapolation capability of Large Language Models (LLMs) based on Rotary Position Embedding is currently a topic of considerable interest. The mainstream approach to addressing extrapolation with LLMs involves modifying RoPE by replacing 10000, the rotary base of $\theta_n={10000}^{-2n/d}$ in the original RoPE, with a larger value and providing longer fine-tuning text. In this work, we first observe that fine-tuning a RoPE-based LLM with either a smaller or larger base in pre-training context length could significantly enhance its extrapolation performance. After that, we propose \textbf{\textit{Scaling Laws of RoPE-based Extrapolation}}, a unified framework from the periodic perspective, to describe the relationship between the extrapolation performance and base value as well as tuning context length. In this process, we also explain the origin of the RoPE-based extrapolation issue by \textbf{\textit{critical dimension for extrapolation}}. Besides these observations and analyses, we achieve extrapolation up to 1 million context length within only 16K training length on LLaMA2 7B and 13B.

Recently pre-trained language representation models such as BERT have shown great success when fine-tuned on downstream tasks including information retrieval (IR). However, pre-training objectives tailored for ad-hoc retrieval have not been well explored. In this paper, we propose Pre-training with Representative wOrds Prediction (PROP) for ad-hoc retrieval. PROP is inspired by the classical statistical language model for IR, specifically the query likelihood model, which assumes that the query is generated as the piece of text representative of the "ideal" document. Based on this idea, we construct the representative words prediction (ROP) task for pre-training. Given an input document, we sample a pair of word sets according to the document language model, where the set with higher likelihood is deemed as more representative of the document. We then pre-train the Transformer model to predict the pairwise preference between the two word sets, jointly with the Masked Language Model (MLM) objective. By further fine-tuning on a variety of representative downstream ad-hoc retrieval tasks, PROP achieves significant improvements over baselines without pre-training or with other pre-training methods. We also show that PROP can achieve exciting performance under both the zero- and low-resource IR settings. The code and pre-trained models are available at //github.com/Albert-Ma/PROP.

Recent work pre-training Transformers with self-supervised objectives on large text corpora has shown great success when fine-tuned on downstream NLP tasks including text summarization. However, pre-training objectives tailored for abstractive text summarization have not been explored. Furthermore there is a lack of systematic evaluation across diverse domains. In this work, we propose pre-training large Transformer-based encoder-decoder models on massive text corpora with a new self-supervised objective. In PEGASUS, important sentences are removed/masked from an input document and are generated together as one output sequence from the remaining sentences, similar to an extractive summary. We evaluated our best PEGASUS model on 12 downstream summarization tasks spanning news, science, stories, instructions, emails, patents, and legislative bills. Experiments demonstrate it achieves state-of-the-art performance on all 12 downstream datasets measured by ROUGE scores. Our model also shows surprising performance on low-resource summarization, surpassing previous state-of-the-art results on 6 datasets with only 1000 examples. Finally we validated our results using human evaluation and show that our model summaries achieve human performance on multiple datasets.

Within the rapidly developing Internet of Things (IoT), numerous and diverse physical devices, Edge devices, Cloud infrastructure, and their quality of service requirements (QoS), need to be represented within a unified specification in order to enable rapid IoT application development, monitoring, and dynamic reconfiguration. But heterogeneities among different configuration knowledge representation models pose limitations for acquisition, discovery and curation of configuration knowledge for coordinated IoT applications. This paper proposes a unified data model to represent IoT resource configuration knowledge artifacts. It also proposes IoT-CANE (Context-Aware recommendatioN systEm) to facilitate incremental knowledge acquisition and declarative context driven knowledge recommendation.