This paper shows that a class of codes such as Reed-Muller (RM) codes have vanishing bit-error probability below capacity on symmetric channels. The proof relies on the notion of `camellia codes': a class of symmetric codes decomposable into `camellias', i.e., set systems that differ from sunflowers by allowing for scattered petal overlaps. The proof then follows from a boosting argument on the camellia petals with second moment Fourier analysis. For erasure channels, this gives a self-contained proof of the bit-error result in Kudekar et al.'17, without relying on sharp thresholds for monotone properties Friedgut-Kalai'96. For error channels, this gives a shortened proof of Reeves-Pfister'23 with an exponentially tighter bound, and a proof variant of the bit-error result in Abbe-Sandon'23. The control of the full (block) error probability still requires Abbe-Sandon'23 for RM codes.
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In this paper we consider an initial-boundary value problem with a Caputo time derivative of order $\alpha\in(0,1)$. The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of convergence of standard schemes. To deal with this singularity, the solution is computed with a fitted difference scheme on a graded mesh. The convergence of this scheme is analysed using a discrete maximum principle and carefully chosen barrier functions. Sharp error estimates are proved, which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes, and also indicate an optimal choice for the mesh grading. This optimal mesh grading is less severe than the optimal grading for the standard L1 scheme. Furthermore, the dependence of the error on the final time forms part of our error estimate. Numerical experiments are presented which corroborate our theoretical results.
Missing data is a common problem in practical settings. Various imputation methods have been developed to deal with missing data. However, even though the label is usually available in the training data, the common practice of imputation usually only relies on the input and ignores the label. In this work, we illustrate how stacking the label into the input can significantly improve the imputation of the input. In addition, we propose a classification strategy that initializes the predicted test label with missing values and stacks the label with the input for imputation. This allows imputing the label and the input at the same time. Also, the technique is capable of handling data training with missing labels without any prior imputation and is applicable to continuous, categorical, or mixed-type data. Experiments show promising results in terms of accuracy.
Motivated by a real failure dataset in a two-dimensional context, this paper presents an extension of the Markov modulated Poisson process (MMPP) to two dimensions. The one-dimensional MMPP has been proposed for the modeling of dependent and non-exponential inter-failure times (in contexts as queuing, risk or reliability, among others). The novel two-dimensional MMPP allows for dependence between the two sequences of inter-failure times, while at the same time preserves the MMPP properties, marginally. The generalization is based on the Marshall-Olkin exponential distribution. Inference is undertaken for the new model through a method combining a matching moments approach with an Approximate Bayesian Computation (ABC) algorithm. The performance of the method is shown on simulated and real datasets representing times and distances covered between consecutive failures in a public transport company. For the real dataset, some quantities of importance associated with the reliability of the system are estimated as the probabilities and expected number of failures at different times and distances covered by trains until the occurrence of a failure.
The Internet of Things (IoT) has grown significantly in popularity, accompanied by increased capacity and lower cost of communications, and overwhelming development of technologies. At the same time, big data and real-time data analysis have taken on great importance and have been accompanied by unprecedented interest in sharing data among citizens, public administrations and other organisms, giving rise to what is known as the Collaborative Internet of Things. This growth in data and infrastructure must be accompanied by a software architecture that allows its exploitation. Although there are various proposals focused on the exploitation of the IoT at edge, fog and/or cloud levels, it is not easy to find a software solution that exploits the three tiers together, taking maximum advantage not only of the analysis of contextual and situational data at each tier, but also of two-way communications between adjacent ones. In this paper, we propose an architecture that solves these deficiencies by proposing novel technologies which are appropriate for managing the resources of each tier: edge, fog and cloud. In addition, the fact that two-way communications along the three tiers of the architecture is allowed considerably enriches the contextual and situational information in each layer, and substantially assists decision making in real time. The paper illustrates the proposed software architecture through a case study of respiratory disease surveillance in hospitals. As a result, the proposed architecture permits efficient communications between the different tiers responding to the needs of these types of IoT scenarios.
This paper aims to front with dimensionality reduction in regression setting when the predictors are a mixture of functional variable and high-dimensional vector. A flexible model, combining both sparse linear ideas together with semiparametrics, is proposed. A wide scope of asymptotic results is provided: this covers as well rates of convergence of the estimators as asymptotic behaviour of the variable selection procedure. Practical issues are analysed through finite sample simulated experiments while an application to Tecator's data illustrates the usefulness of our methodology.
BCH codes form an important subclass of cyclic codes, and are widely used in compact discs, digital audio tapes and other data storage systems to improve data reliability. As far as we know, there are few results on $q$-ary BCH codes of length $n=\frac{q^{m}+1}{q+1}$. This is because it is harder to deal with BCH codes of such length. In this paper, we study $q$-ary BCH codes with lengths $n=\frac{q^{m}+1}{q+1}$ and $n=q^m+1$. These two classes of BCH codes are always LCD codes. For $n=\frac{q^{m}+1}{q+1}$, the dimensions of narrow-sense BCH codes of length $n$ with designed distance $\delta=\ell q^{\frac{m-1}{2}}+1$ are determined, where $q>2$ and $2\leq \ell \leq q-1$. Moreover, the largest coset leader is given for $m=3$ and the first two largest coset leaders are given for $q=2$. The parameters of BCH codes related to the first few largest coset leaders are investigated. Some binary BCH codes of length $n=\frac{2^m+1}{3}$ have optimal parameters. For ternary narrow-sense BCH codes of length $n=3^m+1$, a lower bound on the minimum distance of their dual codes is developed, which is good in some cases.
This study addresses the challenge of extending Large Language Models (LLMs) to non-English languages, specifically those using non-Latin scripts. We propose an innovative approach that utilizes the romanized form of text as an interface for LLMs, hypothesizing that its frequent informal use and shared tokens with English enhance cross-lingual alignment. Focusing on Hindi, we demonstrate through Hindi-to-English translation and sentiment analysis tasks that romanized text not only significantly improves inference efficiency due to its lower fertility compared to native text but also achieves competitive performance with limited pre-training. Additionally, our novel multi-script prompting approach, which combines romanized and native texts, shows promise in further enhancing task performance. These findings suggest the potential of romanization in bridging the language gap for LLM applications, with future work aimed at expanding this approach to more languages and tasks.
We develop a novel multiple hypothesis testing correction with family-wise error rate (FWER) control that efficiently exploits positive dependencies between potentially correlated statistical hypothesis tests. Our proposed algorithm $\texttt{max-rank}$ is conceptually straight-forward, relying on the use of a $\max$-operator in the rank domain of computed test statistics. We compare our approach to the frequently employed Bonferroni correction, theoretically and empirically demonstrating its superiority over Bonferroni in the case of existing positive dependency, and its equivalence otherwise. Our advantage over Bonferroni increases as the number of tests rises, and we maintain high statistical power whilst ensuring FWER control. We specifically frame our algorithm in the context of parallel permutation testing, a scenario that arises in our primary application of conformal prediction, a recently popularized approach for quantifying uncertainty in complex predictive settings.
Emotion recognition in conversation (ERC) has emerged as a research hotspot in domains such as conversational robots and question-answer systems. How to efficiently and adequately retrieve contextual emotional cues has been one of the key challenges in the ERC task. Existing efforts do not fully model the context and employ complex network structures, resulting in limited performance gains. In this paper, we propose a novel emotion recognition network based on curriculum learning strategy (ERNetCL). The proposed ERNetCL primarily consists of temporal encoder (TE), spatial encoder (SE), and curriculum learning (CL) loss. We utilize TE and SE to combine the strengths of previous methods in a simplistic manner to efficiently capture temporal and spatial contextual information in the conversation. To ease the harmful influence resulting from emotion shift and simulate the way humans learn curriculum from easy to hard, we apply the idea of CL to the ERC task to progressively optimize the network parameters. At the beginning of training, we assign lower learning weights to difficult samples. As the epoch increases, the learning weights for these samples are gradually raised. Extensive experiments on four datasets exhibit that our proposed method is effective and dramatically beats other baseline models.
Inequality measures are quantitative measures that take values in the unit interval, with a zero value characterizing perfect equality. Although originally proposed to measure economic inequalities, they can be applied to several other situations, in which one is interested in the mutual variability between a set of observations, rather than in their deviations from the mean. While unidimensional measures of inequality, such as the Gini index, are widely known and employed, multidimensional measures, such as Lorenz Zonoids, are difficult to interpret and computationally expensive and, for these reasons, are not much well known. To overcome the problem, in this paper we propose a new scaling invariant multidimensional inequality index, based on the Fourier transform, which exhibits a number of interesting properties, and whose application to the multidimensional case is rather straightforward to calculate and interpret.
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