In this paper, we introduce a set representation called polynomial logical zonotopes for performing exact and computationally efficient reachability analysis on logical systems. Polynomial logical zonotopes are a generalization of logical zonotopes, which are able to represent up to 2^n binary vectors using only n generators. Due to their construction, logical zonotopes are only able to support exact computations of some logical operations (XOR, NOT, XNOR), while other operations (AND, NAND, OR, NOR) result in over-approximations in the reduced space, i.e., generator space. In order to perform all fundamental logical operations exactly, we formulate a generalization of logical zonotopes that is constructed by dependent generators and exponent matrices. We prove that through this polynomial-like construction, we are able to perform all of the fundamental logical operations (XOR, NOT, XNOR, AND, NAND, OR, NOR) exactly in the generator space. While we are able to perform all of the logical operations exactly, this comes with a slight increase in computational complexity compared to logical zonotopes. We show that we can use polynomial logical zonotopes to perform exact reachability analysis while retaining a low computational complexity. To illustrate and showcase the computational benefits of polynomial logical zonotopes, we present the results of performing reachability analysis on two use cases: (1) safety verification of an intersection crossing protocol and (2) reachability analysis on a high-dimensional Boolean function. Moreover, to highlight the extensibility of logical zonotopes, we include an additional use case where we perform a computationally tractable exhaustive search for the key of a linear feedback shift register.
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In this paper, we study some codes of algebraic geometry related to certain maximal curves. Quantum stabilizer codes obtained through the self orthogonality of Hermitian codes of this error correcting do not always have good parameters. However, appropriate parameters found that the Hermitian self-orthogonal code quantum stabilizer code has good parameters. Therefore, we investigated the quantum stabilizer code at a certain maximum curve and modified its parameters. Algebraic geometry codes show promise for enabling high data rate transmission over noisy power line communication channels.
In this paper, we present an approach for guaranteeing the completion of complex tasks with cyber-physical systems (CPS). Specifically, we leverage temporal logic trees constructed using Hamilton-Jacobi reachability analysis to (1) check for the existence of control policies that complete a specified task and (2) develop a computationally-efficient approach to synthesize the full set of control inputs the CPS can implement in real-time to ensure the task is completed. We show that, by checking the approximation directions of each state set in the temporal logic tree, we can check if the temporal logic tree suffers from the "leaking corner issue," where the intersection of reachable sets yields an incorrect approximation. By ensuring a temporal logic tree has no leaking corners, we know the temporal logic tree correctly verifies the existence of control policies that satisfy the specified task. After confirming the existence of control policies, we show that we can leverage the value functions obtained through Hamilton-Jacobi reachability analysis to efficiently compute the set of control inputs the CPS can implement throughout the deployment time horizon to guarantee the completion of the specified task. Finally, we use a newly released Python toolbox to evaluate the presented approach on a simulated driving task.
In this work, we present the MM-MATH dataset, a novel benchmark developed to rigorously evaluate the performance of advanced large language and multimodal models - including but not limited to GPT-4, GPT-4V, and Claude - within the domain of geometric computation. This dataset comprises 5,929 meticulously crafted geometric problems, each paired with a corresponding image, aimed at mirroring the complexity and requirements typical of ninth-grade mathematics. The motivation behind MM-MATH stems from the burgeoning interest and significant strides in multimodal technology, which necessitates a paradigm shift in assessment methodologies from mere outcome analysis to a more holistic evaluation encompassing reasoning and procedural correctness. Despite impressive gains in various benchmark performances, our analysis uncovers a persistent and notable deficiency in these models' ability to parse and interpret geometric information accurately from images, accounting for over 60% of observed errors. By deploying a dual-focused evaluation approach, examining both the end results and the underlying problem-solving processes, we unearthed a marked discrepancy between the capabilities of current multimodal models and human-level proficiency. The introduction of MM-MATH represents a tripartite contribution to the field: it not only serves as a comprehensive and challenging benchmark for assessing geometric problem-solving prowess but also illuminates critical gaps in textual and visual comprehension that current models exhibit. Through this endeavor, we aspire to catalyze further research and development aimed at bridging these gaps, thereby advancing the state of multimodal model capabilities to new heights.
In this paper, we introduce Neural-ABC, a novel parametric model based on neural implicit functions that can represent clothed human bodies with disentangled latent spaces for identity, clothing, shape, and pose. Traditional mesh-based representations struggle to represent articulated bodies with clothes due to the diversity of human body shapes and clothing styles, as well as the complexity of poses. Our proposed model provides a unified framework for parametric modeling, which can represent the identity, clothing, shape and pose of the clothed human body. Our proposed approach utilizes the power of neural implicit functions as the underlying representation and integrates well-designed structures to meet the necessary requirements. Specifically, we represent the underlying body as a signed distance function and clothing as an unsigned distance function, and they can be uniformly represented as unsigned distance fields. Different types of clothing do not require predefined topological structures or classifications, and can follow changes in the underlying body to fit the body. Additionally, we construct poses using a controllable articulated structure. The model is trained on both open and newly constructed datasets, and our decoupling strategy is carefully designed to ensure optimal performance. Our model excels at disentangling clothing and identity in different shape and poses while preserving the style of the clothing. We demonstrate that Neural-ABC fits new observations of different types of clothing. Compared to other state-of-the-art parametric models, Neural-ABC demonstrates powerful advantages in the reconstruction of clothed human bodies, as evidenced by fitting raw scans, depth maps and images. We show that the attributes of the fitted results can be further edited by adjusting their identities, clothing, shape and pose codes.
In this study, we employ a classification approach to show that different categories of literary "quality" display unique linguistic profiles, leveraging a corpus that encompasses titles from the Norton Anthology, Penguin Classics series, and the Open Syllabus project, contrasted against contemporary bestsellers, Nobel prize winners and recipients of prestigious literary awards. Our analysis reveals that canonical and so called high-brow texts exhibit distinct textual features when compared to other quality categories such as bestsellers and popular titles as well as to control groups, likely responding to distinct (but not mutually exclusive) models of quality. We apply a classic machine learning approach, namely Random Forest, to distinguish quality novels from "control groups", achieving up to 77\% F1 scores in differentiating between the categories. We find that quality category tend to be easier to distinguish from control groups than from other quality categories, suggesting than literary quality features might be distinguishable but shared through quality proxies.
In this work, we introduce DeepIPC, a novel end-to-end model tailored for autonomous driving, which seamlessly integrates perception and control tasks. Unlike traditional models that handle these tasks separately, DeepIPC innovatively combines a perception module, which processes RGBD images for semantic segmentation and generates bird's eye view (BEV) mappings, with a controller module that utilizes these insights along with GNSS and angular speed measurements to accurately predict navigational waypoints. This integration allows DeepIPC to efficiently translate complex environmental data into actionable driving commands. Our comprehensive evaluation demonstrates DeepIPC's superior performance in terms of drivability and multi-task efficiency across diverse real-world scenarios, setting a new benchmark for end-to-end autonomous driving systems with a leaner model architecture. The experimental results underscore DeepIPC's potential to significantly enhance autonomous vehicular navigation, promising a step forward in the development of autonomous driving technologies. For further insights and replication, we will make our code and datasets available at //github.com/oskarnatan/DeepIPC.
In this paper, we study classification and regression error bounds for inhomogenous data that are independent but not necessarily identically distributed. First, we consider classification of data in the presence of non-stationary noise and establish ergodic type sufficient conditions that guarantee the achievability of the Bayes error bound, using universal rules. We then perform a similar analysis for $k$-nearest neighbour regression and obtain optimal error bounds for the same. Finally, we illustrate applications of our results in the context of wireless networks.
In this paper we define a class of polynomial functors suited for constructing coalgebras representing processes in which uncertainty plays an important role. In these polynomial functors we include upper and lower probability measures, finitely additive probability measures, plausibilty measures (and their duals, belief functions), and possibility measures. We give axioms and inference rules for the associated system of coalgebraic modal logic, and construct the canonical coalgebras to prove a completeness result.
In this paper, we present a multidimensional, highly effective method for aggregating data for wireless sensor networks while maintaining privacy. The suggested system is resistant to data loss and secure against both active and passive privacy compromising attacks, such as the coalition attack from a rogue base station and kidnapped sensor nodes. With regard to cluster size, it achieves consistent communication overhead, which is helpful in large-scale WSNs. Due to its constant size communication overhead, the suggested strategy outperforms the previous privacy-preserving data aggregation scheme not only in terms of privacy preservation but also in terms of communication complexity and energy costs.
In this paper, we introduce a new approach to solving the porous medium equation using a moving mesh finite element method that leverages the Onsager variational principle as an approximation tool. Both the continuous and discrete problems are formulated based on the Onsager principle. The energy dissipation structure is maintained in the semi-discrete and fully implicit discrete schemes. We also develop a fully decoupled explicit scheme by which only a few linear equations are solved sequentially in each time step. The numerical schemes exhibit an optimal convergence rate when the initial mesh is appropriately selected to ensure accurate approximation of the initial data. Furthermore, the method naturally captures the waiting time phenomena without requiring any manual intervention.
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