Degree spectra of homeomorphism types of compact Polish spaces

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $0'$-computable low$_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space $X_n$ such that exactly the high$_{n}$-degrees are required to present the homeomorphism type of $X_n$. We also show that no compact Polish space has a least presentation with respect to Turing reducibility. The first version of this article appeared in April 2020. A major update was made in September 2023, with improved proofs and results. This is the final version from January 2024, with more results on \v{C}ech homology groups

Holographic complexity of axion-de Sitter universes

We study the holographic complexity of a pair of asymptotically dS universes in the presence of axion matter, to characterize these observables in more general spacetimes. The system is prepared in a two-copy Hartle-Hawking state by slicing an Euclidean wormhole, which entangles the two universes. We derive the evolution of codimension-1 Complexity=Anything proposals by anchoring the probes to a worldline observer in each of the universes and connecting them through the Euclidean wormhole. We investigate how the axion charge competes with the cosmological constant in the time evolution of complexity. When the complexity proposal equals the volume of an extremal surface, its evolution is determined by the scale factor of the axion-dS universe, and as a result, the observable might increase nearly exponentially for low axion charge, while it decreases to a vanishing value as one approaches the maximal axion charge allowed by de Sitter space.Comment: 16 page

The Earth is Flat? Unveiling Factual Errors in Large Language Models

Large Language Models (LLMs) like ChatGPT are foundational in various applications due to their extensive knowledge from pre-training and fine-tuning. Despite this, they are prone to generating factual and commonsense errors, raising concerns in critical areas like healthcare, journalism, and education to mislead users. Current methods for evaluating LLMs' veracity are limited by test data leakage or the need for extensive human labor, hindering efficient and accurate error detection. To tackle this problem, we introduce a novel, automatic testing framework, FactChecker, aimed at uncovering factual inaccuracies in LLMs. This framework involves three main steps: First, it constructs a factual knowledge graph by retrieving fact triplets from a large-scale knowledge database. Then, leveraging the knowledge graph, FactChecker employs a rule-based approach to generates three types of questions (Yes-No, Multiple-Choice, and WH questions) that involve single-hop and multi-hop relations, along with correct answers. Lastly, it assesses the LLMs' responses for accuracy using tailored matching strategies for each question type. Our extensive tests on six prominent LLMs, including text-davinci-002, text-davinci-003, ChatGPT~(gpt-3.5-turbo, gpt-4), Vicuna, and LLaMA-2, reveal that FactChecker can trigger factual errors in up to 45\% of questions in these models. Moreover, we demonstrate that FactChecker's test cases can improve LLMs' factual accuracy through in-context learning and fine-tuning (e.g., llama-2-13b-chat's accuracy increase from 35.3\% to 68.5\%). We are making all code, data, and results available for future research endeavors

A & B == B & A: Triggering Logical Reasoning Failures in Large Language Models

Recent advancements in large language models (LLMs) have propelled Artificial Intelligence (AI) to new heights, enabling breakthroughs in various tasks such as writing assistance, code generation, and machine translation. A significant distinction of advanced LLMs, such as ChatGPT, is their demonstrated ability to "reason." However, evaluating the reasoning ability of LLMs remains a challenge as most existing evaluations focus on their accuracy on the downstream tasks rather than directly assessing their reasoning processes. Efforts have been made to develop benchmarks and metrics to assess reasoning in LLMs, but they suffer from data leakage or limited scope. In this paper, we introduce LogicAsker, an automatic approach that comprehensively evaluates and improves the logical reasoning abilities of LLMs under a set of atomic reasoning skills based on propositional and predicate logic. The results provide insights into LLMs' reasoning abilities and reveal the logical rules the LLMs did not learn well. We evaluate LogicAsker on six widely deployed LLMs, including GPT-3, ChatGPT, GPT-4, Bard, Vicuna, and Guanaco. The results show that test cases from LogicAsker can find logical reasoning failures in different LLMs with a rate of 25\% - 94\%. In addition, the test cases of LogicAsker can be further used to design demonstration examples for in-context learning, which effectively improves the logical reasoning ability of LLMs, e.g., 10\% for GPT-4. As far as we know, our work is the first to create prompts based on testing results to improve LLMs' formal reasoning ability effectively. All the code, data, and results will be released for reproduction and future research

The semi-analytic theory and computation of finite-depth standing water waves

We propose a semi-analytic Stokes expansion ansatz for finite-depth standing water waves and devise a recursive algorithm to solve the system of differential equations governing the expansion coefficients. We implement the algorithm on a supercomputer using arbitrary-precision arithmetic. The Stokes expansion introduces hyperbolic trigonometric terms that require exponentiation of power series. We handle this efficiently using Bell polynomials. Under mild assumptions on the fluid depth, we prove that there are no exact resonances, though small divisors may occur. Sudden changes in growth rate in the expansion coefficients are found to correspond to imperfect bifurcations observed when families of standing waves are computed using a shooting method. A direct connection between small divisors in the recursive algorithm and imperfect bifurcations in the solution curves is observed, where the small divisor excites higher-frequency parasitic standing waves that oscillate on top of the main wave. A 109th order Pad\'e approximation maintains 25--30 digits of accuracy on both sides of the first imperfect bifurcation encountered for the unit-depth problem. This suggests that even if the Stokes expansion is divergent, there may be a closely related convergent sequence of rational approximations.Comment: 32 pages, 9 figure

Approximation analysis of CNNs from a feature extraction view

Deep learning based on deep neural networks has been very successful in many practical applications, but it lacks enough theoretical understanding due to the network architectures and structures. In this paper we establish some analysis for linear feature extraction by a deep multi-channel convolutional neural networks (CNNs), which demonstrates the power of deep learning over traditional linear transformations, like Fourier, wavelets, redundant dictionary coding methods. Moreover, we give an exact construction presenting how linear features extraction can be conducted efficiently with multi-channel CNNs. It can be applied to lower the essential dimension for approximating a high dimensional function. Rates of function approximation by such deep networks implemented with channels and followed by fully-connected layers are investigated as well. Harmonic analysis for factorizing linear features into multi-resolution convolutions plays an essential role in our work. Nevertheless, a dedicate vectorization of matrices is constructed, which bridges 1D CNN and 2D CNN and allows us to have corresponding 2D analysis

USFM: A Universal Ultrasound Foundation Model Generalized to Tasks and Organs towards Label Efficient Image Analysis

Inadequate generality across different organs and tasks constrains the application of ultrasound (US) image analysis methods in smart healthcare. Building a universal US foundation model holds the potential to address these issues. Nevertheless, the development of such foundational models encounters intrinsic challenges in US analysis, i.e., insufficient databases, low quality, and ineffective features. In this paper, we present a universal US foundation model, named USFM, generalized to diverse tasks and organs towards label efficient US image analysis. First, a large-scale Multi-organ, Multi-center, and Multi-device US database was built, comprehensively containing over two million US images. Organ-balanced sampling was employed for unbiased learning. Then, USFM is self-supervised pre-trained on the sufficient US database. To extract the effective features from low-quality US images, we proposed a spatial-frequency dual masked image modeling method. A productive spatial noise addition-recovery approach was designed to learn meaningful US information robustly, while a novel frequency band-stop masking learning approach was also employed to extract complex, implicit grayscale distribution and textural variations. Extensive experiments were conducted on the various tasks of segmentation, classification, and image enhancement from diverse organs and diseases. Comparisons with representative US image analysis models illustrate the universality and effectiveness of USFM. The label efficiency experiments suggest the USFM obtains robust performance with only 20% annotation, laying the groundwork for the rapid development of US models in clinical practices.Comment: Submit to MedIA, 17 pages, 11 figure

SAR Despeckling via Regional Denoising Diffusion Probabilistic Model

Speckle noise poses a significant challenge in maintaining the quality of synthetic aperture radar (SAR) images, so SAR despeckling techniques have drawn increasing attention. Despite the tremendous advancements of deep learning in fixed-scale SAR image despeckling, these methods still struggle to deal with large-scale SAR images. To address this problem, this paper introduces a novel despeckling approach termed Region Denoising Diffusion Probabilistic Model (R-DDPM) based on generative models. R-DDPM enables versatile despeckling of SAR images across various scales, accomplished within a single training session. Moreover, The artifacts in the fused SAR images can be avoided effectively with the utilization of region-guided inverse sampling. Experiments of our proposed R-DDPM on Sentinel-1 data demonstrates superior performance to existing methods.Comment: 5 pages, 5 figure

Towards 6G MIMO: Massive Spatial Multiplexing, Dense Arrays, and Interplay Between Electromagnetics and Processing

The increasing demand for wireless data transfer has been the driving force behind the widespread adoption of Massive MIMO (multiple-input multiple-output) technology in 5G. The next-generation MIMO technology is now being developed to cater to the new data traffic and performance expectations generated by new user devices and services in the next decade. The evolution towards "ultra-massive MIMO (UM-MIMO)" is not only about adding more antennas but will also uncover new propagation and hardware phenomena that can only be treated by jointly utilizing insights from the communication, electromagnetic (EM), and circuit theory areas. This article offers a comprehensive overview of the key benefits of the UM-MIMO technology and the associated challenges. It explores massive multiplexing facilitated by radiative near-field effects, characterizes the spatial degrees-of-freedom, and practical channel estimation schemes tailored for massive arrays. Moreover, we provide a tutorial on EM theory and circuit theory, and how it is used to obtain physically consistent antenna and channel models. Subsequently, the article describes different ways to implement massive and dense antenna arrays, and how to co-design antennas with signal processing. The main open research challenges are identified at the end.Comment: Submitted to Proceedings of the IEEE, 36 pages, 23 figure

PAHD: Perception-Action based Human Decision Making using Explainable Graph Neural Networks on SAR Images

Synthetic Aperture Radar (SAR) images are commonly utilized in military applications for automatic target recognition (ATR). Machine learning (ML) methods, such as Convolutional Neural Networks (CNN) and Graph Neural Networks (GNN), are frequently used to identify ground-based objects, including battle tanks, personnel carriers, and missile launchers. Determining the vehicle class, such as the BRDM2 tank, BMP2 tank, BTR60 tank, and BTR70 tank, is crucial, as it can help determine whether the target object is an ally or an enemy. While the ML algorithm provides feedback on the recognized target, the final decision is left to the commanding officers. Therefore, providing detailed information alongside the identified target can significantly impact their actions. This detailed information includes the SAR image features that contributed to the classification, the classification confidence, and the probability of the identified object being classified as a different object type or class. We propose a GNN-based ATR framework that provides the final classified class and outputs the detailed information mentioned above. This is the first study to provide a detailed analysis of the classification class, making final decisions more straightforward. Moreover, our GNN framework achieves an overall accuracy of 99.2\% when evaluated on the MSTAR dataset, improving over previous state-of-the-art GNN methods