Ridge surfaces represent important features for the analysis of 3-dimensional (3D) datasets in diverse applications and are often derived from varying underlying data including flow fields, geological fault data, and point data, but they can also be present in the original scalar images acquired using a plethora of imaging techniques. Our work is motivated by the analysis of image data acquired using micro-computed tomography (Micro-CT) of ancient, rolled and folded thin-layer structures such as papyrus, parchment, and paper as well as silver and lead sheets. From these documents we know that they are 2-dimensional (2D) in nature. Hence, we are particularly interested in reconstructing 2D manifolds that approximate the document's structure. The image data from which we want to reconstruct the 2D manifolds are often very noisy and represent folded, densely-layered structures with many artifacts, such as ruptures or layer splitting and merging. Previous ridge-surface extraction methods fail to extract the desired 2D manifold for such challenging data. We have therefore developed a novel method to extract 2D manifolds. The proposed method uses a local fast marching scheme in combination with a separation of the region covered by fast marching into two sub-regions. The 2D manifold of interest is then extracted as the surface separating the two sub-regions. The local scheme can be applied for both automatic propagation as well as interactive analysis. We demonstrate the applicability and robustness of our method on both artificial data as well as real-world data including folded silver and papyrus sheets.
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Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued fields. In particular, we prove that the unit ball with respect to a discrete valuation on a field is a discrete valuation ring and, conversely, that the adic valuation on the field of fractions of a discrete valuation ring is discrete. We define finite extensions of valuations and of discrete valuation rings, and prove some global-to-local results. Building on this general theory, we formalize the abstract definition and some fundamental properties of local fields. As an application, we show that finite extensions of the field $\mathbb{Q}_p$ of $p$-adic numbers and of the field $\mathbb{F}_p(\!(X)\!)$ of Laurent series over $\mathbb{F}_p$ are local fields.
Text-To-Image (TTI) models, exemplified by DALL-E and StableDiffusion, have recently gained prominence for their remarkable zero-shot capabilities in generating images guided by textual prompts. Language, as a conduit of culture, plays a pivotal role in these models' multilingual capabilities, which in turn shape their cultural agency. In this study, we explore the cultural perception embedded in TTI models by characterizing culture across three hierarchical tiers: cultural dimensions, cultural domains, and cultural concepts. We propose a comprehensive suite of evaluation techniques, including intrinsic evaluations using the CLIP space, extrinsic evaluations with a Visual-Question-Answer (VQA) model, and human assessments, to discern TTI cultural perceptions. To facilitate our research, we introduce the CulText2I dataset, derived from four diverse TTI models and spanning ten languages. Our experiments reveal insights into these models' cultural awareness, cultural distinctions, and the unlocking of cultural features, releasing the potential for cross-cultural applications.
We consider relational semantics (R-models) for the Lambek calculus extended with intersection and explicit constants for zero and unit. For its variant without constants and a restriction which disallows empty antecedents, Andreka and Mikulas (1994) prove strong completeness. We show that it fails without this restriction, but, on the other hand, prove weak completeness for non-standard interpretation of constants. For the standard interpretation, even weak completeness fails. The weak completeness result extends to an infinitary setting, for so-called iterative divisions (Kleene star under division). We also prove strong completeness results for product-free fragments.
This paper rigorously shows how over-parameterization changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where $M^* \in \mathbb{R}^{n \times n}$ is a positive semi-definite unknown matrix of rank $r \ll n$, and one uses a symmetric parameterization $XX^\top$ to learn $M^*$. Here $X \in \mathbb{R}^{n \times k}$ with $k > r$ is the factor matrix. We give a novel $\Omega (1/T^2)$ lower bound of randomly initialized GD for the over-parameterized case ($k >r$) where $T$ is the number of iterations. This is in stark contrast to the exact-parameterization scenario ($k=r$) where the convergence rate is $\exp (-\Omega (T))$. Next, we study asymmetric setting where $M^* \in \mathbb{R}^{n_1 \times n_2}$ is the unknown matrix of rank $r \ll \min\{n_1,n_2\}$, and one uses an asymmetric parameterization $FG^\top$ to learn $M^*$ where $F \in \mathbb{R}^{n_1 \times k}$ and $G \in \mathbb{R}^{n_2 \times k}$. Building on prior work, we give a global exact convergence result of randomly initialized GD for the exact-parameterization case ($k=r$) with an $\exp (-\Omega(T))$ rate. Furthermore, we give the first global exact convergence result for the over-parameterization case ($k>r$) with an $\exp(-\Omega(\alpha^2 T))$ rate where $\alpha$ is the initialization scale. This linear convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from $\Omega (1/T^2)$ to linear convergence. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of $\alpha$, recovering the rate in the exact-parameterization case.
This paper proposes an unsupervised method that leverages topological characteristics of data manifolds to estimate class separability of the data without requiring labels. Experiments conducted in this paper on several datasets demonstrate a clear correlation and consistency between the class separability estimated by the proposed method with supervised metrics like Fisher Discriminant Ratio~(FDR) and cross-validation of a classifier, which both require labels. This can enable implementing learning paradigms aimed at learning from both labeled and unlabeled data, like semi-supervised and transductive learning. This would be particularly useful when we have limited labeled data and a relatively large unlabeled dataset that can be used to enhance the learning process. The proposed method is implemented for language model fine-tuning with automated stopping criterion by monitoring class separability of the embedding-space manifold in an unsupervised setting. The proposed methodology has been first validated on synthetic data, where the results show a clear consistency between class separability estimated by the proposed method and class separability computed by FDR. The method has been also implemented on both public and internal data. The results show that the proposed method can effectively aid -- without the need for labels -- a decision on when to stop or continue the fine-tuning of a language model and which fine-tuning iteration is expected to achieve a maximum classification performance through quantification of the class separability of the embedding manifold.
The uniform one-dimensional fragment of first-order logic was introduced a few years ago as a generalization of the two-variable fragment of first-order logic to contexts involving relations of arity greater than two. Quantifiers in this logic are used in blocks, each block consisting only of existential quantifiers or only of universal quantifiers. In this paper we consider the possibility of mixing quantifiers in blocks. We identify a non-trivial variation of the logic with mixed blocks of quantifiers which retains some good properties of the two-variable fragment and of the uniform one-dimensional fragment: it has the finite (exponential) model property and hence decidable, NExpTime-complete satisfiability problem.
Performance analysis is carried out in a near-field multiple-input multiple-output (MIMO) system for both discrete and continuous aperture antennas. The effective degrees of freedom (EDoF) is first derived. It is shown that near-field MIMO systems have a higher EDoF than free-space far-field ones. Additionally, the near-field EDoF further depends on the communication distance. Based on the derived EDoF, closed-form expressions of channel capacity with a fixed distance are obtained. As a further advance, with randomly deployed receivers, ergodic capacity is derived. Simulation results reveal that near-field MIMO has an enhanced multiplexing gain even under line-of-sight transmissions. In addition, the performance of discrete MIMO converges to that of continuous aperture MIMO.
The accurate representation and prediction of physical phenomena through numerical computer codes remains to be a vast and intricate interdisciplinary topic of research. Especially within the last decades, there has been a considerable push toward high performance numerical schemes to solve partial differential equations (PDEs) from the applied mathematics and numerics community. The resulting landscape of choices regarding numerical schemes for a given system of PDEs can thus easily appear daunting for an application expert that is familiar with the relevant physics, but not necessarily with the numerics. Bespoke high performance schemes in particular pose a substantial hurdle for domain scientists regarding their theory and implementation. Here, we propose a unifying scheme for grid based approximation methods to address this issue. We introduce some well defined restrictions to systematically guide an application expert through the process of classifying a given multiphysics problem, identifying suitable numerical schemes and implementing them. We introduce a fixed set of input parameters, amongst them for example the governing equations and the hardware configuration. This method not only helps to identify and assemble suitable schemes, but enables the unique combination of multiple methods on a per field basis. We exemplarily demonstrate this process and its effectiveness using different approaches and systematically show how one should exploit some given properties of a PDE problem to arrive at an efficient compound discretisation.
Current synthetic speech detection (SSD) methods perform well on certain datasets but still face issues of robustness and interpretability. A possible reason is that these methods do not analyze the deficiencies of synthetic speech. In this paper, the flaws of the speaker features inherent in the text-to-speech (TTS) process are analyzed. Differences in the temporal consistency of intra-utterance speaker features arise due to the lack of fine-grained control over speaker features in TTS. Since the speaker representations in TTS are based on speaker embeddings extracted by encoders, the distribution of inter-utterance speaker features differs between synthetic and bonafide speech. Based on these analyzes, an SSD method based on temporal consistency and distribution of speaker features is proposed. On one hand, modeling the temporal consistency of intra-utterance speaker features can aid speech anti-spoofing. On the other hand, distribution differences in inter-utterance speaker features can be utilized for SSD. The proposed method offers low computational complexity and performs well in both cross-dataset and silence trimming scenarios.
Knowledge graph reasoning (KGR), aiming to deduce new facts from existing facts based on mined logic rules underlying knowledge graphs (KGs), has become a fast-growing research direction. It has been proven to significantly benefit the usage of KGs in many AI applications, such as question answering and recommendation systems, etc. According to the graph types, the existing KGR models can be roughly divided into three categories, \textit{i.e.,} static models, temporal models, and multi-modal models. The early works in this domain mainly focus on static KGR and tend to directly apply general knowledge graph embedding models to the reasoning task. However, these models are not suitable for more complex but practical tasks, such as inductive static KGR, temporal KGR, and multi-modal KGR. To this end, multiple works have been developed recently, but no survey papers and open-source repositories comprehensively summarize and discuss models in this important direction. To fill the gap, we conduct a survey for knowledge graph reasoning tracing from static to temporal and then to multi-modal KGs. Concretely, the preliminaries, summaries of KGR models, and typical datasets are introduced and discussed consequently. Moreover, we discuss the challenges and potential opportunities. The corresponding open-source repository is shared on GitHub: //github.com/LIANGKE23/Awesome-Knowledge-Graph-Reasoning.
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