For two real symmetric matrices, their eigenvalue configuration is the arrangement of their eigenvalues on the real line. We study the problem of determining a quantifier-free necessary and sufficient condition for two real symmetric matrices to realize a given eigenvalue configuration as a generalization of Descartes' rule of signs. We exploit the combinatorial properties of our definition for eigenvalue configuration to reduce a two-polynomial root counting problem into several single-polynomial root counting problems of symmetric polynomials. We then leverage the fundamental theorem of symmetric polynomials to derive a final quantifier-free necessary and sufficient condition for two real symmetric matrices to realize a given eigenvalue configuration.
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The presence of faulty or underactuated manipulators can disrupt the end-effector formation keeping of a team of manipulators. Based on two-link planar manipulators, we investigate this end-effector formation keeping problem for mixed fully- and under-actuated manipulators with flexible joints. In this case, the underactuated manipulators can comprise of active-passive (AP) manipulators, passive-active (PA) manipulators, or a combination thereof. We propose distributed control laws for the different types of manipulators to achieve and maintain the desired formation shape of the end-effectors. It is achieved by assigning virtual springs to the end-effectors for the fully-actuated ones and to the virtual end-effectors for the under-actuated ones. We study further the set of all desired and reachable shapes for the networked manipulators' end-effectors. Finally, we validate our analysis via numerical simulations.
The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) G\"odel-L\"ob logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong L\"ob logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic.
Explaining generative diffusion models via visual analysis for interpretable decision-making process
Diffusion models have demonstrated remarkable performance in generation tasks. Nevertheless, explaining the diffusion process remains challenging due to it being a sequence of denoising noisy images that are difficult for experts to interpret. To address this issue, we propose the three research questions to interpret the diffusion process from the perspective of the visual concepts generated by the model and the region where the model attends in each time step. We devise tools for visualizing the diffusion process and answering the aforementioned research questions to render the diffusion process human-understandable. We show how the output is progressively generated in the diffusion process by explaining the level of denoising and highlighting relationships to foundational visual concepts at each time step through the results of experiments with various visual analyses using the tools. Throughout the training of the diffusion model, the model learns diverse visual concepts corresponding to each time-step, enabling the model to predict varying levels of visual concepts at different stages. We substantiate our tools using Area Under Cover (AUC) score, correlation quantification, and cross-attention mapping. Our findings provide insights into the diffusion process and pave the way for further research into explainable diffusion mechanisms.
We examine the moments of the number of lattice points in a fixed ball of volume $V$ for lattices in Euclidean space which are modules over the ring of integers of a number field $K$. In particular, denoting by $\omega_K$ the number of roots of unity in $K$, we show that for lattices of large enough dimension the moments of the number of $\omega_K$-tuples of lattice points converge to those of a Poisson distribution of mean $V/\omega_K$. This extends work of Rogers for $\mathbb{Z}$-lattices. What is more, we show that this convergence can also be achieved by increasing the degree of the number field $K$ as long as $K$ varies within a set of number fields with uniform lower bounds on the absolute Weil height of non-torsion elements.
Deep neural networks used for reconstructing sparse-view CT data are typically trained by minimizing a pixel-wise mean-squared error or similar loss function over a set of training images. However, networks trained with such pixel-wise losses are prone to wipe out small, low-contrast features that are critical for screening and diagnosis. To remedy this issue, we introduce a novel training loss inspired by the model observer framework to enhance the detectability of weak signals in the reconstructions. We evaluate our approach on the reconstruction of synthetic sparse-view breast CT data, and demonstrate an improvement in signal detectability with the proposed loss.
Evaluations of model editing currently only use the `next few token' completions after a prompt. As a result, the impact of these methods on longer natural language generation is largely unknown. We introduce long-form evaluation of model editing (\textbf{\textit{LEME}}) a novel evaluation protocol that measures the efficacy and impact of model editing in long-form generative settings. Our protocol consists of a machine-rated survey and a classifier which correlates well with human ratings. Importantly, we find that our protocol has very little relationship with previous short-form metrics (despite being designed to extend efficacy, generalization, locality, and portability into a long-form setting), indicating that our method introduces a novel set of dimensions for understanding model editing methods. Using this protocol, we benchmark a number of model editing techniques and present several findings including that, while some methods (ROME and MEMIT) perform well in making consistent edits within a limited scope, they suffer much more from factual drift than other methods. Finally, we present a qualitative analysis that illustrates common failure modes in long-form generative settings including internal consistency, lexical cohesion, and locality issues.
We show that a constant number of self-attention layers can efficiently simulate, and be simulated by, a constant number of communication rounds of Massively Parallel Computation. As a consequence, we show that logarithmic depth is sufficient for transformers to solve basic computational tasks that cannot be efficiently solved by several other neural sequence models and sub-quadratic transformer approximations. We thus establish parallelism as a key distinguishing property of transformers.
{We analyze a general Implicit-Explicit (IMEX) time discretization for the compressible Euler equations of gas dynamics, showing that they are asymptotic-preserving (AP) in the low Mach number limit. The analysis is carried out for a general equation of state (EOS). We consider both a single asymptotic length scale and two length scales. We then show that, when coupling these time discretizations with a Discontinuous Galerkin (DG) space discretization with appropriate fluxes, an all Mach number numerical method is obtained. A number of relevant benchmarks for ideal gases and their non-trivial extension to non-ideal EOS validate the performed analysis.
Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer from over-smoothing for problems involving very fine mesh. In this work, we develop a multiscale mesh-based GNN framework mimicking a conventional iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to mitigate challenges with conventional mesh-based GNNs. We use the framework to accelerate phase field (PF) fracture problems involving coupled partial differential equations with a near-singular operator due to near-zero modulus inside the crack. We define the initial graph representation using all mesh resolution levels. We perform a series of downsampling steps using Transformer MP GNNs to reach the coarsest graph followed by upsampling steps to reach the original graph. We use skip connectors from the generated embedding during coarsening to prevent over-smoothing. We use Transfer Learning (TL) to significantly reduce the size of training datasets needed to simulate different crack configurations and loading conditions. The trained framework showed accelerated simulation times, while maintaining high accuracy for all cases compared to physics-based PF fracture model. Finally, this work provides a new approach to accelerate a variety of mesh-based engineering multiphysics problems
In recent years, object detection has experienced impressive progress. Despite these improvements, there is still a significant gap in the performance between the detection of small and large objects. We analyze the current state-of-the-art model, Mask-RCNN, on a challenging dataset, MS COCO. We show that the overlap between small ground-truth objects and the predicted anchors is much lower than the expected IoU threshold. We conjecture this is due to two factors; (1) only a few images are containing small objects, and (2) small objects do not appear enough even within each image containing them. We thus propose to oversample those images with small objects and augment each of those images by copy-pasting small objects many times. It allows us to trade off the quality of the detector on large objects with that on small objects. We evaluate different pasting augmentation strategies, and ultimately, we achieve 9.7\% relative improvement on the instance segmentation and 7.1\% on the object detection of small objects, compared to the current state of the art method on MS COCO.
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