Tight Frame Research Articles

On Approximation by Tight Wavelet Frames on the Field of $$p$$-Adic Numbers

On Approximation by Tight Wavelet Frames on the Field of $$p$$-Adic Numbers

Policy whiplash: How California Title IX coordinators navigated local, state, and federal policy changes during the Trump administration

AbstractFederal Title IX policy requires institutions of higher education (IHEs) to prevent and respond to sexual misconduct. Based on 23 in‐depth interviews, this study explores California Title IX coordinator experiences at a critical policy juncture—the recension of the Obama‐era guidance and the implementation of the Trump administration Title IX rules and regulations—to understand how they responded to, implemented, or resisted federal mandates. Most California Title IX officers reported that the recension of Obama guidelines had limited impact on their daily operations because they lived in a state with more progressive social policy. Participants singled out California case law, state statues, the California Education Code and university system mandates as protective legislation to ensure survivors’ rights. By contrast, the new Title IX regulations that became law in August 2020 were perceived as more consequential. The tight implementation time frame, combined with the complicated and unfunded policy directives, imposed heavy burdens on their administrative offices. Title IX officers foresaw implementation consequences to campus safety, survivors’ willingness to report, and to the credibility of the Title IX office. Taken together, their experiences lend support to progressive state legislative action, restorative justice approaches and for a survivor bill of rights as important counterpoints to federal Title IX policy.

The entanglement criteria based on equiangular tight frames

Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite states. Here we derive some entanglement criteria based on positive operator-valued measurements built from equiangular tight frames. We also present a class of entanglement witnesses based on the equiangular tight frames. At last, we generalize the entanglement criterion for bipartite systems to multipartite systems.

Localization of window functions of dual and tight gabor frames generated by the Gaussian function

Рассматриваются фреймы Габора, порожденные функцией Гаусса. С помощью констант неопределенности оценивается локализация функций двойственных фреймов в зависимости от соотношения параметров частотно-временного окна и степени переполненности. Общий вывод таков: при увеличении диспропорции окна локализация быстро ухудшается. С другой стороны, чем более переопределена исходная система функций, тем лучше локализованы функции двойственного фрейма. Для жесткого фрейма локализация при одном и том же наборе параметров существенно лучше, чем для двойственного фрейма. Рассматриваемая задача тесно связана с задачей интерполяции по равномерным сдвигам функции Гаусса. Построение узловой функции при интерполяции и функции окна двойственного фрейма осуществляется с помощью одних и тех же коэффициентов. Эти коэффициенты играют важную роль и при выводе формул для констант неопределенности. Поэтому в работе изучаются их свойства, связанные со знакочередуемостью и монотонностью убывания по модулю. Библиография: 38 названий.

Fast deep learning with tight frame wavelets

Fast deep learning with tight frame wavelets

Estimation of q for ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization in signal recovery with tight frame

This study aims to reconstruct signals that are sparse with a tight frame from undersampled data by using the ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization method. This problem can be cast as a ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization problem with a tight frame subjected to an undersampled measurement with a known noise bound. We proved that if the measurement matrix satisfies the restricted isometry property with δ2s≤1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\delta _{2s}\\leq 1/2$\\end{document}, there exists a value q0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q_{0}$\\end{document} such that for any q∈(0,q0]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q\\in (0,q_{0}]$\\end{document}, any signal that is s-sparse with a tight frame can be robustly recovered to the true signal. We estimated q0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q_{0}$\\end{document} as q0=2/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q_{0} = 2/3$\\end{document} in the case of δ2s≤1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\delta _{2s}\\leq 1/2$\\end{document} and discussed that the value of q0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q_{0}$\\end{document} can be much higher. We also showed that when δ2s≤0.3317\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\delta _{2s}\\leq 0.3317$\\end{document}, for any q∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q\\in (0,1]$\\end{document}, robust recovery for signals via ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization holds, which is consistent with the case of ℓq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{q}$\\end{document}-minimization without a tight frame.

Piecewise scalable frames in Hilbert spaces

Tight frames are extremely useful in applications. A scalable frame was recently introduced as a frame with the property of generating a tight frame by rescaling its frame vectors. In this paper, we consider piecewise scalable frames. We obtain some characterizations for them, and demonstrate that scalability is stable under unitary operators and isomorphisms between two Hilbert spaces. We further obtain a relation between the piecewise scalable frames in Hilbert spaces, and their tensor product

A metaheuristic MAC protocol for safety applications in cognitive vehicular networks

In a dynamic cognitive vehicular network (CVN) environment, ensuring reliable data delivery of safety messages (sm) is crucial. However, fast-moving vehicles and rapidly changing channel accessibility make sm broadcasting challenging. To address this issue, we propose a cooperative make-up technique, which leverages the 2-Hop Harris Hawks Optimization (HHO) algorithm to choose an optimal helper for sm retransmission in case of failed direct transmission. Our simulations using the Network Simulator version 2 with SUMO demonstrate that the proposed approach achieves cooperative communication within the tight 100ms sm broadcast time frame and outperforms contemporary CVN protocols regarding throughput and packet delivery ratio. Our results suggest that our approach holds great promise for reliable and efficient data delivery in CVNs.

On Kirkwood–Dirac quasiprobabilities and unravelings of quantum channel assigned to a tight frame

On Kirkwood–Dirac quasiprobabilities and unravelings of quantum channel assigned to a tight frame

Problem-Dependent Power of Quantum Neural Networks on Multiclass Classification.

Quantum neural networks (QNNs) have become an important tool for understanding the physical world, but their advantages and limitations are not fully understood. Some QNNs with specific encoding methods can be efficiently simulated by classical surrogates, while others with quantum memory may perform better than classical classifiers. Here we systematically investigate the problem-dependent power of quantum neural classifiers (QCs) on multiclass classification tasks. Through the analysis of expected risk, a measure that weighs the training loss and the generalization error of a classifier jointly, we identify two key findings: first, the training loss dominates the power rather than the generalization ability; second, QCs undergo a U-shaped risk curve, in contrast to the double-descent risk curve of deep neural classifiers. We also reveal the intrinsic connection between optimal QCs and the Helstrom bound and the equiangular tight frame. Using these findings, we propose a method that exploits loss dynamics of QCs to estimate the optimal hyperparameter settings yielding the minimal risk. Numerical results demonstrate the effectiveness of our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and their limitations over convolutional neural networks on image datasets. Our work sheds light on the problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.

Accelerated ISAR Sparse Imaging Based on Tight Frame and Projection Operator

Accelerated ISAR Sparse Imaging Based on Tight Frame and Projection Operator

Challenges for geomechanical laboratory analyses of claystones with regard to the time frame of the site selection process

Abstract. The current programme for the site selection process in Germany schedules 11 months for geomechanical laboratory tests per potential future site (BGE, 2022). For the laboratory analysis of rock salt, this time frame seems realistic as no hydromechanical coupling has to be taken into account. In the case of a crystalline basement, the intact rock matrix is low permeable, but transport processes are dominated by ubiquitous fracture networks, which are hard to capture in laboratory settings. Thus, the effect that hydromechanical coupling in low permeable rock has on the speed of laboratory measurements will probably be no problem. However, for claystones, the geomechanical processes within the rock matrix strongly depend on the hydromechanical coupling, which thus determines the speed with which, for example, loading ramps can be applied in the laboratory. The need to maintain the pore pressure equilibrium during the experiments in order to know the acting effective stresses slows down the speed of the measurements significantly. Given the tight time frame of 11 months and the numerous parameters which have to be analysed according to law (StandAG, 2017), the low speed with which the experiments can be performed represents a severe hindrance. It is thus important to establish measurement protocols which enable the simultaneous determination of several parameters in order to save time. Based on our experience with experiments on Opalinus Clay, we offer a first idea of how such a measurement protocol could look and would like to invite a discussion on how to optimise experiments on claystones further in order to have a chance to meet the very tight time frame required by the site selection process.

Provable deep video denoiser using spatial–temporal information for video snapshot compressive imaging: Algorithm and convergence analysis

Video snapshot compressive imaging (SCI) is a new compressive imaging system that aims to recover multiple video frames from a single measurement. Recent plug-and-play (PnP) imaging methods have been used for solving SCI inverse problems by leveraging pre-trained deep Gaussian denoisers. Under the condition of diminishing noise levels, a necessary assumption for the fixed-point convergence of PnP imaging algorithms is that the plugged denoisers are bounded denoisers. However, it is difficult to prove that existing deep Gaussian denoisers meet this assumption due to the complex network architectures, limiting the convergence analysis of these algorithms. This paper aims to elaborate a bounded deep video denoiser to remedy such a gap. Concretely, we propose a denoiser using double tight frames dubbed as VDTF and plug it into the PnP framework to construct the PnP-VDTF algorithm. In VDTF, a constant network (CNet) is designed, wherein a Swin Transformer-based module and a 3D convolution module extract the spatial information of the global features and the temporal information of the input video, respectively. Theoretically, we provide a strict boundary proof of VDTF, and show that PnP-VDTF can generate fixed-point convergent trajectories. Experiments show that PnP-VDTF can achieve higher-quality reconstructions compared with benchmark SCI algorithms.

Tight wavelet frames on zero-dimensional groups. Construction and approximation

We present a method for constructing tight wavelet frames on locally compact zero-dimensional groups. When constructing frames, we do not use the principle of unitary extension. We also consider the approximate properties of the resulting frames for functions from the Sobolev space with logarithmic weight.

Construction of Mercedes-Benz frames via QR factorization

Finite tight frames are used widely for many applications. An important problem is to construct finite tight frames containing specified elements as needed in real world applications. The question was first studied in details by D. Feng, L. Wang, and Y. Wang in 2007. P. G. Casazza, M. Ficus, D. G. Mixon, J. Peterson, and I. Smalyanau gave a method for constructing a tight frame containing specified elements in 2013. A simple and stable method for constructing equi-norm tight frames from the specified elements was derived by J. Ji and L. Wang in 2022. As the main result, an algorithm based on the QR factorization for completing a Mercedes-Benz frame containing specified elements is presented in this paper and two examples of Mercedes-Benz frames in R4 are constructed to illustrate our approach.

Gradient Flows for Probabilistic Frame Potentials in the Wasserstein Space

In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory, and optimal transport, using the methods of the second to answer questions posed in the first. In particular, we construct gradient flows in the Wasserstein space for a new potential, the tightness potential, which is a modification of the probabilistic frame potential. It is shown that the potential is suited for the application of a gradient descent scheme from optimal transport that can be used as the basis of an algorithm to evolve an existing frame toward a tight probabilistic frame.

Hybrid character sums and near-optimal partial Hadamard codebooks

In this paper, we first study exponential sums of the form∑x∈Fqψ(f(x))χ(g(x)) called hybrid character sums, where f,g are functions over finite field Fq, ψ is a nontrivial multiplicative character and χ is a nontrivial additive character of Fq. The absolute values or explicit values of them are determined under certain conditions. We then present several constructions of complex codebooks which are also called frames. The proposed codebooks are constructed from Hadamard matrices with binary row selection sequences. The hybrid character sums play an important role in determining the maximum magnitude of the inner products between vectors in these codebooks. It is proved that the magnitude of the inner products between distinct code vectors of each codebook is two-valued, and the maximum magnitude of each codebook asymptotically achieves the Welch bound equality. The constructed codebooks are equivalent to nearly equiangular tight frames and have nice application in compressed sensing matrices.

Ensuring quality control in a COVID-19 clinical trial during the pandemic: The experience of the Inserm C20–15 DisCoVeRy study

SettingHealth measures taken during the pandemic deeply modified the clinical research practices. At the same time, the demand for the results of the COVID-19 trials was urgent. Thus, the objective of this article is to share Inserm's experience in ensuring quality control in clinical trials in this challenging context. ObjectivesDisCoVeRy is a phase III randomized study that aimed at evaluating the safety and efficacy of 4 therapeutic strategies in hospitalized COVID-19 adult patients. Between March, 22nd 2020 and January, 20th 2021, 1309 patients were included. In order to guarantee the best quality of data, the Sponsor had to adapt to the current sanitary measures and to their impact on clinical research activity, notably by adapting Monitoring Plan objectives, involving the research departments of the participating hospitals and a network of clinical research assistants (CRAs). ResultsOverall, 97 CRAs were involved and performed 909 monitoring visits. The monitoring of 100% of critical data for all patients included in the analysis was achieved, and despite of the pandemic context, a conform consent was recovered for more than 99% of patients. Results of the study were published in May and September 2021. Discussion/conclusionThe main monitoring objective was met thanks to the mobilization of considerable personnel resources, within a very tight time frame and external hurdles. There is a need for further reflection to adapt the lessons learned from this experience to the context of routine practice and to improve the response of French academic research during a future epidemic.

Entropic uncertainty relations from equiangular tight frames and their applications

Finite tight frames are interesting in various topics including questions of quantum information. Each complex tight frame leads to a resolution of the identity in the Hilbert space. Symmetric informationally complete measurements are a special class of equiangular tight frames. Applications of such frames in quantum physics deserve more attention than they have obtained. We derive uncertainty relations for a quantum measurement assigned to an equiangular tight frame. Main results follow from estimation of the corresponding index of coincidence. State-dependent and state-independent formulations are both addressed. Also, we discuss applications of considered measurements to detect entanglement and other correlations.

Implementing workplace wellness in healthcare settings: lessons learned from the SEED wellness team reflective journaling

Background: This article shares the reflection journey of a newly formed workplace wellness team, whose members documented their experience throughout an eight-week project using reflective journaling. Reflection was seen as a valuable tool for developing greater self-awareness and insight into team dynamics. Sharing reflections among the team also served to strengthen bonds and provided a chance to discuss challenges that were faced. Aim: To demonstrate the importance of promoting and prioritising the practice of reflection amid the high workload intensity of a busy healthcare environment. Conclusion: A diverse group of healthcare workers was able to identify the value of dedicated worktime for reflection while meeting the demands of project targets and tight time frames. Implications for practice: Personal and team reflective practice offers teambuilding opportunities for newly formed teams along with time for confidence building, self-improvement and connecting with others Timely documentation of these reflections can give immediate insight into challenges individuals or the entire team may be experiencing, thus allowing prompt response and adjustments, in addition to creating a record for personal and team learnings over time To ensure engagement in, and sustainability of, wellness-oriented reflective practice, both bottom-up and top-down commitment is required