Optimal Bound Research Articles

Optimal Bounds and Practical Insights: Cantelli’s Inequality Revisited

Optimal Bounds and Practical Insights: Cantelli’s Inequality Revisited

Massive MIMO uplink and downlink joint representation based on couple dictionary learning

AbstractThe challenge of jointly representing both the uplink (UL) and downlink (DL) in massive multiple input multiple output (MIMO) systems have been tackled. Considering the angular reciprocity, a couple dictionary learning (CDL) support to enhance performance and address high complexity has been introduced. This approach minimizes the number of pilots and improves accuracy. Currently, accuracy analysis of UL/DL representation primarily relies on simulation. To bridge this gap, a proportion factor (PF) operator is proposed for CDL to assess accuracy. Specifically, a qualitative analysis formula is provided for accuracy and an optimal upper bound is established. Through theoretical proof, it is demonstrated that the accuracy of CDL for representation is mainly influenced by the cross‐correlation between the pilot matrix and the dictionary matrix. Inspired by PF operator, an optimal couple dictionary learning (OCDL) algorithm using singular value decomposition (SVD) is introduced to obtain dictionary updating, aiming at high‐precision representation. By establishing normalized mean squared error (NMSE), successful representation ratio, bit error rate (BER), and constellation performance, an OCDL algorithm that outperforms existing methods is showcased and channel representation accuracy is enhanced significantly.

Minimization of the lowest positive Neumann-Dirichlet eigenvalue for general indefinite Sturm-Liouville problems

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue λ0ND+ for the general Sturm–Liouville problemy″=q(t)y+λm(t)y, with the Neumann-Dirichlet boundary conditions, where q is a nonnegative potential and another potential m admits to change sign. First, we will study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations to make our results more applicable. Second, based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for the general Sturm–Liouville equation.

Almost Optimal Upper Bound for the Ground State Energy of a Dilute Fermi Gas via Cluster Expansion

AbstractWe prove an upper bound on the energy density of the dilute spin-$$\frac{1}{2}$$ 1 2 Fermi gas capturing the leading correction to the kinetic energy $$8\pi a \rho _\uparrow \rho _\downarrow $$ 8 π a ρ ↑ ρ ↓ with an error of size smaller than $$a\rho ^{2}(a^3\rho )^{1/3-\varepsilon }$$ a ρ 2 ( a 3 ρ ) 1 / 3 - ε for any $$\varepsilon > 0$$ ε > 0 , where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3).

HR-YOLO: A Multi-Branch Network Model for Helmet Detection Combined with High-Resolution Network and YOLOv5

Automatic detection of safety helmet wearing is significant in ensuring safe production. However, the accuracy of safety helmet detection can be challenged by various factors, such as complex environments, poor lighting conditions and small-sized targets. This paper presents a novel and efficient deep learning framework named High-Resolution You Only Look Once (HR-YOLO) for safety helmet wearing detection. The proposed framework synthesizes safety helmet wearing information from the features of helmet objects and human pose. HR-YOLO can use features from two branches to make the bounding box of suppression predictions more accurate for small targets. Then, to further improve the iterative efficiency and accuracy of the model, we design an optimized residual network structure by using Optimized Powered Stochastic Gradient Descent (OP-SGD). Moreover, a Laplace-Aware Attention Model (LAAM) is designed to make the YOLOv5 decoder pay more attention to the feature information from human pose and suppress interference from irrelevant features, which enhances network representation. Finally, non-maximum suppression voting (PA-NMS voting) is proposed to improve detection accuracy for occluded targets, using pose information to constrain the confidence of bounding boxes and select optimal bounding boxes through a modified voting process. Experimental results demonstrate that the presented safety helmet detection network outperforms other approaches and has practical value in application scenarios. Compared with the other algorithms, the proposed algorithm improves the precision, recall and mAP by 7.27%, 5.46% and 7.3%, on average, respectively.

The irregularity strength of dense graphs — on asymptotically optimal solutions of problems of Faudree, Jacobson, Kinch and Lehel

The irregularity strength of a graph G, s(G), is the least k such that there exists a {1,2,…,k}-weighting of the edges of G attributing distinct weighted degrees to all vertices, or equivalently the least k allowing the creation of a multigraph with nonrecurring degrees by blowing each edge e of G to at most k copies of e. In 1991 Faudree, Jacobson, Kinch and Lehel asked for the optimal lower bound for the minimum degree of a graph G of order n which implies that s(G)≤3. More generally, they also posed a similar question regarding the upper bound s(G)≤K for any given constant K. We provide asymptotically tight solutions of these problems by proving that such optimal lower bound is of order 1K−1n for every fixed integer K≥3.

Optimal Error Bounds in the Absence of Constraint Qualifications with Applications to p-Cones and Beyond

We prove tight Hölderian error bounds for all p-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured. Moreover, they illuminate p-cones as a curious example of a class of objects that possess properties in three dimensions that they do not in four or more. Using our error bounds, we analyse least squares problems with p-norm regularization, where our results enable us to compute the corresponding Kurdyka–Łojasiewicz exponents for previously inaccessible values of p. Another application is a (relatively) simple proof that most p-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight. Funding: The second author was partly supported by the Japan Society for the Promotion of Science Grant-in-Aid for Early-Career Scientists [Grants 19K20217, 23K16844] and Grant-in-Aid for Scientific Research [Grant (B)21H03398]. The third author was partly supported by the Hong Kong Research Grants Council [Grant PolyU153000/20p].

Evaluating the quantum optimal biased bound in a unitary evolution process

Seeking the available precision limit of unknown parameters is a significant task in quantum parameter estimation. One often resorts to the widely utilized quantum Cramér–Rao bound (QCRB) based on unbiased estimators to finish this task. Nevertheless, most actual estimators are usually biased in the limited number of trials. For this reason, we introduce two effective error bounds for biased estimators based on a unitary evolution process in the framework of the quantum optimal biased bound. Furthermore, we show their estimation performance by two specific examples of the unitary evolution process, including the phase encoding and the SU(2) interferometer process. Our findings will provide an useful guidance for finding the precision limit of unknown parameters.

Co-designing versatile quadruped robots for dynamic and energy-efficient motions

Abstract This paper presents a concurrent optimization approach for the design and motion of a quadruped in order to achieve energy-efficient cyclic behaviors. Computational techniques are applied to improve the development of a novel quadruped prototype. The scale of the robot and its actuators are optimized for energy efficiency considering the complete actuator model including friction, torque, and bandwidth limitations. This method and the optimal bounding trajectories are tested on the first (non-optimized) prototype design iteration showing that our formulation produces a trajectory that (i) can be easily replayed on the real robot and (ii) reduces the power consumption w.r.t. hand-tuned motion heuristics. Power consumption is then optimized for several periodic tasks with co-design. Our results include, but are not limited to, a bounding and backflip task. It appears that, for jumping forward, robots with longer thighs perform better, while, for backflips, longer shanks are better suited. To explore the tradeoff between these different designs, a Pareto set is constructed to guide the next iteration of the prototype. On this set, we find a new design, which will be produced in future work, showing an improvement of at least 52% for each separate task.

Order Optimal Bounds for One-Shot Federated Learning Over Non-Convex Loss Functions

Order Optimal Bounds for One-Shot Federated Learning Over Non-Convex Loss Functions

Towards Optimal Subsidy Bounds for Envy-Freeable Allocations

We study the fair division of indivisible items with subsidies among n agents, where the absolute marginal valuation of each item is at most one. Under monotone valuations (where each item is a good), it is known that a maximum subsidy of 2(n-1) and a total subsidy of 2(n-1)² are sufficient to guarantee the existence of an envy-freeable allocation. In this paper, we improve upon these bounds, even in a wider model. Namely, we show that, given an EF1 allocation, we can compute in polynomial time an envy-free allocation with a subsidy of at most n-1 per agent and a total subsidy of at most n(n-1)/2. Moreover, we present further improved bounds for monotone valuations.

Crowded pedestrian detection with optimal bounding box relocation

Crowded pedestrian detection with optimal bounding box relocation

Adaptive optimal ℓ∞‐induced robust stabilization of minimum phase SISO plant under bounded disturbance and coprime factor perturbations

AbstractThis paper addresses the problem of optimal robust stabilization of a discrete‐time minimum‐phase plant in the framework of robust control theory in the setup and under poor a priori information. Coefficients of the transfer function of the plant nominal model with stable zeros are unknown and belong to a known bounded polyhedron in the space of coefficients. The gains of coprime factor perturbations of the plant and the upper bound of external disturbance are also unknown. The problem under consideration is to design adaptive controller that minimizes, with the prescribed accuracy, the worst‐case asymptotic upper bound of the output. Solution of the problem is based on set‐membership estimation of unknown parameters and treating the control criterion as the identification criterion. A hard nonconvex problem of on‐line computation of optimal estimates is reduced, under additional nonrestrictive assumption, to a linear‐fractional programming via a nonlinear transformation of estimated parameters. Despite the non‐identifiability of the unknown parameters, the proposed adaptive controller guarantees, with the prescribed accuracy, the same optimal asymptotic upper bound of the output of adaptive system as the optimal controller for the plant with known parameters. In addition to the optimality of adaptive control, the proposed solution provides on‐line verification/validation of current estimates and a priori assumptions.

Notice of Violation of IEEE Publication Principles: Connected Optimal Two-Coverage of Sensor Networks

Notice of Violation of IEEE Publication Principles<br><br>"Connected Optimal Two-Coverage of Sensor Networks,"<br>by Z. Yun, J. Teng, Z. Yu, D. Xuan, B. Chen, W. Zhao,<br>in the IEEE/ACM Transactions on Networking, vol.PP, no.99<br><br>After careful and considered review of the content and authorship of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE's Publication Principles.<br><br>This paper contains significant portions of original text from the paper cited below. The original text was copied without attribution (including appropriate references to the original author(s) and/or paper title) and without permission. One author (Ziqiu Yun) was responsible for the violation, and this violation was done without the knowledge of the other authors.<br><br>"Planar Thinnest Deployment Pattern of Congruent Discs which Achieves 2-coverage"<br>by Ge Jun,<br>in his thesis presented at the School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China, June 2010.<br><br> <br/> In wireless sensor networks, multiple-coverage, in which each point is covered by more than one sensor, is often required to improve detection quality and achieve high fault tolerance. However, finding optimal patterns that achieve multiple-coverage in a plane remains a long-lasting open problem. We first derive the optimal deployment density bound for two-covered deployment patterns where Voronoi polygons generated by sensor nodes are congruent. We then propose optimal two-coverage patterns based on the optimal bound. We further extend these patterns by considering the connectivity requirement and design a set of optimal patterns that achieve two-coverage and one-, two-, and three-connectivity. We also study optimal patterns under practical considerations. To our knowledge, our work is the very first that proves the optimality of multiple-covered deployment patterns.

A UAV-based coverage gap detection and resolution in cellular networks: A machine-learning approach

The use of unmanned aerial vehicles (UAVs) to extend the coverage of terrestrial base stations (BSs) in cellular communication systems has been gaining increasing interest in recent years. This is due to the ease of deploying UAV-mounted BSs within relatively short times with low associated costs. In this work, we formulate the problem of deploying UAV-mounted BSs to mitigate the coverage gaps of the terrestrial BSs of cellular networks in some geographic regions. We then devise a technique that provides the optimal bound of the solution to the coverage gap detection and mitigation problem. We also propose a machine-learning (ML) based technique to provide a real-time solution for deploying UAVs to determine and mitigate the coverage gaps. In both solutions, namely, the optimal and ML-based solutions, the deployment of the UAVs is done in such a way that addresses the coverage gaps at the minimum possible cost. Simulation results show that our ML-based technique performs quite closely to the performance of the optimal solution, at a significantly lower complexity, and hence fulfills the real-time requirements of such deployments. It also provides significantly better performance results than a common benchmark solution from the literature.

Sharp estimates of lowest positive Neumann eigenvalue for general indefinite Sturm-Liouville problems

Given two measures μ,ν and their total variations, we study the minimization of Neumann eigenvalues for measure differential equationdy•=y(t)dμ(t)+λydν(t). By solving the infinitely dimensional minimization problem of the lowest positive Neumann eigenvalue for the measure differential equation, we obtain the optimal lower bound of the lowest positive Neumann eigenvalue for the Sturm-Liouville problemy″=q(t)y+λm(t)y, where q(t) is a nonnegative potential function and the other potential function m(t) admits to change sign.

Optimal diameter estimate of three-dimensional Ricci limit spaces

In this note, we prove that positive scalar curvature can pass to three dimensional Ricci limit spaces of non-negative Ricci curvature when it splits off a line. As a corollary, we obtain an optimal Bonnet-Myers type upper bound. Moreover, we obtain a similar statement in all dimensions for Alexandrov spaces of non-negative curvature.

Fractional strain gradient plasticity and ductile fracture of metals

We present an optimal scaling analysis based on (possibly fractional) strain-gradient plasticity. The analysis yields optimal scaling laws, in the sense of upper and lower bounds of a power-law type with matching exponents, connecting macroscopic fracture properties, such as the critical elongation at failure and the specific fracture energy, to microscopic mechanisms such as cleavage and microplasticity. We show that an optimal upper bound can be derived from an exceedingly simple test deformation that opens up a sheet of parallelepipedic voids. We also show that an optimal lower bound can be obtained by relaxing compatibility between transverse fibers, which effectively renders the analysis one-dimensional. The analysis predicts a ‘gating effect’ of the surface energy. Specifically, a critical surface energy arises from the analysis that marks a sharp transition between brittle and ductile behavior. When the surface energy of the material exceeds the threshold value, the macroscopic specific fracture energy is predicted to rise sharply as a power of the surface energy with an exponent defined precisely by the theory.

Confluence: A Robust Non-IoU Alternative to Non-Maxima Suppression in Object Detection.

Confluence is a novel non-Intersection over Union (IoU) alternative to Non-Maxima Suppression (NMS) in bounding box post-processing in object detection. It overcomes the inherent limitations of IoU-based NMS variants to provide a more stable, consistent predictor of bounding box clustering by using a normalized Manhattan Distance inspired proximity metric to represent bounding box clustering. Unlike Greedy and Soft NMS, it does not rely solely on classification confidence scores to select optimal bounding boxes, instead selecting the box which is closest to every other box within a given cluster and removing highly confluent neighboring boxes. Confluence is experimentally validated on the MS COCO and CrowdHuman benchmarks, improving Average Precision by 0.2--2.7% and 1--3.8% respectively and Average Recall by 1.3--9.3 and 2.4--7.3% when compared against Greedy and Soft-NMS variants. Quantitative results are supported by extensive qualitative analysis and threshold sensitivity analysis experiments support the conclusion that Confluence is more robust than NMS variants. Confluence represents a paradigm shift in bounding box processing, with potential to replace IoU in bounding box regression processes.

FDD Massive MIMO Channel Training: Optimal Rate-Distortion Bounds and the Spectral Efficiency of “One-Shot” Schemes

We study the problem of providing channel state information (CSI) at the transmitter in multi-user “massive” MIMO systems operating in frequency division duplexing (FDD). The wideband MIMO channel is a vector-valued random process correlated in time, space (antennas), and frequency (subcarriers). The base station (BS) broadcasts periodically β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tr</sub> pilot symbols from its <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> antenna ports to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> single-antenna users (UEs). Correspondingly, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> UEs send feedback messages about their channel state using β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fb</sub> symbols in the uplink (UL). Using results from remote rate-distortion theory, we show that, as snr → ∞, the optimal feedback strategy achieves a channel state estimation mean squared error (MSE) that behaves as Θ(1) if β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tr</sub> < <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> and as Θ(snr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">–α</sup> ) when β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tr</sub> ⩾ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> , where α = min(β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fb</sub> / <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> , 1), where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> is the rank of the channel covariance matrix. The MSE-optimal rate-distortion strategy implies encoding of long sequences of channel states, which would yield completely stale CSI and therefore poor multiuser precoding performance. Hence, we consider three practical “one-shot” CSI strategies with minimum one-slot delay and analyze their large-SNR channel estimation MSE behavior. These are: (1) digital feedback via entropy-coded scalar quantization (ECSQ), (2) analog feedback (AF), and (3) local channel estimation at the UEs via compressed sensing and digital feedback. These schemes have different requirements in terms of knowledge of the channel statistics at the UE and at the BS. In particular, the latter strategy requires no statistical knowledge and is closely inspired by a CSI feedback scheme currently proposed in 3GPP standardization. It is shown that ECSQ achieves optimal MSE at the price of a slight increase in feedback rate which vanishes for large SNR. AF achieves the optimal MSE decay rate of Θ(snr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">–1</sup> ) whenever β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tr</sub> , β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fb</sub> ⩾ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> but is sub-optimal if β⩾ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> and β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fb</sub> < <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> . The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3GPP-inspired</i> scheme is shown, via numerical simulations, to achieves performance similar to ECSQ and AF when the multipath channel is sufficiently sparse in the angle-delay domain, but suffers from a large performance gap if this requirement is not met.