Optimal Bound Research Articles

The optimal upper bound of the number of queries for Laplace mechanism under differential privacy

Differential privacy is a state-of-the-art technology for privacy preserving, and Laplace mechanism is a simple and powerful tool to realize differential privacy. However, there is an obvious flaw in differential privacy, which is each query function can only be executed finite times for the reason that adversary can recover the real query result if he executes the same query function many times. Unfortunately, how to set the upper bound for the number of linear queries is still an issue. In this paper, we focus on the linear query function in Laplace-based mechanisms, and we propose a method to set the upper bound for the number of linear queries from the perspective of information theory. The main idea is, firstly we find the most aggressive linear query that leaks the maximum information about the dataset to adversaries, and we set the upper bound of the number of queries so that even if the most aggressive linear query cannot leak the whole self-information about any individual to the adversary. On the other hand, the number of queries is also influenced by the type of dataset (continuous and discrete). In this paper, we also discuss the different upper bound for the number of queries for continuous datasets and discrete datasets. Finally, we conduct the experiments on the continuous dataset and the discrete dataset to prove our result.

Optimal Upper Bounds for the Divergence of Finite-Dimensional Distributions under a Given Variational Distance

We consider the problem of finding the maximum values of divergences D(P‖Q) and D(Q‖P) for probability distributions P and Q ranging in the finite set $$\mathcal{N}=\left\{1,\;2,...,n\right\}$$ provided that both the variation distance V (P,Q) between them and either the probability distribution Q or (in the case of D(P‖Q)) only the value of the minimal component q min of the probability distribution Q are given. Precise expressions for the maximum values of these divergences are obtained. In several cases these expressions allow us to write out some explicit formulas and simple upper and lower bounds for them. Moreover, explicit formulas for the maximum of D(P‖Q) for given V (P,Q) and q min and also for the maximum of D(Q‖P) for given Q and V (P,Q) are obtained for all possible values of these parameters.

IoT Services Allocation at the Edge via Named Data Networking: From Optimal Bounds to Practical Design

Edge computing is a key paradigm to offload the core network and effectively process massive Internet of Things (IoT) raw data without sending them to the cloud. This paradigm normally relies on a set of purpose-built and pre-planned servers, which host storage and processing resources to provide IoT services close to the data sources, thus saving core network resources and offloading the remote cloud infrastructure. In this paper, we propose to turn the network edge into a dynamic, distributed computing environment that supports the provisioning of IoT services, by exploiting the recent evolution of named data networking (NDN), supporting both name-based data retrieval and computation. Specific name structure and novel NDN forwarding mechanisms are designed; a distributed strategy is also engineered to select the service executor among edge nodes, with the objectives to: 1) limit the raw IoT data traffic crossing the network and 2) allocate the service execution according to the nodes’ available processing resources. Numerical analysis shows that the performance of the proposed framework approaches the one of the optimal solution of a formulated integer linear programming problem. System-level ndnSIM simulations confirm that the proposal also outperforms the considered state-of-the-art benchmark solutions in terms of service provisioning time.

The maximum number of systoles for genus two Riemann surfaces with abelian differentials

In this article, we provide bounds on systoles associated to a holomorphic 1-form \omega on a Riemann surface X . In particular, we show that if X has genus two, then, up to homotopy, there are at most 10 systolic loops on (X, \omega) and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus g and a holomorphic 1-form \omega with one zero, we provide the optimal upper bound, 6g-3 , on the number of homotopy classes of systoles. If, in addition, X is hyperelliptic, then we prove that the optimal upper bound is 6g-5 .

Sensor Placement for Optimal Control of Infinite-Dimensional Systems

An important challenge in controlling distributed parameter systems is implementing feedback control laws over an infinite-dimensional space. One widely studied approach is to place sensors at a set of discrete locations and then approximate the state feedback using the sensor outputs. This approach naturally raises the question of where the sensors should be located. In this paper, we investigate the problem of placing a set of sensors on the unit interval in order to minimize the mean-square deviation between a desired infinite-dimensional control law, and an approximate finite-dimensional controller obtained by applying state feedback at the chosen sensor positions. We propose a greedy algorithm and derive optimality bounds on the selected set of sensors. We also present a simplified greedy algorithm in which the incremental improvements from adding each sensor are not updated at each step. We analyze the performance of the approaches under two scenarios. In the case where the sensor placements are constrained such that each of the detection radii of each pair of sensors do not overlap, we show that the two algorithms are equivalent and achieve a 1/2 + ε optimality guarantee, where ε can be made arbitrarily small at the cost of increased computational overhead. When overlaps exist between sensor detection areas, we derive an optimality bound for the simplified algorithm. The value of the optimality bound is determined by the sensor model, cardinality of sensor set, and the allowed minimum sensor distance. Our approach is illustrated through numerical study, in which we compare our proposed greedy algorithm, the current state-of-the-art approach, and the true optimum obtained from exhaustive search.

Amplitude Constrained MIMO Channels: Properties of Optimal Input Distributions and Bounds on the Capacity.

In this work, the capacity of multiple-input multiple-output channels that are subject to constraints on the support of the input is studied. The paper consists of two parts. The first part focuses on the general structure of capacity-achieving input distributions. Known results are surveyed and several new results are provided. With regard to the latter, it is shown that the support of a capacity-achieving input distribution is a small set in both a topological and a measure theoretical sense. Moreover, explicit conditions on the channel input space and the channel matrix are found such that the support of a capacity-achieving input distribution is concentrated on the boundary of the input space only. The second part of this paper surveys known bounds on the capacity and provides several novel upper and lower bounds for channels with arbitrary constraints on the support of the channel input symbols. As an immediate practical application, the special case of multiple-input multiple-output channels with amplitude constraints is considered. The bounds are shown to be within a constant gap to the capacity if the channel matrix is invertible and are tight in the high amplitude regime for arbitrary channel matrices. Moreover, in the regime of high amplitudes, it is shown that the capacity scales linearly with the minimum between the number of transmit and receive antennas, similar to the case of average power-constrained inputs.

Bi-potential and co-rotational formulations applied for real time simulation involving friction and large deformation

This paper concerns mainly numerical aspects for the real-time manipulation of deformable objects in the context of virtual surgery. The bi-potential method in the form of augmented Lagrangian is extended to solve multiple contact problems within real time computing requirements. Nonlinearities of large deformation are also included to describe the mechanical behavior of soft tissues. To obtain both accuracy and realism, the approximated co-rotational finite element method is applied to make a compromise between accuracy and computational cost. It is difficult to detect collisions between virtual objects at interactive rates. In this work, we use an optimal bounding volume hierarchy technique to acquire a good detecting efficiency. Two numerical examples are carried out to illustrate the efficiency, robustness and accuracy of the presented approaches.

Markov factors on average—An [formula omitted] case

The classical Markov polynomial inequality bounds the uniform norm of the derivative of a polynomial on an interval in terms of its degree squared and the norm of the polynomial itself, with the factor of degree squared being the optimal upper bound. Here we study what this factor should be on average, for random polynomials with independent N(0,1) coefficients. In the case of the interval [−1,1] and Jacobi weights defining an L2 space, we show that this average factor is order degree to the 3∕2, as compared to the degree squared worst case upper bound.

On Optimal Bounds in the Local Semicircle Law under Four Moment Condition

We consider symmetric random matrices $${{{\mathbf{X}}}_{n}} = [{{X}_{{jk}}}]_{{j,k = 1}}^{n},n \geqslant 1$$ , whose upper triangular entries are independent random variables with zero mean and unit variance. Under the assumption $$\mathbb{E}{\text{|}}{{X}_{{jk}}}{{{\text{|}}}^{4}} < C$$ , j, k = 1, 2, ..., n, it is shown that the fluctuations of the Stieltjes transform mn(z), $$z = u + i{v},{v} > 0,$$ of the empirical spectral distribution function of the matrix $${{{\mathbf{X}}}_{n}}{\text{/}}\sqrt n $$ about the Stieltjes transform $${{m}_{{{\text{sc}}}}}(z)$$ of Wigner’s semicircle law are of order (n $${v}$$ ) $$^{{ - 1}}\text{ln}n$$ . An application of the result obtained to the convergence rate in probability of the empirical spectral distribution function of $${{{\mathbf{X}}}_{n}}{\text{/}}\sqrt n $$ to Wigner’s semicircle law in the uniform metric is discussed.

Optimized State Estimation of Uncertain Linear Time-Varying Complex Networks With Random Sensor Delay Subject to Uncertain Probabilities

This paper is concerned with the optimal state estimation problem for a class of time-varying uncertain dynamical networks with mixed time-delay under uncertain probabilities. Here, the mixed time-delays include the constant time-delay and the random sensor delay, where the random sensor delay is depicted by a Bernoulli distributed random variable and the occurrence probability of the random sensor delay can be uncertain. The major novelty of the paper lies in that a new time-varying state estimation algorithm is given such that, for all parameter uncertainties, mixed time-delays and uncertain probabilities, a locally optimal upper bound of the estimation error covariance is obtained and the desirable estimator parameter of easy-to-implement feature is designed. Moreover, the performance evaluation problem of the presented estimation algorithm is solved, where the monotonicity analysis is shown regarding the trace of the upper bound and the deterministic occurrence probability of random sensor delay. At last, the simulations are given to show the validity and correctness of the proposed time-varying estimation method. In particular, the comparisons are given to show the relationship of the upper bound and occurrence probability.

An Optimal Upper Bound on the Tail Probability for Sums of Random Variables

Let $s$ be any given real number. An explicit construction is provided of random variables (r.v.'s) $X$ and $Y$ such that $\sup{P}(X+Y\ge s)$ is attained, where the $\sup$ is taken over all r.v.'s ...

Optimal Bounds for Toader-Type Mean in Terms of Arithmetic and Centroidal Means

Optimal Bounds for Toader-Type Mean in Terms of Arithmetic and Centroidal Means

Experimental test of an entropic measurement uncertainty relation for arbitrary qubit observables

The uncertainty principle is an important tenet and active field of research in quantum physics. Information-theoretic uncertainty relations, formulated using entropies, provide one approach to quantifying the extent to which two non-commuting observables can be jointly measured. Recent theoretical analysis predicts that general quantum measurements are necessary to saturate some such uncertainty relations and thereby overcome certain limitations of projective measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation with neutron spin- qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e. positive-operator valued measures) as predicted theoretically. These results showcase experimentally the advantage obtainable by general quantum measurements over projective measurements when probing certain uncertainty relations.

Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization

We consider the generalized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization. Many problems derived from practical applications have showed that usually one of the subproblems in the generalized ADMM is hard to solve, thus a special proximal term is added. In the literature, the proximal term can be indefinite which plays a vital role in accelerating numerical performance. In this paper, we are devoted to deriving the optimal lower bound of the proximal parameter and result in the generalized ADMM with optimal indefinite proximal term. The global convergence and the \begin{document}$ O(1/t) $\end{document} convergence rate measured by the iteration complexity of the proposed method are proved. Moreover, some preliminary numerical experiments on LASSO and total variation-based denoising problems are presented to demonstrate the efficiency of the proposed method and the advantage of the optimal lower bound.

Optimal Online Two-Way Trading with Bounded Number of Transactions

We consider a two-way trading problem, where investors buy and sell a stock whose price moves within a certain range. Naturally they want to maximize their profit. Investors can perform up to k trades, where each trade must involve the full amount. We give optimal algorithms for three different models which differ in the knowledge of how the price fluctuates. In the first model, there are global minimum and maximum bounds m and M. We first show an optimal lower bound of varphi (where varphi =M/m) on the competitive ratio for one trade, which is the bound achieved by trivial algorithms. Perhaps surprisingly, when we consider more than one trade, we can give a better algorithm that loses only a factor of varphi ^{2/3} (rather than varphi ) per additional trade. Specifically, for k trades the algorithm has competitive ratio varphi ^{(2k+1)/3}. Furthermore we show that this ratio is the best possible by giving a matching lower bound. In the second model, m and M are not known in advance, and just varphi is known. We show that this only costs us an extra factor of varphi ^{1/3}, i.e., both upper and lower bounds become varphi ^{(2k+2)/3}. Finally, we consider the bounded daily return model where instead of a global limit, the fluctuation from one day to the next is bounded, and again we give optimal algorithms, and interestingly one of them resembles common trading strategies that involve stop loss limits.

Stabilized lowest equal-order mixed finite element method for the Oseen viscoelastic fluid flow

In this paper, we present a stabilized lowest equal-order mixed finite element (FE) method for the Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law. To approximate the velocity, pressure, and stress tensor, we choose lowest equal-order FE triples p1-p1-p1_{mathrm{dg}} respectively. It is well known that these elements don’t satisfy the inf–sup (or LBB) condition. Owing to the violation of the essential stability condition, the system became unstable. To overcome this difficulty, a standard pressure stabilization term is added to the discrete variational formulation, which ensures the well-posedness of the FE scheme. The existences and uniqueness of the FE scheme are derived. The desired optimal error bound is obtained. Three numerical experiments are executed to illustrate the validity and efficiency of the numerical method. The stabilized method provides attractive computational advantages, such as simpler data structures, parameter-free, no calculations of higher-order derivatives, and fast solver in simulations.

Determination of the economical insulation thickness of building envelopes simultaneously in energy-saving renovation of existing residential buildings

ABSTRACTWhen the energy saving rate of existing residential buildings renovation is determined, the thermal performances of external walls, windows, and roof interact with each other. Therefore, it is necessary to study the determination of economical insulation thickness of building envelopes considering the interaction among building envelope performances. The objective function and its bound of envelope thermal performance optimization in the energy-saving renovation of existing residential buildings in severe cold and cold zones in China were established. It is the conditional extremum problem and can be solved through Lagrange’s method of multipliers to determine the economical insulation thickness of external walls and roofs simultaneously. The method is proved to be feasible by an existing residential building in Beijing. When the same window types are selected, the energy-saving renovation program of the building envelope determined by the Lagrangian optimization method can produce the minimum investment in insulation, minimum investment payback period, and the largest net present value (NPV) of the life cycle.

Modified hybrid decomposition of the augmented Lagrangian method with larger step size for three-block separable convex programming

The Jacobian decomposition and the Gauss–Seidel decomposition of augmented Lagrangian method (ALM) are two popular methods for separable convex programming. However, their convergence is not guaranteed for three-block separable convex programming. In this paper, we present a modified hybrid decomposition of ALM (MHD-ALM) for three-block separable convex programming, which first updates all variables by a hybrid decomposition of ALM, and then corrects the output by a correction step with constant step size alpha in(0,2-sqrt{2}) which is much less restricted than the step sizes in similar methods. Furthermore, we show that 2-sqrt{2} is the optimal upper bound of the constant step size α. The rationality of MHD-ALM is testified by theoretical analysis, including global convergence, ergodic convergence rate, nonergodic convergence rate, and refined ergodic convergence rate. MHD-ALM is applied to solve video background extraction problem, and numerical results indicate that it is numerically reliable and requires less computation.

Minimizing Expected Cycle Time of Stochastic Customer Orders Through Bounded Multi-Fidelity Simulations

This paper considers the scheduling of stochastic customer orders to minimize expected cycle time. Customer orders dynamically arrive at a machine station, and each order consists of multiple product types. Random workloads are required by each product type, and the workloads are assigned to a set of unrelated parallel machines in the station to be processed. The objective is to obtain the minimal long-run expected order cycle time through proper workload assignments. In view of the difficulty in evaluating the objective function, this paper models the targeted problem as a simulation optimization problem and proposes to solve it under the multifidelity model framework. To improve the efficiency in evaluating candidate solutions, a low-fidelity model is constructed to select solutions with better performances for high-fidelity simulations. The effectiveness of this low-fidelity model is demonstrated through a series of theoretical evidences. A simulation optimization algorithm, named Bound-Multi-fidelity Optimization with Ordinal Transformation and Optimal Sampling Bound-(MO2TOS), is developed by taking advantage of the properties of the low-fidelity model. Numerical experiments are conducted to evaluate the performance of the proposed algorithm against three other well-known simulation optimization algorithms in the literature. Results indicate that the Bound-MO2TOS outperforms all the other tested algorithms, and its performance is robust against changes in problem scale. Note to Practitioners —Customer order scheduling models have wide applications in industries where every single order may contain multiple product types and the entire order requires one shipment delivery. In many applications, such as pigments dyes manufacturing, auto repair, and grocery, consideration of customer orders rather than individual jobs is often preferred. Given the stochastic nature and the synchronization constraints of the problem, a multifidelity simulation algorithm named Bound-Multi-fidelity Optimization with Ordinal Transformation and Optimal Sampling Bound-(MO2TOS) is proposed. Experimental results suggest that the proposed algorithm outperforms three other popular simulation optimization algorithms under a variety of scenarios. It is admitted that further improvement can be made in refining the computation resource distribution process of Bound-MO2TOS. In the future research, such an improvement will be more thoroughly elaborated so that practitioners can utilize it to coordinate their productions better.

Blind equalisation in the presence of bounded noise

This study addresses the blind equalisation problem in the presence of bounded noise using an optimal bounding ellipsoid algorithm. This provides an adequate blind equalisation algorithm with an accurate parameter estimation. A fundamental analysis of the involved equaliser is performed to emphasise its underlying properties. This fundamental result is corroborated by promising simulation results.