Logarithmic Cost Research Articles

Analysis of Tree Structure for Secure Group Communication Using LKH Approach

In secure Group communication for applications such as pay per view, teleconferencing, and video conferencing, key management method with logarithmic computation, storage time and bandwidth efficiency are expected. Major aspects with key management are initializing the group with a group key and rekeying the group. The server maintains a key tree. Rekeying is mandatory due to dynamic group nature (join/leave operations) or periodic rekeying. Logical Key Hierarchy has a logarithmic rekeying cost if the key tree is balanced, over multiple rekeying operations key tree is not balanced and LKH yields worst Rekeying cost. To achieve logarithmic cost either keep the key tree balanced or restructure the key tree in such a way that it will still result in the best rekeying cost. This paper explores tree structures using the Logical Key Hierarchy (LKH) approach to find an optimal tree structure for an application.

Inverse reflector design for a point source and far-field target

We present a method for the design of a single freeform reflector that converts the light distribution of a point source to a desired light distribution in the far field. Using the geometrical-optics law of reflection and requiring energy conservation, this optical design problem can be represented by a generalized Monge–Ampère equation for the shape of the reflector with transport boundary condition. We use a generalized least-squares algorithm that can handle a logarithmic cost function in the corresponding optimal transport problem. The algorithm first computes the optical map and subsequently constructs the optical surface. We demonstrate that the algorithm can generate reflector surfaces for a number of complicated target distributions.

Action-Manipulation Attacks Against Stochastic Bandits: Attacks and Defense

Due to the broad range of applications of stochastic multi-armed bandit model, understanding the effects of adversarial attacks and designing bandit algorithms robust to attacks are essential for the safe applications of this model. In this paper, we introduce a new class of attack named action-manipulation attack. In this attack, an adversary can change the action signal selected by the user. We show that without knowledge of mean rewards of arms, our proposed attack can manipulate Upper Confidence Bound (UCB) algorithm, a widely used bandit algorithm, into pulling a target arm very frequently by spending only logarithmic cost. To defend against this class of attacks, we introduce a novel algorithm that is robust to action-manipulation attacks when an upper bound for the total attack cost is given. We prove that our algorithm has a pseudo-regret upper bounded by $\mathcal{O}(\max\{\log T,A\})$, where $T$ is the total number of rounds and $A$ is the upper bound of the total attack cost.

An Assertional Proof of Red–Black Trees Using Dafny

Red–black trees are convenient data structures for inserting, searching, and deleting keys with logarithmic costs. However, keeping them balanced requires careful programming, and sometimes to deal with a high number of cases. In this paper, we present a functional version of a red–black tree variant called left-leaning, due to R. Sedgewick, which reduces the number of cases to be dealt with to a few ones. The code is rather concise, but reasoning about its correctness requires a rather large effort. We provide formal preconditions and postconditions for all the functions, prove their termination, and that the code satisfies its specifications. The proof is assertional, and consists of interspersing enough assertions among the code in order to help the verification tool to discharge the proof obligations. We have used the Dafny verification platform, which provides the programming language, the assertion language, and the verifier. To our knowledge, this is the first assertional proof of this data structure, and also one of the few ones including deletion.

Novel Proportionate Adaptive Filters with Coefficient Vector Reusing

In this paper, a new family of adaptive filtering algorithms is presented, which aims to combine the small misalignment resulting from the reuse of past weight vectors with the fast convergence arising from the proportionate adaptation and logarithmic cost functions. This family of algorithms is obtained as a solution to a deterministic constrained optimization problem, by using the Lagrange multipliers technique, which differs from the traditionally employed stochastic gradient technique. Two special cases are proposed, namely the improved mu-law proportionate least mean logarithmic square with reuse of coefficients (IMPLMLS-RC) algorithm and the improved mu-law proportionate least logarithmic absolute difference with reuse of coefficients (IMPLLAD-RC) algorithm. An energy conservation relationship is established, which can be employed to perform stochastic transient analyses of the proposed algorithms. Simulations in system identification and active noise control applications show the advantages of the IMPLMLS-RC and IMPLLAD-RC algorithms over the traditional LMS and LAD, and the recently proposed LMLS and LLAD, with respect to both steady-state performance and robustness against impulsive noise.

Robust least mean logarithmic square adaptive filtering algorithms

The conventional logarithm cost-based adaptive filters, e.g., the least mean logarithmic square (LMLS) algorithm, cannot combat impulsive noises at the filtering process. To address this issue, we present a novel robust least mean logarithmic square (RLMLS) algorithm by using a generalized logarithmic cost function. The proposed RLMLS algorithm can provide robustness against impulsive noises with the improvement of filtering accuracy. For theoretical analysis, the mean square analysis of RLMLS is provided in terms of the mean square deviation (MSD) and excess mean-square error (EMSE) with a white Gaussian reference. For further performance improvement in different noises, the variable step-size RLMLS (VSSRLMLS) based on the statistics of error is proposed to improve the convergence rate and steady-state mean square error, simultaneously. Analytical results and superiorities of RLMLS and VSSRLMLS in the context of system identification are supported by simulations from the aspects of filtering accuracy and robustness in Gaussian and impulse noises.

Robust Adaptive Algorithm for Smart Antenna System With <inline-formula> <tex-math notation="LaTeX">$\alpha$ </tex-math> </inline-formula>-Stable Noise

One of the main problems facing beamforming in smart antenna system is the $ {\alpha }$ -stable noise. To address this problem, a novel algorithm, named the recursive continuous logarithmic mixed ${p}$ -norm (RCLMP) algorithm, which employs a logarithmic cost, is proposed in this brief. The proposed algorithm combines the logarithmic ${p}$ -norms ${1}{\leq } p \leq 2$ which does not need the parameter selection and prior knowledge of ${\alpha }$ -stable noise, and exhibits good robustness against ${\alpha }$ -stable noise. Moreover, we show some ${H}^{{\infty }}$ norm bounds for the proposed algorithm. Simulation results show that the RCLMP algorithm outperforms the existing algorithms in terms of interference rejection capability and estimation accuracy.

Applications of A Hyper–Graph Grammar System in Adaptive Finite–Element Computations

Abstract This paper describes application of a hyper-graph grammar system for modeling a three-dimensional adaptive finite element method. The hyper-graph grammar approach allows obtaining a linear computational cost of adaptive mesh transformations and computations performed over refined meshes. The computations are done by a hyper-graph grammar driven algorithm applicable to three-dimensional problems. For the case of typical refinements performed towards a point or an edge, the algorithm yields linear computational cost with respect to the mesh nodes for its sequential execution and logarithmic cost for its parallel execution. Such hyper-graph grammar productions are the mathematical formalism used to describe the computational algorithm implementing the finite element method. Each production indicates the smallest atomic task that can be executed concurrently. The mesh transformations and computations by using the hyper-graph grammar-based approach have been tested in the GALOIS environment. We conclude the paper with some numerical results performed on a shared-memory Linux cluster node, for the case of three-dimensional computational meshes refined towards a point, an edge and a face.

Logarithmic divergences from optimal transport and Rényi geometry

Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures. Using the duality in optimal transport, we introduce and study the one-parameter family of $$L^{(\pm \alpha )}$$ -divergences. They extrapolate between the Bregman divergence corresponding to the Euclidean quadratic cost, and the L-divergence introduced by Pal and the author in connection with portfolio theory and a logarithmic cost function. They admit natural generalizations of exponential family that are closely related to the $$\alpha $$ -family and q-exponential family. In particular, the $$L^{(\pm \alpha )}$$ -divergences of the corresponding potential functions are Renyi divergences. Using this unified framework we prove that the induced geometries are dually projectively flat with constant sectional curvatures, and a generalized Pythagorean theorem holds true. Conversely, we show that if a statistical manifold is dually projectively flat with constant curvature $$\pm \alpha $$ with $$\alpha > 0$$ , then it is locally induced by an $$L^{(\mp \alpha )}$$ -divergence. We define in this context a canonical divergence which extends the one for dually flat manifolds.

PRM23 - Development of Medical Cost Prediction Model Based on Statistical Machine Learning Using Health Insurance Claims Data

Japanese government has required insurers to formulate a population health management. A cost prediction is needed to evaluate the plan. Since one insured can have multiple diseases, a traditional linear model is not appropriate for the prediction. We developed a medical cost prediction model using a statistical machine learning method. Claims data of six health insurance societies (1,009,167 insureds, average age: 34.3, female: 54%) covered in 2016 were used. Insureds were randomly divided into those for training data and validation data at a ratio of 7 to 3. Difference in logarithmic value of annual medical cost from that of demographically matched population was defined as excess logarithmic medical cost. A prediction model, a regression model with least absolute shrinkage and selection operator (LASSO), was developed using the excess logarithmic cost as explained valuables, and dummy valuables of each disease and their interaction terms as explanatory valuables. Explanatory valuables were selected by LASSO avoiding overfitting using the validation data. Excess logarithmic costs of extracted individuals (741,067) were almost normally distributed. Of the explanatory valuables, 58% were single diseases and 24%, 14%, and 4% were the interaction terms of two, three, and four diseases, respectively. Among the valuables, constipation had the highest impact on medical cost followed by insomnia, asthma bronchiale, diabetes, and low back pain. Among the interaction terms of two, insomnia and asthma bronchiale had the highest impact. The coefficient of determination of the prediction model (0.42) was higher than that of a traditional linear model (0.25). We developed a cost prediction model using machine learning. The model may be useful to assess a population health management plan. In addition, the prevalence rates of the explanatory valuables of the model may be useful for the comparison among insurers and for planning an effective population health management plan.

Logarithmic Cost Based Constrained Adaptive Filtering Algorithms for Sensor Array Beamforming

This paper introduces a novel family of constrained adaptive filtering algorithms for sensor array beamforming. These algorithms, namely, constrained least mean logarithmic square (CLMLS), $\ell _{1}$ -norm CLMLS ( $\ell _{1}$ -CLMLS) and its weighted version $\ell _{1}$ -WCLMLS, are developed based on a relative logarithmic cost function. The proposed algorithms gracefully adjust cost function depending on the amount of the error thereby achieving better performance compared to constrained least mean square (CLMS) family of algorithms. The transient and steady-state performance analysis of the proposed CLMLS algorithm is presented and these analytical results are validated through extensive simulations. Proposed CLMLS algorithm is then extended to sparse system identification problem by incorporating the $\ell _{1}$ -norm penalty into CLMLS cost function. We show that the resultant $\ell _{1}$ -CLMLS and $\ell _{1}$ -WCLMLS algorithms outperform their CLMS counterparts in sparse system identification. When applied to sparse sensor array synthesis, these algorithms achieve desired beampattern with lesser number of sensor elements compared to state-of-the-art algorithms.

$$L_{0}$$ L 0 -norm constraint normalized logarithmic subband adaptive filter algorithm

With the purpose of identifying sparse unknown system better, a novel sparsity-aware normalized logarithmic subband adaptive filter algorithm is developed by introducing the $$L_{0}$$ -norm constraint of the estimated coefficient vector into the normalized logarithmic cost function. The gradient descent technique is utilized in the derivation of the weight vector updating formula. The proposed algorithm not only acquires a lower steady-state error, but also possesses good robustness against impulsive noise for sparse system. Besides, the reason why its performance is improved is interpreted by rigorous mathematical analysis. Simulation results in the context of sparse system identification have revealed the advantage of the proposed algorithms over other existing algorithms in impulsive noise environments.

Logarithmic cost-based improved constant modulus type blind channel equalisation algorithm for M-QAM/M-PSK signal

In this paper, authors have proposed the blind channel equalisation algorithm based on logarithmic cost function for QAM/PSK signal which is widely used in the digital communication system. Constant modulus algorithm (CMA) and modified constant modulus algorithm (MCMA), the two widely referenced algorithms for blind equalisation of a QAM/PSK signal, exhibit very slow convergence rates and large steady-state mean square error. The proposed algorithm attains faster convergence speed and low steady state mean square error by incorporating the improved logarithmic cost-based error function in the resilient back propagation-based framework. This makes the weight adaption of equaliser simple. Simulation results show that the proposed algorithm has better convergence rate and lower steady state error in comparison to CMA and MCMA algorithms.

Logarithmic cost-based improved constant modulus type blind channel equalisation algorithm for M-QAM/M-PSK signal

In this paper, authors have proposed the blind channel equalisation algorithm based on logarithmic cost function for QAM/PSK signal which is widely used in the digital communication system. Constant modulus algorithm (CMA) and modified constant modulus algorithm (MCMA), the two widely referenced algorithms for blind equalisation of a QAM/PSK signal, exhibit very slow convergence rates and large steady-state mean square error. The proposed algorithm attains faster convergence speed and low steady state mean square error by incorporating the improved logarithmic cost-based error function in the resilient back propagation-based framework. This makes the weight adaption of equaliser simple. Simulation results show that the proposed algorithm has better convergence rate and lower steady state error in comparison to CMA and MCMA algorithms.

Faster All-Pairs Shortest Paths via Circuit Complexity

We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two $n \times n$ matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense $n$-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time $\frac{n^3}{2^{\Omega(\log n)^{1/2}}}$ and is correct with high probability. On the word RAM with wordsize $\log(n)$, the algorithm runs in $n^3/2^{\Omega(\log n)^{1/2}} + n^{2+o(1)}\log(nM)$ time for edge weights in $([0,M] \cap \mathbb{Z})\cup\{\infty\}$. Prior algorithms took either $O(n^3/\log^c n)$ time for various $c \leq 2$, or $O(M^{\alpha}n^{\beta})$ time for various $\alpha > 0$ and $\beta > 2$. The new algorithm applies a tool from circuit complexity, namely the Razborov--Smolensky polynomials for approximately representing ${\sf AC}^0[p]$ circuits, to efficiently reduce a matrix product over the $(\min,+)$ algebra to a relatively small number of rectangular matrix products over $\mathbb{F}_2$, each of which is computable using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in $n^3/2^{\log^{\delta} n}$ time for some $\delta > 0$, which utilizes the Yao--Beigel--Tarui translation of ${\sf AC}^0[m]$ circuits into “nice” depth-two circuits.

A distributed algorithm for demand-side management: Selling back to the grid

Demand side energy consumption scheduling is a well-known issue in the smart grid research area. However, there is lack of a comprehensive method to manage the demand side and consumer behavior in order to obtain an optimum solution. The method needs to address several aspects, including the scale-free requirement and distributed nature of the problem, consideration of renewable resources, allowing consumers to sell electricity back to the main grid, and adaptivity to a local change in the solution point. In addition, the model should allow compensation to consumers and ensurance of certain satisfaction levels. To tackle these issues, this paper proposes a novel autonomous demand side management technique which minimizes consumer utility costs and maximizes consumer comfort levels in a fully distributed manner. The technique uses a new logarithmic cost function and allows consumers to sell excess electricity (e.g. from renewable resources) back to the grid in order to reduce their electric utility bill. To develop the proposed scheme, we first formulate the problem as a constrained convex minimization problem. Then, it is converted to an unconstrained version using the segmentation-based penalty method. At each consumer location, we deploy an adaptive diffusion approach to obtain the solution in a distributed fashion. The use of adaptive diffusion makes it possible for consumers to find the optimum energy consumption schedule with a small number of information exchanges. Moreover, the proposed method is able to track drifts resulting from changes in the price parameters and consumer preferences. Simulations and numerical results show that our framework can reduce the total load demand peaks, lower the consumer utility bill, and improve the consumer comfort level.

Algorithm and Architecture Design of Adaptive Filters With Error Nonlinearities

This paper presents a framework based on the logarithmic number system to implement adaptive filters with error nonlinearities in hardware. The framework is demonstrated through pipelined implementations of two recently proposed adaptive filtering algorithms based on logarithmic cost, namely, least mean logarithmic square (LMLS) and least logarithmic absolute difference (LLAD). To the best of our knowledge, the proposed architectures are the first attempts to implement both LMLS and LLAD algorithms in hardware. We derive error computing algorithms to realize the nonlinear error functions for LMLS and LLAD and map them onto hardware. We also propose a novel variable- $\alpha $ scheme to enhance the original LMLS algorithm and prove its robustness and suitability for VLSI implementations in practical applications. Detailed bit width and error analysis are carried out for the proposed VLSI fixed point implementations. Postlayout implementation results show that with an additional multiplier over conventional least mean square (LMS), 7-dB improvement in steady-state mean square deviation performance can be achieved and with the proposed variable- $\alpha $ scheme, 12-dB improvement can be achieved without compromising the convergence. We will show that LMLS can potentially replace LMS in practical applications, by demonstrating a proof-of-concept by extending the framework to transform domain adaptive filters.

Robust adaptive algorithms for underwater acoustic channel estimation and their performance analysis

We introduce a novel family of adaptive robust channel estimators for highly challenging underwater acoustic (UWA) channels. Since the underwater environment is highly non-stationary and subjected to impulsive noise, we use adaptive filtering techniques based on minimization of a logarithmic cost function, which results in a better trade-off between the convergence rate and the steady state performance of the algorithm. To improve the convergence performance of the conventional first and second order linear estimation methods while mitigating the stability issues related to impulsive noise, we intrinsically combine different norms of the error in the cost function using a logarithmic term. Hence, we achieve a comparable convergence rate to the faster algorithms, while significantly enhancing the stability against impulsive noise in such an adverse communication medium. Furthermore, we provide a thorough analysis for the tracking and steady-state performances of our proposed methods in the presence of impulsive noise. In our analysis, we not only consider the impulsive noise, but also take into account the frequency and phase offsets commonly experienced in real life experiments. We demonstrate the performance of our algorithms through highly realistic experiments performed on accurately simulated underwater acoustic channels.

Indexing quantized ensembles of exemplar-SVMs with rejecting taxonomies

Ensembles of Exemplar-SVMs have been introduced as a framework for Object Detection but have rapidly found a large interest in a wide variety of computer vision applications such as mid-level feature learning, tracking and segmentation. What makes this technique so attractive is the possibility of associating to instance specific classifiers one or more semantic labels that can be transferred at test time. To guarantee its effectiveness though, a large collection of classifiers has to be used. This directly translates in a high computational footprint, which could make the evaluation step prohibitive. To overcome this issue we organize Exemplar-SVMs into a taxonomy, exploiting the joint distribution of Exemplar scores. This permits to index the classifiers at a logarithmic cost, while maintaining the label transfer capabilities of the method almost unaffected. We propose different formulations of the taxonomy in order to maximize the speed gain. In particular we propose a highly efficient Vector Quantized Rejecting Taxonomy to discard unpromising image regions during evaluation, performing computations in a quantized domain. This allow us to obtain ramarkable speed gains, with an improvement up to more than two orders of magnitude. To verify the robustness of our indexing data structure with reference to a standard Exemplar-SVM ensemble, we experiment with the Pascal VOC 2007 benchmark on the Object Detection competition and on a simple segmentation task.

Fractal: An advanced multidimensional range query lookup protocol on nested rings for distributed systems

One of the key issues in large-scale distributed systems such as P2P and grids is the capability of efficient multidimensional range query processing. Although several methods have recently been proposed for solving this problem on distributed systems, these methods have not been able to meet the fundamental necessity of a typical method on large-scale distributed systems, i.e., scalability.This paper presents Fractal, a fully decentralized and highly scalable multidimensional range query lookup protocol for distributed systems. In this work, to organize the available nodes in the system, an n-dimensional space called Key Space is utilized. The available nodes on the nested Fractal rings maintain information about only O(logN) other nodes, and the Fractal lookup protocol discovers the destination node through these nested rings with a logarithmic cost. Because of its flexibility, Fractal allows the system to create a concept called Layering, which minimizes the probability of the wide-area message transfers (WAMTs) during the lookup process. Using several criteria, Fractal is compared with several successful methods that have recently been presented. Simulation results show the efficiency and performance of Fractal in networks of different sizes.