Inner Product Computation Research Articles

Native 2-D-FEM/Mode-Matching Formulation for Second-Order Symmetric Waveguide Devices

In this work we present, taking a holistic approach, the integration between the 2-D finite element method (FEM) and mode-matching (MM) for devices exhibiting second-order rotational symmetry and, optionally, first-order axial symmetry. Using a tailored approach, involving adequate inner products, the formulation allows for the rotationally symmetric modes with complex-valued eigenfunctions to keep their useful physical and mathematical properties. Mode class orthogonality and degeneracy discrimination are kept throughout the pipeline. The approach leads to a more efficient and accurate analysis of devices with these geometrical properties, significantly reducing the dimensionality of the problem, both in the inner-product calculation and in the generalized scattering matrix (GSM) derivation, while providing physical insight regarding modal discrimination.

EFFICIENT AND EXPRESSIVE SEARCH OVER ENCRYPTED DATA WITH MULTI FACTOR AUTHENTICATION SCHEME

More data owners are tempted to outsource their data to cloud services as a result of the development of cloud storage. Sensitive data should be encrypted before being outsourced due to privacy concerns. To assure data availability, a variety of searchable encryption techniques are used. However, the effectiveness of data consumers' queries is not given much consideration by the current search algorithms, particularly in the case of several owners. In this study, linguistic coefficients are used to create an encrypted search index. In particular, the encrypted search index utilizing Lagrange co-efficient makes use of secure inner-product calculation for both search and relevance assessment by taking into account a sizable amount of data in the cloud. A system architecture of preferred search over encrypted data in the cloud scene is built, and the requirements in terms of efficiency and privacy are specified. This enables the clo ud servers to do a secure search without knowing any sensitive data (e.g., keywords and trapdoors).For each data owner, the Lagrange coefficient is used to build indices that can support searches over numerous keyword fields and enable accu rate relevance computation in order to produce an efficient search.To protect user data privacy, a multifactor authenticati on technique might be utilised.After logging in to the system, the user will receive three different sorts of authentications, including an OTP utilizing a number, a captcha, and an OTP image. This picture will serve as a key. In order to accomplish secure search and relevance score calculation, the user's preference and the search query are expressed in vector form. Additionally, Preferred Search over Encrypted Data (PSED) is consistent with cloud computing's scalability and adaptability. Finally, a thorough security study verifies the security of our scheme, and a performance analysis highlights its effectiveness and efficiency. Keywords: Encrypted Data, Inner product calculation, Multifactor authentication, Data privacy

Wave analysis in the complex Fourier transform domain: A new method to obtain the Green's functions of dispersive linear partial differential equations

This paper provides a new analytical method to obtain Green's functions of linear dispersive partial differential equations. The Euler-Bernoulli beam equation and the one-dimensional heat conduction equation (dissipation equation) under impulses in space and time are solved as examples. The complex infinite-domain Green's function of the Euler-Bernoulli beam is derived. A new approach is proposed to obtain the finite-domain Green's function from the infinite-domain Green's function by the reflection and transmission analysis in the complex Fourier transform domain. It is found that the solution obtained by this approach converges much better at short response times compared with that obtained by the traditional modal analysis. Besides, by applying the geometric summation formula for matrix series, a new modal expansion solution requiring no calculation of each mode's inner product is derived, which analytically proves the wave-mode duality and simplifies the calculation. The semi-infinite-domain cases and the coupled-domain cases are also derived by the newly developed method to show its validity and simplicity. It is found that the ‘non-propagating waves’ also possess wave speed, and heat conduction can also be treated as propagating waves.

Secure and Practical Group Nearest Neighbor Query for Location-Based Services in Cloud Computing

Group nearest neighbor (GNN) query enables a group of location-based service (LBS) users to retrieve a point from point of interests (POIs) with the minimum aggregate distance to them. For resource constraints and privacy concerns, LBS provider outsources the encrypted POIs to a powerful cloud server. The encryption-and-outsourcing mechanism brings a challenge for the data utilization. However, as previous work from k − anonymity technique leaks all contents of POIs and returns an answer set with redundant communication cost, the LBS system cannot work properly with those privacy-preserving schemes. In this paper, we illustrate a secure group nearest neighbor query scheme, which is referred to as SecGNN. It supports the GNN query with n n ≥ 3 LBS users and assures the data privacy and query privacy. Since SecGNN only achieves linear search complexity, an efficiency enhanced scheme (named Sec GNN + ) is introduced by taking advantage of the KD-tree data structure. Specifically, we convert the GNN problem to the nearest neighbor problem for their centroid, which can be computed by anonymous veto network and Burmester–Desmedt conference key agreement protocols. Furthermore, the Sec GNN + scheme is introduced from the KD-tree data structure and a designed tool, which supports the computation of inner products over ciphertexts. Finally, we run experiments on a real-database and a random database to evaluate the performance of our SecGNN and Sec GNN + schemes. The experimental results show the high efficiency of our proposed schemes.

A geometric approach to counting norms in cyclic extensions of function fields

In this paper we prove an explicit version of a function field analogue of a classical result of Odoni about norms in number fields in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result of Bary-Soroker, Smilanski, and Wolf which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky , and reproved by the author in his Ph.D thesis using the method of this paper. The main tool is a twisted Grothendieck Lefschetz trace formula, inspired by the work of Church, Farb and Ellenberg on representation stability and asymptotic for point counts on varieties. Using a combinatorial description of the cohomology we obtain a precise quantitative result which works in the $q^n\rightarrow \infty$ regime, and a new type of homological stability phenomena, which arises from the computation of certain inner products of representations.

Inner product free iterative solution and elimination methods for linear systems of a three-by-three block matrix form

Large scale systems of algebraic equations are frequently solved by iterative solution methods, such as the conjugate gradient method for symmetric or a generalized conjugate gradient or generalized minimum residual method for nonsymmetric linear systems. In practice, to get an acceptable elapsed computing time when solving large scale problems, one shall use parallel computer platforms. However, such methods involve orthogonalization of search vectors which requires computation of many inner products and, hence, needs global communication of data, which will be costly in computer times. In this paper, we propose various inner product free methods, such as the Chebyshev acceleration method. We study the solution of linear systems arising from optimal control problems for PDEs, such as the edge element discretization of the time-periodic eddy current optimal control problem. Following a discretize-then-optimize scheme, the resulting linear system is of a three-by-three block matrix form. Various solution methods based on an approximate Schur complement and inner product free iterative solution methods for this linear system are analyzed and compared with an earlier used method for two-by-two block matrices with square blocks. The convergence properties and implementation details of the proposed methods are analyzed to show their effectiveness and practicality. Both serial and parallel numerical experiments are presented to further investigate the performance of the proposed methods compared with some other existing methods.

An Efficient Parallel DA-Based Fixed-Width Design for Approximate Inner-Product Computation

Parallel distributed arithmetic (PDA)-based structures are widely used for high-speed computation of inner product in digital signal processing (DSP) applications. In this article, we have proposed novel PDA-based structures based on an efficient truncation model. To achieve higher bit saving with relatively less truncation error, we present here a novel approach using approximate look-up tables (LUTs), adder trees (ATs), and Wallace-like shift-AT (SAT) with truncated operands to obtain hardware-efficient fixed-width PDA-based inner-product structures. We have three variants of proposed structures based on the proposed truncation approach. We find that the proposed inner-product structure-1 using approximate LUT (ALUT) and approximate AT offers nearly 20% higher bit saving, 20% saving in area-delay product (ADP) and offers relatively less truncation error than the existing structures. The proposed structure-2 using ALUT, ATs, and proposed SAT offers nearly 50% higher bit-saving, 61% ADP saving and offers nearly the same accuracy compared to the existing approximate DA-based structures. Proposed structure-3 offers nearly 60% higher bit saving and calculates outputs with almost the same or marginally less accuracy than the existing structures for higher coefficient word lengths.

Factorization of the translation kernel for fast rigid image alignment

An important component of many image alignment methods is the calculation of inner products (correlations) between an image of pixels and another image translated by some shift and rotated by some angle. For robust alignment of an image pair, the number of considered shifts and angles is typically high, thus the inner product calculation becomes a bottleneck. Existing methods, based on fast Fourier transforms (FFTs), compute all such inner products with computational complexity per image pair, which is reduced to if only distinct shifts are needed. We propose to use a factorization of the translation kernel (FTK), an optimal interpolation method which represents images in a Fourier–Bessel basis and uses a rank-H approximation of the translation kernel via an operator singular value decomposition (SVD). Its complexity is per image pair. We prove that , where 2W is the magnitude of the maximum desired shift in pixels and is the desired accuracy. For fixed W this leads to an acceleration when is large, such as when sub-pixel shift grids are considered. Finally, we present numerical results in an electron cryomicroscopy application showing speedup factors of 3–10 with respect to the state of the art.

Design of parallel and pipelined DA based OBC fir filter for software defined radio

Software Defined Radio (SDR) is a new technology used to implement different wireless communication standard for mobile communication. The Intermediate Frequency (IF) block is the most demanding block in software defined radio. The most important task in intermediate processing block is digital filtering which is carried out by Finite Impulse Response (FIR) filter. One of the major techniques for the calculation of inner product is Distributed Arithmetic (DA) based FIR filter which uses Look Up Table (LUT) to eliminate the need of multiplier. The efficiency of the DA filter is affected with the increasing number of address line and also due to its serial operation. To overcome this problem parallel and pipeline based DA filter using Offset Binary Coding (OBC) for Two Bit At A Time (2-BAAT) is proposed. Our proposed method achieves less area, low power consumption and nominal delay for SDR application.

Learning of Gaussian Processes in Distributed and Communication Limited Systems.

It is of fundamental importance to find algorithms obtaining optimal performance for learning of statistical models in distributed and communication limited systems. Aiming at characterizing the optimal strategies, we consider learning of Gaussian Processes (GP) in distributed systems as a pivotal example. We first address a very basic problem: how many bits are required to estimate the inner-products of some Gaussian vectors across distributed machines? Using information theoretic bounds, we obtain an optimal solution for the problem which is based on vector quantization. Two suboptimal and more practical schemes are also presented as substitutes for the vector quantization scheme. In particular, it is shown that the performance of one of the practical schemes which is called per-symbol quantization is very close to the optimal one. Schemes provided for the inner-product calculations are incorporated into our proposed distributed learning methods for GPs. Experimental results show that with spending few bits per symbol in our communication scheme, our proposed methods outperform previous zero rate distributed GP learning schemes such as Bayesian Committee Model (BCM) and Product of experts (PoE).

Numerical Evaluations of Parallelization Efficiencies of Communication Avoiding Krylov Subspace Method for Large Sparse Linear System

In this study, an improvement technique of convergence property of Communication Avoiding (CA) Kyrlov subspace method is proposed, and parallelization efficiencies of CA Krylov subspace method is numerically investigated. As is known that most of all the procedures of the Krylov subspace method are constituted by addition of vectors, inner products and multiplication of matrices and vectors. These operations are very easy to derive a parallelization efficiency. However, the candidate coefficient matrices of linear system obtained from the numerical analysis such as Finite Element Method are large sparse matrices, and communications occur between Processing Units at short intervals in parallel calculation of inner products of vectors and multiplication of matrices and vectors. These communication times often consume more time than ordinary calculations and become a major cause of deterioration of parallelization efficiency. Although these communications have to be avoided to improve parallelization efficiency, the communication avoiding technique degrades the convergence property of the method. To improve this degradation, reorthogonalization of bases of Krylov subspace is proposed. Results of computation show that a convergence properties of the proposed method are improved. In addition, the parallelization efficiency of the proposed method is about four times faster than that of the serial method.

Multicast Wirelessly Powered Network With Large Number of Antennas via First-Order Method

To prolong the lifetime of energy constrained devices in Internet of Things, devices can harvest wireless energy from the control signal multicast from the access point. Unfortunately, hampered by the path-loss, the efficiency of such multicast wirelessly powered network is low. While large-scale antennas at access point can be used to improve the efficiency, the beamforming design problem in multicast wirelessly powered network is known to be NP-hard, and the traditional difference of convex programming becomes prohibitively time consuming in large-scale settings. On the other extreme, by using the assumption of infinite number of antennas and applying the law of large numbers, simple beamforming solution is possible. However, when applied to scenarios with finite number of antennas, the performance of such asymptotic solution is far from that of difference of convex programming. To resolve this apparent complexity-performance dilemma, this paper develops an algorithm which reduces the computation time by orders of magnitude, while still guaranteeing the same performance compared with the difference of convex programming. In particular, the proposed algorithm consists of two fast-convergent iterative procedures and is guaranteed to obtain a Karush–Kuhn–Tucker solution. Furthermore, in each iteration, the algorithm only requires the computation of inner products between channel vectors and can be run in parallel for all the users. Thus, the complexity scales linearly with the number of antennas at access point. Finally, numerical results validate the performance and the speed of the proposed scheme.

Preferred search over encrypted data

Cloud computing provides elastic data storage and processing services. Although existing research has proposed preferred search on the plaintext files and encrypted search, no method has been proposed that integrates the two techniques to efficiently conduct preferred and privacy-preserving search over large datasets in the cloud. In this paper, we propose a scheme for preferred search over encrypted data (PSED) that can take users’ search preferences into the search over encrypted data. In the search process, we ensure the confidentiality of not only keywords but also quantified preferences associated with them. PSED constructs its encrypted search index using Lagrange coefficients and employs secure inner-product calculation for both search and relevance measurement. The dynamic and scalable property of cloud computing is also considered in PSED. A series of experiments have been conducted to demonstrate the efficiency of the proposed scheme when deploying it in real-world scenarios.

Arbitrary Polynomial Chaos for Uncertainty Propagation of Correlated Random Variables in Dynamic Systems

Dynamic simulation of stochastic systems requires uncertainty propagation. Traditional sample-based uncertainty propagation methods are often computationally intractable for online optimization-based estimation and control applications. Generalized polynomial chaos (gPC) is an efficient uncertainty propagation method that has been used for solving various nonlinear estimation and optimal control problems. However, gPC requires knowledge of the exact probability distribution of the random variables, and does not explicitly account for correlations between these random variables. This paper demonstrates the use of arbitrary polynomial chaos (aPC) for propagation of correlated multivariate random variables. aPC constructs orthogonal polynomial basis functions from only the raw moments of the random variables. Thus, aPC can be used for propagation of uncertainties with arbitrary probability distributions, even if their functional forms are unknown. The main contributions of this paper consist of presenting an algorithm for generating a set of orthogonal polynomial basis functions for correlated multivariate random variables and applying the Galerkin projection to compute closed-form expressions for the dynamics of the aPC expansion coefficients. An algorithm is also presented for efficient calculation of inner products between polynomial basis functions needed in the Galerkin projection. The error convergence properties of aPC are investigated and compared to that of gPC and Monte Carlo using a dynamic simulation case study.

Quasilocal conservation laws in the quantum Hirota model

The extensivity of the quantum Hirota model’s conservation laws on a 1 + 1 dimensional lattice is considered. This model can be interpreted in terms of an integrable many-body quantum Floquet dynamics. We establish the procedure to generate a continuous family of quasilocal conservation laws from the conserved operators proposed by Faddeev and Volkov. The Hilbert–Schmidt kernel which allows the calculation of inner products of these new conservation laws is explicitly computed. This result has potential applications in quantum quench and transport problems in integrable quantum field theories.

Representation and coding of signal geometry

Abstract Approaches to signal representation and coding theory have traditionally focused on how to best represent signals using parsimonious representations that incur the lowest possible distortion. Classical examples include linear and nonlinear approximations, sparse representations and rate-distortion theory. Very often, however, the goal of processing is to extract specific information from the signal, and the distortion should be measured on the extracted information. The corresponding representation should, therefore, represent that information as parsimoniously as possible, without necessarily accurately representing the signal itself. In this article, we examine the problem of encoding signals such that sufficient information is preserved about their pairwise distances and their inner products. For that goal, we consider randomized embeddings as an encoding mechanism and provide a framework to analyze their performance. We also demonstrate that it is possible to design the embedding such that it represents different ranges of distances with different precision. These embeddings also allow the computation of kernel inner products with control on their inner product-preserving properties. Our results provide a broad framework to design and analyze embeddings and generalize existing results in this area, such as random Fourier kernels and universal embeddings.

Multicolumn RBF Network.

This paper proposes the multicolumn RBF network (MCRN) as a method to improve the accuracy and speed of a traditional radial basis function network (RBFN). The RBFN, as a fully connected artificial neural network (ANN), suffers from costly kernel inner-product calculations due to the use of many instances as the centers of hidden units. This issue is not critical for small datasets, as adding more hidden units will not burden the computation time. However, for larger datasets, the RBFN requires many hidden units with several kernel computations to generalize the problem. The MCRN mechanism is constructed based on dividing a dataset into smaller subsets using the k-d tree algorithm. resultant subsets are considered as separate training datasets to train individual RBFNs. Those small RBFNs are stacked in parallel and bulged into the MCRN structure during testing. The MCRN is considered as a well-developed and easy-to-use parallel structure, because each individual ANN has been trained on its own subsets and is completely separate from the other ANNs. This parallelized structure reduces the testing time compared with that of a single but larger RBFN, which cannot be easily parallelized due to its fully connected structure. Small informative subsets provide the MCRN with a regional experience to specify the problem instead of generalizing it. The MCRN has been tested on many benchmark datasets and has shown better accuracy and great improvements in training and testing times compared with a single RBFN. The MCRN also shows good results compared with those of some machine learning techniques, such as the support vector machine and k-nearest neighbors.

VLSI Design and Comparison of DA and LMS Based Reconfigurable FIR Filter

<p>Here, we exhibit the design optimization of one- and two-dimensional fully-pipelined computing structures for area-delay-power-efficient implementation of finite impulse response (FIR) filter by systolic decomposition of distributed arithmetic (DA)-based inner-product computation. This plan is found to offer a flexible choice of the address length of the look-up-tables (LUT) for DA-based computation to determine suitable area-time trade-off. It is seen that by using smaller address-lengths for DA-based computing units, it is possible to decrease the memory-size but on the other side that leads to increase of adder complexity and the latency. For efficient DA-based realization of FIR filters of different orders, the flexible linear systolic design is implemented on a Xilinx Virtex-E XCV2000E FPGA using a hybrid combination of Handel-C and parameterizable VHDL cores. Various key performance metrics such as number of slices, maximum usable frequency, dynamic power consumption, energy density and energy throughput are estimated for different filter orders and address-lengths. Obtained results on analysis shows that performance metrics of the proposed implementation is broadly in line with theoretical expectations. We have seen that the choice of address-length M=4 gives the best of area-delay power-efficient realizations of the FIR filter for different filter orders. Moreover, the proposed FPGA implementation is found to involve significantly less area-delay complexity compared with the existing DA-based implementations of FIR filter.</p>

Pipelined, Flexible Krylov Subspace Methods

We present variants of the Conjugate Gradient (CG), Conjugate Residual (CR), and Generalized Minimal Residual (GMRES) methods which are both pipelined and flexible. These allow computation of inner products and norms to be overlapped with operator and nonlinear or nondeterministic preconditioner application.The methods are hence aimed at hiding network latencies and synchronizations which can become computational bottlenecks in Krylov methods on extreme-scale systems or in the strong-scaling limit. The new variants are not arithmetically equivalent to their base flexible Krylov methods, but are chosen to be similarly performant in a realistic use case, the application of strong nonlinear preconditioners to large problems which require many Krylov iterations. We provide scalable implementations of our methods as contributions to the PETSc package and demonstrate their effectiveness with practical examples derived from models of mantle convection and lithospheric dynamics with heterogeneous viscosity structure. These represent challenging problems where multiscale nonlinear preconditioners are required for the current state-of-the-art algorithms, and are hence amenable to acceleration with our new techniques. Large-scale tests are performed in the strong-scaling regime on a contemporary leadership supercomputer, where speedups approaching, and even exceeding $2\times$ can be observed. We conclude by analyzing our new methods with a performance model targeted at future exascale machines.

Distributed Arithmetic based Low-Power LMS Adaptive FIR Filter Design

filtering forms a significant class of DSP algorithms employed in several hand held mobile devices for applications like echo cancellation, signal de-noising, and channel equalization. This paper presents a different pipelined architecture for low-power implementation of Adaptive filter based on distributed arithmetic (DA). The traditional adder- based shift accumulation for Distributed Arithmetic based computation of inner-product is swapped by conditional signed carry-save accumulation. A fast bit clock is employed only for carry-save accumulation which results in reduction of power consumption in the proposed design, while use of a much slower bit clock is used for rest of the operations. It contains the smaller Look-Up Table (LUT), same quantity of multiplexers and almost half the number of adders in comparison to the existing Distributed Arithmetic-based design. By changing the inner block, a reduction in power consumption is aimed at. So the previous DA-based adaptive filter in average for filter lengths N=4 and N=16 have been implemented.