Number Of Arithmetic Operations Research Articles

Hardware-Aware Evolutionary Explainable Filter Pruning for Convolutional Neural Networks

Filter pruning of convolutional neural networks (CNNs) is a common technique to effectively reduce the memory footprint, the number of arithmetic operations, and, consequently, inference time. Recent pruning approaches also consider the targeted device (i.e., graphics processing units) for CNN deployment to reduce the actual inference time. However, simple metrics, such as the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-norm, are used for deciding which filters to prune. In this work, we propose a hardware-aware technique to explore the vast multi-objective design space of possible filter pruning configurations. Our approach incorporates not only the targeted device but also techniques from explainable artificial intelligence for ranking and deciding which filters to prune. For each layer, the number of filters to be pruned is optimized with the objective of minimizing the inference time and the error rate of the CNN. Experimental results show that our approach can speed up inference time by 1.40× and 1.30× for VGG-16 on the CIFAR-10 dataset and ResNet-18 on the ILSVRC-2012 dataset, respectively, compared to the state-of-the-art ABCPruner.

An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems

We introduce a new numerical method to approximate the solution of a finite horizon deterministic optimal control problem. We exploit two Hamilton-Jacobi-Bellman PDE, arising by considering the dynamics in forward and backward time. This allows us to compute a neighborhood of the set of optimal trajectories, in order to reduce the search space. The solutions of both PDE are successively approximated by max-plus linear combinations of appropriate basis functions, using a hierarchy of finer and finer grids. We show that the sequence of approximate value functions obtained in this way does converge to the viscosity solution of the HJB equation in a neighborhood of optimal trajectories. Then, under certain regularity assumptions, we show that the number of arithmetic operations needed to compute an approximate optimal solution of a d-dimensional problem, up to a precision ε, is bounded by O(Cd|logε|), for some constant C > 1, whereas ordinary grid-based methods have a complexity in O(1/εad) for some constant a > 0.

Adaptive Refinement of the Variational Grid Method

In this paper, we have developed a new approach to the local improvement of an approximate solution which has been obtained with the finite element method. The proposed improvement is adaptive, and it minimizes the energy functional. The discussed algorithm can be implemented analytically as well as numerically. The proposed approach can be applied to the adaptive refinement of the previously calculated variational grid approximation. The proposed method allows you to expand the approximate space by adding new coordinate functions. The implementation of a local approximation requires a small amount of arithmetic operations. The number of arithmetic operations does not depend on the number of nodes of the previously calculated approximation. Examples of the implementation of the method are considered for a problem with strong degeneracy and for a problem without degeneracy.

РАЗРАБОТКА ЭНЕРГЕТИЧЕСКИ ЭФФЕКТИВНОЙ СИСТЕМЫ СВЯЗИ В ТРОПОСФЕРНОМ РАДИОКАНАЛЕ НА БАЗЕ OFDM-СИГНАЛОВ

It is known that inter-symbol interference may occur during data transmission in radio relayand tropospheric communication systems. The presence of multipath propagation and frequencyselectivefading in the tropospheric, radio relay channel significantly reduces the energy efficiencyof the communication system as a whole. The aim of the work was to increase the efficiency ofusing the channel for radio relay and tropospheric communication by using OFDM (orthogonalfrequency-division multiplexing using the methods of reducing the peak factor of the OFDM signaland increasing the linearity of the transmission path. To evaluate digital predistortion algorithmsin the Matlab/Simulink environment, a model was developed for the LMS, NLMS, RLS, RPEMmethods and a power amplifier model with real characteristics. Based on the results of modelingalgorithms, RLS was chosen. In addition, in this work, a modified version of the adaptation algorithmbased on the recursive least squares method (RLSm) was developed. The main result of themodification is: a decrease in the number of arithmetic operations per iteration (more than5 times), an increase in the stability of adaptation algorithms, due to the introduction of regularizationmethods, a decrease in the convergence time, due to the introduction of an exponential dependence.Various algorithms for reducing the peak factor of an OFDM signal were investigated,the best result was achieved by combining Tone reservation (TR) and Active constellation extension(ACE). Simulation in the Matlab/Simulink environment showed that the combination of TRand ACE algorithms reduces the peak factor of OFDM signals by ~5dB for BPSK data stream and~4.5dB for 8-PSK, QAM-16, QAM-64, QAM-128 and QAM- 256. To increase the linearity of thetransmission path, the RLSm digital pre-distortion algorithm was selected and upgraded, it madeit possible to reduce the magnitude of the error vector modulus (EVM) by 13.5dB, and also increasethe modulation/error ratio (MER) by 13.6dB.

An extra-component method for evaluating fast matrix-vector multiplication with special functions

In calculating integral or discrete transforms, use has been made of fast algorithms for multiplying vectors by matrices whose elements are specified as values of special (Chebyshev, Legendre, Jacobi, Laguerre, etc.) functions. To achieve higher efficiency, a convenient general approach for calculating matrix-vector products for some class of problems is proposed. A series of fast simple-structure algorithms developed under this approach can be efficiently implemented with software based on modern microprocessors. Computational experiments show that the method has a precomputation complexity of $$O(N^2 \log N)$$ and an execution complexity of $$O(N \log N)$$ in general. This agrees with the analytical estimates obtained for trigonometric functions. In addition, for trigonometric functions, the number of arithmetic operations at the preliminary stage can be decreased to $$O(N \log N)$$ . The results of computational experiments with the algorithms show that these procedures can decrease the calculation time by several orders of magnitude compared with a conventional direct method of matrix-vector multiplication.

A Comparison of Energy-Efficient Seizure Detectors for Implantable Neurostimulation Devices.

IntroductionAbout 30% of epilepsy patients are resistant to treatment with antiepileptic drugs, and only a minority of these are surgical candidates. A recent therapeutic approach is the application of electrical stimulation in the early phases of a seizure to interrupt its spread across the brain. To accomplish this, energy-efficient seizure detectors are required that are able to detect a seizure in its early stages.MethodsThree patient-specific, energy-efficient seizure detectors are proposed in this study: (i) random forest (RF); (ii) long short-term memory (LSTM) recurrent neural network (RNN); and (iii) convolutional neural network (CNN). Performance evaluation was based on EEG data (n = 40 patients) derived from a selected set of surface EEG electrodes, which mimic the electrode layout of an implantable neurostimulation system. As for the RF input, 16 features in the time- and frequency-domains were selected. Raw EEG data were used for both CNN and RNN. Energy consumption was estimated by a platform-independent model based on the number of arithmetic operations (AOs) and memory accesses (MAs). To validate the estimated energy consumption, the RNN classifier was implemented on an ultra-low-power microcontroller.ResultsThe RNN seizure detector achieved a slightly better level of performance, with a median area under the precision-recall curve score of 0.49, compared to 0.47 for CNN and 0.46 for RF. In terms of energy consumption, RF was the most efficient algorithm, with a total of 67k AOs and 67k MAs per classification. This was followed by CNN (488k AOs and 963k MAs) and RNN (772k AOs and 978k MAs), whereby MAs contributed more to total energy consumption. Measurements derived from the hardware implementation of the RNN algorithm demonstrated a significant correlation between estimations and actual measurements.DiscussionAll three proposed seizure detection algorithms were shown to be suitable for application in implantable devices. The applied methodology for a platform-independent energy estimation was proven to be accurate by way of hardware implementation of the RNN algorithm. These findings show that seizure detection can be achieved using just a few channels with limited spatial distribution. The methodology proposed in this study can therefore be applied when designing new models for responsive neurostimulation.

Fast Real-Time RDFT- and GDFT-Based Direct Fault Diagnosis of Induction Motor Drive

This paper presents the theoretical analysis and experimental verification of a direct fault harmonic identification approach in a converter-fed electric drive for automated diagnosis purposes. On the basis of the analytical model of the proposed real-time direct fault diagnosis, the fault-related harmonic component is calculated using recursive DFT (RDFT) and Goertzel DFT (GDFT), applied instead of the full spectrum calculations required in the most popular FFT algorithm. The simulation model of an inverter sensorlessly controlled induction motor drive is linked with the induction machine rotor fault model for testing the sensitivity of the GDFT- and RDFT-based fault diagnosis to state variable estimation errors. According to the presented simulation results, the accuracy of the direct identification of a fault-related harmonic is sensitive to the quality of fault harmonic frequency estimation. The sensitivity analysis with respect to RDFT and GDFT algorithms is included. Based on the experimental setup with a sensorlessly controlled induction motor drive with the investigated rotor fault, fault diagnosis algorithms were implemented in the microprocessor by integration with the control system in one microcontroller and experimentally verified. The RDFT and GDFT approach has shown accurate and fast direct automated fault identification at a significantly decreased number of arithmetical operations in the microcontroller, which is convenient for the frequency-domain fault diagnosis in electric drives and supports fault-tolerant control system implementation.

Methods of crypto-stable symmetric encryption in the residual number system

This paper proposes new symmetric cryptography algorithms of Residue Number System and its Modified Perfect Form. According to the first method, ciphertext is regarded as a set of residues to the corresponding sets of modules (keys) and decryption or decimal number recovery from its residues takes place according to the Chinese Remainder Theorem. To simplify the calculations, it is proposed to use a Modified Perfect Form of Residue Number System, which leads to a decrease in the number of arithmetic operations (in particular, finding the inverse and multiplying by it) during the decryption process.Another method of symmetric encryption based on the Chinese Remainder Theorem can be applied when fast decryption is required. In this algorithm, the plaintext block is divided into sub-blocks that are smaller than the corresponding module and serve as remainders on dividing some number, which is a ciphertext, by these modules. Plaintext recovery is based on finding the ciphertext remainders to the corresponding modules. Examples of cryptography algorithms implementation and their encryption schemes are given.The security of the proposed methods is estimated on the basis of the Prime number theorem and the Euler function. It is investigated which bitness and a number of modules are required for the developed symmetric security systems to ensure the same security level as the largest length key of the AES algorithm does. It is found that as the number of modules increases, their bitness decreases. Graphical dependencies of cryptanalysis complexity on bitness and a number of modules are built. It is shown that with the increase of specified parameters, the security of the developed methods also increases.

Deep Neural Networks-Based Weight Approximation and Computation Reuse for 2-D Image Classification

Deep Neural Networks (DNNs) are computationally and memory intensive, which present a big challenge for hardware, especially for resource-constrained devices such as Internet-of-Things (IoT) nodes. This paper introduces a new method to improve DNNs performance by fusing approximate computing with data reuse techniques for image recognition applications. First, starting from the pre-trained network, then the DNNs weights are approximated based on the linear and quadratic approximation methods during the retraining phase to reduce the DNN model size and number of arithmetic operations. Then, the DNNs weights are replaced with the linear/quadratic coefficients to execute the inference so that different DNNs weights can be computed using the same coefficients. That leads to a repetition of the weights, which enables the reuse of the DNN sub-computations (computational reuse) and leverages the same data (data reuse) to reduce DNNs computations memory accesses, and improve energy efficiency, albeit at the cost of increased training time. Complete analysis for MNIST, Fashion MNIST, CIFAR 10, CIFAR 100, and tiny ImageNet datasets is presented for image recognition, where different DNN models are used, including LeNet, ResNet, AlexNet, and VGG16. Our results show that the linear approximation achieves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1211.3\times $ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$21.8\times $ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$700\times $ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$19.3\times $ </tex-math></inline-formula> on LeNet-5 MNIST, LeNet Fashion MNIST, VGG16 and ResNet-20. respectively, with small accuracy loss. Compared to the state-of-the-art Row Stationary (RS) method, the proposed architecture saved 54% of the total number of adders and multipliers needed. Overall, the proposed approach is suitable for IoT edge devices as it reduces computing complexity, memory size, and memory access with a small impact on accuracy.

Solving parametric systems of polynomial equations over the reals through Hermite matrices

We design a new algorithm for solving parametric systems of equations having finitely many complex solutions for generic values of the parameters. More precisely, let f=(f1,…,fm)⊂Q[y][x] with y=(y1,…,yt) and x=(x1,…,xn), V⊂Ct×Cn be the algebraic set defined by the simultaneous vanishing of the fi's and π be the projection (y,x)→y. Under the assumptions that f admits finitely many complex solutions when specializing y to generic values and that the ideal generated by f is radical, we solve the following algorithmic problem. On input f, we compute semi-algebraic formulas defining open semi-algebraic sets S1,…,Sℓ in the parameters' space Rt such that ∪i=1ℓSi is dense in Rt and, for 1≤i≤ℓ, the number of real points in V∩π−1(η) is invariant when η ranges over Si.This algorithm exploits special properties of some well chosen monomial bases in the quotient algebra Q(y)[x]/I where I⊂Q(y)[x] is the ideal generated by f in Q(y)[x] as well as the specialization property of the so-called Hermite matrices which represent Hermite's quadratic forms. This allows us to obtain “compact” representations of the semi-algebraic sets Si by means of semi-algebraic formulas encoding the signature of a given symmetric matrix.When f satisfies extra genericity assumptions (such as regularity), we use the theory of Gröbner bases to derive complexity bounds both on the number of arithmetic operations in Q and the degree of the output polynomials. More precisely, letting d be the maximal degrees of the fi's and D=n(d−1)dn, we prove that, on a generic input f=(f1,…,fn), one can compute those semi-algebraic formulas using O˜((t+Dt)23tn2t+1d3nt+2(n+t)+1) arithmetic operations in Q and that the polynomials involved in these formulas have degree bounded by D.We report on practical experiments which illustrate the efficiency of this algorithm, both on generic parametric systems and parametric systems coming from applications since it allows us to solve systems which were out of reach on the current state-of-the-art.

Формализм выписывания уравнения динамики манипуляторов

The problem of cumbersome equations of dynamics for manipulation systems of industrial robots (manipulators) with translational and rotational joints is solved. A new formalism for writing out the equations of dynamics of manipulators by using of guide cosines is proposed. Examples of writing out equations of dynamics of manipulators with guid cosines are given. The equations of dynamics in relative angles of rotation of bodies are obtained with the help of these guide cosines by applying their properties. These manipulators have from three to six degrees of freedom. In their equations of dynamics the geometric, kinematic, static and inertial parameters are explicit. The multipliers for accelerations and products of velocities are optimal in the sense of the minimum of arithmetic operations (additions and multiplications) that are needed for their calculations in the written out equations of dynamics. JS-code and method for verification of the equations of dynamics of manipulators written in analytical form are proposed. The problem is that when the equations are written out manually, errors and oversights in the intermediate entries and the final result are possible. Therefore it is necessary to check the results of writing out for absence of errors, i.e. to perform verification of formulas for calculation of constitutive equations of dynamics. To do this, we can use software designed to calculate the generalized driving forces of manipulators, i.e. to solve the first problem of dynamics. Such software is offered as a web-application, in which JS-function is used for verification of the equations of dynamics of manipulators. The method of verification of formulas to calculate the generalized forces of gravity and multipliers (coefficients) for generalized accelerations and products of generalized velocities in the equations of dynamics is developed. An example of verification of the equations of dynamics of the universal manipulator with six degrees of freedom in space is given. Aim. The aim of research is to develop a formalism for writing out the analytical form of the equations of the manipulators’ dynamics in the guide cosines of the principal axes of the coupled body coordinate systems, whose coefficients contain the minimum number of arithmetic operations. Research methods. The methods of research refer to vector and analytic mechanics of absolutely solid systems, to vector algebra, and to systems analysis and programming in scripting languages. Results. The results contain two proved statements, in which there are the formulas and the methodology that allow us to write manually the equations of dynamics of manipulators with three and six degrees of mobility both in guiding cosines and in generalized coordinates. In both cases it is impossible to simplify the obtained equations. Conclusion. The offered analytical types of the equations of dynamics occupy several lines. By the known classical formalisms (Lagrange, Appel, Nielsen, Newton-Euler, etc.) it is practically impossible to obtain similar results because of the large number of complex mathematical operations in their implementation and the cumbersomeness of the resulting formulas.

Vandermonde Varieties, Mirrored Spaces, and the Cohomology of Symmetric Semi-algebraic Sets

Let $\mathrm{R}$ be a real closed field. We prove that for each fixed $\ell, d \geq 0$, there exists an algorithm that takes as input a quantifier-free first order formula $\Phi$ with atoms $P=0, P > 0, P < 0 \text{ with } P \in \mathcal{P} \subset \mathrm{D}[X_1,\ldots,X_k]^{\mathfrak{S}_k}_{\leq d}$, where $\mathrm{D}$ is an ordered domain contained in $\mathrm{R}$, and computes the ranks of the first $(\ell+1)$ cohomology groups, of the symmetric semi-algebraic set defined by $\Phi$. The complexity of this algorithm (measured by the number of arithmetic operations in $\mathrm{D}$) is bounded by a \emph{polynomial} in $k$ and $\mathrm{card}(\mathcal{P})$ (for fixed $d$ and $\ell$). This result contrasts with the $\mathbf{PSPACE}$-hardness of the problem of computing just the zero-th Betti number (i.e. the number of semi-algebraically connected components) in the general case for $d \geq 2$ (taking the ordered domain $\mathrm{D}$ to be equal to $\mathbb{Z}$). The above algorithmic result is built on new representation theoretic results on the cohomology of symmetric semi-algebraic sets. We prove that the Specht modules corresponding to partitions having long lengths cannot occur with positive multiplicity in the isotypic decompositions of low dimensional cohomology modules of closed semi-algebraic sets defined by symmetric polynomials having small degrees. This result generalizes prior results obtained by the authors giving restrictions on such partitions in terms of their ranks, and is the key technical tool in the design of the algorithm mentioned in the previous paragraph.

On some properties of determinants with complex elements and their practical application

The formulas presented in this paper make it possible to select the real and imaginary part of the determinant value of the n -th order complex quantity, greatly simplifying the process of its deployment. Moreover, its module is given by the determinant of the 2n -th order, the elements of which are the real and imaginary parts of complex numbers. This makes it possible to analyze analytically the process described using determinants with complex numbers. The real and imaginary parts are also determined by the sum of determinants already with n rows and columns, the elements of which make up complex elements. The terms of this sum are solutions of a system of equations represented in closed form using symmetric polynomials, the arguments of which are its coefficients. Part of this combination is expressed by two determinants of the n -th order, the elements of which are the sum and difference of the real and imaginary parts of the elements. This significantly reduces the number of arithmetic operations during the deployment of a complex determinant and the selection of its real and imaginary parts. The given numerical example confirms the feasibility of this approach.

Congruence of Unitary Matrices

C. R. Johnson and S. Furtado have shown that if two unitary matrices are *-congruent, then they are unitarily similar. A counterpart of this statement concerning another type of matrix congruence, namely, the T-congruence is established. In addition, the problem of checking the T-congruence of given matrices using a finite number of arithmetic operations is discussed.

Short Simplex Paths in Lattice Polytopes

The goal of this paper is to design a simplex algorithm for linear programs on lattice polytopes that traces “short” simplex paths from any given vertex to an optimal one. We consider a lattice polytope P contained in $$[0,k]^n$$ and defined via m linear inequalities. Our first contribution is a simplex algorithm that reaches an optimal vertex by tracing a path along the edges of P of length in $$O(n^{4} k\, \hbox {log}\, k).$$ The length of this path is independent from m and it is the best possible up to a polynomial function. In fact, it is only polynomially far from the worst-case diameter, which roughly grows as nk. Motivated by the fact that most known lattice polytopes are defined via $$0,\pm 1$$ constraint matrices, our second contribution is a more sophisticated simplex algorithm which exploits the largest absolute value $$\alpha $$ of the entries in the constraint matrix. We show that the length of the simplex path generated by this algorithm is in $$O(n^2k\, \hbox {log}\, ({nk} \alpha ))$$ . In particular, if $$\alpha $$ is bounded by a polynomial in n, k, then the length of the simplex path is in $$O(n^2k\, \hbox {log}\, (nk))$$ . For both algorithms, if P is “well described”, then the number of arithmetic operations needed to compute the next vertex in the path is polynomial in n, m, and $$\hbox {log}\, k$$ . If k is polynomially bounded in n and m, the algorithm runs in strongly polynomial time.

On Adaptive Haar Approximations of Random Flows

The adaptive approximations for some characteristic of random functions defined on arbitrary irregular grids are discussed in this paper. The mentioned functions can be examined as flows of random real values associated with an irregular grid. This paper considers the question of choosing an adaptive enlargement of the initial grid. The mentioned enlargement essentially depends on the formulation of the criterion in relation to which adaptability is considered. Several criteria are considered here, among which there are several criteria applicable to the processing of random flows. In particular, the criteria corresponding to the mathematical expectation, dispersion, as well as autocorrelation and cross-correlation of two random flows are considered. It is possible to consider criteria corresponding to various combinations of the mentioned characteristics. The number of knots of the initial (generally speaking, irregular) grid can be arbitrary, and the main grid can be any subset of the initial one. Decomposition algorithms are proposed, taking into account the nature of the changes in the initial flow. The number of arithmetic operations in the proposed algorithms is proportional to the length of the initial flow. Sequential processing of the initial flow is possible in real time.

A canonical form for positive definite matrices

We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software. The algorithm runs in a number of arithmetic operations that is exponential in the dimension $n$, but it is practical and more efficient than canonical forms based on Minkowski reduction.

Wavelets in Generalized Haar Spaces

We consider wavelet decompositions of Haar type spaces on arbitrary nonuniform grids by methods of the nonclassical theory of wavelets. The number of nodes of the original (nonuniform) grid can be arbitrary, and the main grid can be any subset of the original one. We proposie decomposition algorithms that take into account the character of changes in the original numerical flow. The number of arithmetical operations is proportional to the length of the original flow, and successive real-time processing is possible for the original flow. We propose simple decomposition and reconstruction algorithms leading to formulas where the coefficients are independent of the grid and are equal to 1 in absolute value.

Algorithms for Inversion Mod $p^k$

This article describes and analyzes all existing algorithms for computing $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) for a prime $p$ p , and also introduces a new algorithm based on the exact solution of linear equations using $p$ p -adic expansions. The algorithm starts with the initial value $c=a^{-1}\pmod {p}$ c = a - 1 ( mod p ) and iteratively computes the digits of the inverse $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) in base $p$ p . The mod 2 version of the algorithm is more efficient than all existing algorithms for small values of $k$ k . Moreover, it stands out as being the only one that works for any $p$ p , any $k$ k , and digit-by-digit. While the new algorithm is asymptotically worse off, it requires the minimal number of arithmetic operations (just a single addition) per step, as compared to all existing algorithms.

Low-Power Non-Binary LDPC Decoder Design via Adaptive Message Length Control Exploiting Domain-Specific Information

With the enhanced performance and convergence speed than their binary counterparts, NB-LDPC codes have been considered for emerging wireless communication and storage applications. However, one challenging issue to apply NB-LDPC codes in low-power embedded applications is the high decoding complexities. In this paper, we propose a comprehensive message length control technique that adaptively truncates decoding messages according to the domain-specific information. The number of arithmetic operations and memory accesses can be greatly reduced with shorter decoding messages. To implement the proposed technique, we propose a sequential decoder architecture which controls the decoding message length. Evaluation results show that the proposed technique can achieve significant power reduction than the conventional techniques while maintaining the same decoding performance.