Walsh Coefficients Research Articles

Time domain model order reduction of discrete-time bilinear systems by discrete Walsh functions

In this paper, the discrete Walsh functions are first used for time domain model order reduction (MOR) of discrete-time bilinear systems. Firstly, in terms of the discrete Walsh functions, the reduced order system is generated by means of the explicit solution formula for the discrete-time bilinear system. Meanwhile, we expand the discrete-time bilinear system under space spanned by the discrete Walsh functions. Then the reduced order system is produced by the orthogonal projection matrix that is defined by discrete Walsh coefficient matrix. Moreover, MOR method via discrete Walsh functions is investigated for discrete-time bilinear systems with nonzero initial conditions, where nonzero initial condition is also utilized to create the orthogonal projection matrix. Theoretical analyses show that the outputs of the resulting reduced order systems can match the first several outputs of the original system in finite intervals. Finally, an example is given to illustrate the effectiveness of the proposed methods.

The solution to an open problem on the bentness of Mesnager's functions

Let n=2m. In the present paper, we study the binomial Boolean functions of the formfa,b(x)=Tr1n(ax2m−1)+Tr12(bx2n−13), where m is an even positive integer, a∈F2n⁎ and b∈F4⁎. We show that fa,b is a bent function if the Kloosterman sumKm(a2m+1)=1+∑x∈F2m⁎(−1)Tr1m(a2m+1x+1x) equals 4, thus settling an open problem of Mesnager. The proof employs tools including computing Walsh coefficients of Boolean functions via multiplicative characters, divisibility properties of Gauss sums, and graph theory.

A new bootstrap assisted test for checking second order stationarity

A new computationally driven method is proposed to check if a possibly nonlinear time series is second order stationary or not, which is important in time series modeling. The new test relies on blocks of blocks bootstrap covariance matrix estimates and Walsh transformations in order to capture the nonlinearity features of time series. The asymptotic normality of the Walsh coefficients and their asymptotic covariance matrix under the null hypothesis are derived for nonlinear processes. In addition, the asymptotic covariance matrix of an increasing dimension is shown to be consistently estimated by a blocks of blocks bootstrap procedure. In the framework of locally stationary nonlinear processes, it is shown that the proposed test is consistent under a sequence of local alternatives. A simulation study is conducted to examine the finite sample performance of the procedure. In many nonlinear time series settings, the proposed test works well while existing methods may have highly inflated type I error rates. The proposed test is applied to an analysis of a financial data set.

Curvature of the Boolean majority function

Abstract The Boolean majority function and the generalized Boolean majority function of an even number n of variables are considered. For these functions exact values of the Walsh coefficients and the curvature are calculated.

The Lattice Geometry of Walsh-Function-Based Adaptive Optics

We show that there is an intrinsic link between the use of Walsh aberration modes in adaptive optics (AO) and the mathematics of lattices. The discrete and binary nature of these modes means that there are infinite combinations of Walsh mode coefficients that can optimally correct the same aberration. Finding such a correction is hence a poorly conditioned optimisation problem that can be difficult to solve. This can be mitigated by confining the AO correction space defined in Walsh mode coefficients to the fundamental Voronoi cell of a lattice. By restricting the correction space in this way, one can ensure there is only one set of Walsh coefficients that corresponds to the optimum correction aberration. This property is used to enable the design of efficient estimation algorithms to solve the inverse problem of finding correction aberrations from a sequence of measurements in a wavefront sensorless AO system. The benefit of this approach is illustrated using a neural-network-based estimator.

A Novel Approach to Designing Surrogate-assisted Genetic Algorithms by Combining Efficient Learning of Walsh Coefficients and Dependencies

Surrogate-assisted evolutionary algorithms have the potential to be of high value for real-world optimization problems when fitness evaluations are expensive, limiting the number of evaluations that can be performed. In this article, we consider the domain of pseudo-Boolean functions in a black-box setting. Moreover, instead of using a surrogate model as an approximation of a fitness function, we propose to precisely learn the coefficients of the Walsh decomposition of a fitness function and use the Walsh decomposition as a surrogate. If the coefficients are learned correctly, then the Walsh decomposition values perfectly match with the fitness function, and, thus, the optimal solution to the problem can be found by optimizing the surrogate without any additional evaluations of the original fitness function. It is known that the Walsh coefficients can be efficiently learned for pseudo-Boolean functions with k -bounded epistasis and known problem structure. We propose to learn dependencies between variables first and, therefore, substantially reduce the number of Walsh coefficients to be calculated. After the accurate Walsh decomposition is obtained, the surrogate model is optimized using GOMEA, which is considered to be a state-of-the-art binary optimization algorithm. We compare the proposed approach with standard GOMEA and two other Walsh decomposition-based algorithms. The benchmark functions in the experiments are well-known trap functions, NK-landscapes, MaxCut, and MAX3SAT problems. The experimental results demonstrate that the proposed approach is scalable at the supposed complexity of O (ℓ log ℓ) function evaluations when the number of subfunctions is O (ℓ) and all subfunctions are k -bounded, outperforming all considered algorithms.

Quantum and Randomised Algorithms for Non-linearity Estimation

Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires Θ (2 n ) queries to an n -bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted λ, with query complexities that depend polynomially on λ. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in n , already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of n . Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique.

Probabilistic characterization of random Max [formula omitted]-Sat

In this paper we provide a probabilistic characterization of the random Max r-Sat problem. We study the variance of the number of clauses satisfied by a random assignment, and the covariance of the numbers of clauses satisfied by a random pair of assignments of an arbitrary distance. Closed-form formulas for the expected value and the variance of these quantities are provided. We asymptotically and probabilistically analyze these formulas and use them to gain insights on the similarity of instances.Based on the above probabilistic characterization, we show that the correlation between the numbers of clauses satisfied by a random pair of assignments of distance d=cn, 0≤c≤1, converges in probability to ((1−c)r−1∕2r)∕(1−1∕2r). Our main result is that the so-called normalized autocorrelation length of Max r-Sat converges in probability to (1−1∕2r)∕r. The latter quantity is of interest in the area of landscape analysis as a way to better understand problems and assess their hardness for local search heuristics. A former result regarding the same quantity only expressed it in terms of Walsh coefficients. All our results apply to random r-Sat as well.

Walsh functions, scrambled [formula omitted]-nets, and negative covariance: Applying symbolic computation to quasi-Monte Carlo integration

We investigate base b Walsh functions for which the variance of the integral estimator based on a scrambled (0,m,s)-net in base b is less than or equal to that of the Monte-Carlo estimator based on the same number of points. First we compute the Walsh decomposition for the joint probability density function of two distinct points randomly chosen from a scrambled (t,m,s)-net in base b in terms of certain counting numbers and simplify it in the special case t is zero. Using this, we obtain an expression for the covariance of the integral estimator in terms of the Walsh coefficients of the function. Finally, we prove that the covariance of the integral estimator is negative when the Walsh coefficients of the function satisfy a certain decay condition. To do this, we use creative telescoping and recurrence solving algorithms from symbolic computation to find a sign equivalent closed form expression for the covariance term.

Quantum algorithms for the Goldreich–Levin learning problem

The Goldreich–Levin algorithm was originally proposed for a cryptographic purpose and then applied to learning. The algorithm is to find some larger Walsh coefficients of an n variable Boolean function. Roughly speaking, it takes a \(poly(n,\frac{1}{\epsilon }\log \frac{1}{\delta })\) time to output the vectors w with Walsh coefficients \(S(w)\ge \epsilon \) with probability at least \(1-\delta \). However, in this paper, a quantum algorithm for this problem is given with query complexity \(O(\frac{\log \frac{1}{\delta }}{\epsilon ^4})\), which is independent of n. Furthermore, the quantum algorithm is generalized to apply for an n variable m output Boolean function F with query complexity \(O(2^m\frac{\log \frac{1}{\delta }}{\epsilon ^4})\).

Quantum cryptographic property testing of multi-output Boolean functions

Compared with Boolean functions, multi-output Boolean functions (a.k.a. vectorial Boolean functions) are commonly used in classical cryptography. More generally, many cryptographic primitives can be treated as multi-output Boolean functions. Hence, the research on property testing of multi-output Boolean functions is meaningful for the design and cryptanalysis of symmetric cryptography. This paper mainly focuses on the cryptographic property testing of multi-output Boolean functions in the quantum world. Firstly, the generalized Deutsch–Jozsa algorithm is proposed to distinguish balanced multi-output Boolean functions from constant ones with a single query. This algorithm has a wider scope of applications with arbitrary ancillary inputs. The first generalized Bernstein–Vazirani algorithm suitable for multi-output Boolean functions is presented to recover the linear coefficients of linear functions. Then, combined with the generalized Deutsch–Jozsa algorithm, the quantum algorithm for estimating Walsh coefficients of multi-output Boolean functions is proposed with the same idea of quantum approximate counting algorithm, accompanied with an algorithm for computing the Walsh coefficient at a specified point based on quantum exact counting algorithm. Finally, with the usage of algorithms mentioned above, the cryptographic property testing of multi-output Boolean functions is studied. In order to describe the distances from having the certain properties, Euclidean distance and Manhattan distance are introduced as complements of Hamming distance. According to the definition, the first balance testing of multi-output Boolean functions is presented by testing the uniformity of images. The second algorithm exploits the relationship between balance and the Walsh coefficients at the point 0 which could be easily extended to k-order resiliency testing. We also briefly analyze the query complexities of strict avalanche criterion testing and k-order propagation criteria testing. The linearity testing algorithm for multi-output Boolean functions based on the generalized Bernstein–Vazirani algorithm can be adapted to Boolean functions achieving a further speedup. The non-junta testing algorithm is proposed with lower query complexity.

Quantum algorithms for learning Walsh spectra of multi-output Boolean functions

In classical cryptography, many cryptographic primitives could be treated as multi-output Boolean functions. The analysis of such functions is of great interest for cryptologists owing to their wide ranges of applications. Since each multi-output Boolean function can be uniquely determined by its Walsh transform, the Walsh spectra could reveal the properties of multi-output Boolean functions. In this paper, several quantum algorithms for learning Walsh spectra of multi-output Boolean functions are proposed. Firstly, with the usage of the amplitude estimation algorithm based on the Monte Carlo method, we present a quantum algorithm that allows one to estimate the Walsh coefficient of a multi-output Boolean function at a specified point with an additive error \(\epsilon \) and probability at least \(1-\delta \). The corresponding query complexity is \({\mathrm O}( {{\epsilon }^{-1}}\log {{\delta }^{-1}} )\). There is an almost quadratic speedup over the classical algorithm. Secondly, we propose a generalized phase kick-back technique for multi-output Boolean functions to encode multiple Walsh coefficients on the amplitudes of states. Based on this generalized technique, a quantum Goldreich–Levin algorithm for arbitrary multi-output Boolean function \(F:{{\{ 0,1 \}}^{n}}\rightarrow {{\{ 0,1 \}}^{m}}\) where \(m,n\in \mathbb {Z}\) is proposed to find those Walsh coefficients satisfying the threshold boundary condition \(\tau \) with probability at least \(1-\delta \). The whole query complexity is \({\mathrm O}\left( \frac{{{2}^{m+5+n/2}}}{{{\tau }^{3}}}\log \frac{{{2}^{m+5}}n}{\delta {{\tau }^{2}}} \right) \). Finally, by using the same idea of the swap-test circuit, the query complexity of the modified quantum Goldreich–Levin algorithm could be lowered to \({\mathrm O}\left( \frac{{{2}^{m+9}}n\pi }{{{\tau }^{4}}}\log \frac{{{2}^{m+3}}n}{\delta {{\tau }^{2}}} \right) \) achieving a further speedup when \(\tau \) is no less than \({\mathrm O}( {{2}^{-n/2+6}}n)\). Those two quantum Goldreich–Levin algorithms have their own advantages in implementation and query complexity.

Changing APN functions at two points

We investigate a construction in which a vectorial Boolean function G is obtained from a given function F over $\mathbb {F}_{2^{n}}$ by changing the values of F at two points of the underlying field. In particular, we examine the possibility of obtaining one APN function from another in this way. We characterize the APN-ness of G in terms of the derivatives and in terms of the Walsh coefficients of F. We establish that changing two points of a function F over $\mathbb {F}_{2^{n}}$ which is plateaued (and, in particular, AB) or of algebraic degree deg(F) < n − 1 can never give a plateaued (and AB, in particular) function for any n ≥ 5. We also examine a particular case in which we swap the values of F at two points of $\mathbb {F}_{2^{n}}$. This is motivated by the fact that such a construction allows us to obtain one permutation from another. We obtain a necessary and sufficient condition for the APN-ness of G which we then use to show that swapping two points of any power function over a field $\mathbb {F}_{2^{n}}$ with n ≥ 5 can never produce an APN function. We also list some experimental results indicating that the same is true for the switching classes from Edel and Pott (Adv. Math. Commun. 3(1):59–81, 2009), and conjecture that the Hamming distance between two APN functions cannot be equal to two for n ≥ 5.

Approximation of Ensemble Boundary Using Spectral Coefficients.

A spectral analysis of a Boolean function is proposed for approximating the decision boundary of an ensemble of classifiers, and an intuitive explanation of computing Walsh coefficients for the functional approximation is provided. It is shown that the difference between the first- and third-order coefficient approximations is a good indicator of optimal base classifier complexity. When combining neural networks, the experimental results on a variety of artificial and real two-class problems demonstrate under what circumstances ensemble performance can be improved. For tuned base classifiers, the first-order coefficients provide performance similar to the majority vote. However, for weak/fast base classifiers, higher order coefficient approximation may give better performance. It is also shown that higher order coefficient approximation is superior to the Adaboost logarithmic weighting rule when boosting weak decision tree base classifiers.

Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration

In this paper we give a new Koksma-Hlawka type inequality for Quasi-Monte Carlo (QMC) integration. QMC integration of a function $f\colon[0,1)^s\rightarrow\mathbb{R}$ by a finite point set $\mathcal{P}\subset[0,1)^s$ is the approximation of the integral $I(f):=\int_{[0,1)^s}f(\mathbf{x})\,d\mathbf{x}$ by the average $I_{\mathcal{P}}(f):=\frac{1}{|\mathcal{P}|}\sum_{\mathbf{x} \in \mathcal{P}}f(\mathbf{x})$. We treat a certain class of point sets $\mathcal{P}$ called digital nets. A Koksma-Hlawka type inequality is an inequality providing an upper bound on the integration error $\text{Err}(f;\mathcal{P}):=I(f)-I_{\mathcal{P}}(f)$ of the form $|\text{Err}(f;\mathcal{P})|\le C\cdot \|f\|\cdot D(\mathcal{P})$. We can obtain a Koksma-Hlawka type inequality by estimating bounds on $|\hat{f}(\mathbf{k})|$, where $\hat{f}(\mathbf{k})$ is a generalized Fourier coefficient with respect to the Walsh system. In this paper we prove bounds on the Walsh coefficients $\hat{f}(\mathbf{k})$ by introducing an operator called ‘dyadic difference’ $\partial_{i,n}$. By converting dyadic differences $\partial_{i,n}$ to derivatives $\frac{\partial }{\partial x_i}$, we get a new bound on $|\hat{f}(\mathbf{k})|$ for a function $f$ whose mixed partial derivatives up to order $\alpha$ in each variable are continuous. This new bound is smaller than the known bound on $|\hat{f}(\mathbf{k})|$ under some instances. The new Koksma-Hlawka type inequality is derived using this new bound on the Walsh coefficients.

Reliable error estimation for Sobol’ indices

A practical problem concerning the estimation of Sobol' indices is how large to choose the number of evaluations to ensure that estimates are accurate enough. This paper addresses this challenge by proposing a reliable error bound for Sobol' indices based on digital sequences. The error bound is defined in terms of the discrete Walsh coefficients of the integrands involved in the Sobol' indices definition. We propose a sequential estimation procedure of first-order and total effect Sobol' indices using the error bound as stopping criterion. The procedure operates under the assumption that all integrands lie inside a particular cone of functions

Universal function for a weighted space $$L^1_{\mu }[0,1]$$ L μ 1 [ 0 , 1

It is shown that there exist such a function \(g\in L^1[0,1]\) and a weight function \(0<\mu (x)\le 1\) that g is universal for the weighted space \(L^1_\mu [0,1]\) with respect to signs of its Fourier–Walsh coefficients.

The sum of modules of Walsh coefficients for some balanced Boolean functions

Рассматриваются следующие сбалансированные булевы функции от $n$ переменных: а) построенные из нормальных бент-функций методом Доббертина, б) мажоритарная, в) функции, у которых в столбце значений единичные значения располагаются подряд, начиная с любого места. Для них вычисляются точные значения сумм модулей коэффициентов Уолша-Адамара и приводятся их оценки.

Universal functions in `correction' problems guaranteeing the convergence of Fourier–Walsh series

We prove the existence of a function with monotone decreasing Fourier–Walsh coefficients which is universal in , , in the sense of modification with respect to the signs of the Fourier coefficients for the Walsh system. In other words, for every function and every one can find a function such that the measure is greater than , the Fourier series of in the Walsh system converges to in the -norm and , . We also prove that for every , , one can find a measurable set of measure and a function with , , such that for every function there is a function with the following properties: coincides with on , the Fourier–Walsh series of converges to in the norm of and the absolute values of all terms in the sequence of the Fourier–Walsh coefficients of the newly obtained function satisfy , .

The sum of modules of Walsh coefficients of Boolean functions

Abstract We obtain achievable lower and upper bounds for the sums of modules of Walsh coefficients of Boolean functions of nvariables. An average value of such sums in the class of all Boolean functions of n variables and in its subclass consisting of all balanced functions is evaluated. We present some classes of nonlinear balanced functions whose sums of modules of Walsh coefficients are close to the obtained lower and upper bounds