Walsh Coefficients Research Articles

Behavior of the Fourier–Walsh coefficients of a corrected function

We prove that, given a sequence {ak}k=1∞ with ak ↓ 0 and {ak}k=1∞ ∉ l2, reals 0 < e < 1 and p ∈ [1, 2], and f ∈ Lp(0, 1), we can find f ∈ Lp(0, 1) with mes{f ≠ f < e whose nonzero Fourier–Walsh coefficients ck(f) are such that |ck(f)| = ak for k ∈ spec(f).

On the universal function for the class L[0,1], p∈ (0,1)

It is shown that for each number p∈(0,1) there exists a function g(x)∈L1[0,1] with monotonically decreasing Fourier–Walsh coefficients, which is universal for the class Lp[0,1] with respect to signs of those coefficients.

Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients

We establish formulas for the b-adic Walsh coefficients of functions in Cα[0,1] for an integer α≥1 and give upper bounds on the Walsh coefficients of these functions. We also study the Walsh coefficients of periodic and non-periodic functions in reproducing kernel Hilbert spaces.

An Explicit Construction of Optimal Order Quasi--Monte Carlo Rules for Smooth Integrands

In a recent paper by the authors, it is shown that there exists a quasi-Monte Carlo (QMC) rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel Hilbert space consisting of smooth functions. In this paper we provide an explicit construction of such an optimal order QMC rule. Our approach is to exploit both the decay and the sparsity of the Walsh coefficients of the reproducing kernel simultaneously. This can be done by applying digit interlacing composition due to Dick to digital nets with large minimum Hamming and Niederreiter-Rosenbloom-Tsfasman metrics due to Chen and Skriganov. To our best knowledge, our construction gives the first QMC rule which achieves the best possible convergence in this function space.

Facial action unit recognition using multi-class classification

Within the context of facial expression classification using the facial action coding system (FACS), we address the problem of detecting facial action units (AUs). Feature extraction is performed by generating a large number of multi-resolution local binary pattern (MLBP) features and then selecting from these using fast correlation-based filtering (FCBF). The need for a classifier per AU is avoided by training a single error-correcting output code (ECOC) multi-class classifier to generate occurrence scores for each of several AU groups. A novel weighted decoding scheme is proposed with the weights computed using first order Walsh coefficients. Platt scaling is used to calibrate the ECOC scores to probabilities and appropriate sums are taken to obtain separate probability estimates for each AU individually. The bias and variance properties of the classifier are measured and we show that both these sources of error can be reduced by enhancing ECOC through bootstrapping and weighted decoding.

Computing Walsh coefficients from the algebraic normal form of a Boolean function

We study the relationship between the Walsh Transform of a Boolean function and its Algebraic Normal Form(ANF), and present algorithms that compute the Walsh coefficients at a small set of points in terms of certain parameters derived from the ANF of a Boolean function. In the first part of this paper, based on the previous result by Gupta and Sarkar, we investigate the formula in Gupta-Sarkar's algorithm in a novel iterative method and obtain a recurrence relation for the Walsh Transform of a Boolean function. The second part is devoted to applying this formula to algorithms to evaluate it. Experimental result shows that for the specified classes of Boolean functions, our algorithms can perform better than Gupta-Sarkar's algorithm. For example, the proposed algorithm "ComputeWalsh" is able to compute the Walsh coefficients of the functions for which the complexity of Gupta-Sarkar's algorithm is impractical. Besides, for functions acting on high number of variables (m>30) and having low number of monomials, the proposed algorithms are advantageous over the Fast Walsh Transform which is a standard method of computing the Walsh Transform with a complexity of O(m2 m ) operations.

Numerical integration of Hölder continuous, absolutely convergent Fourier, Fourier cosine, and Walsh series

We introduce quasi-Monte Carlo rules for the numerical integration of functions f defined on [0,1]s, s≥1, which satisfy the following properties: the Fourier, Fourier cosine or Walsh coefficients of f are absolutely summable and f satisfies a Hölder condition of order α, for some 0<α≤1. We show a convergence rate of the integration error of order max((s−1)N−1/2,sα/2N−α). The construction of the quadrature points is explicit and is based on Weil sums.

Training and making calculations with mixed order hyper-networks

A neural network with mixed order weights, n neurons and a modified Hebbian learning rule can learn any function f:{−1,1}n→R and reproduce its output as the network׳s energy function. The network weights are equal to Walsh coefficients, the fixed point attractors are local maxima in the function, and partial sums across the weights of the network calculate averages for hyperplanes through the function. If the network is trained on data sampled from a distribution, then marginal and conditional probability calculations may be made and samples from the distribution generated from the network. These qualities make the network ideal for optimisation fitness function modelling and make the relationships amongst variables explicit in a way that architectures such as the MLP do not.

On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh---Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.

Counting Boolean functions with specified values in their Walsh spectrum

The problem of counting Boolean functions with specified number s of Walsh coefficients ω in their Walsh spectrum is discussed in this paper. Strategies to solve this problem shall help solving many more problems related to desired cryptographic features of Boolean functions such as nonlinearity, resiliency, algebraic immunity, etc. In an attempt to study this problem, we present a new framework of solutions. We give results for |ω|≥2n−1 and for all s, in line with a previous work of Wu (1998) [12]. We also provide various results such as existence and construction for some s when ω=0, multiplicities for all ω and naive bounds on s for ω>2n/2.

Ensemble pruning using spectral coefficients.

Ensemble pruning aims to increase efficiency by reducing the number of base classifiers, without sacrificing and preferably enhancing performance. In this brief, a novel pruning paradigm is proposed. Two class supervised learning problems are pruned using a combination of first- and second-order Walsh coefficients. A comparison is made with other ordered aggregation pruning methods, using multilayer perceptron base classifiers. The Walsh pruning method is analyzed with the help of a model that shows the relationship between second-order coefficients and added classification error with respect to Bayes error.

Design of Real Time Walsh Transform for Processing of Multiple Digital Signals

This paper presents the design and implementation of multiple digital signals processing using real-time Walsh transforms. The design of real time Walsh transform is done in such a way that it starts producing outputs instantly even before all input data have entered the system. The system consists of Walsh Transform circuit, several Digital Signal Processing (DSP) circuits, and an inverse Walsh transform circuit. The real time Walsh and inverse Walsh transforms are also designed to produce right results for any possible combinations of input data. DSP blocks are able to perform addition, subtraction, and dyadic convolution process of Walsh coefficients of more than one digital signals. Comparisons to the previous methods are briefly presented. It was found that the design of real time Walsh transform structure has better performance than many of the previously reported results in the literature. DOI: http://dx.doi.org/10.11591/ijece.v3i2.1965

Boolean functions with two distinct Walsh coefficients

Are there other Boolean functions having two distinct Walsh coefficients except affine Boolean functions and maximal nonlinear (i.e. bent) Boolean functions? This paper proves that all Boolean functions with exactly two distinct Walsh coefficients are just the two known classes of affine and bent Boolean functions and the Boolean functions obtained by modifying the value of affine or bent Boolean functions at x = 0.

Minimising Added Classification Error Using Walsh Coefficients

Two-class supervised learning in the context of a classifier ensemble may be formulated as learning an incompletely specified Boolean function, and the associated Walsh coefficients can be estimated without the knowledge of the unspecified patterns. Using an extended version of the Tumer-Ghosh model, the relationship between added classification error and second-order Walsh coefficients is established. In this brief, the ensemble is composed of multilayer perceptron base classifiers, with the number of hidden nodes and epochs systematically varied. Experiments demonstrate that the mean second-order coefficients peak at the same number of training epochs as ensemble test error reaches a minimum.

On the approximation of smooth functions using generalized digital nets

In this paper, we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for approximating functions in Walsh Korobov spaces, using digital nets. Here, the key ingredient is the use of generalized digital nets (which have recently been shown to achieve higher order convergence rates for the integration of smooth functions). This allows us to approximate functions with square integrable mixed partial derivatives of order α > 1 in each variable. The approximation error is studied in the worst case setting in the L 2 norm. We also discuss tractability of our proposed approximation algorithm, investigate its computational complexity, and present numerical examples.

THE DECAY OF THE WALSH COEFFICIENTS OF SMOOTH FUNCTIONS

AbstractWe give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and nonperiodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.

Lower and Upper Bounds for Linkage Discovery

For a real-valued function <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> defined on {0,1} <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , the linkage graph of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> is a hypergraph that represents the interactions among the input variables with respect to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> . In this paper, lower and upper bounds for the number of function evaluations required to discover the linkage graph are rigorously analyzed in the black box scenario. First, a lower bound for discovering linkage graph is presented. To the best of our knowledge, this is the first result on the lower bound for linkage discovery. The investigation on the lower bound is based on Yao's minimax principle. For the upper bounds, a simple randomized algorithm for linkage discovery is analyzed. Based on the Kruskal-Katona theorem, we present an upper bound for discovering the linkage graph. As a corollary, we rigorously prove that <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ) function evaluations are enough for bounded functions when the number of hyperedges is <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ), which was suggested but not proven in previous works. To see the typical behavior of the algorithm for linkage discovery, three random models of fitness functions are considered. Using probabilistic methods, we prove that the number of function evaluations on the random models is generally smaller than the bound for the arbitrary case. Finally, from the relation between the linkage graph and the Walsh coefficients, it is shown that, for bounded functions, the proposed bounds are eventually the bounds for finding the Walsh coefficients.

Quick detection of faults in combinational networks designed in minterm format by computing a single novel parameter

This paper reports on the development of a novel scheme as an alternative to Susskind's method [A.K. Susskind, Testing by verifying Walsh coefficients, Proceedings of the 11th Annual Symposium on fault-tolerant computing, June 1981, pp. 206–208.] for detection of struck-at faults in combinational logic circuits. The main idea in the present scheme is to extend an existing design of the combinational network under test in minterm format by some logic checking the correctness of all input–output mappings by computing only one novel parameter after cycling through all 2 n input combinations of a circuit with n inputs. Thus the present scheme uses 2 n n-bit input patterns once to a circuit while Susskind's method uses twice that many. So it provides substantially less work than that needed in Susskind's method. Furthermore, the tester needed to implement the present scheme is a simplified version of that needed in Susskind's method.

A multivariate fast discrete Walsh transform with an application to function interpolation

For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only $O(N \log N)$ operations, where $N$ is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms. This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented.

Higher order functions and Walsh coefficients revisited

The main purpose of this note is to present an alternative, more transparent treatment of the results obtained in [4], which link the epistasis of a function to its Walsh coefficients and its order.