SUFFICIENT CONDITIONS FOR SWITCHING FUNCTIONS TO BE THRESHOLD ONES AND THEIR APPLICATIONS

Threshold Function (TF) is a subset of Boolean function that can be represented with a single linear threshold gate (LTG). In the research about threshold logic, the identification of TF is an important task that determines whether a given function is a TF or not. In this article, we propose a sufficient and necessary condition for a function being a TF. With the proposed sufficient and necessary condition, we devise a TF identification algorithm. The experimental results show that the proposed approach saves 80% CPU time for identifying all the 8-input NP-class TFs as compared with the state-of-the-art. Furthermore, the LTGs corresponding to the identified TFs obtained by the proposed approach have smaller weights and threshold values than the state-of-the-art.

A neuron widely used for learning is based on an inner product of a weight vector and an input vector and a threshold function, which represents a step-like surface in a hyperspace. The proposed neuron decides the position and sharpness of the step by three parameters: the normal vector, the center position vector, and the sharpness. The center position vector makes it easy to control the step position in the input vector space and to produce active neurons. Moreover, the gradient descent method based on those P-neuron parameters achieves efficient use of many neurons. This method modifies the normal vector, not only by the input vector, but also by the difference between the center position and the input vector, and it produces neurons with various normal vectors. As a result, the number of neurons necessary for training is drastically reduced. Simulation results from some basic training problems show the good performance of the proposed neuron. >

Dimension reduction and classification of data is an important issue in the data processing. Reduct in the rough set is a minimal subset of features, which has the same discernible power as the entire features in the higher dimensional scheme. Boolean threshold function was of fundamental interest for classification function. The nearest neighbor relation with minimal distance proposed here has a fundamental information for classification. Further, the nearest neighbor relation plays a crucial role for generation of reducts and threshold functions using the Boolean reasoning on the discernibility and in discernibility matrices, in which the indiscernibility matrix is proposed here to test the sufficient condition for reduct and threshold function. Then, a genetation method for the threshold functions based on the nearest neighbor relation are proposed here using Boolean operations.

It is shown that the problem of determining whether or not a given switching function is realizable with a single threshold gate can be reduced directly to that of determining the value of a two-person, zero-sum game. If this value is $\gt$ 1/2, the function is realizable. If the value is $\geq$ 1/2, the function cannot be realized. In the case of realizability the solution of the game gives the set of weights which accomplishes the realization. The method works equally well with functions having "don't care" conditions. When a function is not realizable a threshold gate is, nevertheless, defined together with a "zone of uncertainty." This gate is such that for a given input, gamma, if the weighted sum is above this "zone of uncertainty" then F(gamma) = 1, within the zone F(gamma;) is uncertain, and below the zone F(gamma) = 0. The larger the value of the game the smaller this "zone of uncertainty." A method for solving two-person zero-sum games is described and several examples are given.

The multi-design multi-response linear regression problem is investigated, in which design matrices are Gaussian with covariance matrices Σ (1:K) = (Σ (1) , ... , 1 (K) ) for K linear regression tasks. Design matrices across tasks are assumed to be independent. The support union of K p-dimensional regression vectors (collected as columns of matrix B*) is recovered using l 1 /l 2 -regularized lasso. Sufficient and necessary conditions on sample complexity are characterized as a sharp threshold to guarantee successful recovery of the support union. This model has been previously studied via l 1 /l ∞ -regularized lassoand via l 1 /l 1 + l 1 /l ∞ -regularized lasso, in which sharp threshold on sample complexity is characterized only for K = 2 and under special conditions. In this paper, using l 1 /l 2 -regularized lasso, sharp threshold on sample complexity is characterized under standard regularization conditions. Namely, if n > c p1 ψ(B*, Σ (1:K) ) log(p - s) where c p1 is a constant, and s is the size of the support set, then l 1 /l 2 -regularized lasso correctly recovers the support union; and if n <; c p2 ψ(B*, Σ (1:K) )log(p - s) where c p2 is a constant, then l 1 /l 2 -regularized lasso fails to recover the support union. In particular, the function ψ(B*, Σ (1:K) ) captures the impact of the sparsity of K regression vectors and the statistical properties of the design matrices on the threshold on sample complexity. Therefore, such threshold function also demonstrates the advantages of joint support union recovery using multitask lasso over individual support recovery using single-task lasso.

We present a novel affine projection algorithm (APA) which dynamically selects input vectors in order to improve convergence performance. The optimum selection of the input vectors is derived by the largest decrease of the mean-square deviation. The experimental results show that the proposed algorithm has a fast convergence speed and a small steady-state error compared to the conventional APA. In addition, the proposed algorithm retains a low overall computational complexity.

Currently the best algorithms for transcription factor binding site predictions are severely limited in accuracy. However, a non-linear combination of these algorithms could improve the quality of predictions. A support-vector machine was applied to combine the predictions of 12 key real valued algorithms. The data was divided into a training set and a test set, of which two were constructed: filtered and unfiltered. In addition, a different of consecutive results was used in the input vector in order to contextualize the neighbouring results. Finally, classification results were improved with the aid of under and over sampling techniques. Our major finding is that we can reduce the False-Positive rate significantly. We also found that the bigger the window, the higher the F-score, but the more likely it is to make a false positive prediction, with the best trade-off being a window size of about 7.

Many voting systems consist of a set of players who can form coalitions; a winning coalition is one that can pass a measure. The winning and losing coalitions can be given by several classes of functions, including switching functions, threshold functions, weighted voting systems, and simple games. We define a zonotope T n T_n related to these functions and describe its coordinates in terms of a measure of voting power. We also give a voting interpretation of the coordinates of the derived zonotope T ¯ n \bar {T}_n .

In an estimation framework, the suspected sparsity of an unknown vector of deterministic parameters is classically accounted for through thresholding functions, some of which being related to Maximum A Posteriori estimates with specific priors. In a detection framework, statistical tests applied to sparse vectors are classically designed so as to limit the focus of the test to a few active components, leading to test statistics of thresholded data. We propose a study of the connections between these two problems. The detection tests we consider are the Generalized Likelihood Ratio Test (GLRT), the Bayes Factor, the Posterior Density Ratio and a LRT using a Maximum A Posteriori estimate with Generalized Gaussian priors. In the case of scalar parameter first, we derive sufficient conditions that any detection test must obey to be equivalent to a GLRT, and prove that the tests above verify them. In the vector case, the LRT with a MAP estimate and the PDR outperform the GLRT and the BF thanks to the thresholding effect of the MAP estimation. The connections between thresholding functions used in estimation and the resulting ”thresholded test statistics” are precisely investigated.

We present formal models and efficient synthesis algorithms for threshold gates of Generalised Threshold Gate (GTG) and Multi Threshold Threshold Gate (MTTG) structures. For GTG synthesis our method does not require to calculate thresholds, that separate function onset and offsets, which greatly simplifies and speeds up the synthesis algorithm. For MTTG we propose a novel synthesis method that determines structure and circuit parameters efficiently. As complexity of MTTG circuit increases with number of thresholds we have developed a dedicated heuristic for efficiently determining the number of required thresholds. The number of resulting thresholds is smaller compared to the number of thresholds calculated using other methods.

An integer polynomial p of n variables is called a threshold gate for the Boolean function f of n variables if for all x∈{0, 1}nf(x)=1 if and only if p(x)≥0. The weight of a threshold gate is the sum of its absolute values. In this paper we study how large weight might be needed if we fix some function and some threshold degree. We prove $2^{\Omega(2^{2n/5})}$ lower bound on this value. The best previous bound was $2^{\Omega(2^{n/8})}$ [12]. In addition we present substantially simpler proof of the weaker $2^{\Omega(2^{n/4})}$ lower bound. This proof is conceptually similar to other proofs of the bounds on weights of nonlinear threshold gates, but avoids a lot of technical details arising in other proofs. We hope that this proof will help to show the ideas behind the construction used to prove these lower bounds.

A classical theorem conjectured by Jacobi asserts that for an odd prime p, the sum of the quadratic residues in the interval (0, p) is less than the sum of the quadratic nonresidues if and only if p 3 (mod 4).We generalize Jacobi's problem to fcth powers (mod p), k > 2, and we consider in some detail a generalization of Jacobi's conjecture to quadratic residues and nonresidues (mod n), n an arbitrary integer > 2. From the set of least positive residues (mod ), let Cq denote the subgroup of quadratic residues (mod n) and let cx, c2.ct be the cosets which can be formed with respect to this subgroup.Computer data supports the following generalized Jacobi conjecture:The sum of the elements in cn is less than or equal to the sum in any of the other cosets for every integer n > 2, a surprising conjecture, especially in view of the fact that counterexamples are easily obtained for k = 4, 6, 8, 10, etc. (The coset sums are identical for odd k and prime modulus.)We resolve the generalized Jacobi conjecture in the affirmative when, for example, n is an integer admitting a primitive root, or n = 2 , a > 3. (Here we give explicit formulae for the four coset sums.)For h = 2p , our proof that the quadratic residues and the quadratic nonresidues (mod n) have the same sum for odd prime p if and only if p ^ 3 (mod 8) is purely, elementary.On the other hand, we need Dirichlet's class number formula for quadratic number fields with discriminant -p = 5 (mod 8) to show that the sum of the quadratic nonresidues strictly exceeds the sum of the quadratic residues (mod 2pa) if p = 3 (mod 8).Computer data gives rise to a host of a. a, a interesting problems we are unable to resolve.For example, if n = 2pj p2 pfr, I < i < r, we conjecture that a sufficient condition that the coset sums not be identical is that we have p = 3 (mod 8) for every i.It is not hard to show that the coset sums are identical if every p = 1 (mod 4).However, the problem of finding a necessary condition is very difficult since, e.g., the coset sums are not identical for n < 1146 when n=2-3-pifp = 23 (mod 24), but the sums are identical if p = 7 (mod 24).

Properties of Facets of Binary Polytopes

Stabilization of switched linear systems using an improved event-triggered control scheme

Sufficient conditions are presented under which the dynamic stochastic approximation estimate converges with probability 1 to the true state vector of a nonlinear stochastic discrete-time dynamic system. These conditions are stated in terms of system dynamics, measurement function, and noise statistics; and are closely related to the concept of on-line stochastic observability.