Generalised Pattern Search with Restarting Fitness Landscape Analysis

Fitness landscape analysis for optimisation is a technique that involves analysing black-box optimisation problems to extract pieces of information about the problem, which can beneficially inform the design of the optimiser. Thus, the design of the algorithm aims to address the specific features detected during the analysis of the problem. Similarly, the designer aims to understand the behaviour of the algorithm, even though the problem is unknown and the optimisation is performed via a metaheuristic method. Thus, the algorithmic design made using fitness landscape analysis can be seen as an example of explainable AI in the optimisation domain. The present paper proposes a framework that performs fitness landscape analysis and designs a Pattern Search (PS) algorithm on the basis of the results of the analysis. The algorithm is implemented in a restarting fashion: at each restart, the fitness landscape analysis refines the analysis of the problem and updates the pattern matrix used by PS. A computationally efficient implementation is also presented in this study. Numerical results show that the proposed framework clearly outperforms standard PS and another PS implementation based on fitness landscape analysis. Furthermore, the two instances of the proposed framework considered in this study are competitive with popular algorithms present in the literature.

OPTIMAL CONSTRUCTION OF THE PATTERN MATRIX FOR PROBABILISTIC NEURAL NETWORKS IN TECHNICAL DIAGNOSTICS BASED ON EXPERT ESTIMATIONS

In the field of technical diagnostics, many tasks are solved by using automated classification. For this, such classifiers like probabilistic neural networks fit best owing to their simplicity. To obtain a probabilistic neural network pattern matrix for technical diagnostics, expert estimations or measurements are commonly involved. The pattern matrix can be deduced straightforwardly by just averaging over those estimations. However, averages are not always the best way to process expert estimations. The goal is to suggest a method of optimally deducing the pattern matrix for technical diagnostics based on expert estimations. The main criterion of the optimality is maximization of the performance, in which the subcriterion of maximization of the operation speed is included. First of all, the maximal width of the pattern matrix is determined. The width does not exceed the number of experts. Then, for every state of an object, the expert estimations are clustered. The clustering can be done by using the k-means method or similar. The centroids of these clusters successively form the pattern matrix. The optimal number of clusters determines the probabilistic neural network optimality by its performance maximization. In general, most results of the error rate percentage of probabilistic neural networks appear to be near-exponentially decreasing as the number of clustered expert estimations is increased. Therefore, if the optimal number of clusters defines a too “wide” pattern matrix whose operation speed is intolerably slow, the performance maximization implies a tradeoff between the error rate percentage minimum and maximally tolerable slowness in the probabilistic neural network operation speed. The optimal number of clusters is found at an asymptotically minimal error rate percentage, or at an acceptable error rate percentage which corresponds to maximally tolerable slowness in operation speed. The optimality is practically referred to the simultaneous acceptability of error rate and operation speed.

Quantum communication complexity of distribution testing

The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $\epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $\tilde{O}(n/(t\epsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.

On the spectrum of noisy blown-up matrices

We study the eigenvalues of large perturbed matrices. We consider a pattern matrix P, we blow it up to get a large block-matrix Bn. We can observe only a noisy version of matrix Bn. So we add a random noise Wn to obtain the perturbed matrix An = Bn + Wn. Our aim is to find the structural eigenvalues of An. We prove asymptotic theorems on this problem and also suggest a graphical method to distinguish the structural and the non-structural eigenvalues of An.

The Bipartite Zero Forcing Set for a Full Sign Pattern Matrix

For an m × n sign pattern P, we define a signed bipartite graph B ( U , V ) with one set of vertices U = { 1 , 2 , … , m } based on rows of P and the other set of vertices V = { 1 ′ , 2 ′ , … , n ′ } based on columns of P. The zero forcing number is an important graph parameter that has been used to study the minimum rank problem of a matrix. In this paper, we introduce a new variant of zero forcing set−bipartite zero forcing set and provide an algorithm for computing the bipartite zero forcing number. The bipartite zero forcing number provides an upper bound for the maximum nullity of a square full sign pattern P. One advantage of the bipartite zero forcing is that it can be applied to study the minimum rank problem for a non-square full sign pattern.