Abstract: One major drawback of deep convolutional neural networks (CNNs) for use in safety critical applications is their black-box nature. This makes it hard to verify or monitor complex, symbolic requirements on already trained computer vision CNNs. In this work, we present a simple, yet effective, approach to verify that a CNN complies with symbolic predicate logic rules which relate visual concepts. It is the first that (1) does not modify the CNN, (2) may use visual concepts that are no CNN in- or output feature, and (3) can leverage continuous CNN confidence outputs. To achieve this, we newly combine methods from explainable artificial intelligence and logic: First, using supervised concept embedding analysis, the output of a CNN is post-hoc enriched by concept outputs.

Data mining techniques can be used to discover useful patterns by exploring and analyzing data and it's feasible to synergitically combine machine learning tools to discover fuzzy classification rules.In this paper, an adaptive Neuro fuzzy network with TSK fuzzy type and an improved quantum subtractive clustering has been developed. Quantum clustering (QC) is an intuition from quantum mechanics which uses Schrodinger potential and time-consuming gradient descent method. The principle advantage and shortcoming of QC is analyzed and based on its shortcomings, an improved algorithm through a subtractive clustering method is proposed. Cluster centers represent a general model with essential characteristics of data which can be use as premise part of fuzzy rules.The experimental results revealed that proposed Anfis based on quantum subtractive clustering yielded good approximation and generalization capabilities and impressive decrease in the number of fuzzy rules and network output accuracy in comparison with traditional methods.

Working with a non-stationary stream of data requires for the analysis system to evolve its model (the parameters as well as the structure) over time. In particular, concept drifts can occur, which makes it necessary to forget knowledge that has become obsolete. However, the forgetting is subjected to the stability-plasticity dilemma, that is, increasing forgetting improve reactivity of adapting to the new data while reducing the robustness of the system. Based on a set of inference rules, Evolving Fuzzy Systems-EFS-have proven to be effective in solving the data stream learning problem. However tackling the stability-plasticity dilemma is still an open question. This paper proposes a coherent method to integrate forgetting in Evolving Fuzzy System, based on the recently introduced notion of concept drift anticipation. The forgetting is applied with two methods: an exponential forgetting of the premise part and a deferred directional forgetting of the conclusion part of EFS to preserve the coherence between both parts. The originality of the approach consists in applying the forgetting only in the anticipation module and in keeping the EFS (called principal system) learned without any forgetting. Then, when a drift is detected in the stream, a selection mechanism is proposed to replace the obsolete parameters of the principal system with more suitable parameters of the anticipation module. An evaluation of the proposed methods is carried out on benchmark online datasets, with a comparison with state-of-the-art online classifiers (Learn++.NSE, PENsemble, pclass) as well as with the original system using different forgetting strategies.

In this paper, a novel stepwise learning approach based on estimating desired premise parts' outputs by solving a constrained optimization problem is proposed. This learning approach does not require backpropagating the output error to learn the premise parts' parameters. Instead, the near best output values of the rules premise parts are estimated and their parameters are changed to reduce the error between current premise parts' outputs and the estimated desired ones. Therefore, the proposed learning method avoids error backpropagation, which lead to vanishing gradient and consequently getting stuck in a local optimum. The proposed method does not need any initialization method. This learning method is utilized to train a new Takagi-Sugeno-Kang (TSK) Fuzzy Neural Network with correlated fuzzy rules including many parameters in both premise and consequent parts, avoiding getting stuck in a local optimum due to vanishing gradient. To learn the proposed network parameters, first, a constrained optimization problem is introduced and solved to estimate the desired values of premise parts' output values. Next, the error between these values and the current ones is utilized to adapt the premise parts' parameters based on the gradient-descent (GD) approach. Afterward, the error between the desired and network's outputs is used to learn consequent parts' parameters by the GD method. The proposed paradigm is successfully applied to real-world time-series prediction and regression problems. According to experimental results, its performance outperforms other methods with a more parsimonious structure.