This paper shows that sum-of-product expression (SOP) minimization produces the generalization ability. We show this in three steps. First, various classes of SOPs are generated. Second, minterms of SOP are randomly selected to generate partially defined functions. And, third, from the partially defined functions, original functions are reconstructed by SOP minimization. We consider Achilles heel functions, majority functions, monotone increasing cascade functions, functions generated from random SOPs, monotone increasing random SOPs, circle functions, and globe functions. As for the generalization ability, the presented method is compared with Naive Bayes, multi-level perceptron, support vector machine, JRIP, J48, and random forest. For these functions, in many cases, only 10% of the input combinations are sufficient to reconstruct more than 90% of the truth tables of the original functions.

For the multi-objective time series search problem, Hasegawa and Itoh [Theoretical Computer Science, Vol.78, pp.58-66, 2018] presented the best possible online algorithm balanced price policy for any monotone function f:Rk→R. Specifically the competitive ratio with respect to the monotone function f(c1,...,ck)=(c1+…+ck)/k is referred to as the arithmetic mean component competitive ratio. Hasegawa and Itoh derived the explicit representation of the arithmetic mean component competitive ratio for k=2, but it has not been known for any integer k≥3. In this paper, we derive the explicit representations of the arithmetic mean component competitive ratio for k=3 and k=4, respectively. On the other hand, we show that it is computationally difficult to derive the explicit representation of the arithmetic mean component competitive ratio for arbitrary integer k in a way similar to the cases for k=2, 3, and 4.

The average performance of a single-user MIMO system under spatially correlated fading and with different types of CSI at the transmitter and with perfect CSI at the receiver was studied in recent work. In contrast to analyzing a single performance metric, e.g. the average mutual information or the average bit error rate, we study an arbitrary representative of the class of matrix-monotone functions. Since the average mutual information as well as the average normalized MSE belong to that class, this universal class of performance functions brings together the information theoretic and signal processing performance metric. We use Lowner's representation of operator monotone functions in order to derive the optimum transmission strategies as well as to characterize the impact of correlation on the average performance. Many recent results derived for average mutual information generalize to arbitrary matrix-monotone performance functions, e.g. the optimal transmit strategy without CSI at the transmitter is equal power allocation. The average performance without CSI is a Schur-concave function with respect to transmit and receive correlation. In addition to this, we derive the optimal transmission strategy with long-term statistics knowledge at the transmitter and propose an efficient iterative algorithm. The beamforming-range is the SNR range in which only one data stream spatially multiplexed achieves the maximum average performance. This range is important since it has a simple receiver structure and well known channel coding. We entirely characterize the beamforming-range. Finally, we derive the generalized water-filling transmit strategy for perfect CSI and characterize its properties under channel correlation.