This paper presents a novel network, called Scale Equalized Higher-order Neural Network (SEHNN) based on concept of Scale Equalization (SE). We show that SE is particularly useful in alleviating the scale divergence problem that plagues higher-order networks. SE comprises two main processes: setting the initial weight vector and conducting the matrix transformation. An illustrative embodiment of SEHNN is built on the Sigma-Pi Network (SPN) applied to task of function approximation. Empirical results verify that SEHNN outperforms other higher-order networks in terms of computation efficiency. Compared to SPN, and Pi-Sigma Network (PSN), SEHNN requires less number of epochs to complete the training process.
We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an O((1/k) log D) rate, where k denotes the number of iterations and D denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is O(D 3 + ND 2 ), where N denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the RρR method [A. I. Lvovsky. Iterative maximumlikelihood reconstruction in quantum homodyne tomography. J. Opt. B: Quantum Semiclass. Opt. 2004] [G. Molina-Terriza et al. Triggered qutrits for quantum communication protocols. Phys. Rev. Lett. 2004.
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