Skewness-Based Projection Pursuit as an Eigenvector Problem in Scale Mixtures of Skew-Normal Distributions

This paper addresses the projection pursuit problem assuming that the distribution of the input vector belongs to the flexible and wide family of multivariate scale mixtures of skew normal distributions. Under this assumption, skewness-based projection pursuit is set out as an eigenvector problem, described in terms of the third order cumulant matrix, as well as an eigenvector problem that involves the simultaneous diagonalization of the scatter matrices of the model. Both approaches lead to dominant eigenvectors proportional to the shape parametric vector, which accounts for the multivariate asymmetry of the model; they also shed light on the parametric interpretability of the invariant coordinate selection method and point out some alternatives for estimating the projection pursuit direction. The theoretical findings are further investigated through a simulation study whose results provide insights about the usefulness of skewness model-based projection pursuit in the statistical practice.

Simultaneous Diagonalization Using Baseband Beamforming in mmWave MU-MIMO Systems

We consider the problem of simultaneous diagonalization of Hermitian matrices for the desired and co-channel interference terms of millimeter-wave (mmWave) multi-user multiple-input multiple-output systems. The joint unitary eigenvectors and the corresponding eigenvalues are known to assist in the mathematical tractability of key performance metrics, such as outage probability, ergodic capacity, and spectral efficiency. We formulate the signal-to-interference-plus-noise ratio in a canonical quadratic form and subsume the digital baseband beamforming vectors in the weight matrices of channels at the transmitter side. Next, a real scalar objective function is defined, which quantifies the correlation loss due to joint-diagonalization. The objective function is then maximized using baseband beamforming under the hardware constraints of the mmWave system. Through simulations, the proposed beamforming algorithm is evaluated by employing several non-linear optimization sub-routines, and it is shown that the “active-set” approach results in improved summary statistics both for the correlation metric and for the time complexity. We also reflect on the effect of optimization on the channel scatterers in mmWave systems.

Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters

Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitioning sets of Paulis into commuting subsets, and show that the proposed approach can help to significantly reduce both the number of CNOT operations and circuit depth for Hamiltonians arising in quantum chemistry. The algorithms for simultaneous diagonalization are also applicable in the context of stabilizer states; in particular we provide novel four- and five-stage representations, each containing only a single stage of conditional gates.

The Bundle of Simultaneously Diagonalizable N-tuples of Matrices

In this paper, a review of the simultaneous diagonalization of n-tuples of matrices for its applications in sciences is presented. For example, in quantum mechanics, position and momentum operators do not have a shared base that can represent the states of the system because they not commute, which is why switching operators form a key element of quantum physics since they define quantities that are compatible, that is, defined simultaneously. We are going to study this kind of family of linear operators using geometric constructions such as the principal bundles and associating them with a cohomology class measuring the deviation of the local product structure from the global product structure.

Spatial blind source separation

Summary Recently a blind source separation model was suggested for spatial data, along with an estimator based on the simultaneous diagonalization of two scatter matrices. The asymptotic properties of this estimator are derived here, and a new estimator based on the joint diagonalization of more than two scatter matrices is proposed. The asymptotic properties and merits of the novel estimator are verified in simulation studies. A real-data example illustrates application of the method.