Related papers: Singularity and Error Analysis of a Simple Quatern…
In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient…
Recent algebraic parametric estimation techniques led to point-wise derivative estimates by using only the iterated integral of a noisy observation signal. In this paper, we extend such differentiation methods by providing a larger choice…
We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and…
One of the goals in scaling sequential machine learning methods pertains to dealing with high-dimensional data spaces. A key related challenge is that many methods heavily depend on obtaining the inverse covariance matrix of the data. It is…
Estimating covariance matrices with high-dimensional complex data presents significant challenges, particularly concerning positive definiteness, sparsity, and numerical stability. Existing robust sparse estimators often fail to guarantee…
We propose a novel estimation framework for quadratic functionals of precision matrices in high-dimensional settings, particularly in regimes where the feature dimension $p$ exceeds the sample size $n$. Traditional moment-based estimators…
Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially…
This article investigates Kak neural networks, which can be instantaneously trained, for complex and quaternion inputs. The performance of the basic algorithm has been analyzed and shown how it provides a plausible model of human perception…
In this work, a quaternion-valued model is proposed in lieu of the Clarke's \alpha, \beta transformation to convert three-phase quantities to a hypercomplex single-phase signal. The concatenated signal can be used for harmonic distortion…
The simultaneous estimation of multiple unknown parameters is the most general scenario in quantum sensing. Quantum multi-parameter estimation theory provides fundamental bounds on the achievable precision of simultaneous estimation.…
A discrete complexified quaternion Fourier transform is introduced. This is a generalization of the discrete quaternion Fourier transform to the case where either or both of the signal/image and the transform kernel are complex…
Quaternions are important for a wide variety of rotation-related problems in computer graphics, machine vision, and robotics. We study the nontrivial geometry of the relationship between quaternions and rotation matrices by exploiting the…
In the field of statistical learning and data analysis, estimating precision matrices (i.e., the inverse of covariance matrices) is a critical task, particularly for understanding dependency structures among variables. However, traditional…
Quaternion-valued signal processing has received increasing attention recently. One key operation involved in derivation of all kinds of adaptive algorithms is the gradient operator. Although there have been some derivations of this…
Relative pose (position and orientation) estimation is an essential component of many robotics applications. Fiducial markers, such as the AprilTag visual fiducial system, yield a relative pose measurement from a single marker detection and…
We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the…
The article concerns compressed sensing methods in the quaternion algebra. We prove that it is possible to uniquely reconstruct - by $\ell_1$-norm minimization - a sparse quaternion signal from a limited number of its linear measurements,…
Electronic health records and other sources of observational data are increasingly used for drawing causal inferences. The estimation of a causal effect using these data not meant for research purposes is subject to confounding and…
We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two…
Bayesian quantum estimation provides a robust framework for quantum technologies, especially in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable Gaussian systems has remained limited and…