Euclidean Quantum Mechanics and Universal Nonlinear Filtering

An important problem in applied science is the continuous nonlinear filtering problem, i.e., the estimation of a Langevin state that is observed indirectly. In this paper, it is shown that Euclidean quantum mechanics is closely related to the continuous nonlinear filtering problem. The key is the configuration space Feynman path integral representation of the fundamental solution of a Fokker-Planck type of equation termed the Yau Equation of continuous-continuous filtering. A corollary is the equivalence between nonlinear filtering problem and a time-varying Schr\"odinger equation.Comment: 19 pages, LaTeX, interdisciplinar

Sparse Volterra and Polynomial Regression Models: Recoverability and Estimation

Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but unfortunately cannot be met by least-squares based or kernel regression methods. To this end, compressed sampling (CS) approaches, already successful in linear regression settings, can offer a viable alternative. The viability of CS for sparse Volterra and polynomial models is the core theme of this work. A common sparse regression task is initially posed for the two models. Building on (weighted) Lasso-based schemes, an adaptive RLS-type algorithm is developed for sparse polynomial regressions. The identifiability of polynomial models is critically challenged by dimensionality. However, following the CS principle, when these models are sparse, they could be recovered by far fewer measurements. To quantify the sufficient number of measurements for a given level of sparsity, restricted isometry properties (RIP) are investigated in commonly met polynomial regression settings, generalizing known results for their linear counterparts. The merits of the novel (weighted) adaptive CS algorithms to sparse polynomial modeling are verified through synthetic as well as real data tests for genotype-phenotype analysis.Comment: 20 pages, to appear in IEEE Trans. on Signal Processin

Parallel Self-Consistent-Field Calculations via Chebyshev-Filtered Subspace Acceleration

Solving the Kohn-Sham eigenvalue problem constitutes the most computationally expensive part in self-consistent density functional theory (DFT) calculations. In a previous paper, we have proposed a nonlinear Chebyshev-filtered subspace iteration method, which avoids computing explicit eigenvectors except at the first SCF iteration. The method may be viewed as an approach to solve the original nonlinear Kohn-Sham equation by a nonlinear subspace iteration technique, without emphasizing the intermediate linearized Kohn-Sham eigenvalue problem. It reaches self-consistency within a similar number of SCF iterations as eigensolver-based approaches. However, replacing the standard diagonalization at each SCF iteration by a Chebyshev subspace filtering step results in a significant speedup over methods based on standard diagonalization. Here, we discuss an approach for implementing this method in multi-processor, parallel environment. Numerical results are presented to show that the method enables to perform a class of highly challenging DFT calculations that were not feasible before

A Digital Predistortion Scheme Exploiting Degrees-of-Freedom for Massive MIMO Systems

The primary source of nonlinear distortion in wireless transmitters is the power amplifier (PA). Conventional digital predistortion (DPD) schemes use high-order polynomials to accurately approximate and compensate for the nonlinearity of the PA. This is not practical for scaling to tens or hundreds of PAs in massive multiple-input multiple-output (MIMO) systems. There is more than one candidate precoding matrix in a massive MIMO system because of the excess degrees-of-freedom (DoFs), and each precoding matrix requires a different DPD polynomial order to compensate for the PA nonlinearity. This paper proposes a low-order DPD method achieved by exploiting massive DoFs of next-generation front ends. We propose a novel indirect learning structure which adapts the channel and PA distortion iteratively by cascading adaptive zero forcing precoding and DPD. Our solution uses a 3rd order polynomial to achieve the same performance as the conventional DPD using an 11th order polynomial for a 100x10 massive MIMO configuration. Experimental results show a 70% reduction in computational complexity, enabling ultra-low latency communications.Comment: IEEE International Conference on Communications 201

Universal Nonlinear Filtering Using Feynman Path Integrals II: The Continuous-Continuous Model with Additive Noise

In this paper, the Feynman path integral formulation of the continuous-continuous filtering problem, a fundamental problem of applied science, is investigated for the case when the noise in the signal and measurement model is additive. It is shown that it leads to an independent and self-contained analysis and solution of the problem. A consequence of this analysis is Feynman path integral formula for the conditional probability density that manifests the underlying physics of the problem. A corollary of the path integral formula is the Yau algorithm that has been shown to be superior to all other known algorithms. The Feynman path integral formulation is shown to lead to practical and implementable algorithms. In particular, the solution of the Yau PDE is reduced to one of function computation and integration.Comment: Interdisciplinary, 41 pages, 5 figures, JHEP3 class; added more discussion and reference