Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.

Dropping the Independence: Singular Values for Products of Two Coupled Random Matrices

Abstract: We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel–Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.

Coherent vortex extraction in 3D homogeneous turbulence: comparison between orthogonal and biorthogonal wavelet decompositions

Abstract: A comparison between two different ways of extracting coherent vortices in three-dimensional (3D) homogeneous isotropic turbulence is performed, using either orthogonal or biorthogonal wavelets. The method is based on a wavelet decomposition of the vorticity field and a subsequent thresholding of the wavelet coefficients. The coherent vorticity is reconstructed from a few strong wavelet coefficients, while the incoherent vorticity is reconstructed from the remaining weak coefficients. The choice of the threshold, which has no adjustable parameters, is motivated for the orthogonal case from the denoising theory. Using only 3 % of the coefficients we show that both decompositions, that is orthogonal and biorthogonal, extract the coherent vortices. They contain most of the energy (around 99 % in both cases) and retain 74 % and 68 % of the enstrophy in the orthogonal and biorthogonal cases, respectively. The incoherent background flow for the orthogonal decomposition, which corresponds to 97 % of the wavelet ...

Theory of rate-distortion-optimal, constrained filterbanks-application to IIR and FIR biorthogonal designs

Abstract: We design filterbanks that are best matched to input signal statistics in M-channel subband coders, using a rate-distortion criterion. Previous research has shown that unconstrained-length, paraunitary filterbanks optimized under various energy compaction criteria are principal-component filterbanks that satisfy two fundamental properties: total decorrelation and spectral majorization. In this paper, we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints. Our results apply to a broad class of rate-distortion criteria, including the conventional coding gain criterion as a special case. A consequence of these properties is that optimal perfect-reconstruction (PR) filterbanks take the form of the cascade of principal-component filterbanks and a bank of pre- and post-conditioning filters. The proof uses variational techniques and is applicable to a variety of constrained design problems. In the second part of this paper, we apply the theory above to practical filterbank design problems. We give analytical expressions for optimal IIR biorthogonal filterbanks; our analysis validates a conjecture by several researchers. We then derive the asymptotic limit of optimal FIR biorthogonal filterbanks as filter length tends to infinity. The performance loss due to FIR constraints is quantified theoretically and experimentally. The optimal filters are quite different from traditional filters. Finally, a sensitivity analysis is presented.

An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations)

Abstract: We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painleve equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonov's elliptic beta integral.