SZU-CHI CHUNGINSTITUTE OF STATISTICSHAO-HSUAN WANGINSTITUTE OF STATISTICPO-YAO NIUINSTITUTE OF STATISTICSU-YUN HUANGINSTITUTE OF STATISTICWEI-HAU CHANGINSTITUTE OF STATISTICI-PING TUINSTITUTE OF STATISTIC
Silver Award Paper in 2020
Annals of Mathematical Sciences and Applicaitons , 5, (2), 2020
Principal component analysis (PCA) is arguably the most widely used
dimension-reduction method for vector-type data. When applied to a
sample of images, PCA requires vectorization of the image data, which
in turn entails solving an eigenvalue problem for the sample covariance matrix. We propose herein a two-stage dimension reduction (2SDR)
method for image reconstruction from high-dimensional noisy image
data. The first stage treats the image as a matrix, which is a tensor of
order 2, and uses multilinear principal component analysis (MPCA) for
matrix rank reduction and image denoising. The second stage vectorizes
the reduced-rank matrix and achieves further dimension and noise reduction. Simulation studies demonstrate excellent performance of 2SDR, for
which we also develop an asymptotic theory that establishes consistency
of its rank selection. Applications to cryo-EM (cryogenic electronic microscopy), which has revolutionized structural biology, organic and medical chemistry, cellular and molecular physiology in the past decade, are
also provided and illustrated with benchmark cryo-EM datasets. Connections to other contemporaneous developments in image reconstruction
and high-dimensional statistical inference are also discussed.
We establish by exact, nonperturbative methods a universality for the correlation functions in Kraichnan's``rapid-change''model of a passively advected scalar field. We show that the solutions for separated points in the convective range of scales are unique and independent of the particular mechanism of the scalar dissipation. Any non-universal dependences therefore must arise from the large length-scale features. The main step in the proof is to show that solutions of the model equations are unique even in the idealized case of zero diffusivity, under a very modest regularity requirement (square-integrability). Within this regularity class the only zero-modes of the global many-body operators are shown to be trivial ones (ie constants). In a bounded domain of size L , with physical boundary conditions, the``ground-state energy''is strictly positive and scales as L with an exponent L .
We study a class of nonlinear nonlocal cochlear models of the transmission line type, describing the motion of basilar membrane (BM) in the cochlea. They are damped dispersive partial differential equations (PDEs) driven by time dependent boundary forcing due to the input sounds. The global well-posedness in time follows from energy estimates. Uniform bounds of solutions hold in the case of bounded nonlinear damping. When the input sounds are multi-frequency tones, and the nonlinearity in the PDEs is cubic, we construct smooth quasi-periodic solutions (multi-tone solutions) in the weakly nonlinear regime, where new frequencies are generated due to nonlinear interaction. When the input consists of two tones at frequencies f 1, f 2 (f 1< f 2), and high enough intensities, numerical results illustrate the formation of combination tones at 2f 1 f 2 and 2f 2 f 1, in agreement with hearing experiments. We visualize
A two-space dimensional active nonlinear nonlocal cochlear model is formulated in the time domain to capture nonlinear hearing effects such as compression, multi-tone suppression and difference tones. The micromechanics of the basilar membrane (BM) are incorporated to model active cochlear properties. An active gain parameter is constructed in the form of a nonlinear nonlocal functional of BM displacement. The model is discretized with a boundary integral method and numerically solved using an iterative second-order accurate finite difference scheme. A block matrix structure of the discrete system is exploited to simplify the numerics with no loss of accuracy. Model responses to multiple frequency stimuli are shown in agreement with hearing experiments. A nonlinear spectrum is computed from the model, and compared with FFT spectrum for noisy tonal inputs. The discretized model is efficient and accurate
Dispersive instability appears in time-domain solutions of classical cochlear models. In this letter, a derivation of optimal initial data is presented to minimize the effect of instability. A second-order accurate implicit boundary integral method is introduced. Numerical solutions of two-dimensional models show that the optimal initial conditions work successfully in time-domain steady-state computations for both the zero Neumann and zero Dirichlet fluid pressure boundary conditions at the helicotrema.