Anomalous Dimension in the Solution of the Barenblatt's Equation

A new method is presented to obtain the anomalous dimension in the solution of the Barenblatt's equation. The result is the same as that in the renormalization group (RG) approach. It gives us insight on the perturbative solution of the Barenblatt's equation in the RG approach. Based on this discussion, an improvement is made to take into account, in more complete way, the nonlinear effect, which is included in the Heaviside function in higher orders. This improved result is better than that in RG approach.Comment: 17 pages, LaTex, no figur

Reconstruction of sparse wavelet signals from partial Fourier measurements

In this paper, we show that high-dimensional sparse wavelet signals of finite levels can be constructed from their partial Fourier measurements on a deterministic sampling set with cardinality about a multiple of signal sparsity

Simultaneous Amplitude and Phase Measurement for Periodic Optical Signals Using Time-Resolved Optical Filtering

Time-resolved optical filtering (TROF) measures the spectrogram or sonogram by a fast photodiode followed a tunable narrowband optical filter. For periodic signal and to match the sonogram, numerical TROF algorithm is used to find the original complex electric field or equivalently both the amplitude and phase. For phase-modulated optical signals, the TROF algorithm is initiated using the craters and ridges of the sonogram.Comment: 10 pages, 5 figure

Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly perturbed version $\mathbf{M}$ is observed. The noise matrix $\mathbf{M}-\mathbf{M}^{\star}$ is composed of zero-mean independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise, for example, when we have two independent samples for each entry of $\mathbf{M}^{\star}$ and arrange them into an {\em asymmetric} data matrix $\mathbf{M}$. The aim is to estimate the leading eigenvalue and eigenvector of $\mathbf{M}^{\star}$. We demonstrate that the leading eigenvalue of the data matrix $\mathbf{M}$ can be $O(\sqrt{n})$ times more accurate --- up to some log factor --- than its (unadjusted) leading singular value in eigenvalue estimation. Further, the perturbation of any linear form of the leading eigenvector of $\mathbf{M}$ --- say, entrywise eigenvector perturbation --- is provably well-controlled. This eigen-decomposition approach is fully adaptive to heteroscedasticity of noise without the need of careful bias correction or any prior knowledge about the noise variance. We also provide partial theory for the more general rank-$r$ case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be beneficial.Comment: accepted to Annals of Statistics, 2020. 37 page