Random band matrices in the delocalized phase, III: Averaging fluctuations

We consider a general class of symmetric or Hermitian random band matrices $H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket^d}$ in any dimension $d\ge 1$, where the entries are independent, centered random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. We assume that $s_{xy}$ vanishes if $|x-y|$ exceeds the band width $W$, and we are interested in the mesoscopic scale with $1\ll W\ll N$. Define the {\it{generalized resolvent}} of $H$ as $G(H,Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix with entries $Z_{xx}\in \mathbb C_+$ for all $x$. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions $d\ge 2$. More precisely, for any fixed $d\ge 2$, we prove that the bulk eigenvectors of $H$ are delocalized in certain averaged sense if $N\le W^{1+\frac{d}{2}}$. This improves the corresponding results in \cite{HeMa2018} under the assumption $N\ll W^{1+\frac{d}{d+1}}$, and in \cite{ErdKno2013,ErdKno2011} under the assumption $N\ll W^{1+\frac{d}{6}}$. For 1D random band matrices, our fluctuation averaging result was used in \cite{PartII,PartI} to prove the delocalization conjecture and bulk universality for random band matrices with $N\ll W^{4/3}$.Comment: 65 page

Difference of modular functions and their CM value factorization

In this paper, we use Borcherds lifting and the big CM value formula of Bruinier, Kudla, and Yang to give an explicit factorization formula for the norm of $\Psi(\frac{d_1+\sqrt{d_1}}2) -\Psi(\frac{d_2+\sqrt{d_2}}2)$, where $\Psi$ is the $j$-invariant or the Weber invariant $\omega_2$. The $j$-invariant case gives another proof of the well-known Gross-Zagier factorization formula of singular moduli, while the Weber invariant case gives a proof of the Yui-Zagier conjecture for $\omega_2$. The method used here could be extended to deal with other modular functions on a genus zero modular curve.Comment: accepted to appear in Trans. AM

Stability of Nonlinear Regime-switching Jump Diffusions

Motivated by networked systems, stochastic control, optimization, and a wide variety of applications, this work is devoted to systems of switching jump diffusions. Treating such nonlinear systems, we focus on stability issues. First asymptotic stability in the large is obtained. Then the study on exponential p-stability is carried out. Connection between almost surely exponential stability and exponential p-stability is exploited. Also presented are smooth-dependence on the initial data. Using the smooth-dependence, necessary conditions for exponential p-stability are derived. Then criteria for asymptotic stability in distribution are provided. A couple of examples are given to illustrate our results

CM fields of Dyhedral type and the Colmez conjecture

In this paper, we consider some CM fields which we call of dihedral type and compute the Artin $L$-functions associated to all CM types of these CM fields. As a consequence of this calculation, we see that the Colmez conjecture in this case is very closely related to understanding the log derivatives of certain Hecke characters of real quadratic fields. Recall that the `abelian case' of the Colmez conjecture, proved by Colmez himself, amounts to understanding the log derivatives of Hecke characters of \Q (cyclotomic characters). In this paper, we also prove that the Colmez conjecture holds for `unitary CM types of signature $(n-1, 1)$' and holds on average for `unitary CM types of a fixed CM number field of signature $(n-r, r)$'.Comment: accepted to appear in Manuscripta Mathematik

Alternating Direction Algorithms for ℓ1\ell_1-Problems in Compressive Sensing

In this paper, we propose and study the use of alternating direction algorithms for several $\ell_1$-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis-pursuit denoising problems of both unconstrained and constrained forms, as well as others. We present and investigate two classes of algorithms derived from either the primal or the dual forms of the $\ell_1$-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an $\ell_1$-problem into one having partially separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the resulting problem. The derived alternating direction algorithms can be regarded as first-order primal-dual algorithms because both primal and dual variables are updated at each and every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical results in comparison with several state-of-the-art algorithms are given to demonstrate that the proposed algorithms are efficient, stable and robust. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points. One point is that algorithm speed should always be evaluated relative to appropriate solution accuracy; another is that whenever erroneous measurements possibly exist, the $\ell_1$-norm fidelity should be the fidelity of choice in compressive sensing

Derivations of Siegel Modular Forms from Connections

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomorphic derivative operator of the Siegel modular forms. In order to get a holomorphic derivative operator, we introduce a weaker notion, called modular connection, on the Siegel upper half plane than a connection in differential geometry. Then we show that on a Siegel upper half plane there exists at most one holomorphic modular connection in some sense, and get a possible holomorphic derivative operator of Siegel modular forms.Comment: 14 page

Local circular law for the product of a deterministic matrix with a random matrix

It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satisfies the circular law. We consider the product $TX$, where $T$ is a deterministic $N\times M$ matrix and $X$ is a random $M\times N$ matrix with independent entries having zero mean and variance $(N\wedge M)^{-1}$. We prove a general local circular law for the empirical spectral distribution (ESD) of $TX$ at any point $z$ away from the unit circle under the assumptions that $N\sim M$, and the matrix entries $X_{ij}$ have sufficiently high moments. More precisely, if $z$ satisfies $||z|-1|\ge \tau$ for arbitrarily small $\tau>0$, the ESD of $TX$ converges to $\tilde \chi_{\mathbb D}(z) dA(z)$, where $\tilde \chi_{\mathbb D}$ is a rotation-invariant function determined by the singular values of $T$ and $dA$ denotes the Lebesgue measure on $\mathbb C$. The local circular law is valid around $z$ up to scale $(N\wedge M)^{-1/4+\epsilon}$ for any $\epsilon>0$. Moreover, if $|z|>1$ or the matrix entries of $X$ have vanishing third moments, the local circular law is valid around $z$ up to scale $(N\wedge M)^{-1/2+\epsilon}$ for any $\epsilon>0$.Comment: 80 pages, 7 figure

Compressive Mechanism: Utilizing Sparse Representation in Differential Privacy

Differential privacy provides the first theoretical foundation with provable privacy guarantee against adversaries with arbitrary prior knowledge. The main idea to achieve differential privacy is to inject random noise into statistical query results. Besides correctness, the most important goal in the design of a differentially private mechanism is to reduce the effect of random noise, ensuring that the noisy results can still be useful. This paper proposes the \emph{compressive mechanism}, a novel solution on the basis of state-of-the-art compression technique, called \emph{compressive sensing}. Compressive sensing is a decent theoretical tool for compact synopsis construction, using random projections. In this paper, we show that the amount of noise is significantly reduced from $O(\sqrt{n})$ to $O(\log(n))$, when the noise insertion procedure is carried on the synopsis samples instead of the original database. As an extension, we also apply the proposed compressive mechanism to solve the problem of continual release of statistical results. Extensive experiments using real datasets justify our accuracy claims.Comment: 20 pages, 6 figure

Mean-Variance Type Controls Involving a Hidden Markov Chain: Models and Numerical Approximation

Motivated by applications arising in networked systems, this work examines controlled regime-switching systems that stem from a mean-variance formulation. A main point is that the switching process is a hidden Markov chain. An additional piece of information, namely, a noisy observation of switching process corrupted by white noise is available. We focus on minimizing the variance subject to a fixed terminal expectation. Using the Wonham filter, we convert the partially observed system to a completely observable one first. Since closed-form solutions are virtually impossible be obtained, a Markov chain approximation method is used to devise a computational scheme. Convergence of the algorithm is obtained. A numerical example is provided to demonstrate the results

Layered BPSK for High Data Rates

This paper proposes a novel transmission strategy, referred to as layered BPSK, which allows two independent symbol streams to be layered at the transmitter on a non-orthogonal basis and isolated from each other at the receiver without inter-stream interference, aiming to achieve high data rates. To evaluate the performance of the proposed scheme, its data rate is formulated over additive white Gaussian noise (AWGN) channels. Based on the theoretical analysis, numerical results are provided for the performance comparisons between the proposed scheme and conventional transmission schemes, which substantiate the validity of the proposed scheme.Comment: 4 pages, 2 figures, submitted to IEEE Wireless Communications Letter