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

Spectral Method and Regularized MLE Are Both Optimal for Top-KK Ranking

This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model --- the Bradley-Terry-Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress towards characterizing the performance (e.g. the mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top-$K$ ranking remains unsettled. We demonstrate that under a natural random sampling model, the spectral method alone, or the regularized MLE alone, is minimax optimal in terms of the sample complexity --- the number of paired comparisons needed to ensure exact top-$K$ identification, for the fixed dynamic range regime. This is accomplished via optimal control of the entrywise error of the score estimates. We complement our theoretical studies by numerical experiments, confirming that both methods yield low entrywise errors for estimating the underlying scores. Our theory is established via a novel leave-one-out trick, which proves effective for analyzing both iterative and non-iterative procedures. Along the way, we derive an elementary eigenvector perturbation bound for probability transition matrices, which parallels the Davis-Kahan $\sin\Theta$ theorem for symmetric matrices. This also allows us to close the gap between the $\ell_2$ error upper bound for the spectral method and the minimax lower limit.Comment: Add discussions on the setting of the general condition numbe

Essays on auction mechanisms and resource allocation in keyword advertising

textAdvances in information technology have created radically new business models, most notably the integration of advertising with keyword-based targeting, or "keyword advertising." Keyword advertising has two main variations: advertising based on keywords employed by users in search engines, often known as "sponsored links," and advertising based on keywords embedded in the content users view, often known as "contextual advertising." Keyword advertising providers such as Google and Yahoo! use auctions to allocate advertising slots. This dissertation examines the design of keyword auctions. It consists of three essays. The first essay "Ex-Ante Information and the Design of Keyword Auctions" focuses on how to incorporate available information into auction design. In our keyword auction model, advertisers bid their willingness-to-pay per click on their advertisements, and the advertising provider can weigh advertisers' bids differently and require different minimum bids based on advertisers' click-generating potential. We study the impact and design of such weighting schemes and minimum-bids policies. We find that weighting scheme determines how advertisers with different click-generating potential match in equilibrium. Minimum bids exclude low-valuation advertisers and at the same time may distort the equilibrium matching. The efficient design of keyword auctions requires weighting advertisers' bids by their expected click-through-rates, and requires the same minimum weighted bids. The revenue-maximizing weighting scheme may or may not favor advertisers with low click-generating potential. The revenue-maximizing minimum-bid policy differs from those prescribed in the standard auction design literature. Keyword auctions that employ the revenue-maximizing weighting scheme and differentiated minimum bid policy can generate higher revenue than standard fixed-payment auctions. The dynamics of bidders' performance is examined in the second essay, "Keyword Auctions, Unit-price Contracts, and the Role of Commitment." We extend earlier static models by allowing bidders with lower performance levels to improve their performance at a certain cost. We examine the impact of the weighting scheme on overall bidder performance, the auction efficiency, and the auctioneer's revenue, and derive the revenue-maximizing and efficient policy accordingly. Moreover, the possible upgrade in bidders' performance levels gives the auctioneer an incentive to modify the auction rules over time, as is confirmed by the practice of Yahoo! And Google. We thus compare the auctioneer's revenue-maximizing policies when she is fully committed to the auction rule and when not, and show that she should give less preferential treatment to low-performance advertisers when she is fully committed. In the third essay, "How to Slice the Pie? Optimal Share Structure Design in Keyword Auctions," we study the design of share structures in keyword auctions. Auctions for keyword advertising resources can be viewed as share auctions in which the highest bidder gets the largest share, the second highest bidder gets the second largest share, and so on. A share structure problem arises in such a setting regarding how much resources to set aside for the highest bidder, for the second highest bidder, etc. We address this problem under a general specification and derive implications on how the optimal share structure should change with bidders' price elasticity of demand for exposure, their valuation distribution, total resources, and minimum bids.Managemen

A uniqueness result for a Schrödinger–Poisson system with strong singularity

In this paper, we consider the following Schrödinger–Poisson system with strong singularity −∆u + φu = f(x)u , x ∈ Ω, −∆φ = u 2 , x ∈ Ω, u > 0, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where Ω ⊂ R3 is a smooth bounded domain, γ > 1, f ∈ L 1 (Ω) is a positive function (i.e. f(x) > 0 a.e. in Ω). A necessary and sufficient condition on the existence and uniqueness of positive weak solution of the system is obtained. The results supplement the main conclusions in recent literature

Multiple nonsymmetric nodal solutions for quasilinear Schrödinger system

In this paper, we consider the quasilinear Schrödinger system in RN (N ≥ 3): −∆u + A(x)u − 1 2 ∆(u 2 )u = 2α |u| α−2u|v| −∆v + Bv − 1 2 ∆(v 2 )v = 2β |u| |v| β−2 v, where α, β > 1, 2 0 is a constant. By using a constrained minimization on Nehari–Pohožaev set, for any given integer s ≥ 2, we construct a nonradially symmetrical nodal solution with its 2s nodal domains

Ground states solution of Nehari-Poho\v{z}aev type for periodic quasilinear Schr\"{o}dinger system

This paper is concerned with a quasilinear Schr\"{o}dinger system in $\mathbb R^{N}$ \left\{\aligned &-\Delta u+A(x)u-\frac{1}{2}\triangle(u^{2})u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &-\Delta v+B(x)v-\frac{1}{2}\triangle(v^{2})v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ & u(x)\to 0\ \hbox{and}\quad v(x)\to 0\ \hbox{as}\ |x|\to \infty,\endaligned\right. where $\alpha,\beta>1$ and $2<\alpha+\beta<\frac{4N}{N-2}$ ($N \geq 3$). $A(x)$ and $B(x)$ are two periodic functions. By minimization under a convenient constraint and concentration-compactness lemma, we prove the existence of ground states solution. Our result covers the case of $\alpha+\beta\in(2,4)$ which seems to be the first result for coupled quasilinear Schr\"{o}dinger system in the periodic situation