The minimal measurement number for low-rank matrix recovery

The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that $4nr?4r^2$ generic measurements are both necessary and sufficient for the recovery of rank-r matrices in ${\mathbb{C}}^{n\times n}$. Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound $4nr?4r^2$ is tight provided $n=2^k+r,k\in {\mathbb{Z}}_{+}$. Motivated by Vinzant's work [19], we construct 11 matrices in ${\mathbb{R}}^{4\times 4}$ by computer random search and prove they define injective measurements on rank-1 matrices in ${\mathbb{R}}^{4\times 4}$. This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than $2n?1$ orthogonal projections are possible for the recovery of $x\in {\mathbb{R}}^n$ from the norm of them, which gives a negative answer for a question raised in [1].     

Wild theories with o-minimal open core

Let T be a consistent o-minimal theory extending the theory of densely ordered groups and let $T^{\prime }$ be a consistent theory. Then there is a complete theory $T^{?}$ extending T such that T is an open core of $T^{?}$, but every model of $T^{?}$ interprets a model of $T^{\prime }$. If $T^{\prime }$ is NIP, $T^{?}$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.     

An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by $X^{(n)}$ the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on $H^1(X^{(n)})$ to define an integrable connection on the trivial vector bundle on $X^{(n)}$ with fiber the universal algebra of the Lie algebra associated with the descending central series of ${\pi }_1$ of $X^{(n)}$. The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.     

General convergence analysis for two-step projection methods and applications to variational problems

First a general model for two-step projection methods is introduced and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let $H$ be a real Hilbert space and $K$ be a nonempty closed convex subset of $H$. For arbitrarily chosen initial points $x^0,y^0\in K$, compute sequences $\left\{x^k\right\}$ and $\left\{y^k\right\}$ such that $x^{k+1}=(1-a^k)x^k+a^k{P}_K[y^k-\rho T\left(y^k\right)]\text{for }\rho >0$$y^k=(1-b^k)x^k+b^k{P}_K[x^k-\eta T\left(x^k\right)]\text{for }\eta >0\text{,}$ where $T:K\to H$ is a nonlinear mapping on $K,P_K$ is the projection of $H$ onto $K$, and $0\le a^k,b^k\le 1$. The two-step model is applied to some variational inequality problems.     

Qualitative property between flows and numerical methods

In this paper, we show that qualitative properties of Axiom A flows with the strong transversality condition are preserved by its discretization mapping obtained via numerical methods. More precisely, if ${\varphi }^t$ is an Axiom A flow with the strong transversality condition and $N$ is a numerical method of order $p⩾1$ for ${\varphi }^t$ on a smooth compact manifold $M$, then there exists $T>1$ such that for all sufficiently large $n\in \mathbb{N}$, there are a continuous onto function $H_n:M\to M$ and a continuous function ${\tau }_n:M\to \mathbb{R}$ such that ${\varphi }^{{\tau }_n(x)}\circ H_n(x)=H_n\circ (N^{T/n}{)}^n(x)$ for all $x\in M$.