We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an mimes n rank mimes n matrix from mimes n number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant mimes n of the sensing operator is less than mimes n, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank mimes n matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.

We classify simply connected compact Sasaki manifolds of dimension 2 n+ 1 with positive transverse bisectional curvature. In particular, the Khler cone corresponding to such manifolds must be bi-holomorphic to C n+ 1\{0}. As an application we recover the theorem of Mori and SiuYau on the Frankel conjecture and extend it to certain orbifold version. The main idea is to deform such Sasaki manifolds to the standard round sphere in two steps, both fixing the complex structure on the Khler cone. First, we deform the metric along the SasakiRicci flow and obtain a limit SasakiRicci soliton with positive transverse bisectional curvature. Then by varying the Reeb vector field which essentially decreases the volume functional, we deform the SasakiRicci soliton to a SasakiEinstein metric with positive transverse bisectional curvature, ie a round sphere. The second deformation is only possible when one treats

Let $(V, 0)$ be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras $L_{k}(V)$ associated to $(V,0)$. Its dimension is denoted as $\lambda_{k}(V)$. The $L_{k}(V)$ is a generalization of the Yau algebra $L(V)$ and $L_{0}(V)=L(V)$. These numbers $\lambda_{k}(V)$ are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute $L_1(V)$ for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of $L_{1}(V)$. We also formulate a sharp upper estimate conjecture for the $L_k(V)$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another conjecture: $\cdots >\lambda^{(i+1)}(V) > \lambda^i(V) \cdots >\lambda^0(V) \geq \lambda_l (V),\; l=1,2,\cdots,n$. We partially prove it for binomial and trinomial singularities.

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