## Portfolio item number 1

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Published in *Github*, 2014

This is our paper for The Mathematical Contest in Modeling where we won Meritorious Winner Awards in 2014

Recommended citation: Zhenzhen Li, Gaolei Li, Long Zhou.

Published in *revised on IEEE Transactions on Information Theory, Arxiv*, 2018

The problem of finding a vector $x$ which obeys a set of quadratic equations. $ abs(a^{T}_k x)^2= y_k$, $k=1,2,⋯,m$, plays an important role in many applications. In this paper we consider the case when both $x$ and $a_k$ are real-valued vectors of length $n$. A new loss function is constructed for this problem, which combines the smooth quadratic loss function with an activation function. Under the Gaussian measurement model, we establish that with high probability the target solution $x$ is the unique local minimizer (up to a global phase factor) of the new loss function provided $m≳n$. Moreover, the loss function always has a negative directional curvature around its saddle points.

Recommended citation: Zhenzhen Li, Jian-Feng Cai, Ke Wei.

Published in *Journal unknown*, 2019

This paper is a working paper about another point of view of such nonconvex problem geometry property.

Published in *Jounal unknown*, 2019

This is my current working paper focus on retrieving sparse signals under optimal sampling complexity. We design an algorithm for sparse phase retrieval and would provide theoretical guarantee for such algorithm, which would be the first practical local algorithm with theoretical guarantee for sparse phase retrieve. Although we may need an good initialization in latter work, we think this at least provide a great first step to such sparse phase retrieval open problem.

** Published:**

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** Published:**

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Undergraduate course, *University 11, Department*, 2014

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Workshop, *University 1, Department*, 2015

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