Fast Rank-One Alternating Minimization Algorithm for Phase Retrieval

Abstract

The phase retrieval problem is a fundamental problem in many fields, which is appealing for investigation. It is to recover the signal vector ${\tilde{{\mathbf {x}}}}\in {\mathbb {C}}^d$ from a set of N measurements $b_n=|{\mathbf {f}}^*_n{\tilde{{\mathbf {x}}}}|^2,\ n=1,\ldots , N$, where $\{{\mathbf {f}}_n\}_{n=1}^N$ forms a frame of ${\mathbb {C}}^d$. Existing algorithms usually use a least squares fitting to the measurements, yielding a quartic polynomial minimization. In this paper, we employ a new strategy by splitting the variables, and we solve a bi-variate optimization problem that is quadratic in each of the variables. An alternating gradient descent algorithm is proposed, and its convergence for any initialization is provided. Since a larger step size is allowed due to the smaller Hessian, the alternating gradient descent algorithm converges faster than the gradient descent algorithm (known as the Wirtinger flow algorithm) applied to the quartic objective without splitting the variables. Numerical results illustrate that our proposed algorithm needs less iterations than Wirtinger flow to achieve the same accuracy.

Larger step size

In this appendix, we demonstrate that, when ${\mathbf {x}}$ and ${\mathbf {y}}$ are sufficiently close, our alternating gradient descent algorithm is roughly 1.5 times faster than the WF algorithm in the real case.

To this end, we let $E({\mathbf {x}},{\mathbf {y}})$ be the function defined in (3.2). Choose $\lambda =0$ and assume ${\mathbf {x}}_k\approx {\mathbf {y}}_k$. Then, by Taylor’s expansion, we obtain

$$\begin{aligned} \begin{aligned}&E({\mathbf {x}}_{k+1},{\mathbf {y}}_k)\\&\quad \approx E({\mathbf {x}}_{k},{\mathbf {y}}_k)+\nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)^*\left[ \begin{matrix}{\mathbf {x}}_{k+1}-{\mathbf {x}}_k\\ \overline{{\mathbf {x}}_{k+1}-{\mathbf {x}}_k}\end{matrix}\right] +\frac{1}{2} \left[ \begin{matrix}{\mathbf {x}}_{k+1}-{\mathbf {x}}_k\\ \overline{{\mathbf {x}}_{k+1}-{\mathbf {x}}_k}\end{matrix}\right] ^*\nabla _{{\mathbf {x}}}^2E({\mathbf {x}}_k,{\mathbf {y}}_k) \left[ \begin{matrix}{\mathbf {x}}_{k+1}-{\mathbf {x}}_k\\ \overline{{\mathbf {x}}_{k+1}-{\mathbf {x}}_k}\end{matrix}\right] \\&\quad =E({\mathbf {x}}_{k},{\mathbf {x}}_k)-\alpha _k\left\| \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\right\| _2^2+\frac{\alpha ^2_k}{2}\mathfrak {R}\left( \nabla _{{\mathbf {x}}}E({\mathbf {x}}_k,{\mathbf {y}}_k)^*\left( \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\right) \\&\quad =E({\mathbf {x}}_{k},{\mathbf {y}}_k)-\alpha _k\left\| (\nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\right\| _2^2 +\frac{\alpha _k^2}{2}\nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)^*\left( \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\\&\quad =E({\mathbf {x}}_{k},{\mathbf {y}}_k)-\alpha _k\left( \left\| \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\right\| _2^2 -\frac{\alpha _k}{2}\nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)^*\left( \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\right) \end{aligned}\nonumber \\ \end{aligned}$$

(A.1)

The last equality hold because $\sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n$ is Hermitian. We choose $\alpha _k>0$. Therefore, $E({\mathbf {x}}_{k+1},{\mathbf {y}}_k)-E({\mathbf {x}}_{k},{\mathbf {y}}_k)\le 0$ as long as

$$\begin{aligned} \frac{2}{\alpha _k}\ge \frac{\nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)^*\left( \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)}{\left\| \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\right\| _2^2}, \end{aligned}$$

which is guaranteed if

$$\begin{aligned} \alpha _k\le \frac{2}{\Vert \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\Vert _2}. \end{aligned}$$

To minimize $E({\mathbf {x}}_{k+1},{\mathbf {y}}_k)-E({\mathbf {x}}_{k},{\mathbf {y}}_k)$, it is easy seen from (A.1) that $\alpha _k$ is chosen as

$$\begin{aligned} \alpha _k=\frac{\left\| \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\right\| _2^2}{\nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)^*\left( \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)}. \end{aligned}$$

In this case,

$$\begin{aligned} E({\mathbf {x}}_{k+1},{\mathbf {y}}_k)-E({\mathbf {x}}_{k},{\mathbf {y}}_k) \approx -\frac{\left\| \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)\right\| _2^4}{2\nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)^*\left( \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \nabla _{{\mathbf {x}}}E({\mathbf {x}}_{k},{\mathbf {y}}_k)}. \end{aligned}$$

(A.2)

Now we consider the WF algorithm, which minimizes $G({\mathbf {x}})=\frac{1}{N}\sum ^N_{n=1}(|{\mathbf {f}}^*_n{\mathbf {x}}|^2-b_n)^2$. Assume we have the same ${\mathbf {x}}_k$ as in the alternating gradient descent algorithm, and the WF algorithm generates the new iterates by

$$\begin{aligned} {\mathbf {x}}_{k+1}={\mathbf {x}}_k-\delta _k\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k). \end{aligned}$$

With the optimal choice of $\delta _k$, an analogous analysis leads to

$$\begin{aligned}&G({\mathbf {x}}_{k+1})-G({\mathbf {x}}_{k}) \nonumber \\&\quad \approx -\frac{1}{2}\frac{\left\| \nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)\right\| _2^4}{\mathfrak {R}\left( \nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)^*H_{11}({\mathbf {x}}_k)\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)\right) +\mathfrak {R}\left( \nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)^TH_{21}({\mathbf {x}}_k)\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)\right) }, \end{aligned}$$

(A.3)

where $\mathfrak {R}$ denotes the real part, and

$$\begin{aligned}&H_{11}({\mathbf {x}}_k) =4\sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {x}}_k{\mathbf {x}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n,\\&H_{21}({\mathbf {x}}_k)=\sum _n\left( \overline{{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {x}}_k}{\mathbf {x}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n+\overline{{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {x}}_k}{\mathbf {x}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) . \end{aligned}$$

Since we assumed ${\mathbf {x}}_k\approx {\mathbf {y}}_k$ and $\lambda =0$,

$$\begin{aligned} \nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)\approx 2\nabla _{{\mathbf {y}}}E({\mathbf {x}}_k,{\mathbf {y}}_k), \end{aligned}$$

(A.4)

which implies

$$\begin{aligned} \begin{aligned}&\mathfrak {R}\left( \nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)^*H_{11}({\mathbf {x}}_k)\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)\right) \\&\quad =\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)^*H_{11}({\mathbf {x}}_k)\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)\\&\quad \approx \left( 2\nabla _{{\mathbf {y}}}E({\mathbf {x}}_k,{\mathbf {y}}_k)\right) ^*\left( 4\sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {x}}_k{\mathbf {x}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \left( 2\nabla _{{\mathbf {y}}}E({\mathbf {x}}_k,{\mathbf {y}}_k)\right) \\&\quad \approx 16\cdot \left( \nabla _{{\mathbf {y}}}E({\mathbf {x}}_k,{\mathbf {y}}_k)\right) ^*\left( \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \nabla _{{\mathbf {y}}}E({\mathbf {x}}_k,{\mathbf {y}}_k). \end{aligned} \end{aligned}$$

(A.5)

If we further assume all vectors involved are real, then we have $H_{21}({\mathbf {x}}_k)=2\sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {x}}_k{\mathbf {x}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n$ and

$$\begin{aligned} \begin{aligned} \mathfrak {R}\left( \nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)^TH_{21}({\mathbf {x}}_k)\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)\right)&=\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)^*H_{21}({\mathbf {x}}_k)\nabla _{{\mathbf {x}}}G({\mathbf {x}}_k)\\&\approx 8\cdot \nabla _{{\mathbf {y}}}E({\mathbf {x}}_k,{\mathbf {y}}_k)^*\left( \sum _n{\mathbf {f}}_n{\mathbf {f}}^*_n{\mathbf {y}}_k{\mathbf {y}}_k^*{\mathbf {f}}_n{\mathbf {f}}^*_n\right) \nabla _{{\mathbf {y}}}E({\mathbf {x}}_k,{\mathbf {y}}_k). \end{aligned}\nonumber \\ \end{aligned}$$

(A.6)

Substituting (A.4), (A.5), and (A.6) into (A.3), we get

$$\begin{aligned} \left( G({\mathbf {x}}_{k+1})-G({\mathbf {x}}_k)\right) \approx \frac{2}{3}\left( E({\mathbf {x}}_{k+1},{\mathbf {y}}_k)-E({\mathbf {x}}_k,{\mathbf {y}}_k)\right) \end{aligned}$$

This means that the alternating gradient descent algorithm is 1.5 times faster than Wirtinger flow in terms of the decreasing of the objective.