On DC Based Methods for Phase Retrieval

Abstract

In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functions (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization subproblems. For each subproblem, the Nesterov accelerated gradient descent algorithm or the Barzilai-Borwein (BB) algorithm is adopted. In addition, we apply the alternating projection method to improve the initial guess in [20] and make it much more closer to the true solution. In the setting of sparse phase retrieval, a standard ℓ ₁ norm term is added to guarantee the sparsity, and the subproblem is solved approximately by a proximal gradient method with the shrinkage-threshold technique directly. Furthermore, a modified Attouch-Peypouquet technique is used to accelerate the iterative computation, which leads to more effective algorithms than the Wirtinger flow (WF) algorithm and the Gauss-Newton (GN) algorithm and etc. Indeed, DC based algorithms are able to recover the solution with high probability when the measurement number m ≈ 2n in the real case and m ≈ 3n in the complex case, where n is the dimension of the true solution. When m ≈ n, the ℓ ₁-DC based algorithm is able to recover the sparse signals with high probability. Our main results show that the DC based methods converge to a critical point linearly. Our study is a deterministic analysis while the study for the Wirtinger flow (WF) algorithm and its variants, the Gauss-Newton (GN) algorithm, the trust region algorithm is based on the probability analysis. Finally, the paper discusses the existence and the number of distinct solutions for phase retrieval problem.

Appendix

In this section we give some deterministic description of the minimizing function F as well as strong convexity of F ₂. We will show that at any global minimizer, the Hessian matrix of F is positive definite in the real case and is nonnegative positive definite in the complex case. These results are used when we apply the KL inequality. For convenience, let $A_\ell = {\mathbf {a}}_\ell \bar {\mathbf {a}}_\ell ^\top$ be the Hermitian matrix of rank one for ℓ = 1, ⋯ , m.

Definition 3

We say a _j, j = 1, ⋯ , m are a ₀-generic if there exists a positive constant a ₀ ∈ (0, 1) such that

$$\displaystyle \begin{aligned}\|({\mathbf{a}}_{j_1}^*\mathbf{y}, \ldots, {\mathbf{a}}_{j_n}^*\mathbf{y})\|\ge a_0 \|\mathbf{y}\|, \quad \forall \mathbf{y}\in \mathbb{C}^n \end{aligned}$$

holds for any 1 ≤ j ₁ < j ₂ < ⋯ < j _n ≤ m.

Theorem 8

Let m ≥ n. Assume a _j, j = 1, ⋯ , m are a ₀ -generic for some constant a ₀ . If there exist n nonzero elements among the measurements b _j, j = 1, ⋯ , m, then for the phase retrieval problem with f(x) = |x|² , F ₂ is positive definite.

Proof

Recall that $F_2= 2\sum _{i=1}^m b_i f({\mathbf {a}}_k^\top \mathbf {x})$. Then the Hessian matrix of F ₂ is

$$\displaystyle \begin{aligned}H_{F_2}(\mathbf{x})= 2\sum_{i=1}^m b_i f''({\mathbf{a}}_k^\top \mathbf{x}) {\mathbf{a}}_i{\mathbf{a}}_i^\top .\end{aligned}$$

Note that f″(x) = 2. Thus we have

$$\displaystyle \begin{aligned} H_{F_2}(\mathbf{x}) = 4 \sum_{\ell=1}^m b_\ell A_\ell. \end{aligned}$$

Let $b_0=\min \{b_{\ell }\neq 0\}$. Then

$$\displaystyle \begin{aligned}{\mathbf{y}}^\top H_{F_2}(\mathbf{x})\mathbf{y} \ge 4 b_0 \|({\mathbf{a}}_{j_1}^*\mathbf{y}, \ldots, {\mathbf{a}}_{j_n}^*\mathbf{y})\|{}^2\ge 4b_0 a_0^2 \|\mathbf{y}\|{}^2.\end{aligned}$$

Thus, F ₂ is strongly convex. □

Theorem 9

Let H _F(x) be the Hessian matrix of the function F(x) and let x ^⋆ be a global minimizer of (2). Suppose that a _j, j = 1, ⋯ , m are a ₀ -generic. Then H _F(x ^⋆) is positive definite in a neighborhood of x ^∗.

Proof

Recall that $A_\ell ={\mathbf {a}}_\ell \bar {\mathbf {a}}_\ell ^\top$ for ℓ = 1, ⋯ , m. It is easy to see

$$\displaystyle \begin{aligned}\nabla F(\mathbf{x})= 2\sum_{\ell=1}^m ({\mathbf{x}}^\top A_\ell\mathbf{x}- b_\ell) A_\ell \mathbf{x}\end{aligned}$$

and the entries h _ij of the Hessian H _F(x) is

$$\displaystyle \begin{aligned}h_{ij} = \frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j} f(\mathbf{x}) = 2\sum_{\ell=1}^m ({\mathbf{x}}^\top A_\ell\mathbf{x}- b_\ell) a_{ij}(\ell) + 4 \sum_{p=1}^n a_{i,p}(\ell) x_p \sum_{q=1}^n a_{j,q}(\ell)x_q, \end{aligned}$$

where $A_\ell = [a_{ij}(\ell )]_{ij=1}^n$. Since (x ^∗)^⊤ A _ℓ x ^∗ = b _ℓ for all ℓ = 1, ⋯ , m, the first summation term of h _ij above is zero at x ^∗. Let M(y) = y ^⊤ H _f(x ^∗)y be a quadratic function of y. Then we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} M(\mathbf{y}) &\displaystyle =&\displaystyle 4 \sum_{\ell=1}^m ({\mathbf{y}}^\top A_\ell {\mathbf{x}}^* ({\mathbf{x}}^*)^\top A_\ell \mathbf{y} = 4\sum_{\ell=1}^m |{\mathbf{y}}^\top A_\ell {\mathbf{x}}^*|{}^2\\ &\displaystyle =&\displaystyle 4 \sum_{\ell=1}^m |{\mathbf{y}}^\top {\mathbf{a}}_\ell|{}^2 |\bar{\mathbf{a}}_\ell^\top {\mathbf{x}}^*|{}^2 \ge 4a_0 \|{\mathbf{x}}^*\|{}^2 \|\mathbf{y}\|{}^2, \end{array} \end{aligned} $$

where the inequality follows from the fact that a _j, j = 1, ⋯ , m are a ₀-generic. It implies that H _F(x ^∗) is positive definite. □

Next, we show that the Hessian H _F(x ^∗) is nonnegative definite at the global minimizer x ^⋆ in the complex case. To this end, we first introduce some notations. Write a _ℓ = a _ℓ + i c _ℓ for ℓ = 1, ⋯ , m. For z = x + iy, we have ${\mathbf {a}}_\ell ^\top {\mathbf {z}}^*= b_\ell$ for the global minimizer z ^∗. Writing $f_\ell (\mathbf {x}, \mathbf {y}) = |{\mathbf {a}}_\ell ^\top \mathbf {z}|{ }^2 -b_\ell =(a_\ell ^\top \mathbf {x} - c_\ell ^\top \mathbf {y})^2 + (c_\ell ^\top \mathbf {x}+ a_\ell ^\top \mathbf {y})^2 -b_\ell$, we consider

$$\displaystyle \begin{aligned} f(\mathbf{x}, \mathbf{y}) = \frac{1}{m} \sum_{\ell=1}^m f_\ell^2. \end{aligned}$$

The gradient of f can be easily computed as follows: ∇f = [∇_x f, ∇_y f] with

$$\displaystyle \begin{aligned} \nabla_{\mathbf{x}} f(\mathbf{x}, \mathbf{y}) = \frac{4}{m}\sum_{\ell=1}^m f_\ell(\mathbf{x}, \mathbf{y}) [(a_\ell^\top \mathbf{x}- c_\ell^\top \mathbf{y}) a_\ell + (c_\ell^\top \mathbf{x} + a_\ell^\top \mathbf{y}) c_\ell] \end{aligned}$$

and

$$\displaystyle \begin{aligned} \nabla_{\mathbf{y}} f(\mathbf{x}, \mathbf{y}) = \frac{4}{m}\sum_{\ell=1}^m f_\ell(\mathbf{x}, \mathbf{y}) [(a_\ell^\top \mathbf{x}- c_\ell^\top \mathbf{y}) (-c_\ell) + (c_\ell^\top \mathbf{x} + a_\ell^\top \mathbf{y}) a_\ell]. \end{aligned}$$

Furthermore, the Hessian of F is given by

$$\displaystyle \begin{aligned} H_F (\mathbf{x}, \mathbf{y}) = \left[ \nabla_{\mathbf{x}} \nabla_{\mathbf{x}} f(\mathbf{x}, \mathbf{y}) \ \nabla_{\mathbf{x}} \nabla_{\mathbf{y}} f(\mathbf{x}, \mathbf{y}); \, \nabla_{\mathbf{y}} \nabla_{\mathbf{x}} f(\mathbf{x}, \mathbf{y} ) \, \nabla_{\mathbf{y}}\nabla_{\mathbf{y}} f(\mathbf{x}, \mathbf{y}) \right], \end{aligned}$$

where the terms ∇_x∇_x f(x, y), ⋯ , ∇_y∇_y f(x, y) are given below.

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \nabla_{\mathbf{x}}\nabla_{\mathbf{x}} f(\mathbf{x}, \mathbf{y}) = \frac{4}{m}\sum_{\ell=1}^m f_\ell(\mathbf{x}, \mathbf{y}) [a_\ell a_\ell^\top + c_\ell c_\ell^\top] \\ &\displaystyle &\displaystyle + \frac{8}{m}\sum_{\ell=1}^m [(a_\ell^\top \mathbf{x}- c_\ell^\top \mathbf{y}) a_\ell + (c_\ell^\top \mathbf{x} + a_\ell^\top \mathbf{y}) c_\ell] [(a_\ell^\top \mathbf{x}- c_\ell^\top \mathbf{y}) a_\ell^\top + (c_\ell^\top \mathbf{x} + a_\ell^\top \mathbf{y}) c_\ell^\top] \end{array} \end{aligned} $$

and

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \nabla_{\mathbf{x}}\nabla_{\mathbf{y}} f(\mathbf{x}, \mathbf{y}) = \frac{4}{m}\sum_{\ell=1}^m f_\ell(\mathbf{x}, \mathbf{y}) [a_\ell (-c_\ell)^\top + c_\ell a_\ell^\top] \\ &\displaystyle &\displaystyle + \frac{8}{m}\sum_{\ell=1}^m [(a_\ell^\top \mathbf{x}- c_\ell^\top \mathbf{y}) a_\ell + (c_\ell^\top \mathbf{x} + a_\ell^\top \mathbf{y}) c_\ell] [(a_\ell^\top \mathbf{x}- c_\ell^\top \mathbf{y}) (-c_\ell)^\top + (c_\ell^\top \mathbf{x} + a_\ell^\top \mathbf{y}) a_\ell^\top]. \end{array} \end{aligned} $$

The terms ∇_y∇_x f(x, y) and ∇_y∇_y f(x, y) can be obtained similarly.

Theorem 10

For phase retrieval problem in the complex case, the Hessian matrix H _f(x ^∗, y ^∗) at any global minimizer z ^∗ := (x ^∗, y ^∗) satisfies H _f(x ^∗, y ^∗) ≥ 0. Furthermore, H _f(x ^∗, y ^∗) = 0 along the direction [−(y ^∗)^⊤, (x ^∗)^⊤]^⊤.

Proof

At the global minimizer z ^∗ = x ^∗ + iy ^∗, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla_{\mathbf{x}}\nabla_{\mathbf{x}} f({\mathbf{x}}^*, {\mathbf{y}}^*) = \frac{8}{m}\sum_{\ell=1}^m &\displaystyle &\displaystyle [(a_\ell^\top {\mathbf{x}}^*- c_\ell^\top {\mathbf{y}}^*) a_\ell + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) c_\ell] \times \\ &\displaystyle &\displaystyle [(a_\ell^\top {\mathbf{x}}^*- c_\ell^\top {\mathbf{y}}^*) a_\ell^\top + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) c_\ell^\top], \end{array} \end{aligned} $$

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla_{\mathbf{x}}\nabla_{\mathbf{y}} f({\mathbf{x}}^*, {\mathbf{y}}^*) = \frac{8}{m}\sum_{\ell=1}^m &\displaystyle &\displaystyle [(a_\ell^\top {\mathbf{x}}^* - c_\ell^\top {\mathbf{y}}^*) a_\ell+ (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) c_\ell] \times \\ &\displaystyle &\displaystyle [(a_\ell^\top {\mathbf{x}}^* - c_\ell^\top {\mathbf{y}}^*) (-c_\ell)^\top + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) a_\ell^\top] \end{array} \end{aligned} $$

and similar for the other two terms. It is easy to check that for any w = u + iv with $\mathbf {u}, \mathbf {v}\in \mathbb {R}^n$, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle [{\mathbf{u}}^\top \, {\mathbf{v}}^\top]^\top H_F({\mathbf{x}}^*, {\mathbf{y}}^*) \left[\mathbf{u} \,; \mathbf{v}\right]\\ &\displaystyle =&\displaystyle \frac{8}{m}\sum_{\ell=1}^m [(a_\ell^\top {\mathbf{x}}^* - c_\ell^\top {\mathbf{y}}^*) a_\ell^\top \mathbf{u} + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) c_\ell^\top \mathbf{u} ]^2 \\ &\displaystyle &\displaystyle + \frac{8}{m}\sum_{\ell=1}^m [(a_\ell^\top {\mathbf{x}}^* - c_\ell^\top {\mathbf{y}}^*) (-c_\ell)^\top\mathbf{v} + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) a_\ell^\top\mathbf{v}]^2\\ &\displaystyle &\displaystyle + \frac{8}{m}\sum_{\ell=1}^m 2[(a_\ell^\top {\mathbf{x}}^* - c_\ell^\top {\mathbf{y}}^*) a_\ell^\top \mathbf{u} + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) c_\ell^\top \mathbf{u} ] \times \\ &\displaystyle &\displaystyle [(a_\ell^\top {\mathbf{x}}^* - c_\ell^\top {\mathbf{y}}^*) (-c_\ell)^\top\mathbf{v} + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) a_\ell^\top\mathbf{v}]\\ &\displaystyle =&\displaystyle \frac{8}{m}\sum_{\ell=1}^m [(a_\ell^\top {\mathbf{x}}^* - c_\ell^\top {\mathbf{y}}^*) a_\ell^\top \mathbf{u} + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) c_\ell^\top \mathbf{u} +(a_\ell^\top {\mathbf{x}}^* - c_\ell^\top {\mathbf{y}}^*) (-c_\ell)^\top\mathbf{v} \\ &\displaystyle &\displaystyle + (c_\ell^\top {\mathbf{x}}^* + a_\ell^\top {\mathbf{y}}^*) a_\ell^\top\mathbf{v}]^2\\ &\displaystyle \ge &\displaystyle 0. \end{array} \end{aligned} $$

It means that H _f(x ^∗, y ^∗) ≥ 0. Furthermore, if we choose u = −y ^∗ and v = x ^∗, then it is easy to show that

$$\displaystyle \begin{aligned}{}[-({\mathbf{y}}^*)^\top \, ({\mathbf{x}}^*)^\top]^\top H_f({\mathbf{x}}^*, {\mathbf{y}}^*) \left[-{\mathbf{y}}^* \,; {\mathbf{x}}^*\right]=0, \end{aligned}$$

which gives that the Hessian H _F along this direction is zero. □