Group Sparse Optimization for Inpainting of Random Fields on the Sphere ¹¹1This work was partly supported by Hong Kong Research Grant Council PolyU15300120 and the Shenzhen Research Institute of Big Data Grant 2019ORF01002.

Let $(\Omega,\mathcal{F},P)$ be a probability space, and let $\mathfrak{B}(\mathbb{S}^{2})$ be the Borel sigma algebra on the unit sphere $\mathbb{S}^{2}:=\{{\rm{x}}\in\mathbb{R}^{3}:\|\rm{x}\|=1\}$, where $\|\cdot\|$ is the $\ell_{2}$ norm. A real-valued random field on $\mathbb{S}^{2}$ is an $\mathcal{F}\otimes\mathfrak{B}(\mathbb{S}^{2})$-measurable function $T:\Omega\times\mathbb{S}^{2}\rightarrow\mathbb{R}$, and it is said to be $2$-weakly isotropic if the expected value and covariance of $T$ are rotationally invariant (see [14]). It is known that a $2$-weakly isotropic random field on the sphere has the following Karhunen-Loève (K-L) expansion [23, 25],

$$T(\omega,{\rm{x}})=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\alpha_{l,m}(\omega)Y_{l,m}({\rm{x}}),\quad{\rm{x}}\in\mathbb{S}^{2},\quad\omega\in\Omega,$$

(1.1)

where $\alpha_{l,m}(\omega)=\int_{\mathbb{S}^{2}}T(\omega,{\rm{x}})\overline{Y_{l,m}({\rm{x}})}d\sigma({\rm{x}})\in\mathbb{C}$ are random spherical harmonic coefficients of $T(\omega,{\rm{x}})$, $Y_{l,m}$, for $m=-l,\ldots,l$, $l=0,1,2,\ldots$ are spherical harmonics with degree $l$ and order $m$, and $\sigma$ is the surface measure on $\mathbb{S}^{2}$ satisfying $\sigma(\mathbb{S}^{2})=4\pi.$ For convenient, let

$$\alpha(\omega)=(\alpha_{0,0}(\omega),\underbrace{\alpha_{1,-1}(\omega),\alpha_{1,0}(\omega),\alpha_{1,1}(\omega)}_{3},\ldots,\underbrace{\alpha_{1,-l}(\omega),\ldots,\alpha_{l,l}(\omega)}_{2l+1},\ldots)^{T}$$

denote the coefficient vector of an isotropic random field $T(\omega,\mathrm{x})$. The infinite-dimensional coefficient vector $\alpha(\omega)$ can be grouped according to degrees $l$ and written in the following group structure

$$\alpha(\omega)=(\alpha_{0\cdot}^{T}(\omega),\alpha_{1\cdot}^{T}(\omega),\ldots,\alpha^{T}_{l\cdot}(\omega),\ldots)^{T},$$

(1.2)

where

$$\alpha_{l\cdot}(\omega)=(\alpha_{l,-l}(\omega),\ldots,\alpha_{l,0}(\omega),\ldots,\alpha_{l,l}(\omega))^{T}\in\mathbb{C}^{2l+1},\quad\quad l\geq 0.$$

From [14], we know that for any given degree $l\geq 1$ and any $\omega\in\Omega$, the sum

$$\sum_{m=-l}^{l}|\alpha_{l,m}(\omega)|^{2}=\|\alpha_{l\cdot}(\omega)\|^{2}$$

is rotationally invariant, while $\sum_{m=-l}^{l}|\alpha_{l,m}(\omega)|$ is not invariant under rotation of the coordinate axes. Moreover, in [14] the angular power spectrum is given by

$$C_{l}=\frac{1}{2l+1}\mathbb{E}[\|\alpha_{l\cdot}(\omega)\|^{2}]$$

if the expected value $\mathbb{E}[T(\omega,{\rm{x}})]=0$ for all ${\rm{x}}\in\mathbb{S}^{2}$. In the study of isotropic random fields [12, 23, 25], the angular power spectrum plays an important role since it contains full information of the covariance of the field and provides characterization of the field. Hence, considering the group structure (1.2) of the coefficients of an isotropic random field is essential.

Sparse representation of a random field $T$ is an approximation of $T$ with few non-zero elements $\alpha_{l,m}(\omega)$ of $\alpha(\omega)$. In group sparse representation, instead of considering elements individually, we seek an approximation of $T$ with few non-zero groups $\alpha_{l\cdot}(\omega)$ of $\alpha(\omega)$, i.e., few non-zero entries of the following vector

$$(\|\alpha_{0\cdot}(\omega)\|,\,\|\alpha_{1\cdot}(\omega)\|,\ldots,\|\alpha_{l\cdot}(\omega)\|,\ldots,)^{T}.$$

In recent years, sparse representation of isotropic random fields has been extensively studied. In [6] the authors studied the sparse representation of random fields via an $\ell_{1}$-regularized minimization problem. In [14], the authors considered isotropic group sparse representation of Gaussian and isotropic random fields on the sphere through an unconstrained optimization problem with a weighted $\ell_{2,1}$ norm, which is an example of group Lasso [34]. In [24], the authors proposed a non-Lipschitz regularization problem with a weighted $\ell_{2,p}$ ($0<p<1$) norm for the group sparse representation of isotropic random fields. The non-Lipschitz regularizer also preserves the rotational invariance property of $\|\alpha_{l\cdot}(\omega)\|^{2}$ for any given degree $l\geq 1$ and any $\omega\in\Omega$.

Isotropic random fields on the sphere have many applications [5, 20, 26, 28, 31], especially in the study of the Cosmic Microwave Background (CMB) analysis [1, 30]. However, the true spherical random field usually presents masked regions and missing observations. Many inpainting methods [1, 13, 30, 32] have been proposed for recovering the true field based on sparse representation. However, they did not take the group structure (1.2) of the coefficients into consideration.

Group sparse optimization models have been successfully used in signal processing, imaging sciences and predictive analysis [3, 11, 18, 17, 27]. In this paper, we propose a constrained group sparse optimization model for inpainting of isotropic random fields on the sphere based on group structure (1.2). Moreover, to recover the true random field by using information only on a subset $\Gamma\subseteq\mathbb{S}^{2}$ which contains an open set, we need the unique continuation property [19] of any realization of a random field, that is, for any fixed $\omega\in\Omega$, if the value of a field equals to zero on $\Gamma$, then the random field is identically zero on the sphere (see Appendix A for more details).

Let $L_{2}(\Omega\times\mathbb{S}^{2})$ be the $L_{2}$ space on the product space $\Omega\times\mathbb{S}^{2}$ with product measure $P\otimes\sigma$ and the induced norm $\|\cdot\|_{L_{2}(\Omega\times\mathbb{S}^{2})}=\mathbb{E}[\|\cdot\|_{L_{2}(\mathbb{S}^{2})}]$, where $L_{2}(\mathbb{S}^{2})$ is the space of square-integrable functions over $\mathbb{S}^{2}$ endowed with the inner product

$$\langle f,g\rangle_{L_{2}(\mathbb{S}^{2})}=\int_{\mathbb{S}^{2}}f({\rm{x}})\overline{g({\rm{x}})}d\sigma({\rm{x}}),\quad\quad\forall f,g\in L_{2}(\mathbb{S}^{2}),$$

and the induced $L_{2}$-norm $\|f\|_{L_{2}(\mathbb{S}^{2})}=(\langle f,f\rangle_{L_{2}(\mathbb{S}^{2})})^{\frac{1}{2}}$. By Fubini’s theorem, for $\omega\in\Omega$, $T(\omega,{\rm{x}})\in L_{2}(\mathbb{S}^{2})$, $P$-a.s., where “$P$-a.s.” stands for almost surely with probability $1$. For brevity, we write $T(\omega,{\rm{x}})$ as $T({\rm{x}})$ or $T$ if no confusion arises.

Let the observed field be given by

$$T^{\circ}:=\mathcal{A}(T^{*})+\Delta,$$

(1.3)

where $T^{*}\in L_{2}(\Omega\times\mathbb{S}^{2})$ is the true isotropic random field that we aim to recover, $\Delta:\Omega\times\mathbb{S}^{2}\rightarrow\mathbb{R}$ is observational noise and $\mathcal{A}:L_{2}(\Omega\times\mathbb{S}^{2})\rightarrow L_{2}(\Omega\times\mathbb{S}^{2})$ is an inpainting operator defined by

$$\mathcal{A}(T({\rm{x}}))=\left\{\begin{array}[]{cl}T({\rm{x}})&\,{\rm if}\ {\rm{x}}\in\Gamma\\ 0&\,{\rm if}\ {\rm{x}}\in\mathbb{S}^{2}\setminus\Gamma,\\ \end{array}\right.$$

(1.4)

where $\mathbb{S}^{2}\setminus\Gamma\subset\mathbb{S}^{2}$ is a nonempty inpainting area and the set $\Gamma\subset\mathbb{S}^{2}$ has an open subset. In our optimization model, we consider the observed field $T^{\circ}$ as one realization of a random field. For notational simplicity, let

$$\alpha=(\alpha^{T}_{0\cdot},\alpha^{T}_{1\cdot},\ldots,\alpha^{T}_{l\cdot},\ldots)^{T}$$

(1.5)

denote the spherical harmonic coefficient vector of an isotropic random field $T$ for a fixed $\omega\in\Omega$, where $\alpha_{l\cdot}=(\alpha_{l,-l},\ldots,\alpha_{l,0},\ldots,\alpha_{l,l})^{T}\in\mathbb{C}^{2l+1}$, $l\in\mathbb{N}_{0}:=\{0,1,2,\ldots\}$.

By Parseval’s theorem, we have

$$\|T\|^{2}_{L_{2}(\mathbb{S}^{2})}=\sum\limits_{l=0}^{\infty}\sum\limits_{m=-l}^{l}|\alpha_{l,m}|^{2}=\sum\limits_{l=0}^{\infty}\|\alpha_{l\cdot}\|^{2}<\infty.$$

Hence the sequence $\{\|\alpha_{l\cdot}\|\}_{l\in\mathbb{N}_{0}}\in\ell^{2}(\mathbb{N}_{0})$, where $\ell^{2}(\mathbb{N}_{0})$ is the space of square-summable sequences. We write $\alpha\in\ell^{2}$ to indicate $\{\|\alpha_{l\cdot}\|\}_{l\in\mathbb{N}_{0}}\in\ell^{2}(\mathbb{N}_{0})$.

The number of non-zero groups of $\alpha$ with group structure (1.5) is calculated by

$$\|\alpha\|_{2,0}=\sum\limits_{l=0}^{\infty}\|\alpha_{l\cdot}\|^{0},$$

where $\|\alpha_{l\cdot}\|^{0}=\left\{\begin{array}[]{cc}1&\textrm{if}\,\,\|\alpha_{l\cdot}\|>0\\ 0&\textrm{otherwise,}\end{array}\right.$ and $\|\alpha\|_{2,0}$ is called $\ell_{2,0}$ norm of $\alpha$. The $\ell_{2,0}$ norm is discontinuous and $\|T\|^{2}_{L_{2}(\mathbb{S}^{2})}<\infty$ does not ensure $\|\alpha\|_{2,0}<\infty$. In [14], the authors used a weighted $\ell_{2,1}$ norm of $\alpha$ defined by

$$\|\alpha\|_{2,1}=\sum\limits_{l=0}^{\infty}\beta_{l}\|\alpha_{l\cdot}\|,$$

where $\beta_{l}>0$. It is easy to see that $\lim_{p\downarrow 0}\|\alpha_{l\cdot}\|^{p}=\|\alpha_{l\cdot}\|^{0}$. The $\ell_{2,p}$ norm $(0<p<1)$ has been widely used in sparse approximation [11, 18]. In this paper, we use a weighted $\ell_{2,p}$ norm of $\alpha$. Let

$$\ell^{p}_{\beta}:=\left\{\alpha\in\ell^{2}:\|\alpha\|_{2,p}^{p}:=\sum_{l=0}^{\infty}\beta_{l}{{\|\alpha_{l\cdot}\|^{p}}}<\infty\right\}$$

be a weighted $\ell^{p}$ ($0<p<1$) space with positive weights $\beta_{0}=1$ and $\beta_{l}=\eta^{l}l^{p}$ for $l\geq 1$, where $\eta>1$ is a constant. For a fixed $\omega\in\Omega$, we consider the following constrained optimization problem

$$\begin{array}[]{cl}\min\limits_{\alpha\in\ell^{p}_{\beta}}&\|\alpha\|_{2,p}^{p}\\ {\rm{s.t.}}&\|\mathcal{A}(T({\rm{x}}))-T^{\circ}({\rm{x}})\|^{2}_{L_{2}(\mathbb{S}^{2})}\leq\varrho,\end{array}$$

(1.6)

where $\varrho>\int_{\mathbb{S}^{2}\setminus\Gamma}|T^{\circ}({\rm{x}})|^{2}d\sigma(\rm{x})$.

A feasible field of problem (1.6) takes the form

$$T({\rm{x}})=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\alpha_{l,m}Y_{l,m}({\rm{x}}),\quad{\rm{x}}\in\mathbb{S}^{2},\quad\alpha\in\ell_{\beta}^{p}.$$

Since $T^{\circ}\in L_{2}(\mathbb{S}^{2})$, for $\epsilon=\varrho-\int_{\mathbb{S}^{2}\setminus\Gamma}|T^{\circ}({\rm{x}})|^{2}d\sigma(\rm{x})$, there is a finite number $L$ such that $\|T^{\circ}(\mathrm{x})-T^{\circ}_{L}(\mathrm{x})\|^{2}_{L_{2}(\mathbb{S}^{2})}<\epsilon$, where $T^{\circ}_{L}(\mathrm{x})=\sum_{l=0}^{L}\sum_{m=-l}^{l}b_{l,m}Y_{l,m}(\mathrm{x})$ and $b_{l,m}=\int_{\mathbb{S}^{2}}T^{\circ}(\mathrm{x})\overline{Y_{l,m}(\mathrm{x})}d\sigma(\mathrm{x})$. Let $b_{l,m}=0$ for $l=L+1,\ldots$, $m=-l,\ldots,l$, then $(b_{0,0},\ldots,b_{L,L},0,0,0,\ldots)^{T}\in\ell_{\beta}^{p}$. From the definition of $\mathcal{A}$, we have

	$\displaystyle\|\mathcal{A}(T_{L}^{\circ}({\rm{x}}))-T^{\circ}({\rm{x}})\|^{2}_{L_{2}(\mathbb{S}^{2})}$	$\displaystyle=$	$\displaystyle\int_{\mathbb{S}^{2}\setminus\Gamma}|T^{\circ}({\rm{x}})|^{2}d\sigma(\mathrm{x})+\int_{\Gamma}|T^{\circ}({\rm{x}})-T_{L}^{\circ}({\rm{x}})|^{2}d\sigma(\rm{x})$
		$\displaystyle<$	$\displaystyle\int_{\mathbb{S}^{2}\setminus\Gamma}|T^{\circ}({\rm{x}})|^{2}d\sigma(\mathrm{x})+\epsilon=\varrho.$

Thus, the feasible set of problem (1.6) is nonempty. Moreover the objective function of problem (1.6) is continuous and level-bounded. Hence an optimal solution of problem (1.6) exists.

Since $\alpha=0$ implies that $\|\alpha\|_{2,p}^{p}=0$, we assume $\|T^{\circ}({\rm{x}})\|^{2}_{L_{2}(\mathbb{S}^{2})}>\varrho$, which implies that $T({\rm{x}})=0$ is not a feasible field of problem (1.6).

Our main contributions are summarized as follows.

• •

  Based on the K-L expansion, the constraint of problem (1.6) can be written in a discrete form with variables $\alpha\in\ell^{p}_{\beta}$. We derive a lower bound for the $\ell_{2}$ norm of nonzero groups of scaled KKT points of (1.6). Based on the lower bound, we prove that the infinite-dimensional problem (1.6) is equivalent to a finite-dimensional optimization problem.

• •

  We propose a penalty method for solving the finite-dimensional optimization problem via unconstrained optimization problems. We establish the exact penalization results regarding local minimizers and $\varepsilon$-minimizers. Moreover, we propose a smoothing penalty algorithm and prove that the sequence generated by the algorithm is bounded and any accumulation point of the sequence is a scaled KKT point of the finite-dimensional optimization problem.

• •

  We give the approximation error of the random field represented by scaled KKT points of the finite-dimensional optimization problem in $L_{2}(\Omega\times\mathbb{S}^{2}).$

The rest of this paper is organised as follows. In Section 2, we prove that the infinite-dimensional discrete optimization problem is equivalent to a finite-dimensional problem. In Section 3, we present the penalty method and give exact penalization results. In Section 4, we discuss optimality conditions of the finite-dimensional optimization problem and its penalty problem. Moreover, we propose a smoothing penalty algorithm and establish its convergence. In Section 5, we give the approximation error in $L_{2}(\Omega\times\mathbb{S}^{2}).$ In Section 6, we conduct numerical experiments on band-limited random fields and images from Earth topography data to compare our approach with some existing methods on the quality of the solutions and inpainted images. Finally, we give conclusion remarks in Section 7.

In this section, we propose the discrete formulation of problem (1.6) and prove that the discrete problem is equivalent to a finite-dimensional problem (2.5).

Based on the definition of $\mathcal{A}$ and the spherical harmonic expansion of $T$, we obtain that

$\displaystyle\begin{split}&\|\mathcal{A}(T({\rm{x}}))-T^{\circ}({\rm{x}})\|^{2}_{L_{2}(\mathbb{S}^{2})}\\ &=\int_{\mathbb{S}^{2}}|\mathcal{A}(T({\rm{x}}))-T^{\circ}({\rm{x}})|^{2}d\sigma({\rm{x}})=\int_{\Gamma}|T({\rm{x}})-T^{\circ}({\rm{x}})|^{2}d\sigma({\rm{x}})+\int_{\mathbb{S}^{2}\setminus\Gamma}|T^{\circ}({\rm{x}})|^{2}d\sigma({\rm{x}})\\ &=\int_{\Gamma}\bigg{|}\sum\limits_{l=0}^{\infty}\sum\limits_{m=-l}^{l}\alpha_{l,m}Y_{l,m}({\rm{x}})\bigg{|}^{2}d\sigma({\rm{x}})-2\textrm{Re}\left(\int_{\Gamma}T^{\circ}({\rm{x}})\left(\sum\limits_{l=0}^{\infty}\sum\limits_{m=-l}^{l}\bar{\alpha}_{l,m}\overline{Y_{l,m}({\rm{x}})}\right)d\sigma({\rm{x}})\right)\\ &\quad+\int_{\mathbb{S}^{2}}|T^{\circ}({\rm{x}})|^{2}d\sigma({\rm{x}})\\ &=\alpha^{T}Y\bar{\alpha}-2\textrm{Re}\left(\sum\limits_{l=0}^{\infty}\sum\limits_{m=-l}^{l}\bar{\alpha}_{l,m}\int_{\Gamma}T^{\circ}({\rm{x}})\overline{Y_{l,m}({\rm{x}})}d\sigma({\rm{x}})\right)+\int_{\mathbb{S}^{2}}|T^{\circ}({\rm{x}})|^{2}d\sigma({\rm{x}})\\ &=\alpha^{H}Y\alpha-2\textrm{Re}(\alpha^{H}\alpha^{\circ})+c,\end{split}$

(2.1)

where $Y$ is an infinite-dimensional matrix with $(Y)_{l^{2}+l+m+1,l^{\prime 2}+l^{\prime}+m^{\prime}+1}=\int_{\Gamma}Y_{l,m}({\rm{x}})\overline{Y_{l^{\prime},m^{\prime}}({\rm{x}})}d\sigma({\rm{x}})\in\mathbb{C},$ $\alpha^{\circ}=((\alpha_{0\cdot}^{\circ})^{T},\ldots,(\alpha^{\circ}_{l\cdot})^{T},\ldots)^{T}$ with $\alpha^{\circ}_{l\cdot}\in\mathbb{C}^{2l+1}$, $\alpha^{\circ}_{l,m}=\int_{\Gamma}T^{\circ}({\rm{x}})\overline{Y_{l,m}({\rm{x}})}d\sigma({\rm{x}})$ and $c=\int_{\mathbb{S}^{2}}|T^{\circ}({\rm{x}})|^{2}d\sigma({\rm{x}}).$

The minimization problem (1.6) can be written as the following optimization problem

$$\begin{array}[]{cl}\min\limits_{\alpha\in\ell_{\beta}^{p}}&\|\alpha\|_{2,p}^{p}\vspace{1mm}\\ {\rm{s.t.}}&\alpha^{H}Y\alpha-2\textrm{Re}(\alpha^{H}\alpha^{\circ})+c\leq\varrho.\end{array}$$

(2.2)

From the setting of $\varrho$ in problem (1.6) and Theorem 8, the feasible set of (2.2) has an interior point and bounded. The following assumption follows from our assumption on problem (1.6).

Note that the objective function and constraint function in (2.2) are real-valued functions with complex variables. Thus, following [24], we apply the Wirtinger calculus (See Appendix B for more details) in this paper.

Since the objective function in problem (2.2) is not Lipschitz continuous at points containing zero groups. We shall consider first-order stationary points of (2.2) and extend the definition of scaled KKT points for finite-dimensional optimization problem with real variables in [8, 10, 29].

From (2.3), for any nonzero vector $\alpha_{l\cdot}^{*}\in\mathbb{C}^{2l+1}$, we have

$$p\beta_{l}\|\alpha^{*}_{l\cdot}\|^{p-2}\alpha^{*}_{l\cdot}+2\nu(Y_{l\cdot}\alpha^{*}-\alpha_{l\cdot}^{\circ})=0,$$

and $\nu>0$. Then,

$$p\beta_{l}\|\alpha^{*}_{l\cdot}\|^{p-1}=2\nu\|Y_{l\cdot}\alpha^{*}-\alpha_{l\cdot}^{\circ}\|\leq 2\nu\|Y\alpha^{*}-\alpha^{\circ}\|.$$

By definition of $Y$ and feasibility of $\alpha^{*}$, there exists $\tilde{c}>0$ such that $\|Y\alpha^{*}-\alpha^{\circ}\|\leq\tilde{c}$. Hence for any nonzero vector $\alpha_{l\cdot}^{*}\in\mathbb{C}^{2l+1}$, we have

$$\|\alpha^{*}_{l\cdot}\|\geq\left(\frac{p\beta_{l}}{2\nu\tilde{c}}\right)^{\frac{1}{1-p}}.$$

(2.4)

By definition of $\beta_{l}$, $l\in\mathbb{N}_{0}$, we obtain that

$\displaystyle\infty>\|\alpha^{*}\|_{2,p}^{p}$

$\displaystyle=$

$\displaystyle\sum\limits_{l=0}^{\infty}\beta_{l}\|\alpha^{*}_{l\cdot}\|^{p}\,=\sum\limits_{\{l\in\mathbb{N}_{0}:\,\|\alpha^{*}_{l\cdot}\|\neq 0\}}\beta_{l}\|\alpha^{*}_{l\cdot}\|^{p}\geq\left(\frac{p}{2\nu\tilde{c}}\right)^{\frac{p}{1-p}}\sum\limits_{\{l\in\mathbb{N}_{0}:\,\|\alpha^{*}_{l\cdot}\|\neq 0\}}(\eta^{l}l^{p})^{\frac{1}{1-p}},$

which implies that $\{l\in\mathbb{N}_{0}:\|\alpha^{*}_{l\cdot}\|\neq 0\}$ is a finite set. Thus, the number of nonzero vectors $\alpha_{l\cdot}^{*}\in\mathbb{C}^{2l+1}$ of a scaled KKT point of problem (2.2) is finite and there exists an $L\in\mathbb{N}_{0}$ such that $L=\max\{l\in\mathbb{N}_{0}:\|\alpha^{*}_{l\cdot}\|\neq 0\}$. Therefore, the infinite-dimensional problem (2.2) can be truncated to a finite-dimensional problem and approximated in a finite-dimensional space.

For notational simplicity, we truncate $\alpha\in\ell_{\beta}^{p}$ to $\alpha=(\alpha_{0,0},\alpha_{1\cdot}^{T},\ldots,\alpha_{L\cdot}^{T})^{T}\in\mathbb{C}^{d}$, where $d:=(L+1)^{2}$. We use $\hat{\alpha}^{\circ}\in\mathbb{C}^{d}$ to denote the truncated vector whose elements are the first $d$ elements of $\alpha^{\circ}$ and $\hat{Y}\in\mathbb{C}^{d\times d}$ to denote the leading principal submatrix of order $d$ of $Y$. Since $\Gamma$ has an open subset, we have $z^{H}\hat{Y}z>0$ for any nonzero $z\in\mathbb{C}^{d}$. Thus, the matrix $\hat{Y}$ is positive definite.

The truncated finite-dimensional problem of problem (2.2) has the following version

$$\begin{array}[]{cl}\min\limits_{\alpha\in\mathbb{C}^{d}}&\Phi(\alpha):=\sum_{l=0}^{L}\beta_{l}\|\alpha_{l\cdot}\|^{p}\vspace{1mm}\\ {\rm{s.t.}}&\alpha^{H}\hat{Y}\alpha-2\textrm{Re}(\alpha^{H}\hat{\alpha}^{\circ})+c\leq\varrho.\end{array}$$

(2.5)

By the definition of $\varrho$, the feasible set of (2.5) is nonempty and has an interior point. The objective function $\Phi(\alpha)$ is continuous, nonnegative with $\Phi(0)=0$, and differentiable except at points containing zero groups. The penalty formulation for (2.5) is

$$\min\limits_{\alpha\in\mathbb{C}^{d}}\quad F_{\lambda}(\alpha):=\Phi(\alpha)+\lambda(\alpha^{H}\hat{Y}\alpha-2\textrm{Re}(\alpha^{H}\hat{\alpha}^{\circ})+c-\varrho)_{+},$$

(2.6)

for some $\lambda>0$, where $(\cdot)_{+}:=\max\{\cdot,0\}$. For any nontrivial solution $\alpha^{*}$ of (2.6), we have

$$0<F_{\lambda}(\alpha^{*})=\min\limits_{\alpha\in\mathbb{C}^{d}}F_{\lambda}(\alpha)<F_{\lambda}(0)=\lambda(c-\varrho)_{+}=\lambda(c-\varrho)<\lambda c.$$

(2.7)

In Sections 3-4, we focus on problems (2.5) and (2.6).

In this section, we consider the relationship between problems (2.5) and (2.6). We first give some notations. For a closed set $S\subset\mathbb{C}^{n}$, ${\rm{dist}}(z,S)=\inf_{z^{\prime}\in S}\|z-z^{\prime}\|$ denotes the distance from a point $z\in\mathbb{C}^{n}$ to $S$ and $\mathbf{B}(b;r)=\{z\in\mathbb{C}^{n}:{{\|z-b\|}}\leq r\}$ denotes a closed ball with radius $r>0$ and center $b\in\mathbb{C}^{n}$. Let $g:\mathbb{C}^{d}\rightarrow\mathbb{R}$ be defined as

$$g(\alpha):=\alpha^{H}\hat{Y}\alpha-2\textrm{Re}(\alpha^{H}\hat{\alpha}^{\circ})+c-\varrho,$$

$S(\alpha):=\{\delta\in\mathbb{R}:g(\alpha)\leq\delta\}$ for $\alpha\in\mathbb{C}^{d}$, and $S^{-1}(\delta):=\{\alpha\in\mathbb{C}^{d}:g(\alpha)\leq\delta\}$. We denote the feasible set of problem (2.5) by $\mathcal{F}_{e}:=\{\alpha\in\mathbb{C}^{d}:g(\alpha)\leq 0\}.$ Let $\mathbb{L}:=\{0,1,\ldots,L\}$.

Since $\hat{Y}$ is positive definite, $g$ is strongly convex and has a unique global minimizer $\hat{Y}^{-1}\hat{\alpha}^{\circ}\neq\hat{\alpha}^{\circ}$ which implies that $g(\hat{\alpha}^{\circ})\in(\inf_{\alpha\in\mathbb{C}^{d}}g(\alpha),\infty).$ Since $g$ is strongly convex and quadratic, there is $C$ such that $\|\alpha-\bar{\alpha}\|\leq C|g(\alpha)-g(\bar{\alpha})|$ for $\alpha,\bar{\alpha}\in\mathbb{C}^{d}$. Choosing $\bar{\alpha}$ such that $g(\bar{\alpha})=0$, from [29, Theorem 9.48], we obtain the following lemma.