# Optimum GSSK Transmission in Massive MIMO Systems Using the Box-LASSO
Decoder

Ayed M. Alrashdi, , Abdullah E. Alrashdi, , Amer Alghadhban, , and Mohamed A.
H. Eleiwa The authors are with the Department of Electrical Engineering,
College of Engineering, University of Ha’il, P.O. Box 2440, Ha’il, 81441,
Saudi Arabia. A. E. Alrashdi is also with Saudi Aramco.Corresponding author:
Ayed M. Alrashdi (e-mail: am.alrashdi@uoh.edu.sa).

###### Abstract

We propose in this work to employ the Box-LASSO, a variation of the popular
LASSO method, as a low-complexity decoder in a massive multiple-input
multiple-output (MIMO) wireless communication system. The Box-LASSO is mainly
useful for detecting simultaneously structured signals such as signals that
are known to be sparse and bounded. One modulation technique that generates
essentially sparse and bounded constellation points is the so-called
generalized space-shift keying (GSSK) modulation. In this direction, we derive
high dimensional sharp characterizations of various performance measures of
the Box-LASSO such as the mean square error, probability of support recovery,
and the element error rate, under independent and identically distributed
(i.i.d.) Gaussian channels that are not perfectly known. In particular, the
analytical characterizations can be used to demonstrate performance
improvements of the Box-LASSO as compared to the widely used standard LASSO.
Then, we can use these measures to optimally tune the involved hyper-
parameters of Box-LASSO such as the regularization parameter. In addition, we
derive optimum power allocation and training duration schemes in a training-
based massive MIMO system. Monte Carlo simulations are used to validate these
premises and to show the sharpness of the derived analytical results.

###### Index Terms:

LASSO, box-constraint, performance analysis, channel estimation, spacial
modulation, power allocation, massive MIMO.

##  I Introduction

The least absolute selection and shrinkage operator (LASSO) [1] is a widely
used method to recover an unknown sparse signal ${\bf s}_{0}$ from noisy
linear measurements ${\bf r}={\bf A}{\bf s}_{0}+{\bf v},$ by solving the
following optimization problem:

$\widehat{{\bf s}}={\rm{arg}}\ \underset{{{\bf
s}\in\mathbb{R}^{n}}}{\operatorname{\min}}\ \|{\bf A}{\bf s}-{\bf
r}\|_{2}^{2}+\gamma\|{\bf s}\|_{1},$ (1)

where $\|\cdot\|_{2}$, and $\|\cdot\|_{1}$ represent the $\ell_{2}$-norm and
$\ell_{1}$-norm of a vector, respectively. Furthermore, ${\bf A}$ is the
measurement matrix, ${\bf v}$ is the noise vector, and $\gamma>0$ is a
regularization parameter that balances between the fidelity of the solution as
controlled by the $\ell_{2}$-norm on one side, and the sparsity of the
solution as enforced by the $\ell_{1}$-norm on the other hand. It allows for
learning a sparse model where few of the entries are non-zero. The LASSO
reduces to the linear regression as $\gamma\to 0$. The LASSO has been widely
used in modern data science and signal processing applications such as in [2,
3, 4].

The LASSO is a special instance of a class of problems called non-smooth
regularized convex optimization problems [5]. In recent years, various forms
of sharp analysis of the asymptotic performance of such optimization problems
have been studied under the assumption of noisy independent and identically
distributed (i.i.d.) Gaussian measurements. They mostly take one of the
following approaches.

The first is the approximate message passing (AMP) framework, which was
utilized in [6, 7, 8] to conduct a sharp asymptotic study of the performance
of compressed sensing problems under the assumptions of i.i.d. Gaussian
sensing matrix.

The authors in [9, 10, 11] undertook a different approach that uses the
replica method from statistical physics, which is a powerful high-dimensional
analysis tool. However, it lacks rigorous mathematical justifications in some
steps. In addition, the high-dimensional error performance of different
regularized estimators has been previously considered using some heuristic
arguments and numerical simulations in [12] and [13].

Another approach based on random matrix theory (RMT) [14] was taken in [15,
16] to analyze the high-dimensional squared error performance of ridge
regression. One major drawback of RMT is that it requires the involved
optimization problems to admit a _closed-form_ solution which is not true for
the general non-smooth convex optimization problems.

The most recent approach is based on a newly developed framework that uses the
convex Gaussian min-max theorem (CGMT) initiated by Stojnic [17] and further
extended by Thrampoulidis _et al._ in [5]. It provides the analysis in a more
natural and elementary way when compared to the previously discussed methods.
The CGMT has been applied to the performance analyses of various optimization
problems. The asymptotic mean square error (MSE) have been analyzed for
various regularized estimators in [18, 19, 20, 21]. Precise analysis of
general performance measures such as the probability of support recovery and
$\ell_{1}$-reconstruction error of the LASSO and Elastic Net was obtained in
[22, 23, 24]. The asymptotic symbol error rate (SER) of the box relaxation
optimization has been derived for various modulation schemes in [25, 26, 27,
28]. CGMT has also been used for the analysis of nonlinear measurements models
(e.g., quantized measurements) in [19] and [29].

###  I-A Contributions

In this paper, instead of the standard LASSO in (1), we will use the so-called
Box-LASSO [20], which is the same as the LASSO but with an additional box-
constraint. We will formally define it in (12). We propose using the Box-LASSO
as a low-complexity decoder in massive MIMO communication systems with modern
modulation methods such as the generalized space-shift keying (GSSK) [30]
modulation and the generalized spatial modulation (GSM) [31]. In such systems,
the transmitted signal vector ${\bf s}_{0}$ is inherently sparse and have
elements belonging to a finite alphabet, which is an excellent setting for
using the Box-LASSO.

Using the CGMT framework, this paper derives sharp asymptotic
characterizations of the mean square error, support recovery probability, and
element error rate of the Box-LASSO under the presence of uncertainties in the
channel matrix that has i.i.d. Gaussian entries. In addition, the analysis
demonstrate that the Box-LASSO outperforms the standard one in all of the
considered metrics. The derived expressions can be used to optimally tune the
involved hyper-parameters of the algorithm. Furthermore, we study the
application of the Box-LASSO to GSSK modulated massive MIMO systems and
optimize their power allocation and training duration.

The additional contributions of this paper against the most related works such
as [20, 26, 23, 24, 32] are summarized as follows:
$\bullet$ This paper considers the more practical and challenging scenario of
_imperfect_ channel state information (CSI), whereas [20, 26, 32] only derived
the analysis for the ideal case of perfect CSI.
$\bullet$ Even when the imperfect CSI case was studied in the previous works
on the LASSO and Box-Elastic Net in [23, 24], only a theoretical imperfect CSI
model (the so-called Gauss-Markov model) was considered. On the other hand,
this work presents the analysis under a more practical model of the imperfect
CSI, which is the linear minimum mean square error (LMMSE) channel estimate in
(II-B) for a massive MIMO application.
$\bullet$ With this massive MIMO application in mind, we derive the
asymptotically optimal power allocation and training duration schemes for GSSK
signal recovery.
$\bullet$ We show that the derived power allocation optimization is nothing
but the well-known scheme that maximizes the effective signal-to-noise ratio
(SNR) in [33].

###  I-B Organization

The rest of this paper is organized as follows. The system model, channel
estimation and the proposed Box-LASSO decoder are discussed in Section II.
Section III provides the main asymptotic results of the work. The application
of Box-LASSO to a massive MIMO system and its power allocation and training
duration optimization is presented in Section IV. Finally, concluding remarks
and future research directions are stated in Section V. The proof of the main
results is deferred to the Appendix.

###  I-C Notations

Here, we gather the basic notations that are used throughout the paper. Bold
face lower case letters (e.g., ${\bf x}$) represent a column vector while
$x_{j}$ is its $j^{th}$ entry. For ${\bf x}\in\mathbb{R}^{n}$, let $\|{\bf
x}\|_{2}=\sqrt{\sum_{j=1}^{n}x_{j}^{2}}$, and $\|{\bf
x}\|_{1}=\sum_{j=1}^{n}|x_{j}|$. Matrices are denoted by upper case letters
such as ${\bf X}$, with ${\bf I}_{n}$ being the $n\times n$ identity matrix.
$(\cdot)^{T}$ and $(\cdot)^{-1}$ are the transpose and inverse operators
respectively. We use the standard notations $\mathbb{E}[\cdot],$ and
$\mathbb{P}[\cdot]$ to denote the expectation of a random variable and
probability of an event respectively. We write $X\sim p_{x}$ to denote that a
random variable $X$ has a probability density/mass function $p_{x}$. In
particular, the notation ${\bf q}\sim\mathcal{N}(\mathbf{0},{\bf R}_{{\bf
q}})$ is used to denote that the random vector ${\bf q}$ is normally
distributed with $\mathbf{0}$ mean and covariance matrix ${\bf R}_{{\bf
q}}=\mathbb{E}[{\bf q}{\bf q}^{T}]$, where $\mathbf{0}$ represent the all-
zeros vector of the appropriate size. The notation $\delta_{\xi}$ is used to
represent a point-mass distribution at $\xi$. We write
$``\overset{P}{\longrightarrow}"$ to denote convergence in probability as
$n\to\infty$. We also use standard notation
${\rm{plim}}_{n\to\infty}\Theta_{n}=\Theta$ to denote that a sequence of
random variables $\Theta_{n},[n=1,2,...]$, converges in probability towards a
constant $\Theta$. When writing $x^{\star}=\mathrm{arg}\min_{x}f(x)$, the
operator $\mathrm{arg}\min$ returns any one of the possible minimizers of $f$.
The function $Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-u^{2}/2}{\rm d}u$
is the $Q$-function associated with the standard normal density.

Finally, for $a,\gamma,l,u\in\mathbb{R}$, such that $\gamma,u\geq 0,l\leq 0,$
we define the following functions:
$\bullet$ The saturated shrinkage function
$\mathcal{H}(a;\gamma,l,u):=\operatorname*{arg\,min}_{l\leq s\leq u}$
$\frac{1}{2}(s-a)^{2}+\gamma|s|$, which is given as:

$\mathcal{H}(a;\gamma,l,u)=\begin{cases}u&,\text{if}\ a\geq u+\gamma\\\
a-\gamma&,\text{if}\ \gamma<a<u+\gamma\\\ 0&,\text{if}\ |a|\leq\gamma\\\
a+\gamma&,\text{if}\ l-\gamma<a<-\gamma\\\ l&,\text{if}\ a\leq
l-\gamma.\end{cases}$ (2)

A plot of this function is depicted in Fig. 1.

Figure 1: Saturated shrinkage function.

$\bullet$ Also, let $\mathcal{J}(a;\gamma,l,u):=\min_{l\leq s\leq
u}\frac{1}{2}(s-a)^{2}+\gamma|s|$, which can be rewritten as:

$\mathcal{J}(a;\gamma,l,u)=\begin{cases}\frac{1}{2}(u-a)^{2}+\gamma
u&,\text{if}\ a\geq u+\gamma\\\ \gamma a-\frac{1}{2}\gamma^{2}&,\text{if}\
\gamma<a<u+\gamma\\\ \frac{1}{2}a^{2}&,\text{if}\ |a|\leq\gamma\\\ -\gamma
a-\frac{1}{2}\gamma^{2}&,\text{if}\ l-\gamma<a<-\gamma\\\
\frac{1}{2}(l-a)^{2}-\gamma l&,\text{if}\ a\leq l-\gamma.\end{cases}$ (3)

##  II Problem Setup

###  II-A System Model

A massive MIMO system with $n$ transmit (Tx) antennas and $m$ receive (Rx)
antennas is considered in this paper. We herein consider a training-based
transmission that consists of a coherence interval with $T=T_{t}+T_{d}$
symbols in which the channel realization is assumed to be constant. During the
first part of this coherence interval, $T_{t}$ symbol intervals are used as
known pilot symbols, with average power, ${P}_{t}$. These pilot symbols are
employed for channel estimation purposes. The remaining $T_{d}$ symbols are
dedicated to transmitting data symbols with average power, $P_{d}$.
Conservation of energy implies that

$P_{t}T_{t}+P_{d}T_{d}={P}\ T,$ (4)

where $P$ is the average total transmission power.

Letting $\nu$ denote the ratio of the total transmission energy allocated to
the data, we may write

$\displaystyle P_{d}T_{d}=\nu TP,\ \ \quad P_{t}T_{t}=(1-\nu)T{P},\
\quad\nu\in(0,1).$ (5)

This system model is illustrated in Fig. 2.

The received signal for the _data_ transmission phase can be modeled as

${\bf r}=\sqrt{\frac{P_{d}}{n}}{\bf H}{\bf s}_{0}+{\bf v}\in\mathbb{R}^{m},$
(6)

where the following model-assumptions hold, except if otherwise stated:

  * •

${\bf H}\in\mathbb{R}^{m\times n}$ is the MIMO channel matrix which has i.i.d.
standard Gaussian entries (i.e., $\mathcal{N}(0,1)$).

  * •

${\bf v}\in\mathbb{R}^{m}$ is the noise vector with i.i.d. standard Gaussian
entries, i.e., ${\bf v}\sim\mathcal{N}(\mathbf{0},{\bf I}_{m})$.

  * •

The unknown signal vector ${\bf s}_{0}$ is assumed to be $k$-sparse, i.e.,
only $k$ of its entries are sampled i.i.d. from a distribution $p_{s_{0}}$
which has zero-mean and unit-variance (i.e.,
$\mathbb{E}[S_{0}^{2}]=\sigma_{s}^{2}=1$), and the remaining entries are
zeros.

Figure 2: A training-assisted massive MIMO system.

###  II-B Estimation of the Channel Matrix

As indicated in the preceding subsection, a training phase in which the
transmitter sends $T_{{t}}$ pilot symbols is used to estimate the channel
matrix ${\bf H}$, which is unknown to the receiver.111In communications
literature, this is known as the “imperfect CSI” case. In this _training
phase_ , the received signal is represented as

${\bf R}_{t}=\sqrt{\frac{{P}_{t}}{n}}{\bf H}{\bf S}_{t}+{\bf V}_{t},$ (7)

where ${\bf R}_{t}\in\mathbb{R}^{m\times T_{t}}$ is the received signal
matrix, ${\bf S}_{t}\in\mathbb{R}^{n\times T_{t}}$ is the matrix of
transmitted pilot symbols, and ${\bf V}_{t}\in\mathbb{R}^{m\times T_{t}}$ is a
zero-mean additive white Gaussian noise (AWGN) matrix with covariance
$\mathbb{E}[{\bf V}_{{t}}{\bf V}_{{t}}^{T}]=T_{{t}}{\bf I}_{m}$.

In this paper, we consider the linear minimum mean square error (LMMSE)
estimate of the channel matrix, which can be derived based on the knowledge of
${\bf R}_{t}$ from (7) as [34]

$\displaystyle\widehat{{\bf H}}$ $\displaystyle=\sqrt{\frac{n}{{P}_{t}}}{\bf
R}_{t}{\bf S}_{{t}}^{T}\left({\bf S}_{{t}}{\bf
S}_{{t}}^{T}+\frac{n}{{P}_{t}}{\bf I}_{n}\right)^{-1},$ $\displaystyle={\bf
H}+\hbox{\boldmath$\Omega$},$ (8)

where $\Omega$ is the channel estimation error matrix, which is uncorrelated
with $\widehat{{\bf H}}$, as per the orthogonality principle of the LMMSE
estimation [34, 33]. For i.i.d. MIMO channels, it has been proven that the
optimal ${\bf S}_{t}$ that minimizes the estimation MSE satisfies [33]

${\bf S}_{t}{\bf S}_{{t}}^{T}=T_{t}{\bf I}_{n}.$ (9)

For the above condition to hold, it is required that

$\displaystyle T_{t}\geq n.$ (10)

Moreover, under (9), it can be shown that the channel estimate $\widehat{{\bf
H}}$ has i.i.d. zero-mean Gaussian entries with variance
$\sigma_{{\widehat{H}}}^{2}=1-\sigma_{{\omega}}^{2}$ [33], where

$\sigma_{{\omega}}^{2}=\frac{1}{1+\frac{{P}_{t}}{n}T_{t}}$ (11)

is the variance of each element in $\Omega$. Furthermore, it can be shown that
the entries of $\Omega$ are i.i.d. $\mathcal{N}(0,\sigma_{\omega}^{2})$
distributed. Note that the training-phase energy ${P}_{t}T_{t}$ controls the
quality of the estimation as it appears from (11). In fact, as
${P}_{t}T_{t}\to\infty$, $\sigma_{\omega}^{2}\to 0$, and $\widehat{{\bf
H}}\to{\bf H}$, which represents the case of perfect CSI.

###  II-C Data Detection via the Box-LASSO

In this work, the problem in (1) is referred to as the standard LASSO, and we
instead introduce the following revised formulation of it termed the Box-
LASSO:

$\widehat{{\bf s}}={\rm{arg}}\ \underset{{{\bf
s}\in\mathbb{B}^{n}}}{\operatorname{\min}}\bigg{\|}\sqrt{\frac{P_{d}}{n}}\widehat{{\bf
H}}{\bf s}-{\bf r}\bigg{\|}_{2}^{2}+\gamma P_{d}\|{\bf s}\|_{1},$ (12)
$\text{where},\mathbb{B}=[\ell,\mu],\ \text{and}\ \ell\leq 0,\mu\geq
0\in\mathbb{R}.$

When compared to (1), we use ${\bf A}=\sqrt{\frac{P_{d}}{n}}\widehat{{\bf H}}$
here. This is due to the fact that ${\bf H}$ is not perfectly known and we
only have its estimate $\widehat{{\bf H}}$ that was obtained by training. In
addition, note that the regularization parameter $\gamma$ is scaled by a
factor of $P_{d}$. This is made such that the two terms grow with the same
pace.

The only difference between (12) and (1), is that (12) now has a “box-
constraint”. However, as we will show later, in cases where the elements of
${\bf s}_{0}$ are bounded or approximately so, this minor modification ensures
a considerable gain in performance. Therefore, the Box-LASSO can be used to
recover simultaneously structured signals [35], for example, signals that are
both bounded and sparse. Such signals appear in various applications including
machine learning [4], wireless communications [36], image processing [2], and
so on. Although the Box-LASSO is not as well-known as the standard LASSO,
there are a few references where it has been applied [37, 38, 39]. Of
particular interest is the application of the Box-LASSO in spatially modulated
MIMO systems such as GSSK modulated signals which we will discuss in Section
IV.

###  II-D Technical Assumptions

In this work, we require the following technical assumptions to hold.

######  Assumption 1.

The analysis requires that the system dimensions ($m$, $n$ and $k$) grow
simultaneously large (i.e., $m,n,k\to\infty$) at fixed ratios:

$\frac{m}{n}\to\eta\in(0,\infty),$

and

$\frac{k}{n}\to\kappa\in(0,1).$

######  Assumption 2.

We assume that the normalized coherence interval, normalized number of pilot
symbols and normalized number data symbols are fixed and given as

$\frac{T}{n}\to\tau\in(1,\infty),$
$\frac{T_{{t}}}{n}\to\tau_{{t}}\in[1,\infty),$

and

$\frac{T_{{d}}}{n}\to\tau_{{d}},$

respectively.

Under these assumptions, the energy conservation in (4) becomes

$P_{t}\tau_{t}+P_{d}\tau_{d}={P}\ \tau,$ (13)

and the channel estimation error variance in (11) reads

$\sigma_{{\omega}}^{2}=\frac{1}{1+P_{t}\tau_{t}}.$ (14)

###  II-E Figures of Merit

We measure the performance of the Box-LASSO using following figures of merit:
Mean Square Error: A widely used figure of merit is the _estimation_ mean
square error (MSE), that measures the divergence of the estimate
$\widehat{{\bf s}}$ from the original signal ${\bf s}_{0}$. Formally, it is
defined as

$\displaystyle{\rm{MSE}}:=\frac{1}{n}\|\widehat{{\bf s}}-{\bf
s}_{0}\|_{2}^{2}.$ (15)

Support Recovery: In sparse recovery problems, a natural performance measure
that is employed in numerous applications is _support recovery_ , that can be
defined as determining whether an element of ${\bf s}_{0}$ is non-zero (i.e.,
on the support), or if it is zero (i.e., off the support). The decision, based
on the Box-LASSO solution $\widehat{{\bf s}}$, proceeds as follows: if
$|\widehat{s}_{j}|\geq\upzeta$, then, $\widehat{s}_{j}$ is on the support,
where $\upzeta>0$ is a user-defined hard threshold on the elements of
$\widehat{{\bf s}}$. Otherwise, $\widehat{s}_{j}$ is off the support.

Essentially, we have two measures: the probability of successful _on_ -support
recovery denoted by $\Uppsi_{\upzeta}^{\text{on}}(\widehat{{\bf s}})$, and the
probability of successful _off_ -support recovery, i.e.,
$\Uppsi_{\upzeta}^{\text{off}}(\widehat{{\bf s}})$. Formally, these quantities
are defined, respectively, as

$\displaystyle\Uppsi_{\upzeta}^{\text{on}}(\widehat{{\bf
s}}):=\frac{1}{k}\sum_{j\in\mathcal{T}({\bf
s}_{0})}\mathbbm{1}_{\\{|\widehat{s}_{j}|\geq\upzeta\\}},$ (16a)
$\displaystyle\Uppsi_{\upzeta}^{\text{off}}(\widehat{{\bf
s}}):=\frac{1}{n-k}\sum_{j\notin\mathcal{T}({\bf
s}_{0})}\mathbbm{1}_{\\{|\widehat{s}_{j}|\leq\upzeta\\}},$ (16b)

where $\mathbbm{1}_{\\{.\\}}$ is the indicator function, and $\mathcal{T}({\bf
s}_{0})$ is the support of ${\bf s}_{0}$, i.e., the set of all non-zero
elements of ${\bf s}_{0}$.

##  III Main Results

###  III-A Performance Characterization

In this subsection, we summarize the main theoretical results regarding the
asymptotic performance of the Box-LASSO. The first theorem gives the sharp
performance analysis of the MSE of the Box-LASSO.

######  Theorem 1.

Let $\widehat{{\bf s}}$ be a minimizer of the Box-LASSO problem in (12), where
${\bf H},{\bf v}$ and ${\bf s}_{0}$ satisfy the model assumptions in Section
II-A. In addition, assume that the optimization problem:
$\max_{\beta>0}\min_{\lambda>0}\ \mathcal{G}(\beta,\lambda)$ has a unique
optimizer $(\beta_{\star},\lambda_{\star})$, where

$\displaystyle\mathcal{G}(\beta,\lambda)$
$\displaystyle:=\frac{\beta\sqrt{\eta}}{2\lambda}+\frac{\beta\lambda\sqrt{\eta}}{2}\left(1+\kappa{P_{d}\sigma_{\omega}^{2}}\right)-\frac{\beta^{2}}{4}-\frac{\beta}{2\lambda\sqrt{\eta}}$
$\displaystyle+\beta\lambda\sqrt{\eta}P_{d}\sigma_{\widehat{H}}^{2}\
\mathbb{E}\left[\mathcal{J}\left(S_{0}+\frac{Z}{\lambda\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}};\frac{\gamma}{\beta\lambda\sqrt{\eta}\sigma_{\widehat{H}}^{2}},\ell,\mu\right)\right],$
(17)

and the expectation is taken over $S_{0}\sim p_{s_{0}}$ and
$Z\sim\mathcal{N}(0,1)$.

Then, under Assumption 1 and Assumption 2, and for a fixed $\gamma>0$, it
holds:

$\displaystyle\underset{n\to\infty}{\rm{plim}}\
{\rm{MSE}}=\frac{1}{P_{d}\sigma_{\widehat{H}}^{2}}\left(\frac{1}{\lambda_{\star}^{2}}-1-\kappa
P_{d}\sigma_{\omega}^{2}\right).$ (18)

###### Proof.

The proof is relegated to the appendix. ∎

######  Remark 1 (Finding optimal scalars).

The scalars $\beta_{\star}$ and $\lambda_{\star}$ can be numerically evaluated
by solving the first-order optimality conditions, i.e.,

$\displaystyle\nabla_{(\beta,\lambda)}\mathcal{G}(\beta,\lambda)=\bf{0}.$ (19)

######  Remark 2 (Roles of $\lambda_{\star}$ and $\beta_{\star}$).

From Theorem 1 above, we can see that the optimal scalar $\lambda_{\star}$ is
related to the asymptotic MSE. However, the role of $\beta_{\star}$ is not
evident from the above theorem.

Based on our derivations, it turns out that $\beta_{\star}$ is related to
another performance metric called the _residual_ [40] between ${\bf r}$ and
the estimate $\widehat{{\bf r}}$ which is also called the prediction error.
Formally, it is defined as

$\displaystyle{\rm{\mathcal{R}}}:=\frac{1}{n}\bigg{\|}\underbrace{\sqrt{\frac{P_{d}}{n}}\widehat{{\bf
H}}\widehat{{\bf s}}}_{:=\widehat{{\bf r}}}-{\bf r}\bigg{\|}_{2}^{2}.$ (20)

Then, as we will prove in the Appendix, under the same assumptions in Theorem
1, it holds

$\displaystyle\underset{n\to\infty}{\rm{plim}}\
\mathcal{R}=\frac{1}{4}\beta_{\star}^{2}.$ (21)

The above expression clearly shows the role $\beta_{\star}$ in predicting the
asymptotic value of the residual, and Fig. 3(b) illustrates its high accuracy.

The residual metric is less relevant to the MIMO application considered in
this paper, but it is of great interest in other practical data science
problems, where you only have an access to the measurement vector ${\bf r}$,
and not the true vector ${\bf s}_{0}$.

######  Remark 3 (Optimal MSE regularizer).

Theorem 1 can be used to find the optimal regularizer
$\gamma_{\star}^{\mathrm{MSE}}$ that minimizes the MSE. See for example Fig.
3a. Particularly, $\gamma_{\star}^{\mathrm{MSE}}$ can be found as follows

$\displaystyle\gamma_{\star}^{\mathrm{MSE}}=$
$\displaystyle\operatorname*{arg\,min}_{\gamma>0}\frac{1}{\lambda_{\star}},$
$\displaystyle=$
$\displaystyle\operatorname*{arg\,max}_{\gamma>0}{\lambda_{\star}}.$ (22)

The above expression can be easily proven, by noting that $\gamma$ appears in
the MSE expression of (18) only implicitly through $\lambda_{\star}$.

In the next theorem, we sharply characterizes the support recovery metrics
introduced earlier in (16).

######  Theorem 2.

Under the same settings of Theorem 1, for any fixed $\upzeta>0$, and under
Assumption 1 and Assumption 2, the on-support and off-support probabilities
converge as

$\underset{n\to\infty}{\rm{plim}}\ \Uppsi_{\upzeta}^{\rm{on}}(\widehat{{\bf
s}})=\mathbb{P}\biggl{[}\biggl{|}\mathcal{H}\biggl{(}S_{0}+\frac{Z}{\lambda_{\star}\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}};\frac{\gamma}{\beta_{\star}\lambda_{\star}\sqrt{\eta}\sigma_{\widehat{H}}^{2}},\ell,\mu\biggr{)}\biggr{|}\geq\upzeta\biggr{]},$
(23)

and

$\underset{n\to\infty}{\rm{plim}}\ \Uppsi_{\upzeta}^{\rm{off}}(\widehat{{\bf
s}})=\mathbb{P}\biggl{[}\biggl{|}\mathcal{H}\biggl{(}\frac{Z}{\lambda_{\star}\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}};\frac{\gamma}{\beta_{\star}\lambda_{\star}\sqrt{\eta}\sigma_{\widehat{H}}^{2}},\ell,\mu\biggr{)}\biggr{|}\leq\upzeta\biggr{]},$
(24)

respectively.

###### Proof.

The proof of Theorem 2 to a great extent follows the proof of Theorem 1, but
is omitted for briefness of the presentation. See [27, 41] for similar proof
techniques. ∎

######  Remark 4 (Regularizer’s strength).

It is easy to see from Theorem 2 that when $\gamma$ grows larger,
$\Uppsi_{\upzeta}^{\rm{off}}(\widehat{{\bf s}})$ converges to $1$ whereas
$\Uppsi_{\upzeta}^{\rm{on}}(\widehat{{\bf s}})$ converges to $0$. When
$\gamma$ approaches $0$, opposite behavior is exhibited. This is expected
since large values of $\gamma$ emphasize the $\ell_{1}$-norm term, resulting
in a sparser solution. This is illustrated clearly in Fig. 4.

######  Remark 5 (Optimal regularizer).

A sensible measure of performance to trade-off between the on-support and off-
support recovery probability is

$\displaystyle\Uppsi_{\upzeta}(\widehat{{\bf s}}):=\theta\
\Uppsi_{\upzeta}^{\rm{on}}(\widehat{{\bf s}})+(1-\theta)\
\Uppsi_{\upzeta}^{\rm{off}}(\widehat{{\bf s}}),\ \text{for}\ \theta\in[0,1].$
(25)

The behavior of this metric as a function of $\gamma$ is sharply characterized
by Theorem 2. As a result, Theorem 2 can also be utilized to determine the
optimal value of $\gamma$ which optimizes $\Uppsi_{\upzeta}(\widehat{{\bf
s}})$.

###  III-B Numerical Illustrations

For the sake of illustration, we will simply look at the instance when ${\bf
s}_{0}$ has elements that are only allowed to take one of two possible values:
$0$, or $E>0$. For a normalized sparsity level $\kappa\in(0,1)$, such prior
knowledge is typically modeled using a sparse-Bernoulli distribution on the
elements of ${\bf s}_{0}$, i.e.,
$S_{0}\sim(1-\kappa)\delta_{0}+\kappa\delta_{E}$. This model is frequently
seen in MIMO communication systems using generalized space-shift keying (GSSK)
modulation [30]; we go over the role of the Box-LASSO in such systems in
Section IV. In this case, setting $\ell=0$, and $\mu=E$ as the box-constraint
values is a natural choice.

Therefore, in our numerical simulations, we assume that ${\bf s}_{0}$ has
elements that are sampled i.i.d. from a sparse-Bernoulli distribution with
$\mathbb{P}[S_{0}=0]=0.8$, $\mathbb{P}[S_{0}=1]=0.2$ (i.e., $\kappa=0.2$) and
$E=1$; to satisfy the unit-variance assumption.

Fig. 3 shows the close match between Theorem 1 asymptotic prediction of the
MSE and residual of the Box-LASSO and the Monte Carlo (MC) simulations. For
the simulations, we used $\eta=1.5,n=100$, $T=500,T_{t}=n,\nu=0.5,$ and $P=15\
{\rm{dB}}$. These results are averaged over $100$ independent realizations of
${\bf s}_{0},{\bf H}$ and ${\bf v}$. From this figure, it can be seen that the
Box-LASSO outperforms the standard LASSO. We can also see in Fig. 3(a) that as
the regularization parameter $\gamma$ is varied, a pronounced minimum for a
certain $\gamma>0$ is observed.

(a) MSE Performance.

(b) Residual Performance.

Figure 3: MSE/Residual performance of the Box-LASSO and the standard LASSO vs.
the regularizer. The analytical prediction is based on Theorem 1 with
$p_{s_{0}}=(1-\kappa)\ \delta_{0}+\kappa\ \delta_{E}$. We used
$\kappa=0.2,\eta=1.5,n=128$, $T=500,T_{t}=n,\nu=0.5,E=1,$ and $P=15\
{\rm{dB}}$.

The analytical expressions of Theorem 2 for the support recovery probabilities
are compared to the MC simulations and displayed in Fig. 4 with the same
simulation settings as in the preceding figure. Once again, this figure
demonstrates the great accuracy of the provided theoretical expressions.

(a) On-Support Probability.

(b) Off-Support Probability.

Figure 4: Probability of successful support recovery of the Box-LASSO. The
analytical prediction is based on Theorem 2 with $p_{s_{0}}=(1-\kappa)\
\delta_{0}+\kappa\ \delta_{E}$. We used $\kappa=0.2,\eta=1.5,n=128$,
$T=500,T_{t}=n,\nu=0.5,E=1,$ and $P=15\ {\rm{dB}}$.

######  Remark 6.

For the previously mentioned sparse-Bernoulli distributed signal, the support
recovery probabilities in (23), and (24) simplify to

$\displaystyle\underset{{n\to\infty}}{\rm{plim}}\
\Uppsi_{\upzeta}^{\text{on}}(\widehat{{\bf s}})=\ $ $\displaystyle
Q\left((\upzeta-E)\lambda_{\star}\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}+\frac{\gamma\sqrt{P_{d}}}{\beta_{\star}\sigma_{\widehat{H}}}\right),$
(26)

and

$\underset{{n\to\infty}}{\rm{plim}}\
\Uppsi_{\upzeta}^{\text{off}}(\widehat{{\bf s}})=\
1-Q\left(\upzeta\lambda_{\star}\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}+\frac{\gamma\sqrt{P_{d}}}{\beta_{\star}\sigma_{\widehat{H}}}\right).$
(27)

These expressions have been used in the numerical simulations above with $E=1$
therein.

######  Remark 7 (Unbounded elements).

In instances when the elements of the original signal are _unbounded_ but take
values in a specific range with high probability, the Box-LASSO can be a
valuable decoder as well. To demonstrate this, let us take the example in
which the elements of ${\bf s}_{0}$ are i.i.d. sparse-Gaussian distributed,
i.e., $S_{0}\sim(1-\kappa)\delta_{0}+\kappa\ \mathcal{N}(0,\sigma_{s}^{2})$.
Fig. 5 illustrates a case in which the Box-LASSO outperforms the standard
LASSO. We used $\ell=-\sigma_{s}$ and $\mu=\sigma_{s}$ as the box-constraints
in this example.

Figure 5: MSE of the Box-LASSO and standard LASSO. The analytical prediction
is based on Theorem 1 with a sparse-Gaussian ${\bf s}_{0}$ signal. We used
$\kappa=0.1,\eta=1.2,n=400$, $T=1000,T_{t}=456,\nu=0.5,\sigma_{s}^{2}=1,$ and
$P=10\ {\rm{dB}}$.

######  Remark 8 (Universality).

Even without the Gaussianity assumption on the elements of the channel matrix
${\bf H}$, our extensive simulations strongly indicate that the statements of
Theorem 1 and Theorem 2 are still valid. This is especially helpful in MIMO
applications where the channel matrix elements can be represented beyond the
typical fading model (Gaussian), such as in the more involved fading models,
e.g., Weibull and Nakagami [42]. The same asymptotic statements appear to hold
regardless of whether the distribution of ${\bf H}$ is Gaussian, Binary, or
Laplacian (as illustrated in Fig. 6). Rigorous proofs, known as universality
results, in [43, 44, 45] support such a claim.

Figure 6: MSE of the Box-LASSO for different measurement matrices. The
theoretical prediction is based on Theorem 1 with a sparse-Bernoulli ${\bf
s}_{0}$ signal. We used $\kappa=0.1,\eta=0.8,n=200$,
$T=700,T_{t}=256,\nu=0.6,P=5\ {\mathrm{dB}}$.

######  Remark 9 (Efficient implementation).

It is worth noting that the Box-LASSO can be efficiently implemented via
quadratic programming (QP) as in [37], where it was used to implement an
efficient algorithm of the constrained LASSO. We applied the same algorithm to
the Box-LASSO decoder in the above numerical simulations utilizing MATLAB
built-in function $\mathsf{quadprog}$.

##  IV GSSK Modulated Massive MIMO Systems

Traditional linear modulation schemes become expensive in modern massive MIMO
systems. This is due to the large required number of radio frequency (RF)
chains needed for the massive number of antennas. One promising modulation
technique is the so-called spatial modulation (SM), where only the antenna’s
location relays information and only a small subset of the antennas is active
at each time [46]. This significantly reduces the system complexity since the
required number of RF chains is less now. This modulation scheme saves energy;
since using fewer RF chains, we have less power dissipation through the power
amplifiers, etc. Modern SM techniques include generalized space-shift keying
(GSSK) modulation [30, 36], and the generalized spatial modulation (GSM) [31].

###  IV-A Box-LASSO for Detecting GSSK Modulated Signals

As mentioned above, recently developed modulation techniques such as GSM and
GSSK modulation, generate signals that are essentially sparse and have
elements belonging to a finite alphabet (i.e., bounded). Hence, when such
modulations are employed, we will use the Box-LASSO as low-complexity decoding
method, instead of the previously proposed standard LASSO decoders [47, 48].

For the sake of simplicity, we will focus on GSSK. Considering a modulation
setup, where a fixed-size set $\mathcal{I}\subset\\{1,\cdots,n\\}$ of active
antennas transmit $s_{0,j}=1,j\in\mathcal{I}$ at each transmission, while the
remaining antennas stay inactive, i.e., $s_{0,j}=0,j\notin\mathcal{I}$. Hence,
only active antennas positions convey information.

To decode ${\bf s}_{0}$, firstly, get a solution $\widehat{{\bf s}}$ of the
Box-LASSO in (12), with $\ell=0$ and $\mu=1$. Then, map $\widehat{{\bf s}}$
into a vector ${{\bf s}}^{\star}$ which has elements either $0$ or $1$. In the
GSSK context, this typically entails sorting $\widehat{{\bf s}}$ and setting
its largest $k$ entries to $1$ and the remaining to $0$ [47].

In order to evaluate the performance of the Box-LASSO in this application, we
will use the so-called _element error rate_ (EER) [20] as a performance
measure. Similar to the support recovery metric, we first hard-thresholding
$\widehat{{\bf s}}$ by a constant $\upzeta\in(0,1)$, in order to map each of
its element to either $0$ or $1$. Then, the EER can be defined as

${\rm{EER}}_{\upzeta}:=\frac{1}{|\mathcal{I}|}\sum_{j\in\mathcal{I}}\mathbbm{1}_{\\{|\widehat{s}_{j}|\leq\upzeta\\}}+\frac{1}{n-|\mathcal{I}|}\sum_{j\notin\mathcal{I}}\mathbbm{1}_{\\{|\widehat{s}_{j}|\geq\upzeta\\}}.$
(28)

The next proposition gives a sharp asymptotic prediction of the EER in the
GSSK modulated MIMO systems application.

######  Proposition 1.

Let ${\rm{EER}}_{\upzeta}$ be as defined in the above equation with
$|\mathcal{I}|=\kappa n$, for $\kappa\in(0,1)$. Also, let $\beta_{\star}$, and
$\lambda_{\star}$ be solutions to the minimax optimization in (1), with
$p_{s_{0}}=(1-\kappa)\delta_{0}+\kappa\delta_{1}$ therein. Then, for a fixed
$\upzeta\in(0,1)$, it holds

$\displaystyle\underset{{n\to\infty}}{\rm{plim}}\ {\rm{EER}}_{\upzeta}=\ $
$\displaystyle Q\left((1-\upzeta)\lambda_{\star}\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}-\frac{\gamma\sqrt{P_{d}}}{\beta_{\star}\sigma_{\widehat{H}}}\right)+Q\left(\upzeta\lambda_{\star}\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}+\frac{\gamma\sqrt{P_{d}}}{\beta_{\star}\sigma_{\widehat{H}}}\right).$
(29)

###### Proof.

The proof follows from Theorem 2 by observing that the EER in (28) may be
rewritten as

${\rm{EER}}_{\upzeta}=2-\Uppsi_{\upzeta}^{\text{on}}(\widehat{{\bf
s}})-\Uppsi_{\upzeta}^{\text{off}}(\widehat{{\bf s}}),$ (30)

with the on/off support probability expressions of the sparse-Bernoulli
distribution derived earlier in (26) and (27) for $E=1$. ∎

On the receiving side of some MIMO systems, there may not be sufficient number
of antennas. This is owing to the receiver’s small size, limited cost or
weight, and low power consumption. Such MIMO systems, where the number of
receive antennas $m$ is less than that of the transmitters $n$ (i.e., $m<n$),
are known as _overloaded_ (or underdetermined) MIMO systems [49]. Fig. 7
illustrates the accuracy of the derived EER expression for a case of an
overloaded system, with $\eta=0.8$. This figure shows that the Box-LASSO
outperforms the standard one in the EER sense as well.

Figure 7: Element Error Rate of the Box-LASSO and the standard LASSO for GSSK
signal recovery. Here, we used
$T=500,m=120,n=T_{t}=150,k=15,\upzeta=0.1,\nu=0.5$, and $P=\mathrm{10\ dB}$.
The data are averaged over $100$ independent iterations.

###  IV-B Power Allocation and Training Duration Optimization

The overall massive MIMO system’s performance can be improved by optimizing
the power allocation between the transmitted pilot and data symbols as oppose
to equal power distribution [33]. Power optimization problems in MIMO systems
have been proposed based on different performance metrics. In [50, 51], the
authors derived a power allocation scheme based on minimizing the MSE, while
minimizing the the bit error rate (BER) and SER was considered in [52, 25,
53]. Training optimization based on maximizing the channel capacity was
addressed in [54, 55, 33]. In addition, the authors in [56, 57, 58] provided
power allocation strategies based on maximizing the sum rates. The above list
of references is not comprehensive, since power allocation optimization
research has very rich literature. However, we cited the most related works to
this paper.

####  IV-B1 Optimal Power Allocation

In this subsection, we will use the previously derived asymptotic results for
the MSE and EER to find an optimal power allocation scheme, in a GSSK
modulated system, that minimizes these error measures.

For fixed $\tau_{t}$ and $\tau$, the power allocation optimization problem can
be caste as

$\displaystyle\min_{P_{t},P_{d}}\rm{MSE}$ $\displaystyle\text{subject to:}\
P_{t}\tau_{t}+P_{d}(\tau-\tau_{t})={P}\tau,$ $\displaystyle
P_{t}=(1-\nu){P}\tau,\ P_{d}=\nu{P}\tau,\ \ 0<\nu<1.$

It can be shown that the above optimization problem boils down to only
optimizing the data energy ratio $\nu$. The results are summarized next.

For fixed $\tau_{t}$ and $\tau$, and using Box-LASSO with an optimal
regularizer $\gamma_{\star}^{\mathrm{MSE}}$ as in (3), the optimal power
allocation that minimizes the MSE is given by

$\displaystyle\nu_{\star}^{\rm{MSE}}=\operatorname*{arg\,min}_{0<\nu<1}\rm{MSE},$
(31)

where $\rm{MSE}$ is the asymptotic MSE expression in (18).

Similarly, when using the EER as a performance metric, we have

$\displaystyle\nu_{\star}^{\rm{EER}}=\operatorname*{arg\,min}_{0<\nu<1}\rm{EER}_{\upzeta}(\gamma_{\star}),$
(32)

where $\rm{EER}_{\upzeta}(\gamma_{\star})$ is the asymptotic EER expression in
(29) with the optimal regularizer $\gamma_{\star}$ that minimizes the EER. For
the Box-LASSO decoder, finding $\nu_{\star}^{\rm{MSE}}$ or
$\nu_{\star}^{\rm{EER}}$ in closed-form expressions seems to be a difficult
task, but by using a bisection method we can numerically find the optimal
power allocation scheme.

In Fig. 8, we plotted the MSE and EER of the Box-LASSO and standard LASSO as
functions of the data energy ratio $\nu$. This figure shows that optimizing
the MSE and EER are equivalent with $\nu_{\star}\approx 0.5373$. Furthermore,
it shows that the optimal power allocation is nothing but the well-known
scheme $\bar{\nu}_{\star}$ which was shown in [25, 33] to maximize the
effective SNR, where $\bar{\nu}_{\star}$ is given as ([25, eq. (35)]):

$\bar{\nu}_{\star}=\begin{cases}\vartheta-\sqrt{\vartheta(\vartheta-1)}&\text{if
$\tau_{d}>1$,}\\\ \frac{1}{2}&\text{if $\tau_{d}=1$,}\\\
\vartheta+\sqrt{\vartheta(\vartheta-1)}&\text{if $\tau_{d}<1$,}\end{cases}$
(33)

where

$\displaystyle\vartheta=\frac{1+P\cdot\tau}{P\cdot\tau(1-\frac{1}{\tau_{d}})}.$
(34)

This result is significant, since it showed again that the optimal power
allocation scheme is nothing but the celebrated one that maximizes the
effective SNR of the MIMO system, i.e., $\bar{\nu}_{\star}$. The power
allocation actually does not depend on the type of the modulation
constellation used, the used detector, or the other problem parameters such as
$\eta$ and $\kappa$.222Provided that we use the LMMSE estimator for the
channel estimation. For example, in [25], the same power allocation scheme was
obtained for a massive MIMO system with $M$-PAM signals and a Box-regularized
least squares (Box-RLS) detector, while this work employs the Box-LASSO
decoder for GSSK signal recovery.

(a) MSE vs. data energy ratio $\nu$.

(b) EER vs. data energy ratio $\nu$.

Figure 8: This figure shows the MSE/EER of Box-LASSO/LASSO as functions of
$\nu$. We used $T=1000,n=400,T_{t}=456,P=12\ \mathrm{dB},\eta=1.5,\kappa=0.1,$
and $\upzeta=0.01$.

####  IV-B2 Optimal Training Duration

In order to optimize the training duration, we introduce the following
performance metric, which is called the goodput. The goodput is calculated by
dividing the amount of useful transmitted data by the time it takes to send it
successfully [59, 25]. Formally, it can be defined as

$\displaystyle{\rm{Goodput}}(\tau_{t},\tau)=\bigg{(}1-\frac{\tau_{t}}{\tau}\bigg{)}(1-\rm{EER}).$
(35)

The optimal value $T_{t}^{\star}$ that maximizes the goodput is determined in
the following Corollary. For a fixed power allocation, the goal is to identify
the optimal number of training symbols out of the total coherence interval
symbols. From (10), we must have $T_{t}^{\star}\geq n$ (or,
$\tau_{t}^{\star}\geq 1$), and obviously, $T_{t}^{\star}<T$ (or,
$\tau_{t}^{\star}<\tau$). Therefore, $\tau_{t}^{\star}$ is a solution to the
maximization problem:

$\displaystyle\tau_{t}^{\star}=\mathrm{arg}\max_{1\leq\tau_{t}<\tau}\overline{\rm{Goodput}}(\tau_{t},\tau),$
(36)

where

$\displaystyle\overline{\rm{Goodput}}(\tau_{t},\tau):=\big{(}1-\frac{\tau_{t}}{\tau}\big{)}[1-{\rm{plim}}_{n\to\infty}\
\rm{EER}_{\upzeta}(\gamma_{\star})]$ (37)

is the asymptotic value of the goodput.

######  Corollary 1 (Optimal Training Duration).

Under imperfect CSI, the optimal training duration that maximizes the goodput
in (36) is given by:

$\displaystyle\tau_{t}^{\star}=1\ (\text{or}\ T_{t}^{\star}=n),$ (38)

for all ${P}$ and $\tau$ (or $T$).

###### Proof.

This result can be proven in a similar manner to [25], details are thus
omitted. ∎

Fig. 9 shows the goodput performance of Box-LASSO simulated versus the
training duration $T_{t}$ which confirms the result of Corollary 1. It shows
that at $T_{t}=n=200$, the goodput is maximized.

Figure 9: Goodput performance vs. $T_{t}$. We used $T=1000,P=12\
\mathrm{dB},n=200,\nu=0.5,\eta=1.5,\kappa=0.1,$ and $\upzeta=0.01$.

##  V Conclusion and Future Work

In this work, we derived sharp asymptotic characterizations of the mean square
error, probability of support recovery and element error rate of the Box-LASSO
under the presence of uncertainties in the channel matrix in the form of
channel estimation errors. The analytical tool used in the analysis is the
recently developed CGMT framework. The derived expressions can be used to
optimally tune the involved hyper-parameters of the algorithm such as the
regularizer. Then, we used the Box-LASSO as a low complexity decoder in an
application of massive MIMO detection using GSSK modulation, and optimize the
power allocation between training and data symbols to minimize the MSE or EER
of the system. Furthermore, we derived the optimal training duration that
maximizes the system’s goodput. Numerical simulations show very close
agreement to the derived theoretical predictions. Moreover, we showed that the
Box-LASSO outperforms standard one in all of the considered performance
metrics.

Finally, we highlight that the generalized spacial modulation (GSM) is more
involved than the considered GSSK modulation since it uses the positions of
active antenna in addition to a constellation symbol (e.g., $M$-QAM, $M$-PSK,
etc.) to encode information [31]. However, we focused in this paper on GSSK
systems since the analysis framework, namely the CGMT, requires real-valued
quantities (signals and channels). An interesting possible future work is to
extend the results of this paper to systems involving complex-valued data such
as GSM and investigate their power allocation optimization. Moreover, this
paper assumes Gaussian channels matrices with i.i.d. entries, but in numerous
wireless communication applications, there are usually spatial correlations
between the antennas. Therefore, another possible extension is to study
correlated massive MIMO systems, where the matrix entries are no longer i.i.d.

## Acknowledgments

This research has been funded by the Scientific Research Deanship at the
University of Ha’il–Saudi Arabia through project number BA-2136.

##  Appendix A Sketch of the Proof

In this appendix, we derive the asymptotic analysis of the considered Box-
LASSO problem’s performance metrics. Our analysis is based on the CGMT, which
is discussed more below.

###  A-A Technical Tool: CGMT

Firstly, we summarize the CGMT framework [5] before the proof of our main
results. For a comprehensive list of technical requirements, please see [5,
27]. Consider the following two optimization problems, which we call the
Primal Optimization (PO) and Auxiliary Optimization (AO) problems,
respectively.

$\displaystyle\Upphi({\bf C}):=\underset{{\bf x}\in\mathcal{S}_{{\bf
x}}}{\operatorname{\min}}\ \underset{{\bf y}\in\mathcal{S}_{{\bf
y}}}{\operatorname{\max}}\ {\bf y}^{T}{\bf C}{\bf x}+\xi({\bf x},{\bf y}),$
(39a) $\displaystyle\upphi({\bf g}_{1},{\bf g}_{2}):=\underset{{\bf
x}\in\mathcal{S}_{{\bf x}}}{\operatorname{\min}}\ \underset{{\bf
y}\in\mathcal{S}_{{\bf y}}}{\operatorname{\max}}\ \|{\bf x}\|{\bf
g}_{1}^{T}{\bf y}+\|{\bf y}\|{\bf g}_{2}^{T}{\bf x}+\xi({\bf x},{\bf y}),$
(39b)

where ${\bf C}\in\mathbb{R}^{\tilde{m}\times\tilde{n}},{\bf
g}_{1}\in\mathbb{R}^{\tilde{m}},{\bf
g}_{2}\in\mathbb{R}^{\tilde{n}},\mathcal{S}_{\bf
x}\subset\mathbb{R}^{\tilde{n}},\mathcal{S}_{\bf
y}\subset\mathbb{R}^{\tilde{m}}$ and
$\xi:\mathbb{R}^{\tilde{n}}\times\mathbb{R}^{\tilde{m}}\mapsto\mathbb{R}$.
Moreover, the function $\xi$ is assumed to be independent of the matrix ${\bf
C}$. Denote by ${\bf x}_{\Upphi}:={\bf x}_{\Upphi}({\bf C})$, and ${\bf
x}_{\upphi}:={\bf x}_{\upphi}({\bf g}_{1},{\bf g}_{2})$ any optimal minimizers
of (39a) and (39b), respectively. Further let $\mathcal{S}_{\bf x},$ and
$\mathcal{S}_{\bf y}$ be convex and compact sets, $\xi({\bf x},{\bf y})$ is
convex-concave continuous on $\mathcal{S}_{\bf x}\times\mathcal{S}_{\bf y}$,
and let ${\bf C},{\bf g}_{1}$ and ${\bf g}_{2}$ all have i.i.d. standard
normal entries.

The PO-AO equivalence is formally stated in the next theorem, the proof of
which can be found in [5].

######  Theorem 3 (CGMT [5]).

Under the above assumptions, the CGMT [5, Theorem 3] shows that the following
statements are true:
(a) For all $c\in\mathbb{R}$ and $t>0$, it holds

$\displaystyle\mathbb{P}\left[\left|\Upphi({\bf C})-c\right|>t\right]\leq
2\mathbb{P}\left[\left|\upphi({\bf g}_{1},{\bf g}_{2})-c\right|>t\right].$
(40)

In words, concentration of the optimal cost of the AO problem around $c$
implies concentration of the optimal cost of the corresponding PO problem
around the same value $c$. According the CGMT, this will finally imply that
the _original_ optimal cost (e.g., the Box-LASSO of (12) in this paper) will
concentrate around $c$ as well. Please see Fig. 10 for a numerical
illustration.

(b) Let $\mathcal{S}$ be any arbitrary open subset of $\mathcal{S}_{\bf x}$,
and $\mathcal{S}^{c}=\mathcal{S}_{\bf x}\setminus\mathcal{S}$. Denote
$\upphi_{\mathcal{S}^{c}}({\bf g}_{1},{\bf g}_{2})$ the optimal cost of the
optimization in (39b), when the minimization over ${\bf x}$ is constrained
over ${\bf x}\in\mathcal{S}^{c}$. Consider the regime
$\tilde{m},\tilde{n}\to\infty$ such that $\frac{\tilde{m}}{\tilde{n}}\to\eta$.
To keep notation short, this regime is denoted by $\tilde{n}\to\infty$.
Suppose that there exist constants $\overline{\phi}$ and
$\overline{\phi}_{\mathcal{S}^{c}}$ such that
(i) $\overline{\phi}<\overline{\phi}_{\mathcal{S}^{c}}$, (ii) $\upphi({\bf
g}_{1},{\bf g}_{2})\overset{P}{\longrightarrow}\overline{\phi}$, and (iii)
$\upphi_{\mathcal{S}^{c}}({\bf g}_{1},{\bf
g}_{2})\overset{P}{\longrightarrow}\overline{\phi}_{\mathcal{S}^{c}}$.

Then, if in addition, $\lim_{\tilde{n}\rightarrow\infty}\mathbb{P}[{\bf
x}_{\upphi}\in\mathcal{S}]=1,$ it also holds that

$\displaystyle\lim_{\tilde{n}\rightarrow\infty}\mathbb{P}[{\bf
x}_{\Upphi}\in\mathcal{S}]=1.$ (41)

When the assumptions of Theorem 3 are met, the CGMT-based proof proceeds in
general as follows:
$\bullet$ Identifying the PO and the associated AO: This step involves
transforming the original optimization problem into the desired minimax PO
form, and then identify its corresponding AO problem.
$\bullet$ Simplifying the AO: In this step, the AO is reduced into a scalar
optimization problem.
$\bullet$ Probabilistic analysis of the AO: In this step, we prove that the AO
converges to a (deterministic) asymptotic optimization problem which involves
only scalar variables.
$\bullet$ Choice of $\mathcal{S}$: The set $\mathcal{S}$ should be selected
properly based on the measure of interest. For instance, if we want to analyze
the MSE or the EER, $\mathcal{S}$ will be the set in which the MSE or the EER
concentrates, respectively.

After introducing the CGMT framework, we prove Theorem 1 next. For clarity,
the steps of the proof are divided into different subsections.

###  A-B PO and AO Identification

To obtain the result of the main theorems using CGMT, we need first to rewrite
the Box-LASSO optimization problem (12) as a PO problem. For convenience, we
consider the error vector

$\displaystyle{\bf e}:={\bf s}-{\bf s}_{0},$ (42)

and the modified Box-set for all $j\in\\{1,2,\cdots,n\\}$:

$\mathbb{D}=\bigg{\\{}e_{j}\in\mathbb{R}:\ell-s_{0,j}\leq e_{j}\leq\mu-
s_{0,j}\bigg{\\}},$ (43)

then the problem in (12) can be reformulated as

$\displaystyle\widehat{{\bf e}}={\rm{arg}}\ \underset{{\bf
e}\in\mathbb{D}^{n}}{\operatorname{\min}}\ $
$\displaystyle\bigg{\|}\sqrt{\frac{P_{d}}{n}}\widehat{{\bf H}}{\bf
e}+\sqrt{\frac{P_{d}}{n}}\hbox{\boldmath$\Omega$}{\bf s}_{0}-{\bf
v}\bigg{\|}_{2}^{2}+\gamma P_{d}\|{\bf e}+{\bf s}_{0}\|_{1}.$ (44)

This minimization is not in the PO form as it is missing the max part. So to
fix this, let us express the _loss_ function using its Legendre-Fenchel
transformation333For any convex function $f$, we can write: $f({\bf
x})=\max_{{\bf y}}{\bf y}^{T}{\bf x}-f^{\star}({\bf y})$, where $f^{\star}$ is
the Fenchel (convex) conjugate of $f$.:

$\|{\bf x}\|_{2}^{2}=\max_{{\bf y}\in\mathbb{R}^{m}}{\bf y}^{T}{\bf
x}-\frac{1}{4}\|{\bf y}\|_{2}^{2}.$ (45)

Hence, the problem above is equivalent to the following

$\displaystyle\underset{{\bf e}\in\mathbb{D}^{n}}{\operatorname{\min}}\
\underset{{\bf y}\in\mathbb{R}^{m}}{\operatorname{\max}}$ $\displaystyle\
\sqrt{\frac{P_{d}}{n}}{\bf y}^{T}\widehat{{\bf H}}{\bf
e}+\sqrt{\frac{P_{d}}{n}}{\bf y}^{T}\hbox{\boldmath$\Omega$}{\bf s}_{0}-{\bf
y}^{T}{\bf v}-\frac{1}{4}\|{\bf y}\|_{2}^{2}+\gamma P_{d}\|{\bf e}+{\bf
s}_{0}\|_{1}.$ (46)

One technical requirement of the CGMT is the compactness of the feasibility
set over ${\bf y}$. This can be handled following the approach in [5, Appendix
A], by introducing a sufficiently large artificial constraint set

$\displaystyle\mathcal{S}_{\bf y}=\biggl{\\{}{\bf y}\in\mathbb{R}^{m}:\|{\bf
y}\|_{2}\leq C_{y}\biggr{\\}},$ (47)

for some sufficiently large constant (independent of $n$) $C_{y}>0$. This will
not asymptotically affect the optimization problem. Then, we obtain

$\displaystyle\underset{{\bf e}\in\mathbb{D}^{n}}{\operatorname{\min}}\
\underset{{\bf y}\in\mathcal{S}_{\bf y}}{\operatorname{\max}}$ $\displaystyle\
\sqrt{\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}}{\bf y}^{T}\widetilde{{\bf
H}}{\bf e}+\sqrt{\frac{P_{d}\sigma_{\omega}^{2}}{n}}{\bf
y}^{T}\widetilde{\hbox{\boldmath$\Omega$}}{\bf s}_{0}-{\bf y}^{T}{\bf
v}-\frac{1}{4}\|{\bf y}\|_{2}^{2}+\gamma P_{d}\|{\bf e}+{\bf s}_{0}\|_{1},$
(48)

where $\widetilde{{\bf H}}$ and $\widetilde{\hbox{\boldmath$\Omega$}}$ are
independent matrices with i.i.d. $\mathcal{N}(0,1)$ entries each. The above
problem is now in a PO form with

$\displaystyle\xi({\bf e},{\bf y})=\ $
$\displaystyle\sqrt{\frac{P_{d}\sigma_{\omega}^{2}}{n}}{\bf
y}^{T}\widetilde{\hbox{\boldmath$\Omega$}}{\bf s}_{0}-{\bf y}^{T}{\bf
v}-\frac{1}{4}\|{\bf y}\|_{2}^{2}+\gamma P_{d}\|{\bf e}+{\bf s}_{0}\|_{1}.$
(49)

Thus, its associated AO is given as

$\displaystyle\underset{{\bf e}\in\mathbb{D}^{n}}{\operatorname{\min}}\
\underset{{\bf y}\in\mathcal{S}_{\bf y}}{\operatorname{\max}}$ $\displaystyle\
\sqrt{\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}}\|{\bf e}\|_{2}{\bf g}^{T}{\bf
y}+\sqrt{\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}}\|{\bf y}\|_{2}{\bf
z}^{T}{\bf e}+\xi({\bf e},{\bf y}),$ (50)

where ${\bf g}\sim\mathcal{N}(\mathbf{0},{\bf I}_{m})$ and ${\bf
z}\sim\mathcal{N}(\mathbf{0},{\bf I}_{n})$ are independent random vectors.

###  A-C AO Simplification

In order to simplify the AO, we first let

$\widetilde{{\bf g}}:=\sqrt{\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}}\|{\bf
e}\|_{2}{\bf g}-{\bf
v}+\sqrt{\frac{P_{d}\sigma_{\omega}^{2}}{n}}\widetilde{\hbox{\boldmath$\Omega$}}{\bf
s}_{0}.$

It can be shown that $\widetilde{{\bf
g}}\sim\mathcal{N}(\mathbf{0},\hbox{\boldmath$\Sigma$}_{\widetilde{{\bf
g}}}),$ where

$\displaystyle\hbox{\boldmath$\Sigma$}_{\widetilde{{\bf
g}}}=\left(\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}\|{\bf
e}\|_{2}^{2}+1+\frac{P_{d}\sigma_{\omega}^{2}}{n}\|{\bf
s}_{0}\|_{2}^{2}\right){\bf I}_{m}.$ (51)

Then, the AO can be rewritten as

$\displaystyle\underset{{\bf e}\in\mathbb{D}^{n}}{\operatorname{\min}}\
\underset{{\bf y}\in\mathcal{S}_{\bf y}}{\operatorname{\max}}\ $
$\displaystyle\widetilde{\bf g}^{T}{\bf
y}+\sqrt{\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}}\|{\bf y}\|_{2}{\bf
z}^{T}{\bf e}-\frac{\|{\bf y}\|_{2}^{2}}{4}+\gamma P_{d}\|{\bf e}+{\bf
s}_{0}\|_{1}.$ (52)

Fixing the norm of ${\bf y}$ to $\alpha:=\|{\bf y}\|_{2}$, we can easily
optimize over its direction. This simplifies the AO to

$\displaystyle\underset{{\bf e}\in\mathbb{D}^{n}}{\operatorname{\min}}\
\max_{\alpha\geq 0}$ $\displaystyle\ \alpha\|\widetilde{\bf
g}\|_{2}+\sqrt{\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}}\ \alpha\ {\bf
z}^{T}{\bf e}-\frac{\alpha^{2}}{4}+\gamma P_{d}\|{\bf e}+{\bf s}_{0}\|_{1}.$
(53)

###  A-D Asymptotic Analysis of the AO

Next, we need to normalize the above optimization problem by $\frac{1}{n}$, to
have all of its terms of the same order, $\mathcal{O}(1)$, and also define

$\displaystyle\beta:=\frac{\alpha}{\sqrt{n}}.$ (54)

Then, after a change of the order of the $\min$-$\max$444[5] shows that
flipping the $\min$-$\max$ order is possible for large dimensions., we obtain:

$\displaystyle\max_{\beta\geq 0}\underset{{\bf
e}\in\mathbb{D}^{n}}{\operatorname{\min}}$ $\displaystyle\
\beta\sqrt{\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}\|{\bf
e}\|_{2}^{2}+1+\frac{P_{d}\sigma_{\omega}^{2}}{n}\|{\bf s}_{0}\|_{2}^{2}}\ \
\frac{\|{\bf g}\|_{2}}{\sqrt{n}}$
$\displaystyle+\sqrt{{P_{d}\sigma_{\widehat{H}}^{2}}}\beta\frac{1}{n}{\bf
z}^{T}{\bf e}-\frac{\beta^{2}}{4}+\frac{\gamma P_{d}}{n}\|{\bf e}+{\bf
s}_{0}\|_{1}.$ (55)

Note the abuse of notation for ${\bf g}$ to represent another standard normal
vector.

To have a separable optimization problem, we use the following variational
identity:

$\displaystyle\sqrt{x}=\min_{\lambda>0}\frac{1}{2\lambda}+\frac{\lambda
x}{2},\ \text{for}\ x>0,$ (56)

with optimum solution $\hat{\lambda}=\frac{1}{\sqrt{x}}$. Using this trick,
with

$\displaystyle x=\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}\|{\bf
e}\|_{2}^{2}+1+\frac{P_{d}\sigma_{\omega}^{2}}{n}\|{\bf s}_{0}\|_{2}^{2},$
(57)

the optimization in (A-D) becomes

$\displaystyle\max_{\beta\geq 0}\min_{\lambda>0}$ $\displaystyle\
\frac{\beta\|{\bf g}\|_{2}}{2\lambda\sqrt{n}}+\frac{\beta\lambda\|{\bf
g}\|_{2}}{2\sqrt{n}}\left(1+\frac{P_{d}\sigma_{\omega}^{2}}{n}\|{\bf
s}_{0}\|_{2}^{2}\right)-\frac{\beta^{2}}{4}$ $\displaystyle+\min_{{\bf
e}\in\mathbb{D}^{n}}\biggl{\\{}\frac{\beta\lambda\|{\bf
g}\|_{2}}{2\sqrt{n}}\frac{P_{d}\sigma_{\widehat{H}}^{2}}{n}\|{\bf
e}\|_{2}^{2}+\sqrt{{P_{d}\sigma_{\widehat{H}}^{2}}}\beta\frac{{\bf z}^{T}{\bf
e}}{n}+\frac{\gamma P_{d}}{n}\|{\bf e}+{\bf s}_{0}\|_{1}\biggr{\\}}.$ (58)

Using the weak law of large numbers (WLLN),

$\displaystyle\frac{\|{\bf
g}\|_{2}}{\sqrt{n}}\overset{P}{\longrightarrow}\sqrt{\eta},$ (59)

and

$\displaystyle\frac{1}{n}\|{\bf
s}_{0}\|_{2}^{2}\overset{P}{\longrightarrow}\kappa\sigma_{s}^{2}=\kappa.$ (60)

Next, using the above convergence results, and working with the original
optimization variable ${\bf s}$ instead of ${\bf e}$, we get

$\displaystyle\max_{\beta\geq 0}\min_{\lambda>0}$ $\displaystyle\
\frac{\beta\sqrt{\eta}}{2\lambda}+\frac{\beta\lambda\sqrt{\eta}}{2}\left(1+{P_{d}\sigma_{\omega}^{2}}\kappa\right)-\frac{\beta^{2}}{4}$
$\displaystyle+\frac{1}{n}\sum_{j=1}^{n}\min_{\ell\leq
s_{j}\leq\mu}\biggl{\\{}\frac{\beta\lambda\sqrt{\eta}}{2}{P_{d}\sigma_{\widehat{H}}^{2}}(s_{j}-s_{0,j})^{2}+\sqrt{{P_{d}\sigma_{\widehat{H}}^{2}}}\beta
z_{j}(s_{j}-s_{0,j})+{\gamma P_{d}}|s_{j}|\biggr{\\}}.$ (61)

After a completion of squares in $s_{j}$, and noting that $\frac{1}{n}{\bf
z}^{T}{\bf s}_{0}\overset{P}{\longrightarrow}0$, the above problem becomes

$\displaystyle\max_{\beta\geq 0}\min_{\lambda>0}$ $\displaystyle\
\frac{\beta\sqrt{\eta}}{2\lambda}+\frac{\beta\lambda\sqrt{\eta}}{2}\left(1+{P_{d}\sigma_{\omega}^{2}}\kappa\right)-\frac{\beta^{2}}{4}-\frac{1}{n}\sum_{j=1}^{n}\frac{\beta}{2\lambda\sqrt{\eta}}z_{j}^{2}$
$\displaystyle+\beta\lambda\sqrt{\eta}P_{d}\sigma_{\widehat{H}}^{2}\
\frac{1}{n}\sum_{j=1}^{n}\min_{\ell\leq
s_{j}\leq\mu}\Biggl{\\{}\frac{1}{2}\Biggl{(}s_{j}-\biggl{(}s_{0,j}+\frac{z_{j}}{\lambda\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}}\biggr{)}\Biggr{)}^{2}+\frac{\gamma}{\beta\lambda\sqrt{\eta}\sigma_{\widehat{H}}^{2}}|s_{j}|\Biggr{\\}}$
$\displaystyle=$ $\displaystyle\max_{\beta\geq
0}\min_{\lambda>0}G(\beta,\lambda,{\bf z},{\bf s}_{0}).$ (62)

The optimization over $s_{j}$ could be obtained in closed-form using the
saturated shrinkage function $\mathcal{H}(a;\gamma,l,u)$ which is defined in
(2). Also, let its optimal objective be $\mathcal{J}(a;\gamma,l,u)=\min_{l\leq
s\leq u}\frac{1}{2}(s-a)^{2}+\gamma|s|$ as defined in (3). Now, again, using
the WLLN, we have

$\displaystyle\frac{1}{n}\sum_{j=1}^{n}z_{j}^{2}\overset{P}{\longrightarrow}1,$
(63)

and for all $\beta>0$, and $\lambda>0$:

$\displaystyle\frac{1}{n}\sum_{j=1}^{n}\mathcal{J}\left(s_{0,j}+\frac{z_{j}}{\lambda\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}};\frac{\gamma}{\beta\lambda\sqrt{\eta}\sigma_{\widehat{H}}^{2}},\ell,\mu\right)\overset{P}{\longrightarrow}\mathbb{E}\left[\mathcal{J}\left(S_{0}+\frac{Z}{\lambda\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}};\frac{\gamma}{\beta\lambda\sqrt{\eta}\sigma_{\widehat{H}}^{2}},\ell,\mu\right)\right],$
(64)

where the expectation is taken over $S_{0}\sim p_{s_{0}}$ and
$Z\sim\mathcal{N}(0,1)$.
Consequently, the objective function in (A-D), i.e., $G(\beta,\lambda,{\bf
z},{\bf s}_{0})$, converges point-wise to the quantity
$\mathcal{G}(\beta,\lambda)$, defined in (1), in the limit of $n\to\infty$.
Afterwards, observe that $G(\beta,\lambda,{\bf z},{\bf s}_{0})$ is convex in
$\lambda$ and concave in $\beta$. With these, and using Theorem 2.7 in [60],
it follows that

$\displaystyle\max_{\beta\geq 0}\min_{\lambda>0}G(\beta,\lambda,{\bf z},{\bf
s}_{0})\overset{P}{\longrightarrow}\max_{\beta\geq
0}\min_{\lambda>0}\mathcal{G}(\beta,\lambda).$ (65)

Finally, the optimization problem in (A-D) simplifies to the following _Scalar
Optimization_ (SO) problem:

$\displaystyle\max_{\beta\geq 0}\min_{\lambda>0}$ $\displaystyle\
\frac{\beta\sqrt{\eta}}{2\lambda}+\frac{\beta\lambda\sqrt{\eta}}{2}\left(1+{P_{d}\sigma_{\omega}^{2}}\kappa\right)-\frac{\beta^{2}}{4}-\frac{\beta}{2\lambda\sqrt{\eta}}$
$\displaystyle+\beta\lambda\sqrt{\eta}P_{d}\sigma_{\widehat{H}}^{2}\
\mathbb{E}\Biggl{[}\mathcal{J}\biggl{(}S_{0}+\frac{Z}{\lambda\sqrt{\eta
P_{d}\sigma_{\widehat{H}}^{2}}};\frac{\gamma}{\beta\lambda\sqrt{\eta}\sigma_{\widehat{H}}^{2}},\ell,\mu\biggr{)}\Biggr{]}$
$\displaystyle=\max_{\beta\geq 0}\min_{\lambda>0}\mathcal{G}(\beta,\lambda).$
(66)

It worth mentioning that in the above equation,
$\mathcal{G}(\beta_{\star},\lambda_{\star})$, where
$(\beta_{\star},\lambda_{\star})$ is the unique solution of (A-D), represents
the the asymptotic value of the objective function in (12) for a minimizer
$\widehat{{\bf s}}$, i.e.,

$\displaystyle\underset{n\to\infty}{\rm{plim}}\
\frac{1}{n}\left(\bigg{\|}\sqrt{\frac{P_{d}}{n}}\widehat{{\bf H}}\widehat{{\bf
s}}-{\bf r}\bigg{\|}_{2}^{2}+\gamma P_{d}\|\widehat{{\bf
s}}\|_{1}\right)=\mathcal{G}(\beta_{\star},\lambda_{\star}).$ (67)

Fig. 10 shows the great accuracy of the above result.

Figure 10: Optimal objective function value of the Box-LASSO vs. the
regularizer for a sparse-Bernoulli vector. We used
$\kappa=0.2,\eta=1.5,n=128$, $T=500,T_{t}=n,\nu=0.5,E=1,$ and $P=15\
{\rm{dB}}$.

After deriving the SO problem, we are now in a position to study the
asymptotic convergence of the MSE. The analysis is given in the next
subsection.

###  A-E Error Analysis via CGMT (Proof of Theorem 1)

In this part, we study the asymptotic convergence of the MSE of the Box-LASSO.
First, using the fact that $\hat{\lambda}=\frac{1}{\sqrt{x}}$ and recalling
from (57) that

$\displaystyle\frac{1}{n}\|\widetilde{\bf
e}\|_{2}^{2}=\frac{1}{P_{d}\sigma_{\widehat{H}}^{2}}\left(\frac{1}{\hat{\lambda}_{n}^{2}}-1-\frac{P_{d}\sigma_{\omega}^{2}}{n}\|{\bf
s}_{0}\|_{2}^{2}\right),$ (68)

where $\widetilde{\bf e}$ is the AO solution in (A-D), and $\hat{\lambda}_{n}$
is the solution to (A-D) in $\lambda$. Using [5, Lemma 10], it can be shown
that $\hat{\lambda}_{n}\overset{P}{\longrightarrow}\lambda_{\star}$, where
$\lambda_{\star}$ is the solution to (A-D). Then, by the WLLN,
$\frac{1}{n}\|{\bf s}_{0}\|_{2}^{2}\overset{P}{\longrightarrow}\kappa$, and
hence

$\displaystyle\frac{1}{n}\|\widetilde{{\bf e}}\|_{2}^{2}$
$\displaystyle\overset{P}{\longrightarrow}\frac{1}{P_{d}\sigma_{\widehat{H}}^{2}}\left(\frac{1}{\lambda_{\star}^{2}}-1-{P_{d}\sigma_{\omega}^{2}}\kappa\right).$
(69)

Recall that $\widetilde{{\bf e}}=\widetilde{{\bf s}}-{\bf s}_{0}$, so the last
step is to use the CGMT to prove that the quantities $\widehat{{\bf s}}-{\bf
s}_{0}$ and $\widetilde{{\bf s}}-{\bf s}_{0}$ are concentrated in the same set
with high probability. Formally, for any fixed $\varepsilon>0$, we define the
set:

$\displaystyle\mathcal{S}_{\varepsilon}=\left\\{\mathbf{q}\in\mathbb{R}^{n}:\left|\frac{\|{\bf
q}\|_{2}^{2}}{n}-\frac{(\frac{1}{\lambda_{\star}^{2}}-1-{P_{d}\sigma_{\omega}^{2}}\kappa)}{P_{d}\sigma_{\widehat{H}}^{2}}\right|<\varepsilon\right\\}.$
(70)

Equation (69) proves that for any $\varepsilon>0$, $\widetilde{{\bf s}}-{\bf
s}_{0}\in\mathcal{S}_{\varepsilon}$ with probability approaching one. Then, we
conclude by applying the CGMT that $\widehat{{\bf s}}-{\bf
s}_{0}\in\mathcal{S}_{\varepsilon}$ with probability approaching one. This
proves the asymptotic prediction of the MSE as summarized in Theorem 1.

The _residual_ prediction in Remark 2, eq. (21), can be proven in a similar
way, by first noting that

$\displaystyle\|\hat{\bf y}\|_{2}^{2}=4n\mathcal{R},$ (71)

where $\hat{\bf y}$ is the PO solution of (46). Using the definition:
$\beta_{n}^{2}=\frac{\|\widetilde{\bf y}\|_{2}^{2}}{n}$, where $\beta_{n},$
and $\widetilde{\bf y}$ are the solutions of (52), and (A-D) respectively, and
following the same steps as in the MSE proof above, one can show that
$\hat{\bf y}$ and $\widetilde{\bf y}$ concentrate in the same set with
probability approaching one, and then apply the CGMT to reach the proof of the
residual result in (21). Details are thus omitted.

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