# Unsupervised Learning Discriminative MIG Detectors in Nonhomogeneous Clutter

Xiaoqiang Hua, Yusuke Ono, Linyu Peng, and Yuting Xu This work was supported
by NSFC (Grant No. 61901479), JSPS KAKENHI (Grant No. JP20K14365), JST CREST
(Grant No. JPMJCR1914), and Keio Gijuku Fukuzawa Memorial Fund. We thank the
editor and the anonymous referees for their constructive comments.
(Corresponding author: Linyu Peng.) X. Hua is with the College of Meteorology
and Oceanography, and the College of Computer Science and Technology, National
University of Defense Technology, Changsha 410073, China (e-mail:
hxq712@yeah.net). Y. Ono and L. Peng are with the Department of Mechanical
Engineering, Keio University, Yokohama 223-8522, Japan (e-mail:
<EMAIL_ADDRESS>l.peng@mech.keio.ac.jp). Y. Xu is with the College of
Physics, Jilin University, Changchun 130012, China (e-mail:
xuyt20@mails.jlu.edu.cn).

###### Abstract

Principal component analysis (PCA) is a commonly used pattern analysis method
that maps high-dimensional data into a lower-dimensional space maximizing the
data variance, that results in the promotion of separability of data. Inspired
by the principle of PCA, a novel type of learning discriminative matrix
information geometry (MIG) detectors in the unsupervised scenario are
developed, and applied to signal detection in nonhomogeneous environments.
Hermitian positive-definite (HPD) matrices can be used to model the sample
data, while the clutter covariance matrix is estimated by the geometric mean
of a set of secondary HPD matrices. We define a projection that maps the HPD
matrices in a high-dimensional manifold to a low-dimensional and more
discriminative one to increase the degree of separation of HPD matrices by
maximizing the data variance. Learning a mapping can be formulated as a two-
step mini-max optimization problem in Riemannian manifolds, which can be
solved by the Riemannian gradient descent algorithm. Three discriminative MIG
detectors are illustrated with respect to different geometric measures, i.e.,
the Log-Euclidean metric, the Jensen–Bregman LogDet divergence and the
symmetrized Kullback–Leibler divergence. Simulation results show that
performance improvements of the novel MIG detectors can be achieved compared
with the conventional detectors and their state-of-the-art counterparts within
nonhomogeneous environments.

###### Index Terms:

Signal detection, matrix information geometry (MIG) detectors, unsupervised
learning, manifold projection, nonhomogeneous clutter.

##  I Introduction

Improving the performance of signal detection in nonhomogeneous clutter is
imperative in many areas, including radar [1, 2, 3, 4, 5], sonar [6, 7, 8],
communication systems [9]. However, the detection performance is often
unsatisfactory as the number of homogeneous sample data is often limited, not
to mention the presence of interferences caused by the heterogeneity. One
effective approach for enhancing the detection performance in nonhomogeneous
clutter is to incorporate a priori clutter information in designing the
detectors, i.e., carrying out a knowledge-aided processing (see, e.g., [10,
11, 12]), and performance analysis confirmed the advantage of such an
architecture over their conventional counterparts (see also [13, 14, 15, 16,
17, 18]). That these knowledge-aided signal detection methods can achieve
significant performance improvements is owing to the sufficient information on
the clutter characteristics which is often not available priorly in practical
applications. Lack of knowledge about the clutter can often yield severe
performance degradation.

In recent years, exploiting matrix information geometry (MIG) to deal with the
problem of signal processing has been attracting extensive attention. MIG, the
geometric study of matrix manifolds, is a relative new extension of the theory
of classical information geometry, which deals with the geometric theory of
probability distributions and its applications [19, 20, 21, 22, 23, 24]. The
reader may refer to [25, 26] for an introduction to classical information
geometry. Many information and signal processing problems can be equivalently
transformed into discriminational problems on matrix differentiable manifolds
with proper distance or divergence functions. For instance, in [27], an MIG-
based clutter covariance matrix (CCM) estimator was proposed in the case of
limited number of sample data, and significant signal-to-interference-plus-
noise ratio gains were achieved over several standard estimators, such as the
loaded sample matrix inversion. In [28], a new direction of arrival (DOA)
estimation approach that employs geodesic distances to estimate the direction
of arrival of several sources was proposed using the MIG theory. The DOA
estimation was reformulated as a single-variable optimization problem on a
Riemannian manifold. Simulation results showed that the proposed method
improved resolution capabilities at low signal-to-noise ratio with respect to
multiple signal classification and minimum variance distortionless response.
In [29], the problem of CCM estimation was treated as computing the geometric
barycenter associated with a geometric distance for a set of secondary basic
Hermitian positive-definite (HPD) matrices that yielded significant
performance improvements. Specially, a geometric detection scheme, which we
call the MIG detector, was developed by Lapuyade-Lahorgue and Barbaresco in
[30]. In MIG detectors, by taking geometric structures of the relevant
manifolds into account, a priori knowledge on the clutter characteristics is
not required.

The performance of MIG detectors is closely related to the discriminative
power of the utilized geometric measures, e.g., [31, 32]. In particular for
HPD manifolds, the affine invariant Riemannian metric (AIRM) and the
corresponding AIRM-MIG detector have been greatly studied and applied [33, 34,
35, 36, 37, 38, 39, 40]. By exploiting discriminative geometric measures, it
is possible to propose MIG detectors of good performances. In [41], they
authors derived the geometric means and medians corresponding to two extended
Kullback–Leibler (KL) divergences, the total KL divergence and the symmetrized
KL divergence (SKLD), and particularly designed two MIG detectors based on the
extended KL divergences. They were applied to target detection in
K-distribution clutter, that evidenced performance gains over their state-of-
the-art counterparts. In [42, 43], the total Bregman divergence (TBD) was
extended to HPD matrix manifolds, and fortunately the geometric means derived
by using the some of the mostly well-known convex functions could be derived
in closed-form. Simulation results showed that the corresponding TBD-MIG
detectors outperformed the AIRM-MIG detector as well as the conventional
detectors in nonhomogeneous clutter. In addition to the discriminative
geometric measures mentioned above, other measures can also be defined in HPD
matrix manifolds. It is worth exploring new discriminative metrics and
designing the corresponding MIG detectors. A major limitation in these MIG
detectors, nevertheless, is that the detection performance is affected by
different clutter characteristics as the discriminative power associated with
a given geometric measure may change as the clutter changes.

To overcome the drawback, in this paper, we develop a projection that maps
higher-order HPD matrices into a lower-dimensional and more discriminative HPD
manifold and enhances the separability of data in the unsupervised learning
scenario; then we propose a type of discriminative MIG detectors, and apply
them to signal detection in nonhomogeneous clutter. Main contributions of the
current study are briefly summarized below.

  1. 1.

Inspired by the principle of principal component analysis (PCA), we propose a
projection subject to an orthonormal constraint for the projection matrix,
that enhances the separability between the target signal and the clutter.
Learning the projection (matrix) by maximizing the variance of data becomes a
two-step mini-max optimization problem in a Stiefel manifold and an HPD
manifold, that can be solved by the Riemannian gradient descent (RGD)
algorithm. Given a set of training HPD matrices that consists of two classes
of data, one containing target signal and another containing only the clutter,
the projection matrix can be obtained in an unsupervised way. One may consider
that the PCA can only reconstruct the data in a better way but cannot lead to
discrimination improvement between the samples. However, during the project,
redundant information originally included in the higher-dimensional HPD
matrices may be reduced during the manifold projection, leading to improvement
of detection performance. From this aspect, the inspiration from PCA is rather
indirect.

  2. 2.

A class of discriminative MIG detectors is designed by incorporating the
manifold projection into the detection architecture. Specifically, the sample
data is modeled as an HPD matrix with the diagonal loading structure, and the
CCM is estimated by the geometric mean about secondary HPD matrices. The CCM
and the HPD matrix in the cell under test (CUT) are transformed into a more
discriminative low-dimensional manifold. Consequently, signal detection is
realized via MIG detector on a lower-dimensional HPD matrix manifold.

  3. 3.

Simulations performed in nonhomogeneous clutter verify the outperformance of
the proposed discriminative MIG detectors in comparison with their state-of-
the-art counterparts as well as the conventional detectors.

The paper is organized as follows. The discriminative MIG detector is
formulated in Section II, and a brief introduction to MIG is given in Section
III. In Section IV, three geometric means are derived for the CCM estimation,
and the problem of learning the projection is formulated as a two-step mini-
max optimization problem in a Stiefel manifold and an HPD manifold. The
performance analysis is presented in Section V, and we conclude finally in
Section VI.

Notations: We use boldface lowercase (uppercase) letters to denote vectors
(matrices). Matrix (or vector) transpose and conjugate transpose are denoted
by the superscripts $(\cdot)^{\operatorname{T}}$ and
$(\cdot)^{\operatorname{H}}$, respectively. Determinant and trace of a matrix
are respectively denoted by $\operatorname{det}(\cdot)$ and
$\operatorname{tr}(\cdot)$. The $N\times N$ identity matrix is denoted by
$\bm{I}_{N}$ or simply $\bm{I}$. The notations $\mathbb{C}^{N}$ and
$\mathbb{C}^{M\times N}$ represent the set of $n$-dimensional complex vectors
and $M\times N$ complex matrices, respectively. The imaginary unit is
$\operatorname{i}$, and finally $\operatorname{E}[\cdot]$ denotes the
statistical expectation.

##  II Problem Formulation

Let $\bm{x}=[x_{0},x_{1},\ldots,x_{N-1}]^{\operatorname{T}}$ be the sample
data collected from $N$ (temporal, spatial, or spatial-temporal) channels. In
general, the problem of signal detection is interpreted as the following
binary hypothesis testing

$\left\\{\begin{aligned} &\mathcal{H}_{0}:\left\\{\begin{aligned}
&\bm{x}=\bm{c},\\\ &\bm{x}_{k}=\bm{c}_{k},\quad
k\in[K],\end{aligned}\right.\\\ &\mathcal{H}_{1}:\left\\{\begin{aligned}
&\bm{x}=\alpha\bm{p}+\bm{c},\\\ &\bm{x}_{k}=\bm{c}_{k},\quad
k\in[K],\end{aligned}\right.\\\ \end{aligned}\right.$ (1)

where $[K]$ denotes the set of $\\{1,2,\ldots,K\\}$ with $K$ the number of
secondary data, $\bm{c}$ and $\bm{c}_{k}$ are the clutter data, and $\bm{x}$
and $\bm{x}_{k}$ denote the observation data. In particular, $\bm{x}$ and
$\bm{c}$ represent data of the CUT. Here, $\mathcal{H}_{0}$ and
$\mathcal{H}_{1}$ denote the null and alternative hypotheses that correspond
to the absence and presence of a target signal, respectively. The unknown
complex parameter $\alpha$ is relevant to the channel propagation effects and
target reflectivity. The known steering vector $\bm{p}$ is given by

$\bm{p}=\frac{1}{\sqrt{N}}\left[1,\exp\left(-\operatorname{i}2\pi
f_{d}\right),\ldots,\exp\left(-\operatorname{i}2\pi
f_{d}(N-1)\right)\right]^{\operatorname{T}},$ (2)

where $f_{d}$ denotes the normalized Doppler frequency.

The observation data is assumed to obey a multivariate complex Gaussian
distribution with zero mean. Therefore, statistical information of the sample
data is closely related to the covariance matrix. The power or correlation of
the sample data, which can be represented as an HPD matrix, is employed for
distinguishing the target signal from the clutter. Various structures can be
specified for the HPD matrix, for instance, the Toeplitz structure [27], the
diagonal loading [44], the shrinkage estimators [45] and the persymmetric
covariance estimators [46]. The HPD manifolds subject to different matrix
structures possess different geometric structures. The resulting differences
in the detection performance cased by different matrix structures will be
analyzed separately. Here, we exploit the HPD matrix with the diagonal loading
structure to model the sample data. The diagonal loading structure has been
successfully applied in signal detection [47, 48, 49]. The diagonal loading
HPD matrix can be expressed by adding diagonal matrix to the sample covariance
matrix (SCM), i.e.,

$\bm{R}=\bm{r}\bm{r}^{\operatorname{H}}+\operatorname{tr}\left(\bm{r}\bm{r}^{\operatorname{H}}\right)\bm{I},$
(3)

where $\bm{r}=[r_{0},r_{1},\dots,r_{N-1}]^{\operatorname{T}}$ denotes
correlation of the sample data, namely

$r_{l}=\operatorname{E}[x_{i}\overline{x}_{i+l}],\quad 0\leq l\leq N-1,1\leq
i\leq N-l-1.$ (4)

where $\overline{x}_{i}$ denotes the conjugate of $x_{i}$. Ergodicity of
stationary Gaussian process allows us to approximate $r_{l}$ by the following
estimator

$\widetilde{r}_{l}=\frac{1}{N}\sum_{i=0}^{N-1-l}{x_{i}\overline{x}_{i+l}},\quad
0\leq l\leq N-1.$ (5)

Using the diagonal loading formalism (3), each sample data can be represented
as an HPD matrix with a diagonal loading structure as the new observation. The
set of all $N\times N$ HPD matrices forms a differentiable manifold; see
Section III for more details. Assuming that $K$ secondary HPD matrices
$\\{\bm{R}_{k}\\}_{k\in[K]}$ are available, we employ the geometric mean
$\bm{R}_{\mathcal{G}}=\mathcal{G}(\bm{R}_{1},\bm{R}_{2},\ldots,\bm{R}_{K})$ to
estimate the CCM. From the viewpoint of MIG, the problem of binary hypothesis
testing Eq. (1) can be rewritten as (see e.g., [43]),

$\left\\{\begin{aligned} \mathcal{H}_{0}:\bm{R}=\bm{R}_{\mathcal{G}},\\\
\mathcal{H}_{1}:\bm{R}\neq\bm{R}_{\mathcal{G}}.\end{aligned}\right.$ (6)

Given the observation $\bm{R}_{D}$ and the CCM estimate
$\bm{R}_{\mathcal{G}}$, by utilizing geometric structure of HPD matrix
manifolds, signal detection can be interpreted as the discrimination of two
HPD matrices $\bm{R}_{D}$ and $\bm{R}_{\mathcal{G}}$ in a differentiable
manifold. Let us consider the null hypothesis
$\mathcal{H}_{0}:\bm{R}=\bm{R}_{\mathcal{G}}$ versus the alternative
hypothesis $\mathcal{H}_{1}:\bm{R}\neq\bm{R}_{\mathcal{G}}$ based on a set of
observations $\\{\bm{R}_{k}\\}_{k\in[K]}$. The problem of signal detection can
be understood as to determine the inner of isosurfaces of the HPD matrix
manifold determined by a distance or divergence, as illustrated in Fig. 1. The
hypothesis $\mathcal{H}_{0}$ is rejected if the observation $\bm{R}_{D}$ of
CUT does not belong to the inner of an isosurface.

Figure 1: The diagram for the geometric interpretation of signal detection

In our detection framework, we learn a projection matrix
$\bm{W}\in\mathbb{C}^{N\times M}(M\leq N)$ of full rank to maximize the
variance of the data by resorting to the training HPD matrices in an
unsupervised scenario, where any $N\times N$ HPD matrix $\bm{R}$ will be
mapped into a more discriminative low-dimensional manifold by

$f_{\bm{W}}(\bm{R})=\bm{W}^{\operatorname{H}}\bm{R}\bm{W}\in\mathbb{C}^{M\times
M},$ (7)

where $\bm{W}$ is conventionally assumed in the (compact and complex) Stiefel
manifold

$\operatorname{St}(M,\mathbb{C}^{N})=\left\\{\bm{A}\in\mathbb{C}^{N\times
M}\mid\bm{A}^{\operatorname{H}}\bm{A}=\bm{I}_{M}\right\\}.$ (8)

Consequently, the problem of signal detection becomes to determining the inner
of an isosurface in an HPD matrix manifold associated to a given distance or
divergence (see Fig. 1), namely

$d\left(f_{\bm{W}}(\bm{R}_{\mathcal{G}}),f_{\bm{W}}(\bm{R}_{D})\right)\mathop{\gtrless}\limits_{\mathcal{H}_{0}}^{\mathcal{H}_{1}}\gamma,$
(9)

where $d(\cdot,\cdot)$ is the distance or divergence that used to measure the
dissimilarity between two points on the HPD matrix manifold. The hypothesis
$\mathcal{H}_{1}$, meaning the presence of a target signal, is accepted if the
observation $f_{\bm{W}}(\bm{R}_{D})$ of CUT lies outside of an isosurface
centered at $f_{\bm{W}}(\bm{R}_{\mathcal{G}})$ with radius $\gamma$, which is
the detection threshold.

##  III Preliminaries of Matrix Information Geometry

Before moving to an MIG solution to the problem formulated in Section II
above, we briefly review the theory of MIG that is relevant to the current
study in this section.

###  III-A HPD manifolds

The general linear group $GL(N,\mathbb{F})$ consists of all $N\times N$
invertible matrices with $\mathbb{F}$ either real $\mathbb{R}$ or complex
$\mathbb{C}$. The Frobenius metric111 It is also called the Hilbert–Schmidt
inner product. is defined by

$\langle\bm{X},\bm{Y}\rangle:=\operatorname{tr}(\bm{X}^{\operatorname{H}}\bm{Y}),\quad\bm{X},\bm{Y}\in
GL(N,\mathbb{F}).$ (10)

In the current paper, our main interest is HPD matrices, that form a subspace
of $GL(N,\mathbb{C})$.

The set of $N\times N$ HPD matrices is denoted by $\mathscr{P}(N,\mathbb{C})$
which is a subset of $GL(N,\mathbb{C})$ and naturally a differentiable
manifold. Each element $\bm{A}\in\mathscr{P}(N,\mathbb{C})$ is Hermitian and
positive-definite, that is

$\bm{A}^{\operatorname{H}}=\bm{A}\text{ and
}\bm{x}^{\operatorname{H}}\bm{A}\bm{x}>0\text{ for all
}\bm{0}\neq\bm{x}\in\mathbb{C}^{N}.$ (11)

The difference of two HPD matrices can be evaluated by a distance, a
divergence or other measures defined in $\mathscr{P}(N,\mathbb{C})$. It is
crucial to specify these measures properly in applications, as different
measures will lead to different isosurfaces.

###  III-B Riemannian structures of HPD manifolds

Except the induced subspace Frobenius metric, the space
$\mathscr{P}(N,\mathbb{C})$ is a Riemannian manifold with the AIRM

$\langle\bm{A},\bm{B}\rangle_{\bm{P}}:=\operatorname{tr}\left(\bm{P}^{-1}\bm{A}\bm{P}^{-1}\bm{B}\right),\quad\bm{A},\bm{B}\in
T_{\bm{P}}\mathscr{P}(N,\mathbb{C}).$ (12)

Under the AIRM, its curvature is non-positive [50, 51, 8, 52]. In the
following, we are going to introduce some of the mostly well-known geometric
measures in the manifold $\mathscr{P}(N,\mathbb{C})$ as either a metric space
equipped with the Frobenius metric or a Riemannian manifold.

In the Riemannian manifold $\mathscr{P}(N,\mathbb{C})$, exponential map and
logarithm map can naturally be defined on the tangent bundle
$T\mathscr{P}(N,\mathbb{C})=\bigcup\limits_{\bm{P}}T_{\bm{P}}\mathscr{P}(N,\mathbb{C})$
by using the geodesics. They are related to matrix exponentials and matrix
logarithms. Matrix exponential for a general matrix $\bm{X}$ is defined by a
Taylor series

$\exp(\bm{X})=\sum_{i=0}^{+\infty}\frac{\bm{X}^{i}}{i!}.$ (13)

Logarithm of a matrix is defined as the inversion of matrix exponential.
Unfortunately, it is not always well-defined as a function. The following
lemma defines the principle logarithm of an invertible matrix together with
some of its important properties, which will be used later.

######  Lemma 1 ([53, 54]).

Let $\bm{X}$ be an invertible matrix and assume that none of its eigenvalues
lie in the closed negative real line. Then, there exists a unique matrix
logarithm of $\bm{X}$ whose eigenvalues lie in the strip

$\\{z\in\mathbb{C}\mid-\pi<\operatorname{Im}(z)<\pi\\}.$

It is referred to as the principle logarithm and denoted by
$\operatorname{Log}\bm{X}$.

The principle (matrix) logarithm satisfies the following properties.

  * (i)

Each pair of the matrices $[(\bm{X}-\bm{I})s+\bm{I}]^{-1}$, $\bm{X}$ and
$\operatorname{Log}\bm{X}$ commutes any real number $s$.

  * (ii)

The following matrix integral is valid:

$\displaystyle\int_{0}^{1}[(\bm{X}-\bm{I})s+\bm{I}]^{-2}\operatorname{d}\\!s$
$\displaystyle\quad\quad\quad=(\bm{I}-\bm{X})^{-1}[(\bm{X}-\bm{I})s+\bm{I}]^{-1}\Big{|}_{s=0}^{1}$
$\displaystyle\quad\quad\quad=\bm{X}^{-1}.$

  * (iii)

Let $\bm{A}(\varepsilon)$ be an invertible matrix satisfying the unique
principal logarithm existence condition above. Furthermore, assume
$\bm{A}(\varepsilon)$ depends on the real parameter $\varepsilon$ smoothly.
Then we have

$\displaystyle\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\operatorname{Log}\bm{A}(\varepsilon)$
$\displaystyle=\int_{0}^{1}[(\bm{A}(\varepsilon)-\bm{I})s+\bm{I}]^{-1}\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\bm{A}(\varepsilon)$
$\displaystyle\quad\quad\quad\times[(\bm{A}(\varepsilon)-\bm{I})s+\bm{I}]^{-1}\operatorname{d}\\!s.$

In the Riemannian manifold $\mathscr{P}(N,\mathbb{C})$ equipped with the AIRM,
the distance of two points $\bm{X},\bm{Y}\in\mathscr{P}(N,\mathbb{C})$ is
given by the length of the local geodesic with them as the endpoints, reading

$\displaystyle d_{R}^{2}(\bm{X},\bm{Y})$
$\displaystyle=\left\lVert\operatorname{Log}\left(\bm{X}^{-1/2}\bm{Y}\bm{X}^{-1/2}\right)\right\rVert^{2}$
(14) $\displaystyle=\sum_{i=1}^{N}\ln^{2}\lambda_{i},$

where $\lambda_{1},\lambda_{2},\ldots,\lambda_{N}$ are the eigenvalues of the
matrix $\bm{X}^{-1/2}\bm{Y}\bm{X}^{-1/2}$. We use
$\left\lVert\cdot\right\rVert$ to denote the Frobenius norm
$\left\lVert\bm{A}\right\rVert^{2}=\operatorname{tr}(\bm{A}^{\operatorname{H}}\bm{A})$
of a matrix $\bm{A}$, induced from the Frobenius metric (10).

Unfortunately, the computational cost of the AIRM distance is often expensive
in practical applications. An alternative choice, the Log-Euclidean metric
(LEM) [55], is defined as follows

$\langle\bm{A},\bm{B}\rangle_{\bm{P}}^{\operatorname{LE}}:=\langle
D_{\bm{A}}\operatorname{Log}\bm{P},D_{\bm{B}}\operatorname{Log}\bm{P}\rangle,$
(15)

where $\bm{A},\bm{B}\in T_{\bm{P}}\mathscr{P}(N,\mathbb{C})$ and
$D_{\bm{A}}\operatorname{Log}\bm{P}$ denotes the directional derivative of the
matrix logarithm along a tangent vector $\bm{A}$ at a point $\bm{P}$. The LEM
distance of two HPD matrices $\bm{X},\bm{Y}\in\mathscr{P}(N,\mathbb{C})$ is
given by the length of the local geodesic as

$d_{L}^{2}(\bm{X},\bm{Y})=\left\lVert\operatorname{Log}\bm{X}-\operatorname{Log}\bm{Y}\right\rVert^{2}.$
(16)

###  III-C Divergences of HPD matrices

By viewing the differentiable manifold $\mathscr{P}(N,\mathbb{C})$ as a metric
space equipped with the Frobenius metric, many other geometric measures can
also be defined. We will be focused on the the Jensen–Bregman LogDet
divergence (JBLD) [56] and the symmetrized Kullback–Leibler divergence (SKLD)
[41] in the current paper. The JBLD and SKLD of two HPD matrices
$\bm{X},\bm{Y}\in\mathscr{P}(N,\mathbb{C})$ are respectively gived by

$d_{J}^{2}(\bm{X},\bm{Y})=\ln\operatorname{det}\left(\frac{\bm{X}+\bm{Y}}{2}\right)-\frac{1}{2}\ln\operatorname{det}(\bm{X}\bm{Y})$
(17)

and

$d_{S}^{2}(\bm{X},\bm{Y})=\frac{1}{2}\operatorname{tr}\left(\bm{Y}^{-1}\bm{X}+\bm{X}^{-1}\bm{Y}-2\bm{I}\right).$
(18)

Note that among all geometric measures introduced above, the AIRM, the JBLD
and the SKLD are invariant with respect to affine transformations.

In the study of optimization problems in $\mathscr{P}(N,\mathbb{C})$, we often
need to compute the gradient of a function $F(\bm{R})$, which is defined by
the covariant/directional derivative associated to a given metric, e.g., a
Riemannian metric or simply the Frobenius metric, as follows

$\langle\nabla
F(\bm{R}),\bm{A}\rangle:=\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}F(\gamma(\varepsilon)),\quad\forall\bm{A}\in
T_{\bm{R}}\mathscr{P}(N,\mathbb{C}),$ (19)

where $\gamma:[0,1]\rightarrow\mathscr{P}(N,\mathbb{C})$ is the unique local
curve satisfying $\gamma(0)=\bm{R}$ and $\dot{\gamma}(0)=\bm{A}$. By taking
the linear part into account, it can be rewritten as

$\langle\nabla
F(\bm{R}),\bm{A}\rangle:=\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}F(\bm{R}+\varepsilon\bm{A}),\quad\forall\bm{A}\in
T_{\bm{R}}\mathscr{P}(N,\mathbb{C}).$ (20)

##  IV Geometric Means and Unsupervised Manifold Projection

###  IV-A Geometric Means

It is well known that the arithmetic mean of a set of $K$ positive real
numbers $\\{x_{k}\\}_{k\in[K]}$ can be calculated by

$\widehat{x}=\frac{1}{K}\sum_{k=1}^{K}x_{k}.$ (21)

In fact, the arithmetic mean is the minimum value of the sum of the squares,
namely

$\widehat{x}:=\underset{x\in\mathbb{R}^{+}}{{\arg\min}}\sum_{k=1}^{K}|x-x_{k}|^{2},$
(22)

where $|x-x_{k}|$ denotes the distance between $x$ and $x_{k}$. Geometric mean
of a set of HPD matrices can similarly be defined.

######  Definition 2.

Given a set of $K$ HPD matrices $\\{\bm{R}_{k}\\}_{k\in[K]}$, the geometric
mean with respect to a geometric measure
$d:\mathscr{P}(N,C)\times\mathscr{P}(N,C)\rightarrow\mathbb{R}$ is obtained
through the following optimization problem

$\bm{\widehat{R}}:=\underset{\bm{R}\in\mathscr{P}(N,\mathbb{C})}{{\arg\min}}\sum_{k=1}^{K}d^{2}\left(\bm{R}_{k},\bm{R}\right).$
(23)

Geometric means of a set of HPD matrices can not always be calculated in
closed form; alternatively, the fixed-point iteration has proven to be
effective for calculating them numerically, e.g., [55, 57, 31]. In the below,
we summarize the algorithms or analytic expressions for computing the
geometric means corresponding to the three measures introduced above, i.e.,
the LEM distance (16), the AIRM geodesic distance (14), the JBLD (17) and the
SKLD (18).

######  Proposition 3.

The LEM mean of HPD matrices $\\{\bm{R}_{k}\\}_{k\in[K]}$ is given by [55]

$\widehat{\bm{R}}=\operatorname{exp}\bigg{(}\frac{1}{K}\sum_{k=1}^{K}\operatorname{Log}\bm{R}_{k}\bigg{)}.$
(24)

######  Proposition 4.

The AIRM mean of $\\{\bm{R}_{k}\\}_{k\in[K]}$ is determined by [54]

$\sum_{k=1}^{K}\operatorname{Log}\left(\bm{R}_{k}^{-1}\bm{\widehat{R}}\right)=0,$
(25)

which can be obtained using the following fixed-point iteration [58]:

$\displaystyle\widehat{\bm{R}}_{t+1}$ $\displaystyle=a\widehat{\bm{R}}_{t}$
(26)
$\displaystyle+(a-1)\sum_{k=2}^{K}\operatorname{Log}\left(\exp\left(\frac{\widehat{\bm{R}}_{t}}{2}\right){\bm{R}}_{k}^{-1}\exp\left(\frac{\widehat{\bm{R}}_{t}}{2}\right)\right),$

where $1-1/K<a<1$, $t$ denotes the iterative index, and the initial value is

$\widehat{\bm{R}}_{0}=\frac{1}{K}\sum_{k=1}^{K}\operatorname{Log}\bm{R}_{k}.$
(27)

######  Proposition 5.

The JBLD mean of HPD matrices $\\{\bm{R}_{k}\\}_{k\in[K]}$ can be obtained
through the fixed-point iteration [57, 31]:

$\bm{\widehat{R}}_{t+1}=\Bigg{(}\frac{1}{K}\sum_{k=1}^{K}\bigg{(}\frac{\bm{\widehat{R}}_{t}+\bm{R}_{k}}{2}\bigg{)}^{-1}\Bigg{)}^{-1}.$
(28)

######  Proposition 6.

The SKLD mean of HPD matrices $\\{\bm{R}_{k}\\}_{k\in[K]}$ is

$\bm{\widehat{R}}=\bm{A}^{-1/2}\left(\bm{A}^{1/2}\bm{B}\bm{A}^{1/2}\right)^{1/2}\bm{A}^{-1/2},$
(29)

where

$\bm{A}=\sum_{k=1}^{K}\bm{R}_{k}^{-1},\quad\bm{B}=\sum_{k=1}^{K}\bm{R}_{k}.$
(30)

###### Proof.

A proof is provided in Appendix A; see also [59]. ∎

###  IV-B Unsupervised Manifold Projection

In this subsection, we introduce the manifold projection that maps HPD
matrices from a high-dimensional manifold to a more discriminative lower-
dimensional one by maximizing the variance of data.

Recall that the variance of a set of vectors $\\{\bm{x}_{k}\\}_{k\in[K]}$ in a
Euclidean space is given by

$\operatorname{Var}=\frac{1}{K}\sum_{k=1}^{K}\left\lVert\bm{x}_{k}-\bm{\widehat{x}}\right\rVert_{2}^{2}\quad\text{with}\quad\bm{\widehat{x}}=\frac{1}{K}\sum_{k=1}^{K}\bm{x}_{k},$
(31)

where $\left\lVert\cdot\right\rVert_{2}$ denotes the $l_{2}$ norm, and
$\bm{\widehat{x}}$ is the mean of the set of vectors.

Given a set of HPD matrices $\\{\bm{R}_{i}\\}_{i\in[J+K]}$ that contains $J$
CCMs and $K$ HPD matrices with a target signal, the variance can similarly be
defined as

$\operatorname{Var}=\frac{1}{J+K}\sum_{i=1}^{J+K}d^{2}\left(\bm{R}_{i},\bm{\widehat{R}}\right).$
(32)

where $\bm{\widehat{R}}$ denotes the mean of $J+K$ HPD matrices, which can be
derived using Eq. (23) with respect to the LEM distance, the JBLD or the SKLD.
Note that the variance (32) can be interpreted as a deterministic counterpart
of the variance function of a probability distribution defined in the HPD
manifold [60].

As briefly introduced in Section II, we propose a manifold projection that
maps HPD matrices into a more lower-dimensional manifold maximizing the data
variance. The projection is defined as

$\displaystyle f_{\bm{W}}:\mathscr{P}(N,\mathbb{C})$
$\displaystyle\rightarrow\mathscr{P}(M,\mathbb{C})$ (33) $\displaystyle\bm{R}$
$\displaystyle\mapsto\bm{W}^{\operatorname{H}}\bm{R}\bm{W},$

where $M\leq N$ and
$\bm{W}\in\operatorname{St}(M,\mathbb{C}^{N})\subset\mathbb{C}^{N\times M}$.
Obviously, $\bm{W}$ is of maximal rank and
$\bm{W}^{\operatorname{H}}\bm{W}=\bm{I}_{M}$. Therefore, for a set of HPD
matrices $\\{\bm{R}_{i}\\}_{i\in[J+K]}$ in $\mathscr{P}(N,\mathbb{C})$,
learning a mapping to achieve maximal variance is equivalent to searching a
projection matrix $\bm{W}$ in the Stiefel manifold. Namely, the problem
becomes solving the optimization problem

$\displaystyle\overline{\bm{W}}$
$\displaystyle:=\underset{\bm{W}\in\operatorname{St}(M,\mathbb{C}^{N})}{\arg\max}\frac{1}{J+K}\sum_{i=1}^{J+K}d^{2}\left(f_{\bm{W}}(\bm{R}_{i}),\bm{\widehat{Z}}\right)$
(34)
$\displaystyle~{}=\underset{\bm{W}\in\operatorname{St}(M,\mathbb{C}^{N})}{\arg\max}\frac{1}{J+K}\sum_{i=1}^{J+K}d^{2}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W},\bm{\widehat{Z}}\right),$

where $\bm{\widehat{Z}}$ is the geometric mean of the set
$\\{\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\\}_{i\in[J+K]}$ in
$\mathscr{P}(M,\mathbb{C})$, namely

$\bm{\widehat{Z}}=\underset{\bm{Z}\in\mathscr{P}(M,\mathbb{C})}{{\arg\min}}\sum_{i=1}^{J+K}d^{2}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W},\bm{Z}\right).$
(35)

######  Remark 7.

Solving the optimization problem (34) is a very complex and nonlinear problem
as $\widehat{\bm{Z}}\in\mathscr{P}(M,\mathbb{C})$ also depends on $\bm{W}$.
Although for the LEM distance and the SKLD, we can obtain the means (see
Propositions 3 and 6) respectively as

$\widehat{\bm{Z}}_{L}=\exp\left(\frac{1}{J+K}\sum_{i=1}^{J+K}\operatorname{Log}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)\right)$
(36)

and

$\widehat{\bm{Z}}_{S}=\bm{A}^{-1/2}\left(\bm{A}^{1/2}\bm{B}\bm{A}^{1/2}\right)^{1/2}\bm{A}^{-1/2},$
(37)

where

$\bm{A}=\sum_{i=1}^{J+K}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)^{-1},\quad\bm{B}=\sum_{i=1}^{J+K}\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}.$
(38)

Since it is difficult to solve the projection matrix $\overline{\bm{W}}$ from
(34) in closed-form, we formulate it as a two-step mini-max optimization
problem as follows:

$\displaystyle\overline{\bm{W}}_{t+1}$
$\displaystyle=\underset{\bm{W}\in\operatorname{St}(M,\mathbb{C}^{N})}{{\arg\max}}\frac{1}{J+K}\sum_{i=1}^{J+K}d^{2}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W},\bm{\widehat{Z}}_{t}\right),$
(39) $\displaystyle\bm{\widehat{Z}}_{t+1}$
$\displaystyle=\underset{\bm{Z}\in\mathscr{P}(M,\mathbb{C})}{{\arg\min}}\sum_{i=1}^{J+K}d^{2}\left(\overline{\bm{W}}_{t+1}^{\operatorname{H}}\bm{R}_{i}\overline{\bm{W}}_{t+1},\bm{Z}\right),$

where $t$ denotes the iterative step. At each step, the minimal problem for
$\widehat{\bm{Z}}$ can either be solved analytically by Remark 7 or
numerically by Proposition 5; the maximal problem can be transformed into a
minimal problem

$\underset{\bm{W}\in\operatorname{St}(M,\mathbb{C}^{N})}{{\arg\min}}\psi(\bm{W}),$
(40)

that can be solved by the RGD algorithm (44) (see also[61, 62]), where at each
step $t$,

$\psi(\bm{W})=-\frac{1}{J+K}\sum_{i=1}^{J+K}d^{2}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W},\bm{\widehat{Z}}_{t}\right).$
(41)

The Riemannian gradient of a function $\psi(\bm{W})$ defined on the Stiefel
manifold $\operatorname{St}(M,\mathbb{C}^{N})$ is given by [63]

$\operatorname{grad}\psi(\bm{W})=\nabla\psi(\bm{W})-\bm{W}\times\operatorname{sym}\left(\bm{W}^{\operatorname{H}}\nabla\psi(\bm{W})\right),$
(42)

where

$\operatorname{sym}(\bm{A})=\frac{\bm{A}+\bm{A}^{\operatorname{H}}}{2}$ (43)

denotes the symmetric part of a matrix $\bm{A}$, and $\nabla\psi(\bm{W})$ is
the Euclidean gradient induced from the Frobenius metric. The RGD algorithm
reads

$\bm{W}_{l+1}=\exp_{\bm{W}_{l}}\left(-\eta_{l}\operatorname{grad}\psi(\bm{W}_{l})\right),$
(44)

where $\eta_{l}$ is the step size, and
$\exp:T\operatorname{St}(M,\mathbb{C}^{N})\rightarrow\operatorname{St}(M,\mathbb{C}^{N})$
is the exponential map associated to the Euclidean metric of the Stiefel
manifold. For more details, the reader may refer to [63, 59].

To compile the RGD algorithm (44), the Euclidean gradient of the function
$\psi(\bm{W})$ is needed. Note that the Frobenius metric (10) can be extended
to $N\times M$ matrices, namely

$\langle\bm{X},\bm{Y}\rangle=\operatorname{tr}\left(\bm{X}^{\operatorname{H}}\bm{Y}\right),\quad\bm{X},\bm{Y}\in\mathbb{C}^{N\times
M}.$ (45)

######  Proposition 8.

The Euclidean gradient of the function $\psi(\bm{W})$ defined by (41)
associated with the LEM is given by

$\displaystyle\nabla\psi(\bm{W})=$
$\displaystyle-\frac{4}{J+K}\sum_{i=1}^{J+K}\bm{R}_{i}\bm{W}\Big{(}\bm{V}^{-1}\operatorname{Log}\bm{V}$
(46)
$\displaystyle\quad-\int_{0}^{1}[(\bm{V}-\bm{I})s+\bm{I}]^{-1}\left(\operatorname{Log}\widehat{\bm{Z}}_{t}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\times[(\bm{V}-\bm{I})s+\bm{I}]^{-1}\operatorname{d}\\!s\Big{)},$

where $\bm{V}=\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}$.

###### Proof.

See Appendix B. ∎

######  Proposition 9.

The Euclidean gradient of the function $\psi(\bm{W})$ with respect to the AIRM
is given by

$\displaystyle\nabla\psi(\bm{W})$
$\displaystyle=\frac{4}{J+K}\sum_{i=1}^{J+K}\bm{R}_{i}\bm{W}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)^{-1}$
(47)
$\displaystyle\quad\quad\quad\quad\quad\quad\times\operatorname{Log}\left(\widehat{\bm{Z}}_{t}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)^{-1}\right).$

###### Proof.

The corresponding function $\psi(\bm{W})$ is

$\psi(\bm{W})=-\frac{1}{J+K}\sum_{i=1}^{J+K}\operatorname{tr}\left(\operatorname{Log}^{2}\left(\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)^{-1}\widehat{\bm{Z}}_{t}\right)\right).$
(48)

Using definition of the Euclidean gradient and following a similar proof of
Appendix B, the result can be directly obtained. Similar computation details
are omitted here.

∎

######  Proposition 10.

The Euclidean gradient of $\psi(\bm{W})$ associated with the JBLD is given by

$\displaystyle\nabla\psi(\bm{W})$
$\displaystyle=-\frac{1}{J+K}\sum_{i=1}^{J+K}\bm{R}_{i}\bm{W}\Big{(}2\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}+\widehat{\bm{Z}}_{t}\right)^{-1}$
(49)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad-\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)^{-1}\Big{)}.$

###### Proof.

See Appendix C. ∎

######  Proposition 11.

The Euclidean gradient of $\psi(\bm{W})$ associated with the SKLD is given by

$\displaystyle\nabla\psi(\bm{W})$
$\displaystyle=-\frac{1}{J+K}\sum_{i=1}^{J+K}\bm{R}_{i}\bm{W}\Big{(}\bm{\widehat{Z}}_{t}^{-1}$
(50)
$\displaystyle\quad\quad-\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)^{-1}\bm{\widehat{Z}}_{t}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)^{-1}\Big{)}.$

###### Proof.

See Appendix D. ∎

###  IV-C Complexity Analysis

In this subsection, we will briefly show the complexity of the calculation of
LEM, AIRM, JBLD, SKLD means given by Propositions 3, 4, 5, 6 and the
arithmetic mean, as well as the Euclidean gradients of $\psi(\bm{W})$ with
respect to the LEM, AIRM, JBLD, SKLD, respectively. The latter are given by
Propositions 8, 9, 10, 11. For simplicity, we only keep the leading terms; for
numerical iterations, we only provide the computational complexity for a
single step.

The complexity figures assume that $K$ number of $N\times N$ HPD matrices are
given and the arithmetic with individual elements has complexity $O(1)$. The
lower HPD manifold is $M$-dimensional. The following facts are used:
$\bm{R}^{-1}\sim O(N^{3})$ and $\operatorname{Log}\bm{R}\sim O(N^{4})$. Matrix
exponential in all algorithms only deal with Hermitian matrices, and one way
to calculate their exponentials is through eigenvalue decomposition, whose
complexity is $O(N^{3})$, same as that of matrix inversion.

TABLE I: Computational complexity of the means Geometric measures | Complexity
---|---
Arithmetic mean | $O(N^{2}(K-1))$
LEM (Proposition 3) | $O(N^{4}K)$
AIRM (Proposition 4, per iteration) | $O(N^{4}(K-1))$
JBLD (Proposition 5, per iteration) | $O(N^{3}(K+1))$
SKLD (Proposition 6) | $O(N^{3}(K+6))$

It is clear from TABLE I that the arithmetic mean costs least time, followed
by the SKLD mean. Although both of them are Riemannian distances, computation
of the LEM mean is much faster than the AIRM mean.

TABLE II: Computational complexity of the Euclidean gradients Geometric measures | Complexity for each step (mod $J+K$)
---|---
LEM (Proposition 8) | $O(2M^{4})+O(N^{2}M)$
AIRM (Proposition 9) | $O(M^{4})+O(2N^{2}M)$
JBLD (Proposition 10) | $O(2M^{3})+O(2NM^{2})+O(2N^{2}M)$
SKLD (Proposition 11) | $O(4M^{3})+O(2NM^{2})+O(2N^{2}M)$

From TABLE II, we notice that computation of gradients of the divergences,
i.e., the JBLD and the SKLD, costs less time compared with the Riemannian
distances, i.e., the LEM and the AIRM. Main reason is again the latter depend
on matrix logarithm.

##  V Simulation Results

In this section, we perform simulations to verify the performance advantage of
the detectors proposed in the current paper, which are compared with the
state-of-the-art counterparts.

###  V-A Environment Setup

The simulations are performed in a non-homogeneous clutter, specifically, in a
Gaussian clutter in the presence of interferences. We generate the sample data
by resorting to an $N$-dimensional complex circular Gaussian distribution with
zero mean and the known covariance matrix

$\bm{C}=\sigma_{c}^{2}\bm{C}_{0}+\sigma_{n}^{2}\bm{I},$ (51)

where $\sigma_{c}^{2}\bm{C}_{0}$ denotes the clutter with $\sigma_{c}^{2}$ the
clutter power while $\sigma_{n}^{2}\bm{I}$ is the thermal noise with
$\sigma_{n}$ the noise power. Therefore, the clutter-to-noise ratio (CNR) is
given by

$CNR=\frac{\sigma_{c}^{2}}{\sigma_{n}^{2}}.$ (52)

The structure of the CCM $\bm{C}_{0}$ is Gaussian shaped with one-lag
correlation coefficient $\rho$, whose entries are given by

$[\bm{C}_{0}]_{i,j}=\rho^{|i-j|}\exp\left(\operatorname{i}2\pi
f_{c}(i-j)\right),\quad i,j=1,2,\ldots,N.$ (53)

Here, $f_{c}$ is the normalized Doppler frequency. $K$ secondary HPD matrices
derived from the diagonal loading formalism (3) are employed to estimate the
CCM matrix as $\bm{R}_{\mathcal{G}}$. The HPD matrix $\bm{R}_{D}$ in the CUT
is computed by the sample data $\bm{x}_{D}$. In the following, the parameters
are chosen as $\sigma_{n}^{2}=1$, CNR$=25$ dB, $\rho=0.95$ and $f_{c}=0.1$.

###  V-B The Training Data

The dimension of the sample data is set to be $N=8$. The normalized Doppler
frequency of target signal is set to $f_{s}=0.2$. Two interferences are
injected into the secondary data with the normalized Doppler frequency
$f=0.22$. The training dataset consists of two subsets with the size of $2000$
each: the set of CCM and the set of HPD matrices containing a signal with
SCR$=25$ dB. Fig. 2 shows the distance between the CCMs and the HPD matrices
with a target signal for different measures. In particular, within the cases
of AIRM, LEM and JBLD, the clutter-clutter distances are more scattered
compared with the clutter-target distances, while conversely the clutter-
target distances are more scattered under the SKLD.

Figure 2: Clutter-target distance for different measures

###  V-C Comparison with Relevant Algorithms

In order to verify the effectiveness of the proposed detectors, we compare the
proposed methods with several well-received algorithms. For convenience, the
following abbreviations are adopted.

  * •

AMF: The traditional adaptive matched filter [64].

  * •

Benchmark: The AMF with known covariance matrix. It is the optimal performance
for the types of AMFs.

  * •

TBD-MIG detectors: The MIG detectors with the total square loss (TSL), the
total von-Neumann (TVN) divergence, and the total log-determinant (TLD)
divergence [42, 43].

Unlike the AMF, the optimal performance of MIG detectors is not the MIG
detectors with the known CCM since the detection performance is closely
related to the discrimination between the target signal and the clutter. To
decrease the computational load, we choose the probability of false alarm as
$P_{fa}=10^{-3}$. A number of $100/P_{fa}$ independent trials are repeated to
estimate the threshold, while $2000$ independent trials are repeated to
estimate the probability of detection $P_{d}$.

###  V-D Simulation Results and Discussions

By using the training dataset, we derive the three projection matrices that
transforms the $N\times N$ HPD matrices to $M\times M$ HPD matrices for $M=8$,
$6$, $4$, and $2$, respectively. We then perform the signal detection on these
manifolds for different size of $K$ secondary data, where $K=M,1.5M$, and
$2M$, respectively. Statistically, as $K$ increases, the estimate accuracy of
the CCM improves, that will certainly affect the detection performance. Figs.
3, 4 and 5 plot the $P_{d}$ vs SCR for the proposed MIG detectors and their
corresponding counterparts as well as the TBD-MIG detectors and the AMF under
different sizes of secondary data. The AMF with the known CCM is also provided
as a benchmark. Figs. 3, 4 and 5 show that the detection performances of all
the considered detectors improve as $K$ becomes larger. In Fig. 3, the MIG
detectors can still work well while the $P_{d}$ of the AMF is very low,
because that the estimate accuracy of the SCM is worse when $K=M$. It should
also be noticed that all the MIG detectors with manifold projection have
better performances compared with their unprojected counterparts, namely the
original MIG detectors, and both the projected and unprojected MIG detectors
outperform the AMF except for the SKLD-MIG detector under $K=2M$. In other
words, the manifold projection can promote the discriminative power of HPD
matrices. Moreover, the TBD-MIG detectors outperform the unprojected AIRM and
LEM MIG detectors and both the projected and unprojected SKLD MIG detectors.

Figure 3: The plots of $P_{d}$ vs SCR for $K=M$ in the nonhomogeneous clutter
with two interferences. Figure 4: The plots of $P_{d}$ vs SCR for $K=1.5M$ in
the nonhomogeneous clutter with two interferences. Figure 5: The plots of
$P_{d}$ vs SCR for $K=2M$ in the nonhomogeneous clutter with two
interferences.

To analyze the difference in the detection performance for different measure-
based MIG detectors. Fig. 6 shows the results of $P_{d}$ vs SCRs for different
measures. It is obvious that the JBLD MIG detector has the best performance.
Detection performance of the AIRM, LEM, and TBD is similar and they are better
than the SKLD when $K>M$. It should be noted that it is probably difficult to
determine detector which is universally better compared with the others since
performance of the detection methods can also depend on features of the
clutter. One important future research would be determining the best detector
against a specific type of clutter.

(a) $K=N$ (b) $K=1.5N$ (c) $K=2N$

Figure 6: $P_{d}$ vs SCR for different measures

##  VI Conclusions

In this paper, we proposed a class of learning discriminative MIG detectors in
the unsupervised scenario, and applied them for signal detection in
nonhomogeneous clutter. The sample data was interpreted as an HPD matrix, and
the secondary HPD matrices were used to estimate the CCM. Inspired by the
principle of PCA, we constructed a manifold projection that maps higher-order
HPD matrices to a more discriminative lower-dimensional HPD manifold with
maximum data variance. Learning the projection with maximum data variance
could be formulated as a two-step mini-max optimization problem in the Stiefel
manifold and the lower-dimensional HPD manifold, respectively, which was
solved by the RGD algorithm. Four discriminative MIG detectors were designed
in the lower-dimensional manifold with respect to the LEM distance, the AIRM
distance, the JBLD and SKLD, respectively. Simulation results showed that the
proposed MIG detectors could outperform their state-of-the-art counterparts
and the AMF in nonhomogeneous clutter.

Potential future research includes the distributed target detection and
further studies of optimization problems in Riemannian manifolds of matrices,
such as the Stiefel manifold in the current study. This is certainly based on
deep understanding of the geometric structures of these manifolds. Practical
applications to radar or sonar via the discriminative MIG detectors should be
interesting as well.

##  Appendix A Proof of Proposition 6: The SKLD mean

The SKLD mean of HPD matrices $\\{\bm{R}_{k}\\}_{k\in[K]}$ is the minimizer of
the function

$\displaystyle F(\bm{R})$
$\displaystyle=\sum_{k=1}^{K}d_{S}^{2}(\bm{R}_{k},\bm{R})$
$\displaystyle=\sum_{k=1}^{K}\operatorname{tr}\left(\bm{R}_{k}^{-1}\bm{R}+\bm{R}^{-1}\bm{R}^{-1}\right)-2NK$

defined in $\mathscr{P}(N,\mathbb{C})$. Using the definition (20), gradient of
the function with respect to the Frobenius metric can be obtained as

$\nabla
F(\bm{R})=\sum_{k=1}^{K}\left(\bm{R}_{k}^{-1}-\bm{R}^{-1}\bm{R}_{k}\bm{R}^{-1}\right).$

The stationary condition $\nabla F(\widehat{\bm{R}})=0$, i.e.,

$\sum_{k=1}^{K}\bm{R}_{k}^{-1}=\widehat{\bm{R}}^{-1}\left(\sum_{k=1}^{K}\bm{R}_{k}\right)\widehat{\bm{R}}^{-1},$

can be rewritten, by multiplying $\widehat{\bm{R}}$ from the left and the
right simultaneously, as

$\widehat{\bm{R}}\left(\sum_{k=1}^{K}\bm{R}_{k}^{-1}\right)\widehat{\bm{R}}=\sum_{k=1}^{K}\bm{R}_{k}.$

It is a special (continuous time) algebraic Riccati equation

$\widehat{\bm{R}}\bm{A}\widehat{\bm{R}}=\bm{B},$ (54)

where $\widehat{\bm{R}}\in\mathscr{P}(N,\mathbb{C})$ is unknown and the known
coefficient matrices $\bm{A}$ and $\bm{B}$ are both HPD:

$\bm{A}=\frac{1}{K}\sum_{k=1}^{K}\bm{R}_{k}^{-1},\quad\bm{B}=\frac{1}{K}\sum_{k=1}^{K}\bm{R}_{k}.$
(55)

Next we will solve the Eq. (54). Multiplying by $\bm{A}^{1/2}$ on both sides,
we have

$\bm{A}^{1/2}\widehat{\bm{R}}\bm{A}\widehat{\bm{R}}\bm{A}^{1/2}=\bm{A}^{1/2}\bm{B}\bm{A}^{1/2}.$

Noticing that its left-hand-side is exactly

$\left(\bm{A}^{1/2}\widehat{\bm{R}}\bm{A}^{1/2}\right)^{2},$

we obtain

$\bm{A}^{1/2}\widehat{\bm{R}}\bm{A}^{1/2}=\left(\bm{A}^{1/2}\bm{B}\bm{A}^{1/2}\right)^{1/2},$

and hence

$\widehat{\bm{R}}=\bm{A}^{-1/2}\left(\bm{A}^{1/2}\bm{B}\bm{A}^{1/2}\right)^{1/2}\bm{A}^{-1/2},$

where $\bm{A}$ and $\bm{B}$ are given by (55). This completes the proof.

##  Appendix B Proof of Proposition 8

At step $t$, the loss function with respect to the LEM reads

$\displaystyle\psi(\bm{W})$
$\displaystyle=-\frac{1}{J+K}\sum_{i=1}^{J+K}\left\lVert\operatorname{Log}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)-\operatorname{Log}\bm{\widehat{Z}}_{t}\right\rVert^{2}.$

It suffices to show the Euclidean gradient of the function $F_{i}(\bm{W})$
with respect to the extended Frobenius metric (45), where

$F_{i}(\bm{W})=\left\lVert\operatorname{Log}\left(\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\right)-\operatorname{Log}\bm{\widehat{Z}}_{t}\right\rVert^{2}.$

Writing

$\bm{A}(\varepsilon)=(\bm{W}+\varepsilon\bm{X})^{\operatorname{H}}\bm{R}_{i}(\bm{W}+\varepsilon\bm{X})$
(56)

and using Eq. (20), for an ${N\times M}$ matrix $\bm{X}$, we have

$\displaystyle\langle\nabla$ $\displaystyle
F_{i}(\bm{W}),\bm{X}\rangle=\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}F_{i}(\bm{W}+\varepsilon\bm{X})$
$\displaystyle=\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}\operatorname{tr}\left(\operatorname{Log}\bm{A}(\varepsilon)-\operatorname{Log}\widehat{\bm{Z}}_{t}\right)^{2}$
$\displaystyle=2\operatorname{tr}\left(\left(\operatorname{Log}\bm{A}(0)-\operatorname{Log}\widehat{\bm{Z}}_{t}\right)\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}\operatorname{Log}\bm{A}(\varepsilon)\right).$

Noticing

$\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}\bm{A}(\varepsilon)=\bm{X}^{\operatorname{H}}\bm{R}_{i}\bm{W}+\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{X}$

and applying Lemma 1, the above equality becomes

$\displaystyle\langle\nabla F_{i}$
$\displaystyle(\bm{W}),\bm{X}\rangle=4\operatorname{tr}\Big{(}\left(\operatorname{Log}\bm{A}(0)-\operatorname{Log}\widehat{\bm{Z}}_{t}\right)$
$\displaystyle\times\int_{0}^{1}[(\bm{A}(0)-\bm{I})s+\bm{I}]^{-1}\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{X}$
$\displaystyle\quad\quad\quad\quad\times[(\bm{A}(0)-\bm{I})s+\bm{I}]^{-1}\operatorname{d}\\!s\Big{)}$
$\displaystyle=$ $\displaystyle
4\operatorname{tr}\Big{(}\Big{(}\bm{A}^{-1}(0)\operatorname{Log}\bm{A}(0)-\int_{0}^{1}[(\bm{A}(0)-\bm{I})s+\bm{I}]^{-1}$
$\displaystyle\times\left(\operatorname{Log}\widehat{\bm{Z}}_{t}\right)[(\bm{A}(0)-\bm{I})s+\bm{I}]^{-1}\operatorname{d}\\!s\Big{)}\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{X}\Big{)}.$

From the defintion $\langle\nabla
F_{i}(\bm{W}),\bm{X}\rangle=\operatorname{tr}\left(\left(\nabla
F_{i}(\bm{W})\right)^{\operatorname{H}}\bm{X}\right)$, we immediately have
that

$\displaystyle\nabla F_{i}(\bm{W})$
$\displaystyle=4\bm{R}_{i}\bm{W}\Big{(}\bm{A}^{-1}(0)\operatorname{Log}\bm{A}(0)$
$\displaystyle\quad-\int_{0}^{1}[(\bm{A}(0)-\bm{I})s+\bm{I}]^{-1}\left(\operatorname{Log}\widehat{\bm{Z}}_{t}\right)$
$\displaystyle\quad\quad\quad\quad\times[(\bm{A}(0)-\bm{I})s+\bm{I}]^{-1}\operatorname{d}\\!s\Big{)},$

where $\bm{A}(0)=\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}$, i.e., the matrix
$\bm{V}$ in Proposition 8. This completes the proof.

##  Appendix C Proof of Proposition 10

The following lemma will be used.

######  Lemma 12.

For any invertible matrix $\bm{B}(\varepsilon)$, we have

$\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\det\bm{B}(\varepsilon)=\det\bm{B}(\varepsilon)\operatorname{tr}\left(\bm{B}^{-1}(\varepsilon)\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\bm{B}(\varepsilon)\right).$

Now the loss function can be written as

$\psi(\bm{W})=-\frac{1}{J+K}\sum_{i=1}^{J+K}F_{i}(\bm{W})$

with

$F_{i}(\bm{W})=\ln\det\left(\frac{\bm{A}(0)+\widehat{\bm{Z}}_{t}}{2}\right)-\frac{1}{2}\ln\det\left(\bm{A}(0)\widehat{\bm{Z}}_{t}\right),$

where $\bm{A}(\varepsilon)$ is given by Eq. (56) and
$\bm{A}(0)=\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}$. Using Lemma 12,
definition of the Euclidean gradient gives

$\displaystyle\langle\nabla F_{i}$
$\displaystyle(\bm{W}),\bm{X}\rangle=\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}\ln\det\left(\frac{\bm{A}(\varepsilon)+\widehat{\bm{Z}}_{t}}{2}\right)$
$\displaystyle\quad\quad\quad-\frac{1}{2}\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}\ln\det\left(\bm{A}(\varepsilon)\widehat{\bm{Z}}_{t}\right)$
$\displaystyle=\operatorname{tr}\left(\left(2\left(\bm{A}(0)+\widehat{\bm{Z}}_{t}\right)^{-1}-\bm{A}^{-1}(0)\right)\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{X}\right).$

Consequently, we have

$\nabla
F_{i}(\bm{W})=\bm{R}_{i}\bm{W}\left(2\left(\bm{A}(0)+\widehat{\bm{Z}}_{t}\right)^{-1}-\bm{A}^{-1}(0)\right)$

and

$\nabla\psi(\bm{W})=-\frac{1}{J+K}\sum_{i=1}^{J+K}\nabla F_{i}(\bm{W}),$

that finishes the proof.

##  Appendix D Proof of Proposition 11

Similarly, we write the loss function as

$\psi(\bm{W})=-\frac{1}{J+K}\sum_{i=1}^{J+K}F_{i}(\bm{W}),$

where

$\displaystyle F_{i}(\bm{W})$
$\displaystyle=\frac{1}{2}\operatorname{tr}\Bigg{(}\bigg{(}\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\bigg{)}^{-1}\bm{\widehat{Z}}_{t}$
$\displaystyle\quad\quad\quad\quad+\bm{\widehat{Z}}_{t}^{-1}\bigg{(}\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}\bigg{)}-2\bm{I}\Bigg{)}.$

Euclidean gradient of the function $F_{i}(\bm{W})$ is given by

$\displaystyle\langle\nabla F_{i}$
$\displaystyle(\bm{W}),\bm{X}\rangle=\frac{1}{2}\frac{\operatorname{d}}{\operatorname{d}\\!\varepsilon}\Big{|}_{\varepsilon=0}\operatorname{tr}\left(\bm{A}^{-1}(\varepsilon)\widehat{\bm{Z}}_{t}+\widehat{\bm{Z}}_{t}^{-1}\bm{A}(\varepsilon)\right)$
$\displaystyle=\frac{1}{2}\operatorname{tr}\Big{(}\left(\widehat{\bm{Z}}^{-1}_{t}-\bm{A}^{-1}(0)\widehat{\bm{Z}}_{t}^{-1}\bm{A}^{-1}(0)\right)$
$\displaystyle\quad\quad\quad\quad\times\left(\bm{X}^{\operatorname{H}}\bm{R}_{i}\bm{W}+\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{X}\right)\Big{)}$
$\displaystyle=\operatorname{tr}\left(\left(\widehat{\bm{Z}}^{-1}_{t}-\bm{A}^{-1}(0)\widehat{\bm{Z}}_{t}\bm{A}^{-1}(0)\right)\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{X}\right),$

and consequently, we obtain

$\nabla
F_{i}(\bm{W})=\bm{R}_{i}\bm{W}\left(\widehat{\bm{Z}}^{-1}_{t}-\bm{A}^{-1}(0)\widehat{\bm{Z}}_{t}\bm{A}^{-1}(0)\right).$

Here, $\bm{A}(\varepsilon)$ is given by (56) and
$\bm{A}(0)=\bm{W}^{\operatorname{H}}\bm{R}_{i}\bm{W}$. This completes the
proof.

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