# Reconstructing Human Pose from Inertial Measurements:
A Generative Model-based Compressive Sensing Approach

Nguyen Quang Hieu, Dinh Thai Hoang, Diep N. Nguyen, and Mohammad Abu Alsheikh
N. Q. Hieu, D. T. Hoang, D. N. Nguyen are with School of Electrical and Data
Engineering, University of Technology Sydney, NSW 2007, Australia, emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>hoang.dinh@uts.edu.au).M. A. Alsheikh is with Faculty of Science & Technology,
University of Canberra, Canberra, ACT 2617, Australia, email:
<EMAIL_ADDRESS>
###### Abstract

The ability to sense, localize, and estimate the 3D position and orientation
of the human body is critical in virtual reality (VR) and extended reality
(XR) applications. This becomes more important and challenging with the
deployment of VR/XR applications over the next generation of wireless systems
such as 5G and beyond. In this paper, we propose a novel framework that can
reconstruct the 3D human body pose of the user given sparse measurements from
Inertial Measurement Unit (IMU) sensors over a noisy wireless environment.
Specifically, our framework enables reliable transmission of compressed IMU
signals through noisy wireless channels and effective recovery of such signals
at the receiver, e.g., an edge server. This task is very challenging due to
the constraints of transmit power, recovery accuracy, and recovery latency. To
address these challenges, we first develop a deep generative model at the
receiver to recover the data from linear measurements of IMU signals. The
linear measurements of the IMU signals are obtained by a linear projection
with a measurement matrix based on the compressive sensing theory. The key to
the success of our framework lies in the novel design of the measurement
matrix at the transmitter, which can not only satisfy power constraints for
the IMU devices but also obtain a highly accurate recovery for the IMU signals
at the receiver. This can be achieved by extending the set-restricted
eigenvalue condition of the measurement matrix and combining it with an upper
bound for the power transmission constraint. Our framework can achieve robust
performance for recovering 3D human poses from noisy compressed IMU signals.
Additionally, our pre-trained deep generative model achieves signal
reconstruction accuracy comparable to an optimization-based approach, i.e.,
Lasso, but is an order of magnitude faster.

###### Index Terms:

Compressive sensing, generative models, inertial measurement units, human pose
estimation, edge computing.

##  I Introduction

###  I-A Motivation

The ability to estimate human body movements plays a key role in emerging
human-computer interaction paradigms such as virtual reality (VR) and extended
reality (XR) [1]. By correctly estimating the 3D position and orientation of
the human body, VR/XR applications such as gaming, virtual offices, and smart
factories can offer a more interactive and immersive experience for users.
Highly accurate solutions for estimating 3D human movements usually rely on
images or videos, which typically require multi-camera calibrated systems [1,
2]. However, the multi-camera systems are limited to capturing human outdoor
activities (e.g., due to sensitive information conveyed in the images/videos)
and severely degraded with poor lightning conditions [1]. Specifically, for
VR/XR applications deployed over wireless systems, e.g., 5G and beyond,
leveraging such images and videos from multi-camera systems for human body
estimation purposes is costly in terms of bandwidth utilization and computing
efficiency [3, 4]. This demands a more effective approach to achieve highly
accurate estimation of human body movements in VR/XR applications deployed
over wireless systems [5].

Fortunately, the inertial measurement unit (IMU) (i.e., accelerometer,
gyroscope, and magnetometer) offers a promising solution to this problem. The
systems based on IMU do not suffer from limitations in camera-based systems.
The IMU sensors can track human movements by measuring the acceleration and
orientation of human body parts, e.g., head orientation or arm/leg movement,
regardless of image sensitivity information and lightning conditions, making
them more suitable for indoor and outdoor VR/XR applications [6, 7]. As the
IMU sensors are typically worn on the body, e.g., wrists, head, or ankles, the
information measured from the IMU can help to track the movement of the body
segments relative to each other. For example, utilizing IMU information such
as the orientation of VR headsets can help the system better predict the user
preferences in VR streaming applications [8, 9]. Moreover, acceleration
readings from the IMU sensors can help to track user step count, thereby
increasing the accuracy of outdoor pedestrian localization[10]. Furthermore,
combining IMU information with a kinematic model of the human body can
simulate the entire body movement of the user in a complete positioning and
sensing system [7, 11]. With such enormous potential, the IMU sensors have
been widely deployed as a standard setting inside mobile phones, tablets, VR
headsets, and VR controllers.

###  I-B Related Works

Unlike solutions for reconstructing movements of independent parts of the
human body, e.g., head or arm, estimating a full body movement of the user
usually requires a set of IMU sensors placed on different parts of the body or
attached to a suit [7]. With a set of IMU sensors, ranging from 3 to 17
sensors, the full body movements can be fully reconstructed with the help of
optimization-based techniques [12, 6] and learning-based techniques [13, 14,
15]. In [12], a Kalman Filter was utilized to correct the kinematics of the 3D
human model, given the joint uncertainties of sensor noise, angular velocity,
and acceleration of the IMU sensors. In [6], the authors proposed a new
optimization approach based on exponential mapping, which transforms the
orientation and acceleration values into equivalent energy functions. After
carefully calibrating between the IMU sensors’ coordinate frames and the 3D
human body’s coordinate frames, the optimization objective can be formulated
as minimizing the set of energy functions over the entire sequence of
collected data.

Different from offline optimization approaches [12, 6], learning-based
approaches can achieve real-time estimation based on pre-trained deep learning
models [13, 14, 15]. In [13], the authors proposed a deep learning approach
based on a recurrent neural network that trains on the entire sequence of data
at the training phase. During the testing phase, the pre-trained model can
estimate the corrected body pose of the user in a shorter time window. In
[14], the authors extended this idea by using a recurrent neural network
combined with a physics-aware motion optimizer that enhances the tracking
accuracy for a longer time window. The authors in [15] reported similar
advantages of using a physics-aware motion optimizer with fewer IMU sensors
being used.

###  I-C Challenges and Proposed Solutions

Although there has been significant effort in improving the precision of human
pose estimation in 3D environments with IMU sensors[12, 7, 6, 13, 14, 15],
there is a lack of human pose estimation approach for VR/XR applications
deployed over wireless networks, where the estimation ability can be strictly
constrained by channel quality, power transmission, and tradeoff between
latency and accuracy of the solution. Deploying the human pose estimation
frameworks over the wireless networks is a non-trivial problem as the
transmitted data is more exposed to channel noise, e.g., due to channel
quality and channel interference. At this point, the existing works overlook
the presence of noise in the received IMU data as the noise may significantly
degrade the reconstruction accuracy of the human pose estimation task,
resulting in a poor quality of experience for the user in a virtual
environment. In addition, the current approaches do not consider the potential
redundancy of the IMU data before transmitting it to the receiver. The
potential redundancy of information from each IMU sensor, such as orientation
and acceleration values at high frequencies (see Fig. 2), can further reduce
the number of data samples that need to be transmitted. As a result,
exploiting the data redundancy can enhance channel utilization and reduce
power consumption for the wireless systems [16].

To the best of our knowledge, there is a lack of studies that address the
above problems, i.e., the noisy IMU data transmission and the potential
redundancy of the IMU data in reconstructing human pose over wireless systems.
To address these problems, we propose a novel framework based on compressive
sensing and generative modeling. On the one hand, the compressive sensing
technique is utilized to down-sample the IMU signals before transmitting the
signals over the wireless channel [17]. Based on rigorous down-sampling (or
projection) techniques, the compressive sensing framework is promising to
reduce the redundant IMU signals. As a result, this approach can not only
reduce the energy consumption associated with IMU data acquisition and
transmission but also enhance channel utilization as the transmitted data is
in a more compressed form. On the other hand, a generative model (e.g., a
variational auto-encoder [18]) deployed at the receiver can help the receiver
make a robust estimation of the human body pose (e.g., through data denoising
and data recovery capabilities), given the noisy compressed IMU measurements
transmitted over a wireless channel. Unlike the optimization-based techniques
[12, 6] and learning-based techniques [13, 14, 15], our proposed generative
model can handle noisy data more effectively and also can exploit the
potential sparsity patterns in the data. To summarize, the main contributions
of our work are as follows:

  * •

We propose an innovative framework based on compressive sensing and generative
modeling techniques for human pose estimation from IMU sensors toward VR/XR
applications deployed over wireless systems. The proposed framework can
accurately recover the original IMU signals from noisy compressed signals
transmitted over a wireless channel. The combination of compressive sensing
and generative modeling generates potential benefits to the system such as
enhancing channel utilization, effective data sensing, and power-efficient
wireless communications.

  * •

We develop a novel design for the measurement matrix at the transmitter, which
helps to transform the high-dimensional signal into a compressed form. Our
measurement matrix design extends the set-restricted eigenvalue condition of
existing generative model-based compressive sensing approaches to a general
setting, which considers the impacts of a wireless communication channel on
data recovery. With rigorous analysis, we prove that our proposed measurement
matrix enables our proposed framework to outperform other deep learning and
optimization approaches, in terms of accuracy and latency of signal
reconstruction process.

  * •

We show that the proposed framework can achieve signal reconstruction accuracy
comparable to the optimization-based approach, i.e., Lasso, with an order of
magnitude faster than Lasso. The fast reconstruction ability of the generative
model makes it a promising solution for VR/XR applications with stringent
latency requirements.

  * •

We demonstrate a practical use case of the generative model that can generate
missing IMU signals, thus creating synthetic body movements for the users
without using input IMU signals. This ability of the generative model is very
useful for potential VR/XR applications over wireless systems as the missing
input data usually happens due to the lossy nature of the wireless
environment.

Figure 1: An illustration of our proposed system model. A set of synchronized
IMU sensors produces a sequence of data, e.g., orientation and acceleration,
and compressive sensing down-samples the data sequence into a shorter
sequence. The down-sampled sequence of IMU data is transmitted over a noisy
channel. The receiver uses a deep generative model to recover the original
data sequence from received signals.

The organization of the paper is as follows. Section II describes the overview
of the system model and preliminaries of compressive sensing and generative
modeling. In Section III, we formulate the problem as a reconstruction error
minimization problem, subject to a power transmission constraint. In Section
IV, we extensively evaluate the performance of the proposed framework with
other baselines, such as an optimization-based approach and a deep learning-
based approach. We also show that the proposed generative model can generate
missing IMU data features, thus directly creating smooth synthetic body
movements for the users. Finally, Section V concludes the paper.

##  II System Overview and Preliminaries

The proposed system model is illustrated in Fig. 1. At the transmitter side,
the user’s body is equipped with a set of 17 IMU sensors placed on standard
positions as in commercial systems [7]. The set of synchronized IMU sensors
produces a sequence of data, e.g., orientation and acceleration, which is
usually aggregated together at a central IMU node (e.g., IMU sensor placed on
the user’s spine). Compressive sensing down-samples the data sequence into a
shorter sequence through matrix multiplication. The down-sampled data sequence
is transmitted over a wireless noisy channel, e.g., a Gaussian channel. On the
receiver side, the edge server uses a deep generative model to recover the
original data sequence from the noisy down-sampled data sequence. From the
recovered IMU data and a kinematic human body model (e.g., SMPL [19]), the
generative model can further generate the 3D avatar model with the corrected
pose.

As described, the proposed framework consists of two main components that are
(i) compressive sensing for the transmitter-receiver communication and (ii) a
generative model for recovering the signals at the receiver, i.e., the edge
server. In the following, we describe the fundamentals of compressive sensing,
generative models, and generative model-based compressive sensing for an end-
to-end learning system.

###  II-A Compressive Sensing

As illustrated in Fig. 1, we have a sequence of data is a real-valued, finite-
length one-dimensional signal $\mathbf{x}^{*}\in\mathbb{R}^{n}$. With
compressing sensing, we want to down-sample the signal $\mathbf{x^{*}}$ before
transmitting it to the receiver. For that, we have a measurement matrix
$\mathbf{A}\in\mathbb{R}^{m\times n}$ to make a linear projection from a
higher dimensional vector $\mathbf{x}^{*}\in\mathbb{R}^{n}$ to a lower
dimensional vector $\mathbf{y}\in\mathbb{R}^{m}$ ($m<n$). Usually, $n$ is
referred to as the length of the original vector and $m$ is the number of
measurements from that vector. In particular, the $m$-dimensional signal being
transmitted over the channel is:

$\mathbf{y}=\mathbf{Ax^{*}}.$ (1)

The received signal at the receiver can be corrupted by noise. In the case of
a Gaussian channel, the received signal at the receiver is [20, 17]:

$\mathbf{\hat{y}}=\mathbf{Ax^{*}}+\boldsymbol{\eta},$ (2)

where $\boldsymbol{\eta}\in\mathbb{R}^{m}$ is a Gaussian noise vector with
zero mean and $\sigma_{N}$ standard deviation, i.e., element $\eta_{i}$
($i=1,2,\ldots,m$) of $\boldsymbol{\eta}$ follows a Gaussian distribution
$\eta_{i}\sim\mathcal{N}(0,\sigma_{N}^{2})$. As observed from equation (2),
the signal $\mathbf{\hat{y}}\in\mathbb{R}^{m}$ is a compressed form of
$\mathbf{x}^{*}\in\mathbb{R}^{n}$. To recover the signal $\mathbf{x}^{*}$ from
the received signal $\mathbf{\hat{y}}$, the receiver needs to solve the
following quadratically constrained optimization problem [20]:

$\displaystyle\mathcal{P}_{0}:\quad$
$\displaystyle\min_{\mathbf{x}}\|\mathbf{x}\|_{1},$ (3a) subject to
$\displaystyle\|\mathbf{Ax}-\mathbf{\hat{y}}\|_{2}\leq\|\boldsymbol{\eta}\|_{2},$
(3b)

where the term $\|\mathbf{x}\|_{p}$ denotes the $l_{p}$ norm
($p=0,1,2,\ldots$) of the vector $\mathbf{x}$, i.e., [20]

$\|\mathbf{x}\|_{p}=(\sum_{j=1}^{n}|x_{j}|^{p})^{1/p}.$ (4)

The problem $\mathcal{P}_{0}$ in (3) forms an underdetermined system, that is,
a system in which there are multiple solutions to the system. To guarantee the
unique recovery of the signal, compressive sensing relies on the two main
assumptions about the signal $\mathbf{x}^{*}$ and the measurement matrix
$\mathbf{A}$. First, the signal $\mathbf{x^{*}}$ has sparsity property.
Second, the measurement matrix $\mathbf{A}$ satisfies specific conditions that
are either Restricted Isometry Property (RIP) or Restricted Eigenvalue
Condition (REC). The definitions of sparsity, RIP, and REC are as follows [20,
17].

######  Definition 1 (Sparsity).

The support of a vector $\mathbf{x}\in\mathbb{R}^{n}$ is the index set of its
nonzero entries, i.e.,

$\sup(\mathbf{x}):=\big{\\{}j\in\\{1,2,\ldots,n\\}:x_{j}\neq 0\big{\\}}.$

The vector $\mathbf{x}\in\mathbb{R}^{n}$ is called $k$-sparse if at most $k$
of its entries are nonzero, i.e., if

$\|\mathbf{x}\|_{0}=\mathbf{card}\big{(}\sup(\mathbf{x})\big{)}\leq k,$

where $\mathbf{card}(\cdot)$ is the cardinality (number of elements) and
$\|\cdot\|_{0}$ is $l_{0}$ norm.

In practice, the sparsity property of the interested signal $\mathbf{x}$ is
usually relaxed to nearly $k$-sparse, meaning that there are $n-k$ entries of
the vector $\mathbf{x}$ are approximately zero. In Fig. 2, we illustrate the
nearly $k$-sparse acceleration data from the IMU dataset in [13]. Note that we
use the Fast Fourier Transform (FFT) in Fig. 2 for illustration purposes only,
and our proposed learning algorithm will not utilize the FFT.

Figure 2: Illustration of acceleration reading from an IMU sensor placed on
the left wrist of the user (top figure) and the Fast Fourier Transform (FFT)
of the x-axis acceleration data (bottom figure). The FFT reveals nearly
$k$-sparse property of the IMU signal in which a few low-frequency
coefficients have dominant values. As a result, the redundancy of the data can
be approximated by considering the $k$ largest coefficients and assuming the
rest coefficients are zero.

######  Definition 2 (Restricted Isometry Property).

Let $S_{k}\subset\mathbb{R}^{n}$ be the set of $k$-sparse vectors. For some
parameter $\delta\in(0,1)$, a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ is
said to satisfy $\text{RIP}(k,\delta)$ if $\forall\mathbf{x}\in S_{k}$,

$(1-\delta)\|\mathbf{x}\|_{2}\leq\|\mathbf{Ax}\|_{2}\leq(1+\delta)\|\mathbf{x}\|_{2}.$

######  Definition 3 (Restricted Eigenvalue Condition).

Let $S_{k}\subset\mathbb{R}^{n}$ be the set of $k$-sparse vectors. For some
parameter $\gamma>0$, a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ is said
to satisfy $\text{REC}(k,\gamma)$ if $\forall\mathbf{x}\in S_{k}$,

$\|\mathbf{Ax}\|_{2}\geq\gamma\|\mathbf{x}\|_{2}.$

Intuitively, RIP implies that $\mathbf{A}$ approximately preserves Euclidean
norms ($l_{2}$ norms) for sparse vectors, and REC implies that sparse vectors
are far from the nullspace of $\mathbf{A}$ [21]. Given the sparsity of
$\mathbf{x}^{*}$ and RIP/REC property of the chosen matrix $\mathbf{A}$, it
has been shown that the recovered signal $\mathbf{\hat{x}}$ is the unique
solution of the problem $\mathcal{P}_{0}$ in (3), i.e.,
$\mathbf{\hat{x}}\approx\mathbf{x}^{*}$ [20]. As the sparsity of the signal
depends on the natural domain of the signal, the solution for
$\mathcal{P}_{0}$ depends on two aspects that are (i) the choice of
measurement matrix $\mathbf{A}$ and (ii) the choice of recovery method, i.e.,
optimization solver for $\mathcal{P}_{0}$. In conventional compressive sensing
methods, the common choices for such aspects are (i) a Gaussian matrix and
(ii) a convex optimization solver like Lasso (Lease absolute shrinkage and
selection operator) [22, 17]. Note that in this work, we do not explicitly
analyze the sparsity of the IMU signal but rely on approximation methods, such
as Lasso and generative models, to solve the optimization problem. In this
way, we do not need to pay the extra cost of signal processing through
transformations, such as Fourier transform or Wavelet transform, at the
transmitter [17, 20, 23, 21].

By using the definition of the $l_{p}$ norm in (4), $\|\mathbf{x}\|_{1}$ is a
convex function, and $\mathcal{P}_{0}$ is a $l_{1}$ minimization problem with
quadratic constraint. The solution of $\mathcal{P}_{0}$ is equivalent to the
output of the Lasso, which consists of solving $\mathcal{P}_{1}$, for some
parameter $\tau\geq 0$ [20]:

$\displaystyle\mathcal{P}_{1}:\quad$
$\displaystyle\min_{\mathbf{x}}\|\mathbf{Ax}-\mathbf{\hat{y}}\|_{2},$ (5a)
subject to $\displaystyle\|\mathbf{x}\|_{1}\leq\tau.$ (5b)

In practice, the solution of Lasso is equivalent to solving the Lagrangian of
the problem $\mathcal{P}_{1}$ above, for some parameter $\lambda\geq 0$, i.e.,
[20, Equation 3.1]:

$\min_{\mathbf{x}}\|\mathbf{Ax}-\mathbf{\hat{y}}\|_{2}^{2}+\lambda\|\mathbf{x}\|_{1}.$
(6)

Intuitively, the $l_{1}$ penalty term $\lambda\|\mathbf{x}\|_{1}$ in (6)
enforces sparsity (Definition 1) by adding penalty proportional to the
absolute values of the coefficients of $\mathbf{x}$. As a result, the sparsity
assumption ($k$-sparse or nearly $k$-sparse) in the structure of the signal
has an impact on the performance of the Lasso solver. Recall that in this
work, we do not explicitly analyze the sparsity of the IMU signal, which
usually causes extra costs through the Fourier/Wavelet transform at the
transmitter. In addition, the solutions relying on the sparsity assumption are
known to yield poor recovery performance when the linear measurement is not
sufficient (i.e., the value of $m$ is too small), or the considered signal has
a small number of dimensions (i.e., the value of $n$ is not sufficiently
large) [23, 21]. This motivates us to utilize generative models, such as
variational auto-encoders (VAEs) [18] and generative adversarial networks
(GANs) [24], as an alternative for the use of sparsity assumption with a
convex optimization solver like Lasso.

###  II-B Generative Models

Generative models are a type of machine learning that can be used for modeling
the complex distribution of large-scale datasets. In the context of
compressive sensing, a generative model can be used to estimate the
distribution of the input signals. After the generative model is trained with
a training set, it can generate a new data sample that is similar to the
samples drawn from the original set [25]. Intuitively, the generative model
can learn and synthesize the underlying distribution of the high dimensional
and complex data, which eliminates the sparsity assumption about the data
structure of conventional compressive sensing techniques.

A generative model describes a probability density function $p$:
$\mathcal{X}\rightarrow\mathbb{R}$ ($\mathcal{X}$ is a finite set) through an
unobserved, or “latent”, variable $\mathbf{z}$. The probability density
function is then calculated by:

$p(\mathbf{x})=\int_{\mathbf{z}}p(\mathbf{x|z})p(\mathbf{z})d\mathbf{z},$ (7)

where $\forall\mathbf{x}\in\mathcal{X}$, the probability $p(\mathbf{z})$ is
the prior, and the forward probability $p(\mathbf{x|z})$ is the likelihood
[25]. In practice, this probability density function is usually parameterized
by a model $\boldsymbol{\theta}$ (e.g., a deep neural network). In such a
case, equation (7) can be rewritten as follows [25]:

$p_{\boldsymbol{\theta}}(\mathbf{x})=\int_{\mathbf{z}}p_{\boldsymbol{\theta}}(\mathbf{x|z})p(\mathbf{z})d\mathbf{z}.$
(8)

The integral in (8) cannot easily be computed as the likelihood
$p(\mathbf{x|z})$ is computationally expensive with conventional methods such
as maximum likelihood, especially for large-scale datasets. In this work, we
develop our generative model based on a popular class of generative models,
which are called variational auto-encoders (VAEs), first introduced in [18].
As opposed to other generative models such as GANs [24], VAEs can generate
more dispersed samples over the data and can learn complex data distributions
[25]. In addition, VAEs are better for data inference, which is suitable for
our generative model that wants to exploit the hidden “sparsity” patterns in
the IMU data.

In VAEs, besides the likelihood parameterized by a decoder (deep neural
network), the probability density function
$p_{\boldsymbol{\theta}}(\mathbf{x})$ is conditioned through an encoder
parameterized by another deep neural network
$q_{\boldsymbol{\phi}}(\mathbf{z|x})$. The encoder approximates the true but
intractable posterior $p_{\boldsymbol{\theta}}(\mathbf{z|x})$. To train a VAE,
we optimize a variational lower bound on $\log
p_{\boldsymbol{\theta}}(\mathbf{x})$, called evidence lower-bound (ELBO). It
is defined as follows [25]:

$\mathcal{L}_{\boldsymbol{\theta},\boldsymbol{\phi}}(\mathbf{x})=\int_{\mathbf{z}}q_{\boldsymbol{\phi}}(\mathbf{z|x})\log\frac{p_{\boldsymbol{\theta}}(\mathbf{x|z})p(\mathbf{z})}{q_{\boldsymbol{\phi}}(\mathbf{z|x)}}d\mathbf{z}.$
(9)

The newly introduced density function $q_{\boldsymbol{\phi}}(\mathbf{z|x})$ is
referred to as the variational (approximate) posterior with
$\boldsymbol{\phi}$ defined as the variational parameters.

As the encoder $q_{\boldsymbol{\phi}}(\mathbf{z|x})$ is used to approximate
the posterior $p_{\boldsymbol{\theta}}(\mathbf{z|x})$, exact sampling from the
posterior is straightforward through an unbiased Monte Carlo estimate of
$\mathcal{L}$ [25]:

$\mathcal{\hat{L}}_{\boldsymbol{\theta,\phi}}(\mathbf{x})=\log\frac{p_{\boldsymbol{\theta}}(\mathbf{x|z})p(\mathbf{z})}{q_{\boldsymbol{\phi}}(\mathbf{z|x)}},\text{
where }\mathbf{z}\leftarrow q_{\boldsymbol{\phi}}(\cdot|\mathbf{x}).$ (10)

The notation $\mathbf{z}\leftarrow q_{\boldsymbol{\phi}}(\cdot|\mathbf{x})$
means that $\mathbf{z}$ is sampled from the approximate posterior distribution
$q_{\boldsymbol{\phi}}(\mathbf{z|x})$. If the process used to generate
$\mathbf{z}$ from $q_{\boldsymbol{\phi}}(\mathbf{z|x})$ is differentiable with
respect to $\boldsymbol{\phi}$, the function $\mathcal{\hat{L}}$ can be
differentiated with respect to $\boldsymbol{\theta}$ and $\boldsymbol{\phi}$
by using a stochastic gradient decent estimator. Once $\mathcal{L}$ in (9) is
optimized, we can approximate the true probability density function
$p(\mathbf{x})$ through the learned neural network with parameters
$\boldsymbol{\theta}$, i.e., $p_{\boldsymbol{\theta}}(\mathbf{x})\approx
p(\mathbf{x})$. In other words, we can generate the new data samples from the
learned probability density function.

Figure 3: The proposed CS-VAE learning algorithm with a novel measurement
matrix at the transmitter and the generative model, i.e., a VAE, at the
receiver. The transmitted signal at the transmitter is the $m$-dimensional
vector $\mathbf{y}$, which is a compressed version of the original
$n$-dimensional vector $\mathbf{x}^{*}$. At the receiver, the VAE recovers the
original signal, i.e., $\mathbf{\hat{x}}\approx\mathbf{x}^{*}$, from a noisy
and compressed measurement $\mathbf{\hat{y}}$.

###  II-C Generative Model-based Compressive Sensing

In the context of compressive sensing, the data sample $\mathbf{x}$ from the
training set is, however, not fully observable, i.e., our model can only
observe the noisy compressed or down-sampled version $\mathbf{\hat{y}}$.
Replacing $\mathbf{x}$ with $\mathbf{y}=\mathbf{Ax}$, the unbiased Monte Carlo
estimation of the ELBO in (10) is rewritten as:

$\mathcal{\hat{L}}_{\boldsymbol{\theta,\phi}}(\mathbf{y})=\log\frac{p_{\boldsymbol{\theta}}(\mathbf{Ax|z})p(\mathbf{z})}{q_{\boldsymbol{\phi}}(\mathbf{z|Ax})},\text{
where }\mathbf{z}\leftarrow
q_{\boldsymbol{\phi}}(\cdot|\mathbf{y}=\mathbf{\hat{y}}).$ (11)

As observed from the above equation, the generative model cannot directly
generate the data sample $\mathbf{x}$ from the compressed observation
$\mathbf{\hat{y}}$ without prior knowledge of the measurement matrix
$\mathbf{A}$. In other words, the measurement matrix $\mathbf{A}$ is assumed
to be known at the generative model [23, 21]. Recall that the generative model
is deployed at the receiver, therefore, the assumption about sharing prior
information, e.g., a codebook, between the transmitter and receiver, is
commonly used in the source and channel coding methods [26, 27]. Given the
setting above, the solution of (5) is equivalent to the output of the
generator $G(\mathbf{z})$ of the problem $\mathcal{P}_{2}$, for some
$\upsilon\geq 0$, as follows [21]:

$\displaystyle\mathcal{P}_{2}:\quad$
$\displaystyle\min_{\mathbf{z}}\|\mathbf{A}G(\mathbf{z})-\mathbf{\hat{y}}\|_{2},$
(12a) subject to $\displaystyle\|G(\mathbf{z})\|_{1}\leq\upsilon.$ (12b)

The generator $G(\mathbf{z})$ is defined as a function
$G:\mathbb{R}^{k}\rightarrow\mathbb{R}^{n}$ mapping a latent vector
$\mathbf{z}$ to the mean of the conditional distribution
$p_{\boldsymbol{\theta}}(\mathbf{x|z})$. Given the observation
$\mathbf{\hat{y}}$ as the input of the model, the latent vector $\mathbf{z}$
is obtained by sampling from the posterior distribution
$q_{\boldsymbol{\phi}}(\cdot|\mathbf{y})$ in (11). After that, the generator
$G(\mathbf{z})$ produces the output vector $\mathbf{\hat{x}}$ from this latent
vector $\mathbf{z}$, i.e., $G(\mathbf{z})=\mathbf{\hat{x}}$. As a result, the
minimizer $\mathbf{z}^{*}$ of the optimization problem of $\mathcal{P}_{2}$ in
(12) makes $G(\mathbf{z}^{*})\approx\mathbf{x}^{*}$ [23].

In comparison with (5), the variable vector $\mathbf{x}$ is now replaced by
the generative function $G(\mathbf{z})$ in (12). As a result, when one can
optimize the objective $\mathcal{P}_{2}$ in (12), the generator
$G(\mathbf{z})$ can generate the new samples which are similar to the original
vector $\mathbf{x}^{*}$. Recall that under the compressive sensing setting,
our generative model does not observe the full observation of the signals as
in the conventional setting of generative modeling, i.e., learning directly
$p_{\boldsymbol{\theta}}(\mathbf{x})\approx p(\mathbf{x})$. In the compressive
sensing setting, the generative model can only observe the noisy and
compressed signal $\mathbf{\hat{y}}$. Therefore, the optimization objective
defined in (12) is to indirectly optimize the generative model via the
observation $\mathbf{\hat{y}}$, given the measurement matrix $\mathbf{A}$.
Thereafter, the space of signals that can be recovered with the generative
model is given by the range of the generator function, i.e.,

$S_{G}=\\{G(\mathbf{z}):\mathbf{z}\in\mathbb{R}^{k}\\}.$ (13)

As the range of the signals is now transformed into the latent space
$\mathbf{z}\in\mathbb{R}^{k}$, the RIP and REC properties of the measurement
matrix $\mathbf{A}$ no longer guarantee the accuracy of the recovered signals.
With the generative model-based compressive sensing, the measurement matrix
$\mathbf{A}$ is required to satisfy a Set-Restricted Eigenvalue Condition
(S-REC), which is a generalized version of REC [23], i.e.,

######  Definition 4 (Set-Restricted Eigenvalue Condition).

Let $S\subseteq\mathbb{R}^{n}$, for some parameters $\gamma>0$ and $\kappa\geq
0$, a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ is said to satisfy the
$\text{S-REC}(S,\gamma,\kappa)$ if $\forall\mathbf{x}_{1},\mathbf{x}_{2}\in
S$,

$\|\mathbf{A(x_{1}-x_{2})}\|_{2}\geq\gamma\|\mathbf{x_{1}-x_{2}}\|_{2}-\kappa.$

Intuitively, the S-REC property generalizes the REC property to an arbitrary
set of vectors $S$ instead of considering the set of approximately sparse
vectors $S_{k}$ [21]. This generalization makes S-REC a nice property for
solving a compressive sensing problem with a stochastic gradient estimator via
deep neural networks.

##  III Problem Formulation and Proposed Learning Algorithm

In this section, we utilize the generative model-based compressive sensing
framework for our system model in Fig. 1. The presence of the communication
channel between the transmitter and the receiver makes the reconstruction of
IMU signals with generative model-based compressive sensing much more
challenging. In particular, using the measurement matrices in [23] and [21]
cannot guarantee the power constraint of the transmitter. Conventional
normalization techniques like $l_{2}$ normalization [28] are not applicable as
they yield nonlinear projection from $\mathbf{x}$ to $\mathbf{y}$, thus making
the receiver cannot recover the original signal. For this, we propose a new
measurement matrix that (i) ensures the power constraint for the transmitter
and (ii) satisfies the S-REC property of generative model-based compressive
sensing. The learning algorithm with the newly designed measurement matrix is
described in the following.

The proposed learning process, which we refer to as “CS-VAE” (Compressive
Sensing-based Variational Auto-Encoder), is illustrated in Fig. 3. At the
transmitter, we have the vector $\mathbf{x}^{*}\in\mathbb{R}^{n}$ and a
measurement matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$. The output of the
measurement matrix is the signal $\mathbf{y}\in\mathbb{R}^{m}$ in which $m<n$.
The signal $\mathbf{y=Ax^{*}}$ is subjected to the power constraint at the
transmitter, i.e., $\frac{1}{m}\|\mathbf{y}\|_{2}^{2}\leq P_{T}$, where
$P_{T}$ is the transmission power constraint on a single channel use [27, 29].
Details of the power constraint for our framework are further discussed in
Appendix A. In particular, the optimization problem of the proposed learning
model is similar to (12) with an additional power constraint as follows:

$\displaystyle\mathcal{P}_{3}:\quad$
$\displaystyle\min_{\mathbf{z}}\|\mathbf{A}G(\mathbf{z})-\mathbf{\hat{y}}\|_{2},$
(14a) subject to $\displaystyle\frac{1}{m}\|\mathbf{y}\|_{2}^{2}\leq P_{T},$
(14b) $\displaystyle\|G(\mathbf{z})\|_{1}\leq\upsilon.$ (14c)

The power constraint in (14b) poses additional challenges in designing the
measurement matrix $\mathbf{A}$ to ensure that the recovered signal is unique
and similar to the original signal. Specifically, this is a very challenging
quadratically constrained problem [20], and designing a measurement matrix
that satisfies the duo-constraint, i.e., S-REC property and power constraint,
has not been investigated in the literature. Existing generative model-based
compressive sensing approaches [23, 21, 30, 31] cannot be directly applied to
this problem. To address this duo-constraint optimization problem
$\mathcal{P}_{3}$ in (14), we first design a new measurement matrix
$\mathbf{A}$ in Proposition 1 that makes $\mathbf{y=Ax}$ satisfy the power
constraint $\frac{1}{m}\|\mathbf{y}\|_{2}^{2}\leq P_{T}$. After the power
constraint is eliminated, we use the Lagrangian of $\mathcal{P}_{3}$ as a loss
function to train the generative model in a similar manner as in [23].

The proposed measurement matrix for the problem $\mathcal{P}_{3}$ is stated as
follows.

######  Proposition 1 (S-REC with power constraint).

The recovered signal obtained by the generative model-based compressive
sensing method under the power constraint is guaranteed to be a unique
solution if

  * •

$\mathbf{A}$ satisfies S-REC property, and

  * •

Each element $A_{ij}$ (element $j$-th of the $i$-th row) of $\mathbf{A}$ is
drawn i.i.d from a Gaussian distribution with zero mean and variance
$\sigma_{a}^{2}=\frac{P_{T}}{n^{2}d^{2}(d\sigma_{x}+\mu_{x})^{2}}$, i.e.,

$A_{ij}\sim\mathcal{N}\Big{(}0,\frac{P_{T}}{n^{2}d^{2}(d\sigma_{x}+\mu_{x})^{2}}\Big{)},$

where $\sigma_{x}^{2}$ and $\mu_{x}$ are the statistical variance and mean of
the source signals $\mathbf{x}\in\mathbb{R}^{n}$, respectively, and $d>0$ is a
real number derived from the Chebyshev’s inequality.

The proof of Proposition 1 can be found in Appendix A.

######  Remark.

The normal distribution used to generate the random matrix $\mathbf{A}$ in
Proposition 1 contains the mean and variance values of the source signals
$\mathbf{x}$. This assumption of knowing statistical variance and mean values
of the source signals is common in source-channel coding schemes [26]. For
example, in the case of i.i.d Gaussian source with power constraint $P$, these
values are $\sigma_{x}^{2}\approx P$ and $\mu_{x}=0$ [26, Section 9.1]. In the
view of compressive sensing as a source-channel coding scheme, $\mathbf{A}$
can be considered as an encoding function [27]. In addition, as training deep
learning models usually requires access to the training set for pre-processing
and learning, the assumption of knowing the mean and variance values of the
signals is more reasonable and practical than using the i.i.d Gaussian source.
Another parameter in Proposition 1 is $d>0$, a real number that restricts the
random variable $x_{j}$ (element $j$-th of the source signal $\mathbf{x}$) to
the interval $[\mu_{x}-d\sigma_{x},\mu_{x}+d\sigma_{x}]$ with probability is
at least $1-\frac{1}{d^{2}}$. Details of the parameters are discussed in
Appendix A.

As the measurement matrix $\mathbf{A}$ is designed based on the Proposition 1,
the power constraint in (14) is guaranteed and thus can be reduced. As a
result, the optimization problem in (14) is equivalent to solving the
following problem:

$\mathcal{P}_{4}:\min_{\mathbf{z}}\|\mathbf{A}G(\mathbf{z})-\mathbf{\hat{y}}\|_{2}^{2}+\lambda\|G(\mathbf{z})\|_{1},$
(15)

where $\lambda$ is the Lagrange multiplier. As $\mathbf{z}$ is differentiable
with respect to the generative model’s parameters (e.g., using
reparameterization trick [25]), one can use the loss function based on (15) to
train the generative model [21]. Once the generative model is trained to
obtain the solution for (14), denoted by $\mathbf{z}^{*}$, the reconstruction
error can be bounded with probability $1-e^{-\Omega(m)}$ by [23]:

$\|G(\mathbf{z}^{*})-\mathbf{x}^{*}\|_{2}\leq
6\min_{\mathbf{z}\in\mathbb{R}^{k}}\|G(\mathbf{z})-\mathbf{x}^{*}\|_{2}+3\|\boldsymbol{\eta}\|_{2}+2\epsilon,$
(16)

where $\epsilon$ is an additive error term caused by the use of gradient
decent-based optimizers.

Pseudo codes of our training algorithm are in Algorithm 1, which can be
described as follows. We first initialize the measurement matrix $\mathbf{A}$
following the Proposition 1, together with random parameters for the inference
network, i.e., encoder, $q_{\boldsymbol{\phi}}(\mathbf{z|y})$, and the
generative model, i.e., decoder, $p_{\boldsymbol{\theta}}(\mathbf{x|z})$
(i.e., lines 1-3 of the Algorithm 1). For each training loop, a batch of
sample $\mathbf{x}^{*}$ is i.i.d sampled from the training set (line 5). The
input of the VAE’s encoder is $\mathbf{\hat{y}}$ obtained by using (2) (line
6). The latent vector $\mathbf{z}$ is obtained in line 7, and the output of
the VAE’s decoder is $\mathbf{\hat{x}}$ in line 8. After that, the training
loss $L(\mathbf{z})$ of the VAE can be computed as in line 9, which optimizes
the problem $\mathcal{P}_{4}$ using the Adam optimization solver. After
training, the pre-trained VAE can reconstruct signal
$\mathbf{\hat{x}}=G(\mathbf{z}^{*})$ ($\mathbf{z}^{*}$ is fixed during testing
with the test set) with reconstruction error bounded by (16) (line 14).

1 Input: Initialize measurement matrix $\mathbf{A}$ that satisfies Proposition
1.

2 Initialize encoder of the VAE with inference model
$q_{\boldsymbol{\phi}}(\mathbf{z|y})$.

3 Initialize decoder of the VAE with generative model
$p_{\boldsymbol{\theta}}(\mathbf{x|z})$.

4 for  _t = 0, 1, 2 , $\ldots$_ do

5 Sample $\mathbf{x}^{*}$ from the training set.

6 Obtain $\mathbf{\hat{y}}$ from (2).

7 Obtain latent vector $\mathbf{z}\leftarrow
q_{\boldsymbol{\phi}}(\cdot|\mathbf{y})$, with $\mathbf{y}=\mathbf{\hat{y}}$.

8 Obtain $\mathbf{\hat{x}}\leftarrow p_{\boldsymbol{\theta}}(\mathbf{x|z})$ at
the output of the generator $G(\mathbf{z})$.

9 Compute the loss based on (15), i.e.,
$L(\mathbf{z})=\|\mathbf{A}G(\mathbf{z})-\mathbf{\hat{y}}\|_{2}^{2}+\lambda\|G(\mathbf{z})\|_{1},$
(17)

10 Update the neural network parameters by using backpropagation with the loss
$L(\mathbf{z})$.

11 end for

12Output: $\mathbf{z}\rightarrow\mathbf{z}^{*}$.

13 Reconstructed signal $\mathbf{\hat{x}}=G(\mathbf{z}^{*})$.

Reconstruction error: $\|G(\mathbf{z}^{*})-\mathbf{x}^{*}\|_{2}$ bounded by
(16).

Algorithm 1 CS-VAE: Training VAE to reconstruct signals from noisy compressed
measurements

##  IV Performance Evaluation

###  IV-A Dataset and Simulation Settings

####  IV-A1 Dataset and VAE’s Parameters

We use the IMU data from the DIP-IMU dataset [13], designed specifically for
capturing 3D body human motion with calibrated IMU sensors. The dataset
contains acceleration and orientation information of 17 IMU sensors placed on
the participants. The entire dataset consists of 64 data sequences of 10
participants, equivalent to 330,178 frames of motion under various activities.
The frames are recorded at the rate of 60 frames per second. The output of the
$j$-th IMU is a combination of orientation information, denoted by
$\mathbf{o}^{(j)}_{t}\in\mathbb{R}^{9}$, and acceleration information, denoted
by $\mathbf{a}^{(j)}_{t}\in\mathbb{R}^{3}$. One frame in the dataset at time
step $t$ is denoted by
$\mathbf{x}_{t}=\big{[}\mathbf{o}^{(1)}_{t},\mathbf{a}^{(1)}_{t},\mathbf{o}^{(2)}_{t},\mathbf{a}^{(2)}_{t},\ldots,\mathbf{o}^{(17)}_{t},\mathbf{a}^{(17)}_{t}\big{]}$.
As a result, one frame $\mathbf{x}_{t}$ in the dataset has $17\times
9+17\times 3=204$ features. We use the data sequences collected from 8
participants as the training set, and the data sequences collected from the
other 2 participants as the test set. After removing all the data samples that
have missing features (i.e., NaN values), the training set and test set
contain 220,076 and 56,990 data samples, respectively. To stabilize the
training process, we normalize the data $\mathbf{x}_{t}$ in the training and
test sets within the range $(-1,1)$.

By following the training process illustrated in Fig. 3, the desired signal
$\mathbf{x}^{*}$ is represented as $\mathbf{x}_{t}\in\mathbb{R}^{204}$.
Hereafter, we remove the time step $t$ notation for the sake of simplicity.
The signal $\mathbf{x}^{*}$ is then multiplied with the measurement matrix
$\mathbf{A}$ to get the signal $\mathbf{y}$ with fewer features, i.e.,
$m<204$. The signal $\mathbf{y}$ is then passed through a simulated channel
with Gaussian noise $\boldsymbol{\eta}\in\mathbb{R}^{m}$. The noise vector
$\boldsymbol{\eta}$ has $m$ elements, and each element follows a Gaussian
distribution $\mathcal{N}(0,\sigma_{N}^{2})$.

The noisy signal $\mathbf{\hat{y}}\in\mathbb{R}^{m}$ is then used as the input
of the VAE. The reconstructed signal at the output of the VAE is
$\mathbf{\hat{x}}\in\mathbb{R}^{204}$. For this, we design the network
architecture of the VAE as follows. The encoder is a fully connected network
with an input layer having $m$ neurons, one hidden layer having 64 neurons,
and one latent layer having 10 neurons. The decoder is a fully connected
network that has two hidden layers, each of which has 64 neurons. Finally, the
output layer has 204 neurons. The activation function used for the hidden
layers is ReLu and the activation function used for the output layer is Tanh.
We train the model for 50 epochs with a batch size of 60. We then use the
trained model to evaluate the performance on the test set with the same batch
size. All the parameter settings are described in Table I.

####  IV-A2 3D Avatar Model

Based on the reconstructed signals from the proposed CS-VAE model, we further
transform the signals into a 3D human avatar. For this, we use the non-
commercial Skinned Multi-Person Linear Model (SMPL) model in [19]. SMPL is a
parametrized model of a 3D human body template that takes 72 pose parameters
and 10 shape parameters, denoted by $\mathbf{p}\in\mathbb{R}^{72}$ and
$\mathbf{s}\in\mathbb{R}^{10}$, respectively, and returns a mesh with 6,890
vertices in a 3D space. By adjusting the pose and shape parameters, i.e.,
$\mathbf{p}$ and $\mathbf{s}$, we can animate the 3D avatars that mimic the
physical shapes and movements of human users. Further details of the SMPL
model can be found in [19].

To transform the IMU signals $\mathbf{\hat{x}}\in\mathbb{R}^{204}$ into a pose
parameter $\mathbf{p}\in\mathbb{R}^{72}$, we use another VAE to learn the
mapping function
$F(\mathbf{\hat{x}}):\mathbb{R}^{204}\rightarrow\mathbb{R}^{72}$. This mapping
function helps us to transform IMU signals into the input of the SMPL model
[13]. For example, the changes of acceleration and orientation of the IMU data
(i.e., $\mathbf{\hat{x}}$ varies) from the left wrist and left elbow of the
user will make the 3D avatar move its left arm (i.e., $\mathbf{p}$ varies). In
other words, given the reconstructed signals, we can create a human avatar in
a virtual 3D space with a specific pose. Note that we keep the shape
parameters $\mathbf{s}$ of the SMPL model as constant numbers for the sake of
simplicity as body shape modeling is not our focus. In the following, we first
present the performance evaluation of the signal reconstruction with our
proposed framework. Secondly, we show that our reconstructed signals are more
robust in creating 3D human avatar poses. Lastly, we show a simple but
effective method to make smooth animated motions for the avatars without using
input signals, thanks to the ability of the generative model.

####  IV-A3 Baseline Approaches

Notation | IMU Parameters | Values
---|---|---
$n$ | Dimension of signals |  $204$ features
$m$ | Number of linear | $[48:192]$
| measurements | features
| Frame per seconds | 60 fps [32]
$P_{T}$ | Transmit power |  $0.1$ Watt
|  | [32, 33]
$\sigma_{N}$ | Standard deviation of noise | $[1:500]$
|  | $\times 10^{-4}$
$d$ | Parameter of $\mathbf{A}$ | 2
| in Proposition 1 |
| Algorithmic Parameters | Values
| Learning rate | $10^{-4}$
| (Adam optimizer) |
| KL divergence weight | $10^{-5}$
$\lambda$ |  $l_{1}$ penalty | $10^{-5}$
| Number of training epochs | 50
TABLE I: Parameter settings.

We evaluate the performance of our proposed framework, denoted by CS-VAE, and
other baseline approaches. The considered baseline approaches are (i) Lasso
(Lease absolute shrinkage and selection operator) [22] and (ii) DIP (Deep
Inertial Poser) [13].

Lasso is a widely adopted algorithm for solving $l_{1}$ penalized optimization
problem in compressive sensing [20, 17]. It is a regression analysis technique
that incorporates an $l_{1}$ penalty term into the optimization problem. In
particular, the solution of Lasso is obtained by solving the optimization
problem in (5), which is equivalent to solving the Lagrangian in (6). As
observed from equation (6), Lasso solves an optimization problem that involves
the $l_{1}$ penalty term $\lambda\|\mathbf{x}\|_{1}$, which is similar to the
term $\lambda\|G(\mathbf{z})\|_{1}$ in the loss function $L(\mathbf{z})$ in
the proposed Algorithm 1. For a fair comparison, we use the same value for the
$l_{1}$ penalty $\lambda=10^{-5}$ for both Lasso and CS-VAE approaches.

Note that by using the measurement matrix $\mathbf{A}$ that follows the
Proposition 1, the setting of Lasso in (6) is now constrained with transmit
power in equation (14b). In our later experiments, we empirically show that
the power constraint in (14b) makes the optimization problem much more
challenging, resulting in Lasso’s failures in reconstructing the original
signals. For a more comprehensive evaluation, we also introduce a relaxed
version of Lasso, denoted by the notation “Lasso w.o.$P_{T}$” (Lasso without
power constraint $P_{T}$). In this relaxed version of Lasso, we remove the
power constraint in (14b) and use Lasso to recover the original signals. This
can be done by replacing $\mathbf{A}$ in (6) with another unconstrained
measurement matrix $\mathbf{B}$ with Gaussian entries. We use a similar
measurement matrix as in [23, 21] in which elements of
$\mathbf{B}\in\mathbb{R}^{m\times n}$ are $B_{ij}\sim\mathcal{N}(0,1/m)$.
Without the power constraint, “Lasso w.o.$P_{T}$” is expected to produce the
optimal results as upper bounds for our evaluation. As shown later, our
proposed CS-VAE approach achieves comparable results to the optimal solutions
obtained by “Lasso w.o.$P_{T}$”. Notably, the competitive results of our
approach also come with an order of magnitude faster than Lasso in terms of
decoding time, i.e., the time to find the solution for the optimization
problem given a batch of input data.

DIP is a deep learning approach for reconstructing body pose from a fixed set
of IMU sensors. The main idea of DIP and other approaches in this line of
work, e.g., [13, 14], is to reconstruct the full body pose, e.g., using SMPL
model, from measurements of IMU sensors placed on the important joints of the
human body, i.e., head, spine, two wrists, and two knees. The common
approaches of such frameworks are using recurrent neural networks to access
the entire training set during the training time and using a shorter time
window at the test time. The number of IMU sensors can be reduced to 3
sensors, e.g., sensors placed on the head and two wrists, but such an approach
needs an extensive motion database and physics-based simulation engine [15].

In comparison with our proposed CS-VAE framework, the aforementioned works in
[13, 14, 15] can be viewed as an underdetermined system in which the deep
learning models try to recover the full body pose given signals from a few IMU
sensors. As such, instead of using a matrix multiplication for downsampling
the data, we can impose the training process of DIP by manually selecting the
IMU sensors from the 17 IMU sensors in the dataset. Note that in the following
results, we use the same VAE’s architecture and training loss in Fig. 3 for
the DIP baseline approach, rather than using the recurrent neural network in
[13]. The main reason is that the recurrent neural network needs to see the
entire training set during the training process, which may yield advantages
and inappropriate comparisons with CS-VAE and Lasso. With similar architecture
and training with batches of signals, it is a more reasonable comparison for
DIP against CS-VAE and Lasso. The details of simulation parameters are
described in Table I.

###  IV-B Simulation Results

####  IV-B1 Impacts of the number of measurements

Figure 4: Mean square error of reconstructed signals when the number of
measurements $m$ increases.

We evaluate the performance of the proposed framework when the number of
measurements $m$ increases from 48 measurements to 192 measurements, which are
approximately $23\%$ and $94\%$ of the total number of the IMU orientation and
acceleration features, respectively. We select the number of measurements in
Fig. 4 to be divisible by 12. The only reason for this selection is that it is
easier for the DIP baseline framework as the DIP framework needs to work with
the set of IMU sensors in which each IMU has 12 features of measurements (9
features for the orientation and 3 features for the acceleration). Unlike DIP,
the CS-VAE and Lasso approaches are flexible with any arbitrary number of
measurements. The error bars in Figs. 4, 5, and 6 are equivalent to half of
the standard deviations from the mean values.

As observed from Fig. 4, the Mean Square Error (MSE) values in most approaches
decrease when the number of measurements increases, except for Lasso as it
fails to recover the signals. This observation about Lasso shows that the
power constraint makes the setting much more difficult to obtain the results.
When we remove the power constraint, the relaxed optimization problem can be
effectively solved with Lasso, as illustrated by the Lasso w.o.$P_{T}$
baseline. Recall that we use Lasso w.o.$P_{T}$ as an upper bound for
comparison. The results show that our CS-VAE approach achieves the highest
performance, i.e., low MSE values, which is closest to the performance of the
upper bound solution, i.e., Lasso w.o.$P_{T}$.

The reason for the inferior performance of DIP to CS-VAE can be explained as
follows. The linear projection of DIP from 204 measurements into a lower
number of measurements only preserves the completeness of orientation and
acceleration features. For example, the linear projection of DIP with $m=120$
measurements makes it equivalent to the data from 10 IMU sensors in which each
IMU preserves the full 12 features of orientation and acceleration. With
compressive sensing technique in other approaches, the completeness of such 12
features no longer holds as the linear projection is performed by the
measurement matrix $\mathbf{A}$, which may preserve the features with the
sparse values rather than keeping the full orientation and acceleration values
of certain IMU sensors. We observe that our newly proposed design of the
measurement matrix $\mathbf{A}$ is the key factor making the CS-VAE approach
work well with noisy and sparse signals, given a simple neural network’s
architecture of VAE.

####  IV-B2 Impacts of the channel noise

Figure 5: Mean square error of reconstructed signals when channel noise power
increases.

Next, we evaluate the performance of the approaches under different channel
noise power values. As we consider the Gaussian channel, the noise’s power is
equivalent to the variance of the Gaussian noise, i.e., $\sigma_{N}^{2}$ in
(2) [26]. We fix the power $P_{T}$ and increase the standard deviation of the
noise from $10^{-4}$ to $500\times 10^{-4}$ to obtain the results in Fig. 5.
As observed from Fig. 5, the proposed CS-VAE approach achieves better
performance under most of the considered scenarios, compared to the Lasso and
DIP approaches. Similar to the observation in the previous setting, Lasso
fails to reconstruct the signals regardless of the noise level. The relaxed
version of Lasso, i.e., Lasso w.o.$P_{T}$ achieves the highest performance as
it is not constrained by the power transmission.

We observe that under high noise’s power value, i.e., $\sigma_{N}=500\times
10^{-4}$, CS-VAE performs worse than the DIP approach. The reason is that the
design of matrix $\mathbf{A}$ can bound the signal $\mathbf{y}=\mathbf{Ax}$
with ${\frac{1}{m}\|\mathbf{y}\|_{2}^{2}\leq P_{T}}$, but this also restricts
the signal $\mathbf{y}$ into suboptimal power region. With DIP, we adopt the
power normalization strategy in [28] in which the value of $\mathbf{y}$ is
normalized by its $l_{2}$ norm $\|\mathbf{y}\|_{2}$. As the results suggest,
this power normalization can be effective with high noise levels but it
becomes less robust with low noise values. This is also the reason we design a
new measurement matrix $\mathbf{A}$ rather than following this power
normalization scheme as it yields a nonlinear projection from $\mathbf{x}$ to
$\mathbf{y}$, which is in contrast to the linear projection idea in
compressive sensing. Nevertheless, the CS-VAE achieves more robust and better
performance than Lasso and DIP, and the results are closest to Lasso
w.o.$P_{T}$.

####  IV-B3 Decoding latency with respect to the number of input samples

Figure 6: Decoding time at the receiver when the number of input samples
increases.

Next, we investigate the decoding time at the receiver under different sizes
of the input samples in Fig. 6. The number of input samples $\mathbf{\hat{y}}$
fed into the VAE is equivalent to the number of measurements $m$ multiplied by
the batch size. Note that in Fig. 6, we use the approximate values for the
number of input samples in the x-axis for ease of illustration. The exact
values are calculated by $m\times b$, where $m$ is the number of measurements
and $b$ is the batch size. For example, with $10\times 10^{3}$ input samples
in Fig. 6, we use $m=168$ and $b=60$ to produce $10,080$ input samples. These
$10,080$ input samples are actually $14\times 12\times 60$ samples, where $14$
is the number of IMU sensors, $12$ is the number of features generated by each
IMU sensor per frame, and $60$ is the number of frames per second. In other
words, a sequence of $10\times 10^{3}$ input samples is equivalent to data
generated by the IMU sensors in one second. Similarly, a sequence of $70\times
10^{3}$ input samples is equivalent to data generated by the IMU sensors in 7
seconds.

The transmission of a batch of data samples is equivalent to the case when we
collect a batch of samples after a period of time and then transmit them to
the receiver. For Lasso and Lasso w.o.$P_{T}$, this batch of samples can be
effectively used as the input of the optimization solver as the Lasso model
can handle matrix-like input data. The decoding time is measured as the single
forward operation from the input to the output of the pre-trained VAE models
to obtain the reconstructed signals. With Lasso, the decoding time is measured
as the time to find the solution for the optimization problem. We use the same
central processing unit (CPU) to compute the decoding time for all approaches.

As observed from Fig. 6, the decoding time values of the CS-VAE and DIP
approaches are significantly lower than that of the Lasso approaches. The main
reason is that the search space of Lasso’s solver increases as the number of
input samples increases. In contrast, the decoding time values of CS-VAE and
DIP slowly increase with the number of input samples because the pre-trained
VAE models only need to forward the received signals through the deep neural
network layers to obtain the reconstructed signals. Notably, the cost of
finding high accuracy for reconstructed signals of Lasso w.o.$P_{T}$ makes it
significantly slower than the deep learning approaches.

It is worth mentioning that the CS-VAE and DIP approaches need to be pre-
trained before being applied to get the results in Fig. 6 while the Lasso
approach does not have this pre-training process. However, the pre-training
process can be greatly facilitated through modern graphics processing unit
(GPU) training. The observation in this experiment also suggests the real-time
decoding capability of the CS-VAE approach with lower decoding latency. For
example, with a number of input samples of $10^{4}$, which is equivalent to a
sequence data of one second, the decoding time of the CS-VAE approach is
approximately $8\times 10^{-2}$ second. We observe that although sharing the
same network architecture with the CS-VAE approach, the DIP approach
experiences slightly higher decoding time. The main reason is that the
implementation of the linear projection matrix of DIP requires a few extra
matrix transformation steps and de-transformation steps. However, this
difference in the decoding time is not significant.

####  IV-B4 3D pose estimation from reconstructed signals

Figure 7: Reconstructed 3D poses from noisy and compressed $m=168$
measurements and noise $\sigma_{N}=10\times 10^{-4}$.

In Fig. 7, we draw random poses from the reconstructed signals in the test
set. The poses are generated from the SMPL model and the pre-trained VAE model
(modeled as a function
$F(\mathbf{\hat{x}}):\mathbb{R}^{204}\rightarrow\mathbb{R}^{72}$) as described
in Section IV-A2. In this experiment, we reconstruct the poses from 168
measurements (i.e., 82% of the total signals) and the channel noise power
$\sigma_{N}=10\times 10^{-4}$. As observed from Fig. 7, our reconstructed
poses are more accurate compared with the Lasso and DIP approaches. As Lasso
w.o.$P_{T}$ is considered as the upper bound of all the signal reconstruction
scenarios, Fig. 7 clearly shows that the reconstructed 3D poses by Lasso
w.o.$P_{T}$ (green poses in the fourth column) achieve the most similarity to
the ground truth reference poses (red poses in the first column). The 3D poses
obtained by CS-VAE can accurately mimic the body poses of the references with
slight errors in arm movements, e.g., the pose in the second row. With Lasso,
as suggested by the poor signal reconstruction performance, the 3D poses
obtained by Lasso fail to mimic the reference poses. In our experiment, the
received signals are down-sampled and contain additive noise and the models do
not have access to the full training set as in [13], resulting in poor
performance of the DIP approach.

####  IV-B5 Pose interpolation without input signals

Figure 8: Interpolation between key poses (red avatars) without using input
IMU signals. The gray avatars represent the poses generated by the VAE when
the input IMU signals are missing, e.g., due to transmission loss.

We further use the pre-trained latent space and the decoder of the VAE to
generate novel synthesis poses to animate the avatar. The ability to generate
synthesis data is one of the most important features of generative models,
which has not been well demonstrated in the literature of wireless systems. In
particular, we consider a simple pose interpolation task as follows. Given two
key poses, illustrated by the left pose and right pose in red color in Fig. 8,
we aim to create a smooth transition between these two key poses by generating
the immediate light gray colored poses. The need for creating the intermediate
poses is important as the IMU signals might be lost during the transmission
over the severe lossy wireless channels. In such a case, if the receiver
cannot fill the missing poses or the transmitter does not retransmit the data,
the user may experience motion sickness in the virtual 3D environment, thus
decreasing the quality of experience.

Given the above setting, as illustrated in Fig. 8, we obtain the two key poses
by reconstructing the IMU signals similar to the previous experiments. Let’s
denote the received signals $\mathbf{\hat{y}}_{1}$ and $\mathbf{\hat{y}}_{2}$
corresponding to the two key poses, respectively. The VAE’s encoder can map
the signals into two corresponding vectors in the latent space that are
$\mathbf{z}_{1}=q_{\boldsymbol{\phi}}(\mathbf{z}|\mathbf{y}=\mathbf{\hat{y}}_{1})$
and
$\mathbf{z}_{2}=q_{\boldsymbol{\phi}}(\mathbf{z}|\mathbf{y}=\mathbf{\hat{y}}_{2})$.
As a result, the reconstructed signals corresponding to the two key poses are
$\mathbf{\hat{x}}_{1}=p_{\boldsymbol{\theta}}(\mathbf{x|z}=\mathbf{z}_{1})$
and
$\mathbf{\hat{x}}_{2}=p_{\boldsymbol{\theta}}(\mathbf{x|z}=\mathbf{z}_{2})$.
To simply interpolate between the two latent vectors $\mathbf{z}_{1}$ and
$\mathbf{z}_{2}$, we use the intermediate latent vectors
$\mathbf{z}_{j}=\varrho\mathbf{z}_{2}+(1-\varrho)\mathbf{z}_{1}$ where
$\varrho\in[0,1]$ is the interpolation parameter. Intuitively, $\varrho=1$
makes the intermediate latent vector $\mathbf{z}_{j}=\mathbf{z}_{2}$,
resulting in the pose similar to the key pose on the right of the Fig. 8.
Similarly, $\varrho=0$ imposes the vector $\mathbf{z}_{j}=\mathbf{z}_{1}$. The
arbitrary value of $\varrho\in[0,1]$ creates a latent vector that is a linear
combination of the two vectors $\mathbf{z}_{1}$ and $\mathbf{z}_{2}$. As a
result, we can make a smooth transition of the avatar poses by simply
increasing the value of $\varrho$, as shown in Fig. 8. As described, there is
no need for the input signals when interpolation takes place in the learned
latent space $\mathbf{z}$. Similar linear interpolation techniques have been
explored to create synthesized data in the various domains [34]. Our
experiment shows the potential extension of the proposed framework to future
VR/XR applications in conjunction with generative modeling where the
synthesized data can be utilized.

##  V Conclusion

In this paper, we have developed a novel framework for 3D human pose
estimation from IMU sensors with generative model-based compressive sensing.
The proposed framework helps the IMU sensors reduce the amount of information
exchanged with the receiver, thus further enhancing channel utilization and
encoding-decoding efficiency. At the receiver’s side, we have employed a deep
generative model, i.e., a VAE, that can recover the original signals from
noisy compressed samples. With the ability of the generative model at the
receiver, we have achieved an order of magnitude faster than Lasso in terms of
decoding latency. We have further demonstrated that the proposed framework can
learn a latent representation space and generate synthetic data samples,
making it possible to fulfill missing data features (e.g., due to lossy
transmissions) without using input data from the IMU sensors. Interesting
findings suggest that the proposed generative model-based compressive sensing
framework can achieve state-of-the-art performance in challenging scenarios
with severe noise and fewer number of measurements, compared with other
optimization and deep learning approaches.

##  Appendix A Proof of Proposition 1

The purpose of the Proposition 1 is to construct a measurement matrix that
satisfies the S-REC property in Definition 4 and the power constraint in (14)
so that the recovered signal of the generative model is unique and accurate
with high probability. In the following, we prove that if each element
$A_{ij}$ (element $j$-th of the $i$-th row of matrix $\mathbf{A}$) follows a
normal distribution
$A_{ij}\sim\mathcal{N}\Big{(}0,\frac{P_{T}}{n^{2}d^{2}(d\sigma_{x}+\mu_{x})^{2}}\Big{)}$,
the matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ satisfies the S-REC property
and power constraint. The first part of this section proves the guarantee of
the power constraint. The second part proves the S-REC property of the
measurement matrix.

###  A-A Proof of $\mathbf{y}=\mathbf{Ax}$ guaranteeing the power constraint
of a Gaussian channel

First, we need prove that the matrix $\mathbf{A}$ satisfies that power
constraint

$\frac{1}{m}\|\mathbf{y}\|_{2}^{2}\leq P_{T}.$ (18)

By using $l_{2}$ norm definition in (4), the power constraint can be rewritten
as [26, Equation 9.2, Chapter 9]:

$\frac{1}{m}\sum_{i=1}^{m}|y_{i}|^{2}\leq P_{T}.$ (19)

Recall that we have a $n$-dimensional vector
$\mathbf{x}=[x_{1},x_{2},\ldots,x_{j},\ldots,x_{n}]^{\top}$ and the
$m$-dimensional vector
$\mathbf{y}=[y_{1},y_{2},\ldots,y_{i},\ldots,y_{m}]^{\top}$, where the
superscript $\top$ denotes the transpose. The measurement matrix
$\mathbf{A}\in\mathbb{R}^{m\times n}$ is defined by:

$\mathbf{A}=\left[\begin{array}[]{cccc}A_{11}&A_{12}&\cdots&A_{1n}\\\
A_{21}&A_{22}&\cdots&A_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
A_{m1}&A_{m2}&\cdots&A_{mn}\end{array}\right].$

Using matrix calculation, we have $y_{i}=\sum_{j=1}^{n}A_{ij}x_{j}$ with
$i=1,2,\ldots,m$. Following the definition of $l_{p}$ norm in (4), we have

$\displaystyle\frac{1}{m}\|\mathbf{y}\|_{2}^{2}$
$\displaystyle=\frac{1}{m}\sum_{i=1}^{m}|y_{i}|^{2}$ (20a)
$\displaystyle=\frac{1}{m}\sum_{i=1}^{m}\Big{(}\sum_{j=1}^{n}A_{ij}x_{j}\Big{)}^{2}$
(20b)
$\displaystyle\leq\frac{1}{m}\sum_{i=1}^{m}\Big{[}\big{(}\sum_{j=1}^{n}A_{ij}^{2}\big{)}\big{(}\sum_{j=1}^{n}x_{j}^{2}\big{)}\Big{]},$
(20c)

where (20c) directly applies Cauchy–Schwarz inequality, i.e.,
$\Big{(}\sum_{j=1}^{n}A_{ij}x_{j}\Big{)}^{2}\leq\big{(}\sum_{j=1}^{n}A_{ij}^{2}\big{)}\big{(}\sum_{j=1}^{n}x_{j}^{2}\big{)}$.
In the following, we derive the bounds for the two terms inside (20c) which
are $\sum_{j=1}^{n}A_{ij}^{2}$ and $\sum_{j=1}^{n}x_{j}^{2}$. For this
purpose, we use Chebyshev’s inequality [26, Chapter 3, Equation 3.32], which
can be stated as follows.

######  Definition 5 (Chebyshev’s inequality).

Let $X$ be a random variable with mean $\mu$ and variance $\sigma^{2}$. For
any $\varepsilon>0$,

$\mathbb{P}\big{(}|X-\mu|>\varepsilon\big{)}\leq\frac{\sigma^{2}}{\varepsilon^{2}}.$

In our setting, we are more interested in the central limits, i.e., the
distances away from the mean values, of the random variables, i.e., $x_{j}$
and $A_{ij}$. Let’s $\varepsilon=d\sigma$ for real number $d>0$, Chebyshev’s
inequality can be rewritten as

$\mathbb{P}\big{(}|X-\mu|>d\sigma\big{)}\leq\frac{1}{d^{2}}.$ (21)

Equivalently, we can bound the absolute value of $|X-\mu|\leq d\sigma$ with
probability is at least $1-\frac{1}{d^{2}}$, i.e.,
$\mathbb{P}\big{(}|X-\mu|\leq d\sigma\big{)}>1-\frac{1}{d^{2}}$. By choosing
the value of $d$, we can bound the value of $X$ within a certain distance away
from its mean with known probability. For sufficiently large $d$, we have the
following inequalities

$\mu-d\sigma\leq X\leq\mu+d\sigma,$ (22)

with high probability.

Now, by using inequalities in (22) for random variables $x_{j}$ associated
with mean $\mu_{x}$ and variance $\sigma_{x}^{2}$, and $A_{ij}$ associated
with zero-mean and variance $\sigma_{a}^{2}$, we have the following
inequalities

$\displaystyle\quad-d\sigma_{a}\leq A_{ij}\leq d\sigma_{a},$ (23a)
$\displaystyle\mu_{x}-d\sigma_{x}\leq x_{j}\leq\mu_{x}+d\sigma_{x},$ (23b)

with high probabilities. By taking the square of $A_{ij}$ and $x_{j}$, we have

$\displaystyle A_{ij}^{2}\leq d\sigma_{a}^{2},$ (24a) $\displaystyle
x_{j}^{2}\leq\max\big{[}(d\sigma_{x}^{2}+\mu_{x})^{2},(-d\sigma_{x}+\mu_{x})^{2}\big{]},$
(24b)

with high probabilities. Taking the sum over $n$ samples, we have

$\sum_{j=1}^{n}A_{ij}^{2}\leq nd\sigma_{a}^{2},$ (25)

and

$\sum_{j=1}^{n}x_{j}^{2}\leq\max\big{[}n(d\sigma_{x}^{2}+\mu_{x})^{2},n(-d\sigma_{x}+\mu_{x})^{2}\big{]},$
(26)

with high probabilities. As we empirically observe from the dataset that
$\mu_{x}>0$, the bound for the above equation can be simplified as

$\sum_{j=1}^{n}x_{j}^{2}\leq n(d\sigma_{x}^{2}+\mu_{x})^{2}.$ (27)

Replacing (25) and (27) in (20), we finally have

$\displaystyle\frac{1}{m}\|\mathbf{y}\|_{2}^{2}$
$\displaystyle\leq\frac{1}{m}\sum_{i=1}^{m}nd^{2}\sigma_{a}^{2}n(d\sigma_{x}+\mu_{x})^{2}$
(28a) $\displaystyle=n^{2}d^{2}(d\sigma_{x}+\mu_{x})^{2}\sigma_{a}^{2}.$ (28b)

As we want to have a power constraint $\frac{1}{m}\|\mathbf{y}\|_{2}^{2}\leq
P_{T}$, by choosing $n^{2}d^{2}(d\sigma_{x}+\mu_{x})^{2}\sigma_{a}^{2}=P_{T}$,
we have the variance of $A_{ij}$ is

$\sigma_{a}^{2}=\frac{P_{T}}{n^{2}d^{2}(d\sigma_{x}+\mu_{x})^{2}}.$ (29)

The derivation above for $\sigma_{a}$ proves the Proposition 1. In other
words, by generating the measurement matrix $\mathbf{A}$ from the normal
distribution with zero-mean and variance $\sigma_{a}^{2}$, the transmit power
$\frac{1}{m}\|\mathbf{y}\|_{2}\leq P_{T}$ can be achieved with high
probability.

###  A-B Proof of $\mathbf{Ax}$ satisfying S-REC property

Next, we prove that given the measurement matrix $\mathbf{A}$ that follows the
Proposition 1 will satisfy the S-REC property in Definition 4. In particular,
the matrix $\mathbf{A}$ is said to satisfy the
$\text{S-REC}(S_{G},\gamma,\kappa)$, i.e., set-restricted eigenvalue condition
of the set $S_{G}$ (defined in (13)) with the parameters $\gamma>0$ and
$\kappa\geq 0$, if $\forall\mathbf{x}_{1},\mathbf{x}_{2}\in S_{G}$,

$\|\mathbf{A}\big{(}G(\mathbf{z}_{1})-G(\mathbf{z}_{2})\big{)}\|_{2}\geq\gamma\|G(\mathbf{z}_{1})-G(\mathbf{z}_{2})\|_{2}-\kappa.$
(30)

The range of the generator $G(\mathbf{z})$ can be easily bounded by the output
layer of the deep neural network. In our experiments, we use the Tanh
activation function as the output of the neural network. Therefore, we have a
simple bound $-1\leq G(\mathbf{z})\leq 1$, which is similar to the bound of
processed signal vectors $\mathbf{x}^{*}$.

Let’s define a vector $\mathbf{v}=G(\mathbf{z}_{1})-G(\mathbf{z}_{2})$ with
the bound of each element $v_{j}$ ($j=1,2,\ldots,n$) of $\mathbf{v}$ is
$-2\leq v_{j}\leq 2$, (30) can be rewritten as

$\|\mathbf{Av}\|_{2}\geq\gamma\|\mathbf{v}\|_{2}-\kappa.$ (31)

By using the definition of $l_{p}$ norm in (4), we have

$\displaystyle\gamma\|\mathbf{v}\|_{2}$
$\displaystyle=\gamma\sqrt{\sum_{j=1}^{n}v_{j}^{2}}$ (32a)
$\displaystyle\leq\gamma\sqrt{4n}$ (32b) $\displaystyle=2\gamma\sqrt{n},$
(32c)

where the inequality in the second line is obtained by the bound $-2\leq
v_{j}\leq 2$. As a result, we have the following inequality for the term on
the right-hand side of (30):

$\gamma\|\mathbf{v}\|_{2}-\kappa\leq 2\gamma\sqrt{n}-\kappa.$ (33)

To find a possible lower bound for $\|\mathbf{Av}\|_{2}$, we use the
inequalities between the $l_{1}$ and $l_{2}$ norm, and then find the
probabilistic lower bound of the $l_{1}$ norm based on Bernstein inequality.
In particular, we apply the following inequality (see equations (A.3) and
(A.4) in Definition A.2 of [20]):

$\|\mathbf{Av}\|_{2}\geq\frac{1}{\sqrt{m}}\|\mathbf{Av}\|_{1}.$ (34)

Using the definition of $l_{1}$ norm in (4), (34) can be rewritten as

$\displaystyle\|\mathbf{Av}\|_{2}$
$\displaystyle\geq\frac{1}{\sqrt{m}}\sum_{j=1}^{n}\big{|}A_{ij}v_{j}\big{|}$
(35a)
$\displaystyle\geq\frac{1}{\sqrt{m}}\Big{|}\sum_{j=1}^{n}A_{ij}v_{j}\Big{|},$
(35b)

where the inequality in the second line is obtained by using the generalized
triangle inequality. Next, we use a probabilistic lower bound for (35b) which
applies Bernstein inequality (see Theorem 7.27 - Chapter 7 of [20]), i.e.,
given the measurement matrix $\mathbf{A}$, which has elements $A_{ij}$ are
zero-mean sub-gaussian random variables, we have the following probabilistic
lower bound of

$\mathbb{P}\Big{(}\big{|}\sum_{j=1}^{n}A_{ij}v_{j}\big{|}\geq t\Big{)}\leq
2\exp\Big{(}{\frac{-t^{2}}{4c\|\mathbf{A}\|_{2}^{2}}}\Big{)},$ (36)

for $\forall t>0$, where $c$ is a subgaussian parameter. Let’s
$t=t_{0}\sqrt{m}$ with $\forall t_{0}>0$, (36) can be rewritten as

$\displaystyle\mathbb{P}\Big{(}\big{|}\sum_{j=1}^{n}A_{ij}v_{j}\big{|}\geq\sqrt{m}t_{0}\Big{)}\leq
2\exp\Big{(}\frac{-t_{0}^{2}m^{2}}{4c\|\mathbf{A}\|_{2}^{2}}\Big{)}$ (37a)
$\displaystyle\Rightarrow$
$\displaystyle\mathbb{P}\Big{(}\frac{1}{\sqrt{m}}\big{|}\sum_{j=1}^{n}A_{ij}v_{j}\big{|}\geq
t_{0}\Big{)}\leq
2\exp\Big{(}\frac{-t_{0}^{2}m^{2}}{4c\|\mathbf{A}\|_{2}^{2}}\Big{)}.$ (37b)

Using the inequality in (35), (37b) becomes

$\mathbb{P}\Big{(}\|\mathbf{Av}\|_{2}\geq t_{0}\Big{)}\leq
2\exp\Big{(}\frac{-t_{0}^{2}m^{2}}{4c\|\mathbf{A}\|_{2}^{2}}\Big{)}.$ (38)

By choosing $t_{0}=2\gamma\sqrt{n}-\kappa$, (31) can be written as

$\|\mathbf{Av}\|_{2}\geq 2\gamma\sqrt{n}-\kappa,$ (39)

with probability $1-2\exp\Big{(}\frac{(-2\gamma
m\sqrt{n}-m\kappa)^{2}}{4c\|\mathbf{A}\|_{2}^{2}}\Big{)}$. Applying inequality
in (33), i.e., $2\gamma\sqrt{n}-\kappa\geq\gamma\|\mathbf{v}\|_{2}-\kappa$, we
have

$\|\mathbf{Av}\|_{2}\geq\|\mathbf{v}\|_{2}-\kappa$ (40)

with probability is at least $1-2\exp\Big{(}\frac{(-2\gamma
m\sqrt{n}-m\kappa)^{2}}{4c\|\mathbf{A}\|_{2}^{2}}\Big{)}$. As $\mathbf{v}$ is
defined by $\mathbf{v}=G(\mathbf{z}_{1})-G(\mathbf{z}_{2})$, finally, we have

$\|\mathbf{A}\big{(}G(\mathbf{z}_{1})-G(\mathbf{z}_{2})\big{)}\|_{2}\geq\gamma\|G(\mathbf{z}_{1})-G(\mathbf{z}_{2})\|_{2}-\kappa$
(41)

with probability $1-2\exp\Big{(}\frac{(-2\gamma
m\sqrt{n}-m\kappa)^{2}}{4c\|\mathbf{A}\|_{2}^{2}}\Big{)}$. This proves the
S-REC($S_{G},\gamma,\kappa$) property in (30).

By combining (41) and (29), the proof is now completed.

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