# Maximum Likelihood Uncertainty Estimation: Robustness to Outliers

Deebul S. Nair,1 Nico Hochgeschwender, 1 2 Miguel A. Olivares-Mendez 3

###### Abstract

We benchmark the robustness of maximum likelihood based uncertainty estimation
methods to outliers in training data for regression tasks. Outliers or noisy
labels in training data results in degraded performances as well as incorrect
estimation of uncertainty. We propose the use of a heavy-tailed distribution
(Laplace distribution) to improve the robustness to outliers. This property is
evaluated using standard regression benchmarks and on a high-dimensional
regression task of monocular depth estimation, both containing outliers. In
particular, heavy-tailed distribution based maximum likelihood provides better
uncertainty estimates, better separation in uncertainty for out-of-
distribution data, as well as better detection of adversarial attacks in the
presence of outliers.

## Introduction

The ability to estimate the uncertainty along with network prediction has
become a relevant feature for the adoption of deep neural networks (DNN) in
safety-critical and autonomous systems (Borg et al. 2019; Schwalbe and Schels
2020). Accurate and calibrated uncertainties can be used to gain confidence
for making decisions in autonomous systems (Jha et al. 2018; Serban, Poll, and
Visser 2020). Uncertainty estimation is a challenging problem, especially in
high dimensional data because of the lack of ground truth regarding
uncertainty. The presence of noisy labels or outliers in the training data
elevates the challenge of uncertainty estimation. Robust uncertainty
estimation is the capability of learning algorithms to correctly learn
uncertainty by ignoring the outliers. In this work we focus on the following
research question: For a regression problem, given a training dataset, where
an $\eta$-fraction of data are outliers, how can we robustly estimate
uncertainty in the predictions?

No Outliers |  |  |
---|---|---|---
10% Outliers |  |  |
| (a) Gaussian | (b) Ensemble | (c) Laplace
Figure 1: Depiction of the contrast in performance of uncertainty estimation
on one dimensional regression data containing outliers. Row 1 shows data with
no outliers while row 2 data contains 10% outliers. The blue dots are clean
data while red dots are outliers. The shaded blue region is the estimated
uncertainty(three standard deviations). (a) Gaussian loss function (Nix and
Weigend 1994) (b) Ensemble method (Lakshminarayanan, Pritzel, and Blundell
2017) (c) Laplace loss function (ours)

Uncertainty estimation approaches can be broadly classified in three
categories:  1. Bayesian methods (Kendall and Gal 2017; Dusenberry et al.
2020b);  2. Sampling based methods (Gal and Ghahramani 2016; Lakshminarayanan,
Pritzel, and Blundell 2017);  3. Single-model methods (Nix and Weigend 1994;
Amini et al. 2020; Sensoy, Kaplan, and Kandemir 2018) . Although, Bayesian and
sampling based methods provide state-of-the-art uncertainty estimates
(Dusenberry et al. 2020a; Wen, Tran, and Ba 2020) they have come at a cost:
increased training time, increased prediction time and increased memory
requirements. For example, the Bayesian neural learning method by converting
weights to distributions (Blundell et al. 2015) requires a doubling in
parameters, MC-Dropout (Gal and Ghahramani 2016) with 50 Monte Carlo samples
will require 50x forward passes and Deep ensembles (Lakshminarayanan, Pritzel,
and Blundell 2017) with an ensemble size of 5 will require 5x the number of
weights. These become an operational challenge for their adoption in real-time
autonomous systems. On the contrary, single-model based uncertainty estimation
methods predict the uncertainty with a single network, no increase in weights
and a single forward pass (Nix and Weigend 1994; Malinin and Gales 2019;
Sensoy, Kaplan, and Kandemir 2018). In this work, we focus on single-model
methods because of their applicability to autonomous systems.

All the uncertainty estimation methods have worked exclusively with clean
datasets, however getting clean annotated datasets in real world applications
is a difficult task. As a result, real world datasets collected even after
human annotation contain large label noise (Algan and Ulusoy 2021). Learning
in the presence of noisy labels is a well studied problem for the
classification task (Arazo et al. 2019; Xia et al. 2019; Li, Soltanolkotabi,
and Oymak 2020). Also learning uncertainty in addition to the classification
in the presence of label noise is now gaining importance because of safety
(Neverova, Novotný, and Vedaldi 2019; Goel and Chen 2021; Northcutt, Jiang,
and Chuang 2021). Although all these methods have been evaluated for their
prediction and uncertainty quantification for the classification task, their
performance for the regression task in the presence of outliers (noisy labels)
is not a well studied problem. In this work, we focus on the problem of
robustly learning accurate uncertainty in the presence of outliers.

To handle outliers in statistics, heavy-tailed distribution (Student-t
distribution, mixture of Gaussians and Laplace) has been introduced (Lange,
Little, and Taylor 1989; Tak, Ellis, and Ghosh 2019; Pasi, Jarno, and Aki
2011). They model the noise in the data using a heavy-tailed distribution. As
the heavy-tailed distribution assumes some data can be present away from the
mean, the outlier data produces comparatively less loss and have minimum
impact while training. In this paper, we study the use of heavy-tailed
distributions for maximum likelihood based uncertainty estimation and
benchmark their robustness to outliers in labels. In particular, we use the
Laplace distribution as a heavy-tailed distribution because of its high
breakaway point (Bosse, Agamennoni, and Gilitschenski 2016), indicating the
high proportion of outlier data it can handle.

We formulate the loss function by taking the negative log-likelihood of the
Laplace distribution. This loss function provides protection against outliers
by reducing their effects while training. This can be demonstrated with the
help of a toy example as shown in Fig. 1. The dataset is a one dimensional
(1D) regression problem where the ground truth $y$ is a sine-wave. The dataset
contains heteroskedatic noise with varying levels of uncertainty across $x$.
In addition to the noisy data we also add 10% outliers to the data. The
addition of outliers results in a complete mis-calibration of uncertainty
estimated by Gaussian loss function (Nix and Weigend 1994) and Evidential
(Amini et al. 2020) however, the proposed Laplace loss function uncertainty
estimates are not that degraded.

We benchmark the robustness of uncertainty estimation methods when trained
with noisy data. Specifically, this work makes the following contributions:

  1. 1.

we improve the robustness of uncertainty estimation by modeling the loss
function using a heavy-tailed distribution;

  2. 2.

we evaluate the robustness of the loss function on a standard regression
datasets benchmark and complex vision regression task of depth estimation; and

  3. 3.

we evaluate the use of predicted uncertainty for out-of-distribution (OOD)
detection and adversarial attacks detection.

## Uncertainty Estimation in Regression

In a regression learning problem we have a training dataset $D=(x,y)$ drawn
from a joint distribution $D(X,Y)$, where for each co-variate sample pair,
$x\in X$ is the input data and $y\in Y$ is the label data. In addition we
define a neural network $f_{\theta}$ parameterized by $\theta$. We train the
parameters $\theta$ of the neural network by minimizing the empirical loss
function $\ell$.

$\hat{\theta}=\underset{\theta}{\text{minimize}}R(\theta);\quad
R(\theta)=\underset{<x,y>\in D}{E}[\ell(x,y,\theta)]$

In a typical (i.e. deterministic and non-robust) regression problem, the loss
function can be the sum of squared residuals.

$\ell(x,y,\theta)=\frac{1}{2}(y-f_{\theta}(x))^{2}$

The above problem formulation can only do point estimates and can be
considered as estimating the mean of a probability distribution. A simple
extension is to estimate the entire conditional probability distribution by
predicting variance $\sigma^{2}$ (in case of Gaussian distribution assumption)
in addition to the point estimate $\mu$ (Nix and Weigend 1994). However, a
Gaussian distribution is not robust to outliers in training data.

## Improving Robustness with Heavy-Tailed Distribution

Figure 2: Negative log likelihood (NLL) (left) and probability density
function (PDF) (right) of the Laplace loss function

We define the problem of robust uncertainty estimation.

######  Definition 1 (Robustly learning the output of a neural network with
bounded second moments under additive outliers).

Let $D$ be the set of $n$ samples so that $D=D_{clean}\cup D_{outlier}$, where
$D_{clean}$ is a subset of $D$ and $D_{outlier}$ satisfies $|D_{outlier}|<\eta
n$. Given an $\eta$-outlier set of samples from $D(x,y)$ and learn the output
of a neural network $\hat{y}$ and $var$ such that it minimizes
$|y_{clean}-\hat{y}|$ and $|var(y_{clean})-var(\hat{y})|$

Thus the goal of a robust loss function will be to accurately predict the
output and its uncertainty by ignoring outliers in the data. In this paper we
have selected a heavy tail distribution, Laplace distribution for modeling the
uncertainty in the output.

Regression models using heavy-tailed error distributions to accommodate
outliers are studied in statistics by (West 1984). The heavy tail of the
distribution reduces the impact of the outlier while finding the maximum
likelihood and thus has limited impact on the predicted value and the
uncertainty. We choose Laplace as the heavy-tailed distribution because of its
well-defined moments and it has a concave loss function, as is required for
the uncertainty estimation.

#### Laplace Maximum Likelihood

The probability density function of a Laplace distribution is given by:

$p(y|\mu,s)=\frac{1}{2s}\exp(-\frac{|y-\mu|}{s})$ (1)

where $p(y|\mu,s)$ is only defined for $s>0$ and $\mu\in R$. We use the
Negative Log likelihood (NLL) of the Laplacian distribution $-log(p(.|\mu,s))$
for training the neural networks

$\ell_{NLL}(x,y,\theta)=-\log(p(y|f_{\theta}^{\mu},f_{\theta}^{s})=\log(2f_{\theta}^{s})+\frac{|y-f_{\theta}^{\mu}|}{f_{\theta}^{s}}$
(2)

The probability density function (PDF) and the NLL for different values of
$f_{\theta}^{s}$ are plotted in Fig. 2. The loss function has several
properties that makes it suitable for gradient-based optimization. The loss
function is a convex function. The convexity ensures that the loss is minimum
when $|y-\mu|$ is minimum and increases monotonically with respect to
$|y-\mu|$ .

$min(-\log(p(y|f_{\theta}^{\mu},f_{\theta}^{s})))\Rightarrow
f_{\theta}^{\mu}=y\quad\quad\frac{\partial\ell}{\partial|y-f_{\theta}^{\mu}|}\geq
0$ (3)

The loss monotonically increases with respect to $f_{\theta}^{s}$:

$\frac{\partial\ell}{\partial
f_{\theta}^{s}}(|y-f_{\theta}^{\mu}|,f_{\theta}^{s})\geq 0$ (4)

This property helps to learn the correct aleatoric uncertainty. The
monotonicity guaranteed by the loss function with respect to $s$ forces the
optimization algorithm to reduce $f_{\theta}^{s}$ when there is less aleatoric
noise in the data. As can be observed in Fig. 2, when $f_{\theta}^{s}$ is
reduced the minimum of NLL reduces, thus ensuring that the correct
uncertainties are learned.

## Related Work

#### Uncertainty Estimation

Nix and Weigend (1994) estimated the uncertainty for regression problems by
modeling the output of the neural network as a Gaussian distribution and
learning the output using a Gaussian NLL loss function. This was the first
demonstration of how the neural network can learn the uncertainty in addition
to the predicted value. In the deep learning era, uncertainty estimation in
large-scale vision tasks was demonstrated by Blundell et al. (2015) which
represented the weights of network with distribution. Gal and Ghahramani
(2016) estimated the uncertainty by using dropouts and multiple forward
passes. The current state-of-the-art in uncertainty estimation is based on
ensemble methods (Lakshminarayanan, Pritzel, and Blundell 2017; Dusenberry et
al. 2020a; Wen, Tran, and Ba 2020) which use a set of models under a single
one. Even though all these methods provide good uncertainty estimates, they
still require multiple forward passes from the model.

#### Single-Model Approaches to Uncertainty Estimation

Uncertainty estimation using a single-model can be achieved by replacing loss
function based approaches (Malinin and Gales 2018; Sensoy, Kaplan, and
Kandemir 2018; Amini et al. 2020), computing closed-form posterior for the
output layer (Riquelme, Tucker, and Snoek 2018; Snoek et al. 2015), changing
the output layer (Calandra et al. 2016; Tagasovska and Lopez-Paz 2019),
spectral normalization (Liu et al. 2020), or by two-sided gradient penalty
(Van Amersfoort et al. 2020). Laplace maximum likelihood builds on these
approaches by replacing the loss function and changing the output layer.

#### Robust Training in Neural Networks

Robust training in the presence of outliers is a classical statistics problem
and is dominated by M-estimator methods by Huber (2004). In neural networks, a
generalized M-estimator loss function was proposed by Barron (2019), which
uses the negative log of the density function to improve robustness while
training. Lathuilière et al. (2018) proposed the use of Gaussian-uniform
mixture model as a loss function which continuously adapts as per the outliers
in data.

## Experiments

In this section we benchmark performance of the proposed loss function. We
will show that the proposed loss function improves robustness to outliers in
data. These benchmarks are not intended to represent the state-of-the-art for
any particular task; on the contrary they are intended to demonstrate the
capability of our loss function when learning uncertainty with data containing
outliers. We compare the following uncertainty estimation methods:  1.
Gaussianrefers to the model introduced in (Nix and Weigend 1994) 2.
Ensemblecorresponds to the ensemble of deep learning method proposed by
(Lakshminarayanan, Pritzel, and Blundell 2017) 3. Evidentialrefers to the
Normal-inverse-Gamma distribution based uncertainty estimation by (Amini et
al. 2020) 4. Laplacerefers to the method we propose.  We benchmark these
methods in terms of uncertainty prediction in a 1D outlier regression dataset,
real world regression datasets and NyuV2 monocular depth estimation dataset
with outliers (Silberman et al. 2012).

We first discuss the need for new metrics for uncertainty comparison and
select an appropriate metric to benchmark the predicted uncertainty. In the
subsequent section we focus on benchmarking the breakaway point, which is the
percentage of outliers that can be present in the data without any significant
change in the predicted output. We use a synthetic 1D dataset for adding
different percentage of outliers. As we will show, our proposed loss function
is particularly effective even with a high percentage of outliers. We further
benchmark the methods on real world datasets of regression to demonstrate how
the proposed loss function performs compared to other methods. Finally, we
focus on vision based learning task of monocular depth estimation. Here we
test the robustness of the methods to outliers by using the original depth
dataset with outlier sensor noise. The results demonstrate the capability of
our proposed loss function to learn correct depth even in the presence of
outliers.

A

B

Figure 3: Breakaway Point estimation: (A) The change in RMSE and Interval
score with respect to outlier percentage in the data for the 1D regression
dataset. (B): The uncertainty of the neural network output with different
methods learned at different outlier percentages $\eta$. The shaded blue
region is three standard deviations.

### Assessing Uncertainty Estimation Techniques

As this work focuses on the robustness capacity of DNNs to correctly estimate
the uncertainty, we needed metrics which access the quality of the estimated
uncertainty separately from the quality of prediction. _Scoring Rules_ are a
group of such metrics which assess the quality of predicted uncertainties, by
assigning a numerical score based on both the prediction and uncertainty
(Gneiting and Raftery 2007). Most of the literature in uncertainty estimation
for regression use root mean square error (RMSE) for comparing performance of
model and NLL for comparing performance of uncertainty. Although NLL is a
proper Scoring Rule it is not comparable across different distributions. In
this work we use _Interval score_ Scoring Rule.

#### Interval score

uses the prediction interval, with lower and upper endpoints represented by
predictive quantiles at levels $\frac{\alpha}{2}$ and $1-\frac{\alpha}{2}$.
The Interval score is defined as :

$S_{\alpha}^{int}(l,u;y)=(u-l)+\frac{2}{\alpha}(l-y)\mathbf{1}\\{y<l\\}+\frac{2}{\alpha}(y-u)\mathbf{1}\\{y>u\\}$
(5)

The score rewards narrow predictions and penalizes prediction outside the
interval. For our experiments we fix the $\alpha$ to 95%. Based on the mean
and variance predicted by the neural network, we first calculate the 95%
quantile prediction interval ($l$, $u$) which is then used to calculate the
Interval score. As the Interval score compares the 95% quantile prediction
interval it becomes comparable across different distributions (in our case
Gaussian and Laplace distribution).

### Empirical Breakaway Point Benchmark

In this experiment, we empirically benchmark the breakaway point for all the
methods. Breakaway point is the percentage of outliers that can be present in
the dataset after which the estimator predicts statistically wrong outputs
(Huber 2004). Here we interpret the outputs as both the predicted output and
the predicted uncertainty. The regression problem we consider is the 1D
regression problem as shown in Fig. 1 which is a sine wave with increasing
aleatoric uncertainties along the x-axis. The particular dataset was selected
such that the network has to learn the underlying aleatoric uncertainty even
in the presence of outliers. The neural network is 4 layer fully connected
network with 100 neurons in each layer with rectified linear unit (ReLU)
activation function. An illustrative scatter plot of the dataset with outliers
and the predicted uncertainty is shown in Fig. 3-B. In this benchmark we start
with no outliers and increase the outliers upto 50% of the original dataset
and record RMSE and Interval score for each dataset. Fig. 3-A plots the RMSE
and Interval score for different levels of outliers. Gaussian and Ensemble
methods have breakaway points at 10% outliers for the output prediction (Fig.
3-A left plot), while for the uncertainties the breakaway point is 10% for
Evidential learning (Fig. 3-A right plot). In this benchmark, we can conclude
that the breakaway points are different for the predicted output value and its
uncertainty. Some methods are better at output prediction and worse at
uncertainty prediction, while Laplace method performs best on both the
scenarios in the given example.

Figure 4: Benchmark Regression Tasks: RMSE and Interval score for different
methods and datasets. Our method achieves statistical comparable results in
eight of the nine datasets in RMSE. While for Interval score it provides
comparable in all the nine datasets

A

B

C

Figure 5: Monocular Depth Estimation Trained on the NYU-Depth-v2-dataset
containing outliers. (A) Example of estimated depth images and corresponding
uncertainty per method. (B) Uncertainty calibration plot (C) Performance
comparison between models trained on clean data and outlier data.

### Benchmark Regression Tasks

In addition to the toy datasets we validate the uncertainty estimation with
respect to real world datasets as used in the Lakshminarayanan, Pritzel, and
Blundell (2017); Gal and Ghahramani (2016); Amini et al. (2020). We replicate
the experiment and compare RMSE and the Interval score. The purpose of the
experiment is to compare the uncertainty quality obtained from different
methods to that of the proposed method on real world datasets. Each experiment
is executed 20 times each with random sampling of the train and test dataset.
The network used is a fully-connected network with 3 hidden layers containing
100 neurons, and was trained to convergence. All models were trained with
learning rate $\eta=5e^{-3}$ and batch size of 512. The results are shown in
Fig. 4. As seen in the results the proposed Laplace method performs comparable
to other uncertainty estimation methods both in RMSE and Interval score. The
Evidential method shows some deviation in the predicted uncertainty from the
other methods in the Interval score plot for all four datasets.

### Monocular Depth Estimation with Outliers

Figure 6: Outliers in the NYU-Depth-v2-dataset (Silberman et al. 2012). (A)
Input RGB image (B) Depth image captured from Kinect camera. The dark spots
are outliers where no depth information is available because of sensor noise
(C) The histogram of the depth where the outliers can be separated

Monocular depth estimation is a supervised regression task, where the model is
trained to predict depth of each corresponding pixel. The input data is a RGB
image of size HxWx3 and the label is a depth image with size HxW whose pixel
values represent the depth of the scene in meters. We evaluate monocular depth
estimation on the NYU-Depth-v2-dataset which is collected in an indoor
environment using a Kinect v1 camera.

In fields of robotics, the datasets collected are noisy and always
contaminated with outliers. For example, in Fig. 6-B we can observe black
spots (outliers) which contain no depth information, in the collected depth
labels because of sensor noise in the Kinect v1 camera. In the NYU-
Depth-v2-dataset the outliers in the images were removed by using the Levin
Colorization method. For our work, we have not removed the outliers and used
the depth labels along-with the outliers. In order for the results to be
comparable to Amini et al. (2020), rather than directly using the outlier
labels, we simulate the outliers in the cleaned depth data using simulated
noise model of a Kinect v1 camera (Handa et al. 2014; Barron and Malik 2013;
Bohg et al. 2014).

The training dataset consists of 27K RGB-depth image pairs. For comparison, we
use the architecture similar to Amini et al. (2020) which is a U-Net
(Ronneberger, Fischer, and Brox 2015) with spatial dropout. The goal of the
experiment is not to prove state-of-the-art depth estimation but comparison of
uncertainty estimation, hence we have limited to the popular depth estimation
architecture. The U-Net model consists of five convolutional and pooling
blocks in both the down-sampling and up-sampling parts. The input image shape
is (160,128). The final layer outputs HxWx$n$ layers, where $n=2$ for Gaussian
and Laplace regression while $n=4$ for Evidential regression. We perform three
independent runs for all the methods and report the distribution in all the
experiments. The hyper-parameters used are: Adam optimizer with learning rate
$5e^{-5}$, batch size of 32 and over 70000 iterations. The evaluation is done
using a disjoint dataset with non-outlier depth labels.

#### Clean Data vs Outlier Data

We report the comparison of RMSE and Interval score between models trained on
clean depth data and depth data containing outliers in Fig. 5-C. Clean depth
data here we refer to the NYU-Depth-v2-dataset in which the outliers are
removed by using the Levin Colorization method, while outlier data is the
depth data containing outliers. The goal of the experiment is to showcase the
degradation in the uncertainty estimation when the depth data contains
outliers. In Fig. 5-C we can observe the change in performance and uncertainty
estimation when trained with outliers in data. First we observe that adding
outliers degrades performance of all the three methods which is as expected.
For Evidential, we observe there is large shift in RMSE but less in Interval
score, while for Gaussian the Interval score shift is highest. The change
between both the metrics is minimal for Laplace regression indicating
robustness to outliers.

#### Uncertainty Calibration

Here we measure uncertainty calibration of each method using calibration
curves, where the expected confidence (inverse of uncertainty) is plotted with
respect to observed confidence. The expected confidence is calculated based on
the error; for samples with minimum error we expect maximum confidence and
vice versa. For a well calibrated method, the curve will follow the line
$y=x$. In Fig. 5-B we plot the calibration curves for three methods. The
results show calibration curves over three independent trials represented by
the shaded region in different color. Here we observe that the Gaussian method
underestimates confidence in the low confidence region and overestimates
confidence in the high confidence region. Evidential shows calibrated
predictions, however, there is large spread between indicating lack of
reliability in learning. Overall Laplace has the comparative best calibration
over multiple trials.

#### Adversarial Attack Detection

A

B

Figure 7: Adversarial detection (A) Error, (B) Entropy as a function to
adversarial perturbation $\epsilon$ on the NYU-Depth-v2-dataset.

The models which are trained in the above section are evaluated on an
adversarial dataset generated using the Fast Gradient Sign Method (FGSM)
(Goodfellow, Shlens, and Szegedy 2015), using various values of the
adversarial perturbation coefficient $\epsilon$. Adversarial images are
artificial generated images with $\epsilon$ intentional feature perturbation
which causes the model to give wrong predictions. The adversarial attack
results for different values of $\epsilon$ are shown in Fig. 7. We observe in
Fig. 7-A, that the error increases with increase in adversarial attack for all
the methods. In Fig. 7-B, we plot the entropy over the range of $\epsilon$.
The desirable result is that the entropy should increase for increasing level
of $\epsilon$ (Amini et al. 2020), indicating higher uncertainty for higher
adversarial inputs. We observe Laplace and Gaussian show an increase in
entropy as adversarial input increases. For Gaussian, we also observe that for
lower $\epsilon$ it has higher spread of entropy, thus indicating high
uncertainty for no adversarial input images. The rise of uncertainty is
minimum for Evidential. Thus we can conclude that the Laplace loss function
learns to estimate appropriate uncertainty while learning from data containing
outliers.

#### Out-of-Distribution Detection

We also evaluated our trained models on an OOD dataset. The ApolloScape
dataset by Huang et al. 2018 was selected as the OOD dataset. The ApolloScape
dataset is also a depth estimation dataset consisting of images from outdoor
settings, thus making it appropriate for testing OOD. The desirable result is
an increased uncertainty for OOD data predictions as compared to In-
distribution (ID) predictions. The predicted uncertainty for OOD data should
also be statistically different than the ID predictions. This clear separation
helps in defining a threshold value for making informed decisions on trusting
the predictions. Fig. 8-A plots the interquartile box plots of entropy which
shows that only the Laplace method is able to statistically distinguish
between ID and OOD data. The separation is minimum for the Evidential method.
This is corroborated in Fig. 8-B where we plot the distribution of entropy for
both ID and OOD data from the Laplace method.

A

B

Figure 8: Out of distribution Entropy comparison for in-distribution (ID) data
and out-of-distribution (OOD) data (A) Comparison Box plot of entropy between
ID and OOD data (B) Entropy distribution for the Laplace method

## Conclusion and Outlook

We have benchmarked maximum likelihood based uncertainty estimation for deep
regression. We observed that using Gaussian-based loss functions are not
robust to outliers in the training data and consequently provide inaccurate
uncertainty estimates. We proposed a heavy-tailed distribution based loss
function as an alternative in order to improve robustness to outliers. The
heavy-tailed Laplace loss function accurately estimates uncertainty in
predictions and has a high breakaway point compared to the other methods. When
applied to high dimensional datasets containing outliers, such as depth
estimation datasets, the Laplace loss function is able to better estimate the
uncertainties. Our proposed loss function improves state-of-the-art in three
uncertainty estimating benchmarks:  1. high breakaway point;  2. detection of
OOD ;  3. detection of adversarial inputs.  In the future work, we would
benchmark performance by modeling the loss function with other heavy-tailed
distributions. The proposed robust loss function could benefit in building
software which uses uncertainty from the neural network for safe deployment of
deep neural networks in autonomous systems.

## Acknowledgments

Deebul Nair gratefully acknowledges the ongoing support of the Bonn-Aachen
International Center for Information Technology and a PhD scholarship from the
Graduate Institute of the Bonn-Rhein-Sieg University. This work was supported
by the European Union’s Horizon 2020 project SciRoc (grant agreement No
780086), SESAME (grant agreement No 101017258) and DLR CASSy project. All the
code is available at https://github.com/deebuls/uncertainty-robustness-deep-
regression-benchmark,

## References

  * Algan and Ulusoy (2021) Algan, G.; and Ulusoy, I. 2021.  Image classification with deep learning in the presence of noisy labels: A survey.  _Knowledge Based Systems_ , 215: 106771.
  * Amini et al. (2020) Amini, A.; Schwarting, W.; Soleimany, A.; and Rus, D. 2020.  Deep Evidential Regression.  In _Advances in Neural Information Processing Systems_ , volume 33, 14927–14937.
  * Arazo et al. (2019) Arazo, E.; Ortego, D.; Albert, P.; O’Connor, N.; and McGuinness, K. 2019.  Unsupervised label noise modeling and loss correction.  In _International Conference on Machine Learning_ , 312–321.
  * Barron (2019) Barron, J. T. 2019.  A general and adaptive robust loss function.  In _IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 4331–4339.
  * Barron and Malik (2013) Barron, J. T.; and Malik, J. 2013.  Intrinsic Scene Properties from a Single RGB-D Image.  _IEEE/CVF Conference on Computer Vision and Pattern Recognition_.
  * Blundell et al. (2015) Blundell, C.; Cornebise, J.; Kavukcuoglu, K.; and Wierstra, D. 2015.  Weight uncertainty in neural network.  In _International Conference on Machine Learning_ , 1613–1622.
  * Bohg et al. (2014) Bohg, J.; Romero, J.; Herzog, A.; and Schaal, S. 2014.  Robot arm pose estimation through pixel-wise part classification.  _International Conference on Robotics and Automation_.
  * Borg et al. (2019) Borg, M.; Englund, C.; Wnuk, K.; Duran, B.; Levandowski, C.; Gao, S.; Tan, Y.; Kaijser, H.; Lönn, H.; and Törnqvist, J. 2019.  Safely Entering the Deep: A Review of Verification and Validation for Machine Learning and a Challenge Elicitation in the Automotive Industry.  _Journal of Automotive Software Engineering_ , 1(1): 1–19.
  * Bosse, Agamennoni, and Gilitschenski (2016) Bosse, M.; Agamennoni, G.; and Gilitschenski, I. 2016.  Robust Estimation and Applications in Robotics.  _Found. Trends Robotics_ , 4(4): 225–269.
  * Calandra et al. (2016) Calandra, R.; Peters, J.; Rasmussen, C. E.; and Deisenroth, M. P. 2016.  Manifold Gaussian processes for regression.  In _2016 International Joint Conference on Neural Networks_ , 3338–3345. IEEE.
  * Dusenberry et al. (2020a) Dusenberry, M.; Jerfel, G.; Wen, Y.; Ma, Y.; Snoek, J.; Heller, K.; Lakshminarayanan, B.; and Tran, D. 2020a.  Efficient and scalable bayesian neural nets with rank-1 factors.  In _International conference on machine learning_ , 2782–2792.
  * Dusenberry et al. (2020b) Dusenberry, M. W.; Tran, D.; Choi, E.; Kemp, J.; Nixon, J.; Jerfel, G.; Heller, K.; and Dai, A. M. 2020b.  Analyzing the role of model uncertainty for electronic health records.  In _ACM Conference on Health, Inference, and Learning_ , 204–213.
  * Gal and Ghahramani (2016) Gal, Y.; and Ghahramani, Z. 2016.  Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning.  In _International Conference on Machine Learning_ , volume 48 of _JMLR Workshop and Conference Proceedings_ , 1050–1059.
  * Gneiting and Raftery (2007) Gneiting, T.; and Raftery, A. E. 2007.  Strictly proper scoring rules, prediction, and estimation.  _Journal of the American statistical Association_ , 102(477): 359–378.
  * Goel and Chen (2021) Goel, P.; and Chen, L. 2021.  On the Robustness of Monte Carlo Dropout Trained with Noisy Labels.  In _IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2219–2228.
  * Goodfellow, Shlens, and Szegedy (2015) Goodfellow, I.; Shlens, J.; and Szegedy, C. 2015.  Explaining and Harnessing Adversarial Examples.  In _International Conference on Learning Representations_.
  * Handa et al. (2014) Handa, A.; Whelan, T.; McDonald, J.; and Davison, A. J. 2014.  A benchmark for RGB-D visual odometry, 3D reconstruction and SLAM.  In _International Conference on Robotics and Automation_ , 1524–1531. IEEE.
  * Huang et al. (2018) Huang, X.; Cheng, X.; Geng, Q.; Cao, B.; Zhou, D.; Wang, P.; Lin, Y.; and Yang, R. 2018.  The ApolloScape Dataset for Autonomous Driving.  In _2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops_ , 1067–10676.
  * Huber (2004) Huber, P. J. 2004.  _Robust statistics_ , volume 523.  John Wiley & Sons.
  * Jha et al. (2018) Jha, S.; Raman, V.; Sadigh, D.; and Seshia, S. A. 2018.  Safe Autonomy Under Perception Uncertainty Using Chance-Constrained Temporal Logic.  _Journal of Automated Reasoning_ , 60(1): 43–62.
  * Kendall and Gal (2017) Kendall, A.; and Gal, Y. 2017.  What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?  In _Advances in Neural Information Processing Systems_ , volume 30.
  * Lakshminarayanan, Pritzel, and Blundell (2017) Lakshminarayanan, B.; Pritzel, A.; and Blundell, C. 2017.  Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles.  In _Advances in Neural Information Processing Systems 30, Long Beach, CA, USA_ , 6402–6413.
  * Lange, Little, and Taylor (1989) Lange, K. L.; Little, R. J. A.; and Taylor, J. M. 1989.  Robust Statistical Modeling Using the t Distribution.  _Journal of the American Statistical Association_ , 84: 881–896.
  * Lathuilière et al. (2018) Lathuilière, S.; Mesejo, P.; Alameda-Pineda, X.; and Horaud, R. 2018.  Deepgum: Learning deep robust regression with a gaussian-uniform mixture model.  In _European Conference on Computer Vision_ , 202–217.
  * Li, Soltanolkotabi, and Oymak (2020) Li, M.; Soltanolkotabi, M.; and Oymak, S. 2020.  Gradient descent with early stopping is provably robust to label noise for overparameterized neural networks.  In _International conference on artificial intelligence and statistics_ , 4313–4324.
  * Liu et al. (2020) Liu, J. Z.; Lin, Z.; Padhy, S.; Tran, D.; Bedrax-Weiss, T.; and Lakshminarayanan, B. 2020.  Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness.  In _Advances in Neural Information Processing Systems 33:_.
  * Malinin and Gales (2018) Malinin, A.; and Gales, M. 2018.  Predictive uncertainty estimation via prior networks.  In _International Conference on Neural Information Processing Systems_ , 7047–7058.
  * Malinin and Gales (2019) Malinin, A.; and Gales, M. J. F. 2019.  Reverse KL-Divergence Training of Prior Networks: Improved Uncertainty and Adversarial Robustness.  In _Advances in Neural Information Processing Systems_.
  * Neverova, Novotný, and Vedaldi (2019) Neverova, N.; Novotný, D.; and Vedaldi, A. 2019.  Correlated Uncertainty for Learning Dense Correspondences from Noisy Labels.  In _Advances in Neural Information Processing Systems 32_ , 918–926.
  * Nix and Weigend (1994) Nix, D. A.; and Weigend, A. S. 1994.  Estimating the mean and variance of the target probability distribution.  In _International Conference on Neural Networks_ , volume 1, 55–60. IEEE.
  * Northcutt, Jiang, and Chuang (2021) Northcutt, C.; Jiang, L.; and Chuang, I. 2021.  Confident learning: Estimating uncertainty in dataset labels.  _Journal of Artificial Intelligence Research_ , 70: 1373–1411.
  * Pasi, Jarno, and Aki (2011) Pasi, J.; Jarno, V.; and Aki, V. 2011.  Robust Gaussian Process Regression with a Student-t Likelihood.  _Journal of Machine Learning Research_.
  * Riquelme, Tucker, and Snoek (2018) Riquelme, C.; Tucker, G.; and Snoek, J. 2018.  Deep Bayesian Bandits Showdown: An Empirical Comparison of Bayesian Deep Networks for Thompson Sampling.  In _6th International Conference on Learning Representations_.
  * Ronneberger, Fischer, and Brox (2015) Ronneberger, O.; Fischer, P.; and Brox, T. 2015.  U-net: Convolutional networks for biomedical image segmentation.  In _International Conference on Medical image computing and computer-assisted intervention_ , 234–241. Springer.
  * Schwalbe and Schels (2020) Schwalbe, G.; and Schels, M. 2020.  A Survey on Methods for the Safety Assurance of Machine Learning Based Systems.  In _10th European Congress on Embedded Real Time Software and Systems_.
  * Sensoy, Kaplan, and Kandemir (2018) Sensoy, M.; Kaplan, L. M.; and Kandemir, M. 2018.  Evidential Deep Learning to Quantify Classification Uncertainty.  In _Advances in Neural Information Processing Systems 31_ , 3183–3193.
  * Serban, Poll, and Visser (2020) Serban, A.; Poll, E.; and Visser, J. 2020.  Towards Using Probabilistic Models to Design Software Systems with Inherent Uncertainty.  In _Software Architecture - 14th European Conference_ , volume 12292 of _Lecture Notes in Computer Science_ , 89–97. Springer.
  * Silberman et al. (2012) Silberman, N.; Hoiem, D.; Kohli, P.; and Fergus, R. 2012.  Indoor Segmentation and Support Inference from RGBD Images.  In _European Conference on Computer Vision_.
  * Snoek et al. (2015) Snoek, J.; Rippel, O.; Swersky, K.; Kiros, R.; Satish, N.; Sundaram, N.; Patwary, M. M. A.; Prabhat; and Adams, R. P. 2015.  Scalable Bayesian Optimization Using Deep Neural Networks.  In _International Conference on Machine Learning_ , volume 37 of _JMLR Workshop and Conference Proceedings_ , 2171–2180.
  * Tagasovska and Lopez-Paz (2019) Tagasovska, N.; and Lopez-Paz, D. 2019.  Single-Model Uncertainties for Deep Learning.  _Advances in Neural Information Processing Systems_ , 32: 6417–6428.
  * Tak, Ellis, and Ghosh (2019) Tak, H.; Ellis, J. A.; and Ghosh, S. K. 2019.  Robust and Accurate Inference via a Mixture of Gaussian and Student’s t Errors.  _Journal of Computational and Graphical Statistics_ , 28: 415 – 426\.
  * Van Amersfoort et al. (2020) Van Amersfoort, J.; Smith, L.; Teh, Y. W.; and Gal, Y. 2020.  Uncertainty estimation using a single deep deterministic neural network.  In _International Conference on Machine Learning_ , 9690–9700.
  * Wen, Tran, and Ba (2020) Wen, Y.; Tran, D.; and Ba, J. 2020.  Batchensemble: an alternative approach to efficient ensemble and lifelong learning.  In _International Conference on Learning Representations_.
  * West (1984) West, M. 1984.  Outlier models and prior distributions in Bayesian linear regression.  _Journal of the Royal Statistical Society: Series B (Methodological)_ , 46(3): 431–439.
  * Xia et al. (2019) Xia, X.; Liu, T.; Wang, N.; Han, B.; Gong, C.; Niu, G.; and Sugiyama, M. 2019.  Are anchor points really indispensable in label-noise learning?  _Advances in Neural Information Processing Systems_ , 32: 6838–6849.


# A 3D Positioning-based Channel Estimation Method for RIS-aided mmWave
Communications

Yaoshen Cui, Haifan Yin, Li Tan, and Marco Di Renzo, This work was supported
by the National Natural Science Foundation of China under grant No. 62071191.
The corresponding author is Li Tan.Yaoshen Cui, Haifan Yin, and Li Tan are
with the School of Electronic Information and Communications, Huazhong
University of Science and Technology, Wuhan, 430074, China (email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>ltan@hust.edu.cn). M. Di Renzo is
with Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux
et Systèmes, 3 Rue Joliot-Curie, 91192 Gif-sur-Yvette, France. (marco.di-
renzo@universite-paris-saclay.fr)A part of this work [1] was presented in the
10th IEEE/CIC International Conference on Communications in China (IEEE/CIC
ICCC 2021).

###### Abstract

A fundamental challenge in millimeter-wave (mmWave) communication is the
susceptibility to blocking objects. One way to alleviate this problem is the
use of reconfigurable intelligent surfaces (RIS). Nevertheless, due to the
large number of passive reflecting elements on RIS, channel estimation turns
out to be a challenging task. In this paper, we address the channel estimation
for RIS-aided mmWave communication systems based on a localization method. The
proposed idea consists of exploiting the sparsity of the mmWave channel and
the topology of the RIS. In particular, we first propose the concept of
reflecting unit set (RUS) to improve the flexibility of RIS. We then propose a
novel coplanar maximum likelihood-based (CML) 3D positioning method based on
the RUS, and derive the Cramer-Rao lower bound (CRLB) for the positioning
method. Furthermore, we develop an efficient positioning-based channel
estimation scheme with low computational complexity. Compared to state-of-the-
art methods, our proposed method requires less time-frequency resources in
channel acquisition as the complexity is independent to the total size of the
RIS but depends on the size of the RUSs, which is only a small portion of the
RIS. Large performance gains are confirmed in simulations, which proves the
effectiveness of the proposed method.

###### Index Terms:

Reconfigurable intelligent surface, intelligent reflecting surface,
millimeter-wave communication, 3D-localization, Cramer-Rao lower bound

##  I Introduction

The commercialization of the fifth-generation (5G) of communications is in
full swing and is facilitated by several key technologies, including massive
multiple-input multiple-output (MIMO) and millimeter-wave (mmWave)
communications. These technologies leads to significantly increase of the
spectral efficiency and alleviate the shortage of spectrum in wireless
communication networks. However, mmWave communication systems are much more
susceptible to blockage effects compared with lower frequency band
communication systems[2], which may undermine the potential of mmWave
communications in some network deployments. This problem at mmWave frequencies
calls for solutions to make full use of the large available bandwidth.

A possible solution to counteract the coverage problem of mmWave systems is
the deployment of reconfigurable intelligent surfaces (RIS) [3, 4, 5](also
known as intelligent reflecting surfaces (IRSs)[6] and metasurfaces[7]). RIS
is an emerging technology with the capability of intelligently manipulating
the electromagnetic waves, which has lately drawn much attention from academia
and industry as it has the potential of controlling the wireless propagation
environment at a low cost and energy consumption. The authors of [8, 9]
developed free-space path loss channel models for RIS-aided systems based on
the electromagnetic characteristics of the RIS and validated them through
simulations. In [10], the authors developed a new type of high-gain and low-
cost RIS with 256 reflecting elements, which operates at 2.3 GHz and 28.5 GHz.
The authors of [11] demonstrated a prototype of RIS-aided communication system
and showed some indoor and outdoor field trials with a fabricated RIS endowed
with 1100 reflecting elements. The authors of [12] conceived a system for
serving paired power-domain non-orthogonal multiple access (NOMA) users by
designing the passive beamforming coefficients at the RIS.

In theory, the full potential of RIS may be achieved when the channel state
information (CSI) is known. However, the RIS is usually not equipped with
power amplifiers, digital signal processing units, multiple radio frequency
chains and the number of the reflecting elements is usually large, thus making
the CSI estimation a challenging problem.

In literature, several methods have been proposed to solve estimate the
channel information in RIS-aided systems[13]. These include the cascaded
channel decomposition approach [14, 15, 16, 17], and methods based on the
sparsity of channels[18, 19, 20], the quasi-static property [21, 22], etc. In
a multiple-input single-output (MISO) system, the authors of [14, 15] proposed
to decompose the cascaded RIS channel to sub-channels, each of them
corresponding to an RIS element, which are easier to be estimated. In a multi-
user system, the authors of [16] proposed a three-phase pilot-based channel
estimation protocol by activating users one by one. The authors of [17]
developed two iterative estimation algorithms based on the parallel factor
(PARAFAC) decomposition method. By exploiting the angular sparsity of the
channels, the authors of [18, 19] formulated the cascaded CSI estimation as a
sparse signal recovery problem, which is solved using compressive sensing (CS)
algorithms. The authors of [20] proposed a two-stage channel estimation
algorithm by bilinear sparse matrix factorization method. Since the positions
of the base station (BS) and the RIS are usually fixed, the BS-RIS channel
varies slowly and therefore can be assumed quasi-static. The paper [21, 22]
developed a matrix-calibration-based method and a two-timescale channel
estimation framework, respectively. These methods require less training
overhead by exploiting the quasi-static property of the BS-RIS channel.
However, these traditional channel estimation methods generally require huge
time-frequency resources or high computational complexity, especially when the
number of the reflecting elements is very large.

RIS has several applications for positioning as well[23]. In [24], the authors
studied the possibility of RIS-aided localization and derived the
corresponding Fisher information matrix (FIM) and the CRLB in closed-form
expressions. The authors of [25] utilized an RIS to facilitate the positioning
in conjunction with an ultra-wideband (UWB) indoor systems. The authors of
[26] considered an RIS-aided mmWave MIMO system when the direct line of sight
(LOS) is obstructed, and proposed an adaptive phase adjustment method based on
hierarchical codebooks and feedbacks from mobile stations to enable joint
localization and communication. Furthermore, conventional localization
methods, e.g., those based on the received signal strength (RSS), the time of
arrival (TOA), the time difference of arrival (TDOA) and the angle of arrival
(AOA), can also be utilized for improving the localization performance in
mmWave and massive MIMO systems[27].

In this paper, we propose a low-complexity channel estimation scheme that is
based on a novel localization method. The method is designed for application
to RIS-aided mmWave communication systems, and it exploits 1) the geometry of
the RIS, 2) the structural information of the wideband channels, and 3) a new
low-complexity coplanar maximum likelihood-based 3D positioning method. The
main contributions of this paper are summarized below:

  * •

We introduce the concept of _Reflecting Unit Set_ (RUS), which is a set of
neighboring reflecting units in a specific part of the RIS. A joint minimum
mean square estimator (JMMSE) is proposed to calculate the distances between
the RUSs and the user equipment (UE) based on the low-rank property of the
wideband channel covariance matrix in the frequency domain. We utilize the
RUSs as anchors to compute the 3D position of the UE by exploiting the spatial
structure of the wideband channel.

  * •

We propose a coplanar maximum likelihood-based (CML) three-dimensional
localization scheme that offers a high accuracy. Since the reflecting units of
the RIS are coplanar, determining the 3D position of a target is a challenging
problem. Closed-form expressions are derived for the localization of the
target with the help of the RUSs based on the maximum likelihood (ML)
criterion and the principles of Euclidean geometry.

  * •

We derive a closed-form expression of the CRLB for the proposed scheme. We
also show that its complexity does not depend on the number of reflecting
units on the RIS, but it depends only on the number and size of the RUSs. As
the number of the reflecting elements increases, the complexity of our
proposed scheme remains low.

With the aid of numerical results, we show that the proposed method requires
much lower computational complexity and uses much fewer time-frequency
resources for CSI acquisition compared with the state-of-the-art methods[14,
15, 16, 17, 18, 19, 20, 21, 22]. The complexity and the pilot overhead of the
state-of-the-art methods depend on the total size of the RIS, which grows
rapidly when increasing the size of the RIS. In contrast, the required
complexity and the time-frequency resources of the proposed method are just
related to the size of a small portion of the RIS. Specially, we show that the
proposed method needs around 20 orthogonal frequency division multiplexing
(OFDM) symbols for channel estimation (or 0.167 ms with 120 kHz of subcarrier
spacing in mmWave systems) [28], even with thousands reflecting units on the
RIS. To the best of our knowledge, this is the first low-complexity channel
estimation method for RIS-aided systems based on the RUS concept.

The remainder of this paper is organized as follows. In Section II, we
describe the considered RIS-aided mmWave system and channel model. In Section
III, we propose the channel estimation method based on 3D localization. In
Section IV, we present numerical results for the proposed method. Finally,
Section V concludes the paper.

_Notation:_ Bold-face symbols denote vectors and matrices.
$\mathbb{C}^{x\times y}$ is the space of $x\times y$ complex-valued matrices.
$\odot$, $\lfloor\cdot\rfloor$ and $\mod$ stand for the Hadamard product, the
rounding down and the remainder operation, respectively. For a vector
$\bm{x}$, $\|\bm{x}\|$ is its Euclidean norm and $\text{diag}(\bm{x})$ denotes
a diagonal matrix with each diagonal element being the corresponding element
in $\bm{x}$. $\bm{X}^{T}$, $\bm{X}^{*}$, $\bm{X}^{H}$ and $\text{tr}(\bm{X})$
are the transpose, conjugate, conjugate transpose and trace of a matrix
$\bm{X}$, respectively. $\bm{I}$ denotes the identity matrix.

##  II RIS-aided Wireless Communications and Channel modeling

Figure 1: RIS-assisted wireless communication system

Consider an RIS-aided communication system where the LOS path between the BS
and the UE is blocked, as shown in Fig. 1. Due to the high transmission
frequency of mmWave, the number of paths is much smaller than in sub-6 GHz
bands. The RIS contains $N$ reflecting elements and is deployed on the y-z
plane of a Cartesian coordinate system. Without loss of generality, we assume
the bottom left corner of the RIS coincides with the origin of the coordinate
system. The RIS is modeled as a uniform planar array (UPA) with $N_{h}$
columns and $N_{v}$ rows. The area of each element is $A_{u}=D_{h}\times
D_{v}$, where $D_{h}$ and $D_{v}$ are the element spacing in the horizontal
and vertical directions, respectively. For each reflecting element, the gain
is denoted by $G_{u}$, and the normalized power radiation pattern111The power
radiation pattern quantifies the amount of power radiated or received by an
antenna as a function of the direction of observation with respect to the
center of the antenna. is $F(\vartheta,\varphi)$, with $\vartheta$ and
$\varphi$ being the zenith and azimuth angles, respectively.

$F_{n}^{t}=F(\vartheta_{n}^{t},\varphi_{n}^{t})$ and
$F_{n}^{r}=F(\vartheta_{n}^{r},\varphi_{n}^{r})$ are the normalized power
radiation pattern from the $n$-th reflecting unit of the RIS to the BS and
from the $n$-th reflecting unit of the RIS to the UE, respectively, where
$\vartheta_{n}^{t}$, $\varphi_{n}^{t}$, $\vartheta_{n}^{r}$ and
$\varphi_{n}^{r}$ are the corresponding zenith and azimuth angles.

We introduce the vectors $\bm{\phi}=[\phi_{1},\ldots,\phi_{\emph{N}}]$ and
$\bm{\beta}=[\beta_{1},\ldots,\beta_{\emph{N}}]$, where $\phi_{n}\in[0,2\pi)$
and $\beta_{n}\in[0,1],(n=1,\ldots,N)$ denote the phase shift and the
amplitude of the reflection coefficient of the $n$-th reflecting unit of the
RIS, respectively. The vector of reflection coefficients of the RIS, denoted
by $\bm{\theta}\in\mathbb{C}^{N\times 1}$, is expressed as

$\begin{split}\bm{\theta}&={[\theta_{1},\cdots,\theta_{\emph{N}}]}^{T}\\\
&={[\beta_{1}e^{j\phi_{1}},\ldots,\beta_{\emph{N}}e^{j\phi_{\emph{N}}}]}^{T},\end{split}$
(1)

where $\theta_{n}=\beta_{n}e^{j\phi_{n}}(n=1,\ldots,N)$ is the reflection
coefficient of the $n$-th reflection unit of the RIS.

We mainly consider the LOS links between the transceivers and the RIS, which
are usually much stronger than other multipath components in the mmWave band
[29, 30]. Without loss of generality, we assume the BS and the UE both have
single antenna, yet the proposed method can be readily generalized to
multiple-antenna BS/UE cases. The channel between the BS and the $n$-th
reflecting unit is modeled as[8]

$h_{n}=\sqrt{\frac{A_{u}F^{tx}_{n}F^{t}_{n}}{4\pi d_{t,n}^{2}}}e^{\frac{-j2\pi
d_{t,n}}{\lambda}},$ (2)

where $\lambda$ is the wavelength,
$F^{tx}_{n}=F^{tx}(\vartheta_{n}^{tx},\varphi_{n}^{tx})$ is the radiation
pattern of the transmit antenna from the BS to the $n$-th reflecting element
of the RIS with $\vartheta_{n}^{tx}$ and $\varphi_{n}^{tx}$ denoting the
corresponding zenith and azimuth angles, and $d_{t,n}$ denoting the distance
between the BS and the $n$-th reflecting element of the RIS.

The channel vector between the BS and the RIS, which is denoted by
$\bm{h}\in\mathbb{C}^{N\times 1}$, is thus expressed as

$\begin{split}\bm{h}&=[h_{1},\cdots,h_{\emph{N}}]^{T}\\\
&=\leavevmode\resizebox{346.89731pt}{}{$\bigg{[}\sqrt{\frac{A_{u}F^{tx}_{1}F^{t}_{1}}{4\pi
d_{t,1}^{2}}}e^{\frac{-j2\pi
d_{t,1}}{\lambda}},\cdots,\sqrt{\frac{A_{u}F^{tx}_{N}F^{t}_{N}}{4\pi
d_{t,N}^{2}}}e^{\frac{-j2\pi d_{t,N}}{\lambda}}\bigg{]}^{T}$}.\end{split}$ (3)

Similarly, the channel between the UE and the $n$-th reflecting element is
expressed as

$g_{n}=\sqrt{\frac{A_{r}F^{rx}_{n}F_{n}^{r}}{4\pi d_{r,n}^{2}}}e^{\frac{-j2\pi
d_{r,n}}{\lambda}},$ (4)

where $A_{r}$ is the aperture of the receiving antenna,
$F^{rx}_{n}=F^{rx}(\vartheta_{n}^{rx},\varphi_{n}^{rx})$ is the normalized
radiation pattern of the receive antenna from the UE to the $n$-th reflecting
element of the RIS with $\vartheta_{n}^{rx}$ and $\varphi_{n}^{rx}$ being the
zenith and azimuth angles, and $d_{r,n}$ being the distance between the UE and
the $n$-th reflecting element of the RIS.

Based on (4), the LOS channel vector between the UE and all the elements on
the RIS is expressed as:

$\begin{split}\bm{g}&=[g_{1},\cdots,g_{\emph{N}}]^{T}\\\
&=\leavevmode\resizebox{346.89731pt}{}{$\bigg{[}\sqrt{\frac{A_{r}F^{rx}_{1}F^{r}_{1}}{4\pi
d_{r,1}^{2}}}e^{\frac{-j2\pi
d_{r,1}}{\lambda}},\cdots,\sqrt{\frac{A_{r}F^{rx}_{N}F^{r}_{N}}{4\pi
d_{r,N}^{2}}}e^{\frac{-j2\pi d_{r,N}}{\lambda}}\bigg{]}^{T}$}.\end{split}$ (5)

Given the BS-RIS channel and the RIS-UE channel in (3) and (5), the cascaded
BS-RIS-UE channels are expressed as:

$\begin{split}w&=\left((\sqrt{G_{r}}\bm{g})\odot(\sqrt{G_{t}}\bm{h})\right)^{T}(\sqrt{G_{u}}\bm{\theta})\\\
&=\sqrt{G_{r}G_{u}G_{t}}(\bm{g}\odot\bm{h})^{T}\bm{\theta}\\\
&=\frac{\sqrt{G_{r}G_{u}G_{t}A_{r}A_{u}}}{4\pi}\sum_{n=1}^{N}\frac{\sqrt{\tilde{F}_{n}}\theta_{n}}{d_{t,n}d_{r,n}}e^{\frac{-j2\pi(d_{t,n}+d_{r,n})}{\lambda}}\end{split}$
(6)

where $G_{t}$ denotes the transmit antenna gain, $G_{r}$ denotes the receive
antenna gain and $\tilde{F}_{n}=F^{tx}_{n}F^{t}_{n}F^{rx}_{n}F^{r}_{n}$
accounts for the effect of the normalized power radiation patterns on the
received signal power.

The signal received by the UE is

$\begin{split}r&=sw+\eta\\\
&=s\frac{\sqrt{G_{r}G_{u}G_{t}A_{r}A_{u}}}{4\pi}\sum_{n=1}^{N}\frac{\sqrt{\tilde{F}_{n}}\theta_{n}}{d_{t,n}d_{r,n}}e^{\frac{-j2\pi(d_{t,n}+d_{r,n})}{\lambda}}+\eta\end{split}$
(7)

where $s$ is the transmitted signal by the BS and
$\eta\sim\mathcal{CN}(0,\sigma^{2})$ denotes the additive white Gaussian noise
(AWGN) at the UE whose variance is $\sigma^{2}$.

Our target is to maximize the power of the UE received signal $P_{r}$ by
designing the reflection coefficient vector $\bm{\theta}$ based on the
estimated CSI. The problem can be formulated as

$\check{\bm{\theta}}=\operatorname*{argmax}\limits_{\bm{\theta}}\\{|(\bm{g}\odot\bm{h})^{T}\bm{\theta}|^{2}\\}.$
(8)

In principle, if the CSI is perfectly known, the optimal reflection
coefficient of the $n$-th reflecting element of the RIS is

$\check{\theta}_{n}=\frac{g_{n}^{*}h_{n}^{*}}{|g_{n}h_{n}|},n=1,\cdots,N.$ (9)

However, the CSI has to be acquired beforehand. Due to the huge number of RIS
reflecting elements and the unavailability of signal processing units at the
RIS, the CSI acquisition process is a challenging task. Moreover, the mobility
of the UE make it imperative to acquire timely CSI. In the following section,
we will show an efficient channel estimation method to address these issues.

##  III Proposed CSI acquisition Scheme based on 3D localization

In this section, we show the localization-based channel estimation method. The
proposed approach capitalizes on two properties of mmWave channels: 1) The
structural characteristics of the mmWave channel that is dominated by the LOS
path; 2) The different distances between the reflecting elements and the UE.
Since the positions of the RIS and the BS are generally fixed, the BS-RIS
channel is quasi-static and is assumed known. Our target is to estimate the
RIS-UE channel by a two-step 3D positioning method, which is based on a novel
concept named reflecting unit set (RUS), with an illusition shown in Fig. 2.
These RUSs serve as _anchor nodes_ to locate the position of the UE. More
details are given below.

###  III-A Reflecting Unit Set

Figure 2: The model of RIS with RUSs

RUS is a subset of the reflecting elements in a certain part of the RIS. An
RUS can beamform the incident signal to the receiver, provided that a proper
set of phase shifts is applied to the elements of the RUS. Without loss of
generality, all $M$ RUSs are assumed to have the same shape, that is, a
rectangular structure with $\tilde{N}=\tilde{N}_{h}\times\tilde{N}_{v}$
reflecting units, where $\tilde{N}_{h}$ and $\tilde{N}_{v}$ denote the numbers
of columns and rows of the reflecting elements within one RUS, respectively.

The key properties of an RUS are explained follows:

####  III-A1 Active State

An RUS is in the _active_ state when the reflecting elements inside the RUS
reflect the incident signals. If the state is _inactive_ , on the other hand,
the elements do not reflect the signal of interest, i.e., the electromagnetic
waves are absorbed. Examples of metamaterials with these property are given in
[31, 32]. In addition, the elements of the RUS may be equipped with amplifiers
in order to enhance the reflected signal[33].

####  III-A2 Codebook and Codeword

In order to beamform the incident wave to a certain direction, a codebook is
introduced, which contains a set of codewords. Each codeword serves as a
vector of the reflection coefficients. Since the shape of the RUS is an UPA,
we adopt the 3GPP codebook [34], which is based on discrete Fourier transform
(DFT)

$\displaystyle\bm{u}_{p}$ $\displaystyle=\begin{bmatrix}1\quad e^{j\frac{2\pi
p}{O_{1}\tilde{N}_{v}}}\quad\cdots\quad e^{j\frac{2\pi
p(\tilde{N}_{v}-1)}{O_{1}\tilde{N}_{v}}}\end{bmatrix},$ (10)
$\displaystyle\bm{v}_{l,p}$ $\displaystyle=\begin{bmatrix}\bm{u}_{p}\quad
e^{j\frac{2\pi l}{O_{2}\tilde{N}_{h}}}\bm{u}_{p}\quad\cdots\quad
e^{j\frac{2\pi
l(\tilde{N}_{h}-1)}{O_{2}\tilde{N}_{h}}}\bm{u}_{p}\end{bmatrix}^{T}.$ (11)

The vectors $\bm{v}_{l,p}\in\mathbb{C}^{\tilde{N}_{h}\tilde{N}_{v}\times 1}$
are the codewords, where $p=1,\cdots,\tilde{N}_{v}$ and
$l=1,\cdots,\tilde{N}_{h}$ are the indices of the vertical and horizontal beam
directions, respectively. The positive integers $O_{1}$ and $O_{2}$ are the
oversampling factors in the vertical and horizontal directions respectively.
We set the order of the reflecting units of the RUS as follows: starting from
the lower left corner of the RUS, the number increases along the first column,
then the second, etc, until the $\tilde{N}_{h}$-th column. The relationship
between the indices of the vertical and horizontal beam directions and the
index of the codeword (denoted by $\tilde{n}\in[1,\tilde{N}]$) can be
expressed as:

$\tilde{n}=(l-1)\tilde{N}_{v}+p,$ (12)

and then the $\tilde{n}$-th codeword can be defined as
$\tilde{\bm{\theta}}_{\tilde{n}}\triangleq\bm{v}_{l,p}$. The corresponding
codebook, denoted by $\tilde{\bm{\Theta}}$, can be expressed as

$\tilde{\bm{\Theta}}=[\tilde{\bm{\theta}}_{1},...,\tilde{\bm{\theta}}_{\tilde{N}}].$
(13)

####  III-A3 Selection of Codeword

The RUS can be seen as a small RIS. According to (6), the cascaded BS-RUS-UE
channel corresponding to the $\tilde{n}$-th codeword is expressed as

$\tilde{w}_{\tilde{n}}=\sqrt{G_{r}G_{u}G_{t}}(\tilde{\bm{g}}\odot\tilde{\bm{h}})^{T}\tilde{\bm{\theta}}_{\tilde{n}},$
(14)

where $\tilde{\bm{h}}$ and $\tilde{\bm{g}}$, derived from (2)-(3) and (4)-(5)
respectively, denote the BS-RUS channel and the RUS-UE channel. Based on (7),
the received signal at the UE via the RUS is

$\tilde{r}_{\tilde{n}}=s\tilde{w}_{\tilde{n}}+\eta.$ (15)

The codeword that maximizes the power of the UE received signal is chosen, and
the index of the selected codeword can be expressed as

$\tilde{n}_{opt}=\operatorname*{argmax}\limits_{\tilde{n}}\\{\tilde{r}_{\tilde{n}}\\}.$
(16)

Note that several trials are needed to find the optimal codeword. An ordinary
strategy is traversing all codewords in the codebook $\tilde{\bm{\Theta}}$ to
select the codeword. The same selected codeword can be used in a certain time
interval, and even re-used by all the RUSs. This will help reduce the
complexity at the cost of a weaker pilot signal reception at the UE.

Note also that the shape of the RUS is not limited to UPA, and the size of the
RUS can be customized. Smaller RUS may provide better accuracy when the
reflected signal is strong enough for the UE to acquire good channel
estimation quality. However, when the desired reflected signal is weak, for
example due to large pathloss, the area of the RUS should be larger, or the
RUSs should be equipped with power amplifiers.

We then estimate the RUS-UE distances based on the angle-delay structure of
the wideband mmWave channels.

###  III-B RUS-UE Distance Estimations

Due to the sparse distribution of the multipath delays, the channel covariance
matrix in the frequency domain has a low rank. In this paper, we exploit this
low-rankness property to increase the accuracy of the delay estimation. With
ultra-wide bandwidth, mmWave communications can offer higher resolution of the
delay and improve the accuracy of the distance estimation. In this section, we
propose a distance estimation method exploiting the low rank of the frequency-
domain channel covariance matrix.

Denote the central frequency by $F_{c}$. The whole bandwidth contains $K$ sub-
bands, where the center frequency and the wavelength of the $k$-th sub-band
are $f_{k}$ and $\lambda_{k}=\frac{c}{f_{k}}$ respectively ($c$ denotes the
speed of light). The vector of frequencies is denoted as
$\bm{f}=[f_{1},...,f_{K}]$. The power spectral density of the noise is
$\bar{\sigma}^{2}$, and the power of the noise at each sub-band is
$\sigma_{n}^{2}=\bar{\sigma}^{2}f_{d}$, where $f_{d}$ is the bandwdith of each
sub-band. Without loss of generality, we assume equal power allocation in all
sub-bands, and the transmitted power of a sub-band is
$\sigma_{t}^{2}=P_{t}/K$, where $P_{t}$ is the total transmit power.

The received signal that is reflected by the $m$-th RUS (denoted as
$\text{RUS}_{m}$) at sub-band $k$ is

$\tilde{r}_{m,k}=s_{m,k}\tilde{w}_{m,k}+\eta_{m,k},$ (17)

where $\tilde{s}_{m,k}$, $\tilde{w}_{m,k}$ and
$\eta_{m,k}\sim\mathcal{CN}(0,\sigma_{n}^{2})$ denote the transmitted signal,
the cascaded channel via $\text{RUS}_{m}$ at $k$-th sub-band and the AWGN at
the UE, respectively. We show more details of several distance estimation
methods below.

####  III-B1 Matrix Pencil (MP)-based Method

According to the low-rank property and the wideband setting, we develop a
Matrix Pencil [35] based distance estimation method. Considering the cascaded
channel via the $m$-th RUS with $\tilde{N}$ units and the proper codeword, the
relationship between the wavelength of the $K$ sub-bands can be expressed as

$\lambda_{k}=\frac{c}{f_{1}+(k-1)f_{d}},k=1,\ldots,K.$ (18)

Then based on the one-dimensional MP method[36], the estimated distance
between the UE and the $m$-th RUS can be expressed as

$\hat{d}_{m}=\frac{1}{\tilde{N}}\sum_{\tilde{n}=1}^{\tilde{N}}\frac{c\log\ddot{z}_{\tilde{n}}}{-j2\pi
f_{d}}-d_{bs,m}$ (19)

where

$\ddot{z}_{\tilde{n}}=e^{\frac{-j2\pi(d_{t,\tilde{n}}+d_{r,\tilde{n}})f_{d}}{c}},\tilde{n}=1,\dots,\tilde{N},$
(20)

denotes the eigenvalue of the Hankel matrix product for matrix pencil method
and $d_{bs,m}$ denotes the BS-$\text{RUS}_{m}$ distance.

####  III-B2 Conventional MMSE-based Method

Adopting the conventional MMSE-based estimation criterion, we show the MMSE
channel estimation independently for each sub-band, when the $\text{RUS}_{m}$
is activated and a proper codeword is chosen.

Based on (17), with the MMSE method, the estimated cascaded channel of the
$k$-th sub-band via the $m$-th RUS is expressed as

$\hat{w}_{m,k}=R_{m,k}s_{m,k}^{*}(\sigma_{t}^{2}R_{m,k}+\sigma_{n}^{2})^{-1}\tilde{r}_{m,k},$
(21)

where $R_{m,k}=\mathbb{E}[\tilde{w}_{m,k}\tilde{w}_{m,k}^{*}]$ denotes the
covariance coefficient of the cascaded channel of the $k$-th sub-band and the
$m$-th RUS. The mean squared error (MSE) of the estimation is

$\begin{split}\mathcal{M}_{m,k}&=\mathbb{E}\\{\|\hat{w}_{m,k}-\tilde{w}_{m,k}\|^{2}\\}\\\
&=R_{m,k}(1+\frac{\sigma_{t}^{2}}{\sigma_{n}^{2}}R_{m,k})^{-1}\\\
&=\frac{R_{m,k}\sigma_{n}^{2}}{\sigma_{t}^{2}R_{m,k}+\sigma_{n}^{2}}\end{split},$
(22)

Based on (21), the estimated cascaded channel vector of the $m$-th RUS is
expressed as

$\hat{\bm{w}}_{m}=[\hat{w}_{m,1},\hat{w}_{m,2},\ldots,\hat{w}_{m,K}]^{T}.$
(23)

and the distance between the UE and the $\text{RUS}_{m}$ can be acquired by
Bartlett’s method[37].

####  III-B3 Proposed Joint MMSE (JMMSE)-based Method

In order to give full play to the role of ultra-wideband, we develop a JMMSE-
based distance estimation method which is able to exploit the low-rankness
structure of the covariance matrix in the frequency domain[38]. When
$\text{RUS}_{m}$ is in the active state with a proper codeword selected, we
estimate the corresponding RUS-UE distance with the JMMSE channel estimation
of all sub-bands.

The BS-$\text{RUS}_{m}$-UE channel at all $K$ sub-bands is written as

$\tilde{\bm{w}}_{m}=[\tilde{w}_{m,1},\tilde{w}_{m,2},\ldots,\tilde{w}_{m,K}]^{T},$
(24)

where $\tilde{w}_{m,k}(k=1,\cdots,K)$ denotes the BS-$\text{RUS}_{m}$-UE
channel at sub-band $k$.

The received signal via $\text{RUS}_{m}$ at all $K$ sub-bands is

$\tilde{\bm{r}}_{m}=[\tilde{r}_{m,1},\ldots,\tilde{r}_{m,K}]^{T}=\bm{S}_{m}\tilde{\bm{w}}_{m}+\bm{\eta}_{m},$
(25)

where $\bm{S}_{m}=\text{diag}([s_{m,1},\ldots,s_{m,K}]^{T})$ and
$\bm{\eta}_{m}=[\eta_{m,1},\ldots,\eta_{m,K}]^{T}$ denote the transmitted
signal vector of all $K$ sub-bands via $\text{RUS}_{m}$ and the noise vector
respectively.

With the JMMSE estimation method, the estimated cascaded channel via
$\text{RUS}_{m}$ is given by

$\hat{\bm{w}}_{m}=\bm{R}_{m}(\bm{R}_{m}+\frac{\alpha\sigma_{n}^{2}}{\sigma_{t}^{2}}\bm{I})^{-1}\bm{S}_{m}^{-1}\tilde{\bm{r}}_{m}$
(26)

where $\bm{R}_{m}=\mathbb{E}[\tilde{\bm{w}}_{m}\tilde{\bm{w}}_{m}^{H}]$ is the
frequency-domain channel covariance matrix and $\alpha$ is a parameter of
regularization. Denote the true path delay by $t_{m}$, the estimated path
delay of the BS-$\text{RUS}_{m}$-UE channel is computed by

$\begin{split}\hat{t}_{m}&=\operatorname*{argmax}_{t_{m}}\\{\|\bm{\hat{w}}_{m}^{T}\bm{b}(t_{m})\|^{2}\\}\\\
&=t_{m}+\Delta t_{m},\end{split}$ (27)

where $\Delta t_{m}$ is the estimation error and
$\bm{b}(t)\in\mathbb{C}^{K\times 1}$ is the delay response vector defined as

$\bm{b}(t)=[e^{j2\pi f_{1}t},\cdots,e^{j2\pi f_{K}t}]^{T}.$ (28)

The distance between the UE and $\text{RUS}_{m}$ is $d_{m}=c(t_{m}-t_{bs,m})$,
and the estimated one is expressed as

$\begin{split}\hat{d}_{m}&=c(\hat{t}_{m}-t_{bs,m})\\\ &=d_{m}+\Delta
d_{m},\end{split}$ (29)

where $\Delta d_{m}=c\Delta t_{m}$ denotes the estimated error of the distance
of UE-$\text{RUS}_{m}$ channel, and $t_{bs,m}$ denotes the true delay of
BS-$\text{RUS}_{m}$ channel. Since the BS and the RIS are generally fixed,
their locations and the delay $t_{bs,m}$ can be assumed known.

###  III-C Proposed Coplanar ML-based 3D Localization Method

Since all the anchors (i.e., RUSs) of the RIS are on the same plane (y-z
plane), most trilateration localization methods fail to work in this case,
such as the least squares matrix based method [39]. The authors of [40] and
[41] gave alternate ML-based solutions for TDOA and TOA estimation in 2D
settings with col-linear anchors. In this section, we propose a novel coplanar
maximum likelihood-based (CML) 3D localization method which is dedicated to
channel estimation in RIS-aided mmWave communication systems.

Let $\bm{p}=(x,y,z)$ denote the position of the UE. The vector of the
estimated distances between all $M$ RUSs and the UE is written as

$\hat{\bm{d}}=[\hat{d}_{1},\hat{d}_{2},\ldots,\hat{d}_{M}]^{T}=\bm{d}+\bm{e},\\\
$ (30)

where $\bm{d}=[d_{1},d_{2},\ldots,d_{M}]^{T}$ is the vector of the
corresponding true distances, and $\bm{e}=[e_{1},\ldots,e_{M}]^{T}$ is the
vector of the additive estimation errors.

Based on (25) and (26), the estimated error of the BS-$\text{RUS}_{m}$-UE
channel is

$\begin{split}\bm{e}_{wm}&=\hat{\bm{w}}_{m}-\bm{w}_{m}\\\
&=(\bm{A}-\bm{I})\bm{w}_{m}+\bm{A}\bm{S}^{-1}\bm{\eta}_{m}\end{split}$ (31)

where
$\bm{A}=\bm{R}_{m}(\bm{R}_{m}+\frac{\alpha\sigma_{n}^{2}}{\sigma_{t}^{2}}\bm{I})^{-1}$.
When the SNR is high, $\bm{A}$ is close to $\bm{I}$, and then $\bm{e}_{wm}$
follows a zero-mean Gaussian distribution. Based on this fact, the entry of
$\bm{e}$ is modeled as a Gaussian distribution
$e_{m}\sim\mathcal{CN}(0,\sigma_{d}^{2}),m=1,\ldots,M$ where $\sigma_{d}^{2}$
is the variance. The conditional probability density function (PDF) of the
estimated distance $\hat{\bm{d}}$ is thus

$P(\hat{\bm{d}}|\bm{p})=(2\pi\sigma_{d}^{2})^{-\frac{M}{2}}\text{exp}\bigg{(}-\frac{J}{2\sigma_{d}^{2}}\bigg{)},$
(32)

where

$J=\sum_{m=1}^{M}(d_{m}-\hat{d}_{m})^{2}.$ (33)

According to the maximum likelihood criterion, finding $\bm{p}$ from
$\hat{\bm{d}}$ is to equivalent to minimizing (33).

The coordinates of the four given coplanar RUSs are $\bm{p}_{1}=(0,0,0)$,
$\bm{p}_{2}=(0,a,0)$, $\bm{p}_{3}=(0,a,b)$ and $\bm{p}_{4}=(0,0,b)$, where $a$
and $b$ are real. The estimated distances between each anchor node and the UE
to be localized are $\hat{d}_{m}(m=1,\ldots,4)$. Then, the true distances are
calculated by

$\displaystyle d_{1}$ $\displaystyle=(x^{2}+y^{2}+z^{2})^{\frac{1}{2}},$ (34)
$\displaystyle d_{2}$ $\displaystyle=[x^{2}+(y-a)^{2}+z^{2}]^{\frac{1}{2}},$
(35) $\displaystyle d_{3}$
$\displaystyle=[x^{2}+(y-a)^{2}+(z-b)^{2}]^{\frac{1}{2}},$ (36) $\displaystyle
d_{4}$ $\displaystyle=[x^{2}+y^{2}+(z-b)^{2}]^{\frac{1}{2}}.$ (37)

The closed-form solutions for minimizing (33) are shown in Theorem 1.

###### Theorem 1

The maximum likelihood (ML) estimation of the distances are

$\displaystyle d_{1}$
$\displaystyle=\frac{\hat{d}_{1}\bigg{(}\hat{d}_{1}^{2}+\hat{d}_{3}^{2}\pm\sqrt{(\hat{d}_{1}^{2}+\hat{d}_{3}^{2})(\hat{d}_{2}^{2}+\hat{d}_{4}^{2})}\bigg{)}}{2(\hat{d}_{1}^{2}+\hat{d}_{3}^{2})},$
(38) $\displaystyle d_{2}$
$\displaystyle=\frac{\hat{d}_{2}d_{1}}{2d_{1}-\hat{d}_{1}},$ (39)
$\displaystyle d_{3}$ $\displaystyle=\frac{\hat{d}_{3}d_{1}}{\hat{d}_{1}},$
(40) $\displaystyle d_{4}$
$\displaystyle=\frac{\hat{d}_{4}d_{1}}{2d_{1}-\hat{d}_{1}},$ (41)

###### Proof.

To minimize (33), we differentiate $J$ with respect to $x$, $y$ and $z$ and
set the results to zero

$\displaystyle\frac{\partial J}{\partial x}$
$\displaystyle=\sum_{m=1}^{4}\frac{2x(d_{m}-\hat{d}_{m})}{d_{m}}$ (42)
$\displaystyle\frac{\partial J}{\partial y}$
$\displaystyle=\sum_{m=1}^{4}\frac{2y(d_{m}-\hat{d}_{m})}{d_{m}}-\sum_{m=2,3}\frac{2a(d_{m}-\hat{d}_{m})}{d_{m}}$
(43) $\displaystyle\frac{\partial J}{\partial z}$
$\displaystyle=\sum_{m=1}^{4}\frac{2z(d_{m}-\hat{d}_{m})}{d_{m}}-\sum_{m=2,4}\frac{2b(d_{m}-\hat{d}_{m})}{d_{m}}$
(44)

Combining (42) - (44), $d_{2},d_{3}\ \text{and}\ d_{4}$ can be formulated in
terms of $d_{1}$

$d_{2}=\frac{\hat{d}_{2}d_{1}}{2d_{1}-\hat{d}_{1}},\
d_{3}=\frac{\hat{d}_{3}d_{1}}{\hat{d}_{1}},\
d_{4}=\frac{\hat{d}_{4}d_{1}}{2d_{1}-\hat{d}_{1}}$ (45)

From (34)-(37), the relationship among $d_{1},d_{2},d_{3}\ \text{and}\ d_{4}$
can be expressed as

$d_{1}^{2}+d_{3}^{2}-d_{2}^{2}-d_{4}^{2}=0.$ (46)

Then, substituting (45) into (46), we have

$\frac{4K_{13}d_{1}^{4}}{\hat{d}_{1}^{2}}-\frac{4K_{13}d_{1}}{\hat{d}_{1}^{3}}+(K_{13}-K_{24})d_{1}^{2}=0$
(47)

where $K_{13}=\hat{d}_{1}^{2}+\hat{d}_{3}^{2}>0$ and
$K_{24}=\hat{d}_{2}^{2}+\hat{d}_{4}^{2}>0$. Since $d_{m}>0$, (47) can be
solved by

$\begin{split}d_{1}&=\frac{\hat{d}_{1}\big{(}K_{13}\pm\sqrt{K_{13}K_{24}}\big{)}}{2K_{13}}\\\
&=\frac{\hat{d}_{1}\bigg{(}\hat{d}_{1}^{2}+\hat{d}_{3}^{2}\pm\sqrt{(\hat{d}_{1}^{2}+\hat{d}_{3}^{2})(\hat{d}_{2}^{2}+\hat{d}_{4}^{2})}\bigg{)}}{2(\hat{d}_{1}^{2}+\hat{d}_{3}^{2})}.\end{split}$
(48)

Finally, $d_{2},d_{3},d_{4}$ can be derived from (45) and (48). ∎

Since at least one root of $d_{1}$ is positive, we choose the correct root as
follows. If only one root is positive, it is the value we needed. If all roots
are positive, the one corresponding to the smaller $J$ in (33) is selected.

According to (34)-(37), Theorem 1, and Euclidean geometry, the 3D position
$\bm{p}=(x,y,z)$ of the UE can be solved by

$\displaystyle y$
$\displaystyle=\frac{\bar{d}_{1}^{2}-\bar{d}_{2}^{2}-\bar{d}_{3}^{2}+\bar{d}_{4}^{2}+2a^{2}}{4a},$
(49) $\displaystyle z$
$\displaystyle=\frac{\bar{d}_{1}^{2}+\bar{d}_{2}^{2}-\bar{d}_{3}^{2}-\bar{d}_{4}^{2}+2b^{2}}{4b},$
(50) $\displaystyle x$
$\displaystyle=\sqrt{\bar{d}_{m}^{2}-\big{(}y-\bm{p}_{m}(y)\big{)}^{2}-\big{(}z-\bm{p}_{m}(z)\big{)}^{2}},$
(51)

where $m=1,2,3,4$. We assume the UE and the BS are in the front side of the
RIS, which means $\bar{x}>0$, as shown in Fig. 1. The CML method selects the
solution of $\bm{p}=(x,y,z)$ that gives the minimum $J$.

Note that the constraints on the positions of the RUSs can be relaxed, i.e.,
they are not necessarily on the four corners of the RIS, and our method can be
extended easily if we transform the basis of the y-z plane according to the
properties of the linear transformation space.

The closed-form Fisher Information Matrix and the Cramer-Rao lower bounds
(CRLBs) of our 3D positioning method are shown in Theorem 2.

###### Theorem 2

The elements of Fisher Information Matrix (FIM) denoted by $\bm{\Psi}$ is
expressed as

$\begin{split}\bm{\Psi}_{ij}&=-\mathbb{E}\bigg{[}\frac{\partial^{2}\ln
P(\hat{\bm{d}}|\bm{p})}{\partial\bm{p}(i)\partial\bm{p}(j)}\bigg{]}\\\
&=\sum_{m=1}^{M}\frac{1}{\sigma_{d}^{2}}\frac{\big{(}\bm{p}(i)-\bm{p}_{m}(i)\big{)}\big{(}\bm{p}(j)-\bm{p}_{m}(j)\big{)}}{d_{m}^{2}},\end{split}$
(52)

where $i,j=1,2,3,$ and $\bm{p}_{m}=(x_{m},y_{m},z_{m})$ denotes the location
of the $m$-th anchor node.

Then, the CRLB matrix of the CML 3D localization method is

$\bm{C}=\left[\begin{matrix}C_{11}&C_{12}&C_{13}\\\ C_{21}&C_{22}&C_{23}\\\
C_{31}&C_{32}&C_{33}\end{matrix}\right]=\bm{\Psi}^{-1},$ (53)

where $C_{11}$ , $C_{22}$ and $C_{33}$ denote the CRLBs of the estimation of
the (x, y, z) coordinates respectively.

###### Proof.

See Appendix A. ∎

Overall, with the position of the UE estimated by our localization scheme that
includes the distance estimation phase and CML-based 3D localization phase, we
may obtain the estimation of the channel $\bm{g}$ and further compute the
reflecting coefficient vector $\bm{\theta}$, according to Sec. II. Taking the
JMMSE-based method as an example, we summarize the procedure in Algorithm 1.

Algorithm 1 The Proposed JMMSE and CML based Channel Estimation Scheme

1:$\bm{h},\bm{S},\bm{p}_{bs},\bm{p}_{m},m=1,\ldots,M,\bm{p}_{n},n=1...,N$

2:for $m=1,\ldots,M$ do

3: Activate the $m$-th RUS to reflect the transmitted signal;

4: % Choose the optimal codeword of the $m$-th RUS.

5: for $\tilde{n}=1,\ldots,\tilde{N}$ do

6: Select the $\tilde{n}$-th codeword $\tilde{\bm{\theta}}_{\tilde{n}}$ for
the RUS;

7: UE receives the wideband pilot signal:

8:
$\tilde{\bm{r}}_{\tilde{n}}=[\tilde{r}_{\tilde{n},1},\ldots,\tilde{r}_{\tilde{n},K}]^{T}$

9: end for

10:
$\tilde{n}_{opt}=\operatorname*{argmax}\limits_{\tilde{n}}(\|\tilde{\bm{r}}_{\tilde{n}}\|)$;

11: Estimate the BS-$\text{RUS}_{m}$-UE wideband channel based on JMMSE (26);

12: Estimate the channel delay based on (27) and (28);

13:end for

14:% Localization of the UE with the CML method.

15:Determine the position of the UE $\bm{p}$ based on Theorem 1 and (49)-(51);

16:% Channel reconstruction.

17:Calculate the distance between the UE and the elements of the RIS:
$d_{r,n}=\|p-p_{n}\|,n=1,...,N$;

18:Reconstruct the RIS-UE channel $\bm{g}$ based on (5);

19:$\bm{p}$, $\bm{g}$

Note that the radiation patterns are not needed when computing the optimal
reflection coefficients $\bm{\theta}$, only the path delay
$d_{r,n}=\|p-p_{n}\|,n=1,...,N$ is sufficient. The complexity order of our
method is $\mathcal{O}(M\tilde{N})$, which does not depend on the total number
of the reflecting units on the RIS. It helps to keep the complexity low,
especially for a large RIS. As the number of the reflecting elements
increases, the complexity and training overhead of most traditional methods
will increase significantly. For a large RIS, our proposed scheme has a better
performance due to the exploitation of the differences of the RUS-UE
distances. In reality, the number of the RUSs can be set to $M=4$ according to
our CML method, and the number of reflecting units on each RUS is much smaller
than that on the overall RIS, i.e., ($\tilde{N}\ll N$).

Consider for example an RIS with four RUSs, each of the RUS has four rows and
four columns of reflecting units. A DFT codebook containing 16 codewords is
used. Assuming the same codeword is shared by all RUSs, the searching of the
codeword takes 16 OFDM symbols. Four more symbols are needed to complete
activating all RUSs. Thus the total number of OFDM symbols needed can be
limited to 20. In general, the above proposed method can be completed
efficiently with low time-frequency resource utilization.

##  IV Numerical Results

In this section, we verify the proposed method in simulations and analyze its
performance and robustness. We consider an RIS-aided mmWave communication
system as shown in Fig. 1. The coordinates of the BS are $(5,-5,2)$ (meters),
and the LOS path between the BS and the UE is assumed to be negligible due to
the presence of blocking objects. The main simulation parameters are shown in
Table I.

TABLE I: Simulation Parameters Symbol | Value | Definition
---|---|---
$N$ | 8192 | Number of reflecting units on the RIS
$N_{v}$ | 64 | Number of rows of reflecting units on the RIS
$N_{h}$ | 128 | Number of columns of reflecting units on the RIS
$D_{v}$(m) | 0.005 | Row spacing between adjacent units
$D_{h}$(m) | 0.005 | Column spacing between adjacent units
$M$ | 4 | Number of RUSs on the RIS
$\tilde{N}$ | 16 | Number of reflecting units on an RUS
$\tilde{N}_{v}$ | 4 | Number of rows of reflecting units on an RUS
$\tilde{N}_{h}$ | 4 | Number of columns of reflecting units on an RUS
$F_{c}$(GHz) | 28 | Center frequency
$f_{d}$(Mhz) | 3.6 | Bandwidth of a subband (5 resource blocks)
$K$ | 128 | Number of subbands
$\bar{\sigma}^{2}$(dBm/Hz) | -170 | Noise power spectral density
$\alpha$ | $10^{4}$ | Coefficient of regularization

The RIS is located in the y-z plane with the center at
$\bm{p}_{c}=(0,0.32,0.16)$. The RIS contains $N=N_{v}\times N_{h}$ reflecting
units. $M=4$ RUSs are at the four corners of RIS with size
$\tilde{N}=\tilde{N}_{h}\times\tilde{N}_{v}$, as shown in Fig. 2. The total
transmit power is $P_{t}=30$ dBm. The gain of each reflecting unit is
$G_{u}=9.03$ dBi[8]. The antenna gains of the transmit antenna and receive
antenna are $G_{t}=G_{r}=21$ dBi, the aperture of the receive antenna
$A_{r}=\frac{\lambda^{2}}{4\pi}$, and the radiation patterns of the transmit
and receive antennas are $F_{n}^{tx}=F_{n}^{rx}=1,n=1,\ldots,N$. The radiation
pattern of each reflecting unit is:

$F(\vartheta,\varphi)=\left\\{\begin{aligned}
\cos^{3}(\vartheta)&\quad\vartheta\in[0,\frac{\pi}{2}],\varphi\in[0,2\pi]\\\
0&\quad\vartheta\in(0,\frac{\pi}{2}],\varphi\in[0,2\pi]\end{aligned}\right.$
(54)

Figure 3: The RMSE performance of distance estimation vs SNR of the received
signal via the RUSs.

For a fixed UE with coordinate $\bm{p}=(5,0.32,0.16)$, Fig. 3 shows the root
mean squared error (RMSE) of the estimated distances, defined as
$\sqrt{\mathbb{E}[\|\bm{d}-\hat{\bm{d}}\|^{2}]}$, versus the signal to noise
radio (SNR) of the received signal via the RUSs, which is defined as

$\text{SNR}=\frac{\|\bm{S}\tilde{\bm{w}}\|^{2}}{\|\bm{\eta}\|^{2}}=\frac{P_{t}\|\tilde{\bm{w}}\|^{2}}{\bar{\sigma}^{2}Kf_{d}},$
(55)

according to (25). It varies as a function of the total transmit power
$P_{t}$. The distance estimated by the conventional method based on Time of
Flight (ToF) is drawn from the Gaussian distribution
$\mathcal{CN}(0,\sigma_{\text{ToF}}^{2})$ where $\sigma_{\text{ToF}}^{2}$ is 1
ns[27]. For the estimated distance, it is observed that the RMSE under the
proposed JMMSE method is much smaller than the other three schemes. The RMSEs
under MMSE and MP are both smaller than the ToF scheme when the SNR is greater
than -5 dB.

Figure 4: The RMSE performance of the localization vs SNR of the received
signal via the RUSs.

Combining the aforementioned distance estimation methods and the CML
localization method, Fig. 4 shows the RMSE of the localization of the UE,
which is defined as $\sqrt{\mathbb{E}[\|\bm{p}-\hat{\bm{p}}\|^{2}]}$. Similar
to the result of the distance estimation, the RMSE obtained by combining JMMSE
and CML is much smaller than the other three schemes and achieves centimeter
level positioning accuracy.

Figure 5: The MSE performance of the 3D coplanar localization method versus
the distance error $\sigma_{d}^{2}$.

To analyze the performance of the proposed CML localization algorithm
theoretically, we consider a UE at a fixed position $\bm{p}=(5,0.32,0.16)$,
which is unknown to the system, and assume that all the estimated errors of
the $\text{UE-RUS}_{m}$ distances share the same noise variance
$\sigma_{d}^{2}$. That is, each error of the measured distance is drawn from
$\mathcal{CN}(0,\sigma_{d}^{2})$. The sum of the MSEs is defined as

$\tilde{\mathcal{M}}=\mathcal{M}(x)+\mathcal{M}(y)+\mathcal{M}(z),$ (56)

where $\mathcal{M}(x)$, $\mathcal{M}(y)$ and $\mathcal{M}(z)$ are the MSEs of
the estimated x, y, z-coordinate of the UE respectively, e.g.,
$\mathcal{M}(x)=\mathbb{E}[(\hat{x}-\bar{x})^{2}]$. According to Theorem 2,
the sum of CRLBs is defined as

$\tilde{\mathcal{C}}=C_{11}+C_{22}+C_{33}.$ (57)

Fig. 5 compares the sum of the MSE for the CML method with the corresponding
CRLB and the traditional weighted least squares (WLS) algorithm [42]. The
figure shows the proposed CML is much closer to the CRLB than the traditional
WLS.

Figure 6: The normalized MSE of the channel estimation versus the distance
error $\sigma_{d}^{2}$.

Further, we illustrate the performance of the proposed channel estimation
scheme against the WLS algorithm and the conventional trilateration method
[43]. The normalized MSE (NMSE) of the channel estimation is defined as

$\text{NMSE}=\mathbb{E}\Big{[}\frac{|w-\hat{w}|^{2}}{|w|^{2}}\Big{]},$ (58)

where $w$ and $\hat{w}$ denote the true cascaded BS-RIS-UE channel and the
estimated channel, respectively. Fig. 6 shows the NMSE of channel estimation
based on the aforementioned methods, and indicates that the proposed scheme
can achieve lower NMSE than -15 dB when $\sigma_{d}^{2}$ is smaller than
$10^{-7}$, which outperforms the other two methods.

We let the UE move on the central perpendicular line (CPL) of the RIS. The
total transmit power of the BS is fixed to $P_{t}=30$ dBm. We combine the
JMMSE distance estimation method and CML localization algorithm, which is
labeled as “Proposed Method”. The curve obtained with the true position of the
UE is marked as “Upper Bound”, meaning that perfect CSI is known. In the
“Uniform Random Case”, the phase shift coefficient of each reflecting element
is randomly and uniformly distributed, i.e.,
$\phi_{n}\sim\mathcal{U}(0,2\pi)$. The performance of the JBF algorithm [20]
is also shown as a baseline.

Figure 7: The SNR of the received signal via the optimized RIS versus the
X-coordinate of the UE.

Fig. 7 shows the SNR of the received signal at the UE as a function of the
x-coordinate of the UE. The proposed method achieves a higher SNR than the JBF
method, as it approaches the upper bound and has around 40 dB gain over the
random phase case.

Figure 8: The throughput at the receiver versus X-coordinate of the UE.

Fig. 8 shows the throughput versus the x-coordinate of the UE. The throughput
is defined as $\log_{2}(1+\text{SNR})$. It is observed that the throughput of
the proposed method is much higher than the other three schemes and approaches
the upper bound. Moreover, the proposed method provides around 13 bps/Hz
higher throughput than the random case.

##  V Conclusions

In this paper, we considered the channel estimation of an RIS-aided mmWave
communication system. We first proposed the RUS concept for the purpose of
positioning the UE based on the structure of the mmWave channel. We then
proposed a CML 3D localization algorithm with a JMMSE distance estimation
method. The CRLBs of the distance estimation error of our proposed method are
derived. The CSI is reconstructed with the positioning results. By exploiting
the the sparse multipath angle-delay distribution of the mmWave channel, our
method obtains accurate channel estimate we at the cost of very few training
resources. Simulation results demonstrated significant performance gain in
terms of the SNR over traditional methods.

##  Appendix A  Proof of Theorem 2

The distance between the UE and the $m$-th RUS is defined as:

$\begin{split}d_{m}&=\|\bm{p}-\bm{p}_{m}\|\\\
&=[(\bar{x}-x_{m})^{2}+(\bar{y}-y_{m})^{2}+(\bar{z}-z_{m})^{2}]^{\frac{1}{2}},\end{split}$
(59)

the partial derivative of which with respect to $\bm{p}(i),i=1,2,3$, can be
calculated as

$\frac{\partial
d_{m}}{\partial\bm{p}(i)}=\frac{\bm{p}(i)-\bm{p}_{m}(i)}{d_{m}}$ (60)

The log-likelihood of (32) is

$\ln
P(\hat{\bm{d}}|\bm{p})=-\sum_{m=1}^{M}\frac{(\hat{d}_{m}-d_{m})^{2}}{2\sigma_{d}^{2}}-\sum_{m=1}^{M}\ln\sqrt{2\pi\sigma_{d}^{2}}$
(61)

and the partial derivative of (61) with respect to $\bm{p}(i)$ is

$\begin{split}\frac{\partial\ln
P(\hat{\bm{d}}|\bm{p})}{\partial\bm{p}(i)}&=-\sum_{m=1}^{M}\frac{\partial}{\partial\bm{p}(i)}\left[\frac{(\hat{d}_{m}-d_{m})^{2}}{2\sigma_{d}^{2}}\right]\\\
&=\sum_{m=1}^{M}\frac{(\hat{d}_{m}-d_{m})[\bm{p}(i)-\bm{p}_{m}(i)]}{\sigma_{d}^{2}d_{m}}\\\
\end{split}$ (62)

Further, the second partial derivative of (32) with respect to $\bm{p}(i)$ and
$\bm{p}(j)$ is

$\begin{split}\frac{\partial^{2}\ln
P(\hat{\bm{d}}|\bm{p})}{\partial\bm{p}(i)\partial\bm{p}(j)}&=\sum_{m=1}^{M}\frac{1}{\sigma_{d}^{2}}\frac{\partial}{\partial\bm{p}(j)}\Big{[}\frac{(\hat{d}_{m}-d_{m})[\bm{p}(i)-\bm{p}_{m}(i)]}{d_{m}}\Big{]}\\\
&=\sum_{m=1}^{M}\frac{1}{\sigma_{d}^{2}}\Big{[}-\frac{[\bm{p}(i)-\bm{p}_{m}(i)]\cdot[\bm{p}(j)-\bm{p}_{m}(j)]}{d_{m}^{2}}\\\
&+\frac{(\hat{d}_{m}-d_{m})[\bm{p}(i)-\bm{p}_{m}(i)]\cdot[\bm{p}(j)-\bm{p}_{m}(j)]}{d_{m}^{3}}\Big{]}\end{split}$
(63)

Since $\triangle d_{m}$ follows a Gaussian distribution,
$\mathbb{E}[\hat{d}_{m}-d_{m}]=0$. The element of the FIM is expressed as:

$\begin{split}\bm{\Psi}_{ij}&=-\mathbb{E}\bigg{[}\frac{\partial^{2}\ln
P(\hat{\bm{d}}|\bm{p})}{\partial\bm{p}(i)\partial\bm{p}(j)}\bigg{]}\\\
&=\sum_{m=1}^{M}\frac{1}{\sigma_{d}^{2}}\frac{[\bm{p}(i)-\bm{p}_{m}(i)]\cdot[\bm{p}(j)-\bm{p}_{m}(j)]}{d_{m}^{2}},\end{split}$
(64)

Finally, the CRLB matrix can be expressed as:

$\bm{C}=\left[\begin{matrix}C_{11}&C_{12}&C_{13}\\\ C_{21}&C_{22}&C_{23}\\\
C_{31}&C_{32}&C_{33}\end{matrix}\right]=\bm{\Psi}^{-1}.$ (65)

## References

  * [1] Y. Cui and H. Yin, “Channel estimation for RIS-aided mmWave communications via 3D positioning,” in _Proc. IEEE/CIC Int. Conf. Commun. China (ICCC)_ , Jul. 2021, pp. 399–404.
  * [2] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” _IEEE Access_ , vol. 1, pp. 335–349, 2013.
  * [3] M. Di Renzo, K. Ntontin, J. Song, F. H. Danufane, X. Qian, F. Lazarakis, J. De Rosny, D.-T. Phan-Huy, O. Simeone, R. Zhang, M. Debbah, G. Lerosey, M. Fink, S. Tretyakov, and S. Shamai, “Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison,” _IEEE Open J. Commun. Soc._ , vol. 1, pp. 798–807, 2020.
  * [4] M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. de Rosny, and S. Tretyakov, “Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 11, pp. 2450–2525, 2020.
  * [5] M. Di Renzo, F. H. Danufane, and S. Tretyakov, “Communication models for reconfigurable intelligent surfaces: From surface electromagnetics to wireless networks optimization,” _arXiv:2110.00833_ , 2021. [Online]. Available: https://arxiv.org/abs/2110.00833
  * [6] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Trans. Wireless Commun._ , pp. 1–1, Aug. 2019.
  * [7] W. Tang, X. Li, J. Y. Dai, S. Jin, Y. Zeng, Q. Cheng, and T. J. Cui, “Wireless communications with programmable metasurface: Transceiver design and experimental results,” _China Commun._ , vol. 16, no. 5, pp. 46–61, May. 2019.
  * [8] W. Tang, M. Z. Chen, X. Chen, J. Y. Dai, Y. Han, M. Di Renzo, Y. Zeng, S. Jin, Q. Cheng, and T. J. Cui, “Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimental measurement,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 1, pp. 421–439, 2021.
  * [9] W. Tang, X. Chen, M. Z. Chen, J. Y. Dai, Y. Han, M. Di Renzo, S. Jin, Q. Cheng, and T. J. Cui, “Path loss modeling and measurements for reconfigurable intelligent surfaces in the millimeter-wave frequency band,” _arXiv:2101.08607_ , 2021. [Online]. Available: https://arxiv.org/abs/2101.08607
  * [10] L. Dai, B. Wang, M. Wang, X. Yang, J. Tan, S. Bi, S. Xu, F. Yang, Z. Chen, M. Di Renzo, C. B. Chae, and L. Hanzo, “Reconfigurable intelligent surface-based wireless communications: Antenna design, prototyping, and experimental results,” _IEEE Access_ , vol. 8, pp. 45 913–45 923, 2020.
  * [11] X. Pei, H. Yin, L. Tan, L. Cao, Z. Li, K. Wang, K. Zhang, and E. Björnson, “RIS-aided wireless communications: Prototyping, adaptive beamforming, and indoor/outdoor field trials,” _IEEE Trans. Commun._ , vol. 69, no. 12, pp. 8627–8640, Dec. 2021.
  * [12] T. Hou, Y. Liu, Z. Song, X. Sun, Y. Chen, and L. Hanzo, “Reconfigurable intelligent surface aided NOMA networks,” _IEEE J. Select. Areas Commun._ , vol. 38, no. 11, pp. 2575–2588, Nov. 2020.
  * [13] C. Pan, G. Zhou, K. Zhi, S. Hong, T. Wu, Y. Pan, H. Ren, M. Di Renzo, A. L. Swindlehurst, R. Zhang _et al._ , “An overview of signal processing techniques for ris/irs-aided wireless systems,” _arXiv:2112.05989_ , 2021\. [Online]. Available: https://arxiv.org/abs/2112.05989
  * [14] D. Mishra and H. Johansson, “Channel estimation and low-complexity beamforming design for passive intelligent surface assisted MISO wireless energy transfer,” in _Proc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP)_ , May 2019, pp. 4659–4663.
  * [15] T. L. Jensen and E. D. Carvalho, “An optimal channel estimation scheme for intelligent reflecting surfaces based on a minimum variance unbiased estimator,” in _Proc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP)_ , May 2020, pp. 5000–5004.
  * [16] Z. Wang, L. Liu, and S. Cui, “Channel estimation for intelligent reflecting surface assisted multiuser communications: Framework, algorithms, and analysis,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 10, pp. 6607–6620, Oct. 2020.
  * [17] L. Wei, C. Huang, G. C. Alexandropoulos, C. Yuen, Z. Zhang, and M. Debbah, “Channel estimation for RIS-empowered multi-user MISO wireless communications,” _IEEE Trans. Commun._ , pp. 1–1, 2021.
  * [18] J. Chen, Y.-C. Liang, H. V. Cheng, and W. Yu, “Channel estimation for reconfigurable intelligent surface aided multi-user MIMO systems,” _arXiv:1912.03619_ , 2019. [Online]. Available: http://arxiv.org/abs/1912.03619
  * [19] P. Wang, J. Fang, H. Duan, and H. Li, “Compressed channel estimation for intelligent reflecting surface-assisted millimeter wave systems,” _IEEE Signal Process. Lett._ , vol. 27, pp. 905–909, 2020.
  * [20] Z. He and X. Yuan, “Cascaded channel estimation for large intelligent metasurface assisted Massive MIMO,” _IEEE Wirel. Commun. Lett._ , vol. 9, no. 2, pp. 210–214, 2020.
  * [21] H. Liu, X. Yuan, and Y. J. A. Zhang, “Matrix-calibration-based cascaded channel estimation for reconfigurable intelligent surface assisted multiuser MIMO,” _IEEE J. Sel. Areas Commun._ , pp. 1–1, 2020.
  * [22] C. Hu, L. Dai, S. Han, and X. Wang, “Two-timescale channel estimation for reconfigurable intelligent surface aided wireless communications,” _IEEE Trans. Commun._ , vol. 69, no. 11, pp. 7736–7747, 2021.
  * [23] A. Abrardo, D. Dardari, and M. Di Renzo, “Intelligent reflecting surfaces: Sum-rate optimization based on statistical position information,” _IEEE Trans. Commun._ , vol. 69, no. 10, pp. 7121–7136, 2021.
  * [24] H. Wymeersch and B. Denis, “Beyond 5G wireless localization with reconfigurable intelligent surfaces,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , 2020, pp. 1–6.
  * [25] T. Ma, Y. Xiao, X. Lei, W. Xiong, and Y. Ding, “Indoor localization with reconfigurable intelligent surface,” _IEEE Commun. Lett._ , pp. 1–1, 2020.
  * [26] J. He, H. Wymeersch, T. Sanguanpuak, O. Silven, and M. Juntti, “Adaptive beamforming design for mmWave RIS-aided joint localization and communication,” in _Proc. IEEE Wireless Commun. Net. Conf. Work. (WCNCW)_ , 2020, pp. 1–6.
  * [27] F. Lemic, J. Martin, C. Yarp, D. Chan, V. Handziski, R. Brodersen, G. Fettweis, A. Wolisz, and J. Wawrzynek, “Localization as a feature of mmWave communication,” in _Proc. Int. Wireless Commun. Mobile Compu. Conf. (IWCMC)_ , 2016, pp. 1033–1038.
  * [28] 3GPP, _NR; Physical Channels and Modulation (Release 15)_. Technical Specification TS 38.211, available: http://www.3gpp.org.
  * [29] B. Di, H. Zhang, L. Li, L. Song, Y. Li, and Z. Han, “Practical hybrid beamforming with finite-resolution phase shifters for reconfigurable intelligent surface based multi-user communications,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 4, pp. 4565–4570, 2020.
  * [30] E. Basar and I. Yildirim, “SimRIS channel simulator for reconfigurable intelligent surface-empowered communication systems,” in _2020 IEEE Latin-American Conf. Commun. (LATINCOM)_ , 2020, pp. 1–6.
  * [31] M. F. Imani, D. R. Smith, and P. Hougne, “Perfect absorption in a disordered medium with programmable meta-atom inclusions,” _Adv. Funct. Mater._ , vol. 30, no. 52, p. 2005310, Dec. 2020.
  * [32] S. Li, F. Xu, X. Wan, T. J. Cui, and Y. Q. Jin, “Programmable metasurface based on substrate-integrated waveguide for compact dynamic-pattern antenna,” _IEEE Trans. Antennas Propag._ , pp. 1–1, 2020\.
  * [33] Z. Zhang, L. Dai, X. Chen, C. Liu, F. Yang, R. Schober, and H. V. Poor, “Active RIS vs. passive RIS: Which will prevail in 6G?” _arXiv:2103.15154_ , 2021. [Online]. Available: https://arxiv.org/abs/2103.15154
  * [34] 3GPP, _5G; NR; Physical Layer Procedures for Data (Release 15)_. Technical Specification TS 38.214, available: http://www.3gpp.org.
  * [35] Y. Hua and T. K. Sarkar, “Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,” _IEEE Trans. Acoust., Speech, Signal Process._ , vol. 38, no. 5, pp. 814–824, 1990.
  * [36] A. Gaber and A. Omar, “A study of wireless indoor positioning based on joint TDOA and DOA estimation using 2-D matrix pencil algorithms and IEEE 802.11ac,” _IEEE Trans. Wireless Commun._ , vol. 14, no. 5, pp. 2440–2454, 2015.
  * [37] E. H. Gismalla and E. Alsusa, “On the performance of energy detection using Bartlett’s estimate for spectrum sensing in cognitive radio systems,” _IEEE Trans. Signal Process._ , vol. 60, no. 7, pp. 3394–3404, Jul. 2012.
  * [38] H. Yin, D. Gesbert, M. Filippou, and Y. Liu, “A coordinated approach to channel estimation in large-scale multiple-antenna systems,” _IEEE J. Sel. Areas Commun._ , vol. 31, no. 2, pp. 264–273, Feb. 2013.
  * [39] K. Cheung, H. So, W.-K. Ma, and Y. Chan, “Least squares algorithms for time-of-arrival-based mobile location,” _IEEE Trans. Signal Process._ , vol. 52, no. 4, pp. 1121–1130, 2004.
  * [40] Y. T. Chan and K. C. Ho, “A simple and efficient estimator for hyperbolic location,” _IEEE Trans. Signal Process._ , vol. 42, no. 8, pp. 1905–1915, 1994.
  * [41] Y.-T. Chan, H. Yau Chin Hang, and P. chung Ching, “Exact and approximate maximum likelihood localization algorithms,” _Trans. Veh. Technol._ , vol. 55, no. 1, pp. 10–16, 2006.
  * [42] D. Ni, O. A. Postolache, C. Mi, M. Zhong, and Y. Wang, “UWB indoor positioning application based on Kalman filter and 3-D TOA localization algorithm,” in _Proc. Int. Symp. Adv. Topics Electr. Eng. (ATEE)_ , 2019, pp. 1–6.
  * [43] E. Doukhnitch, M. Salamah, and E. Ozen, “An efficient approach for trilateration in 3D positioning,” _Computer Communications_ , vol. 31, no. 17, pp. 4124–4129, 2008.