# Unsupervised Learning for Human Sensing Using Radio Signals

Tianhong Li Lijie Fan∗ Yuan Yuan∗ Dina Katabi
MIT CSAIL Indicates equal contribution. This work was supported by the GIST-
MIT Research Collaboration grant funded by GIST.

###### Abstract

There is a growing literature demonstrating the feasibility of using Radio
Frequency (RF) signals to enable key computer vision tasks in the presence of
occlusions and poor lighting. It leverages that RF signals traverse walls and
occlusions to deliver through-wall pose estimation, action recognition, scene
captioning, and human re-identification. However, unlike RGB datasets which
can be labeled by human workers, labeling RF signals is a daunting task
because such signals are not human interpretable. Yet, it is fairly easy to
collect unlabelled RF signals. It would be highly beneficial to use such
unlabeled RF data to learn useful representations in an unsupervised manner.
Thus, in this paper, we explore the feasibility of adapting RGB-based
unsupervised representation learning to RF signals. We show that while
contrastive learning has emerged as the main technique for unsupervised
representation learning from images and videos, such methods produce poor
performance when applied to sensing humans using RF signals. In contrast,
predictive unsupervised learning methods learn high-quality representations
that can be used for multiple downstream RF-based sensing tasks. Our empirical
results show that this approach outperforms state-of-the-art RF-based human
sensing on various tasks, opening the possibility of unsupervised
representation learning from this novel modality.

##  1 Introduction

RF-based vision has emerged as an attractive research direction that uses
Radio Frequency (RF) signals to “see” human poses, body shape, and activities
through walls and in dark settings [3, 12, 11, 27, 20, 44, 46, 23, 33, 37, 24,
21, 4, 38, 40, 35, 31]. While visible light can be easily blocked by walls and
opaque objects, RF signals in the WiFi range, can traverse such occlusions. RF
signals reflect off the human body, provide information to track people, and
capture their shape and actions. Previous work has leveraged those properties
to detect people and track their walking speed [2]. More recent advances in
RF-based learning have demonstrated the feasibility of using neural networks
to perform various computer vision tasks like pose estimation [46], action
recognition [24], scene captioning [12], and human re-identification [11]
using RF signals as the sole input. These systems are capable of working
through walls and in dark scenarios, and thus go beyond the limitations faced
by RGB-based systems.

Such RF-based tasks have typically used supervised learning. Yet, unlike RGB
datasets, which can be labeled by human workers, labeling RF signals is a
daunting task because such signals are not human interpretable. To label RF
data, a synchronized human-interpretable stream, like video, must be present
to assist the annotator. Specifically, when collecting an RF dataset,
researchers deploy an RF device and a camera system, synchronize the data
streams from the two systems, and calibrate their views and positions with
respect to each other. The annotator then generates labels based on the RGB
data. However, using RGB data as assistance could only label a small portion
of RF data. In particular, it is hard to use this approach to generate
labelled RF datasets of natural living at home since most people would have
privacy concerns about deploying cameras in their homes. Also, a single camera
has a limited field of view; thus, users typically need to deploy synchronize
and calibrate a multi-camera system, introducing significant overhead.
Moreover, cameras do not work well in dark settings and with occlusions, which
are common indoor scenarios.

The above limitations motivate the need for learning from unlabelled RF
signals. Unsupervised or self-supervised representation learning has attracted
much recent interest and is a rapidly growing research area in computer vision
[16, 10, 39, 19, 14, 5, 48, 25, 15]. It refers to learning data
representations that capture potential labels of interest and doing so without
human supervision. Most unsupervised representation learning methods are
designed for RGB data; and no past work has shown the feasibility of
unsupervised learning from RF signals.

Figure 1: Illustration of RF signals and corresponding RGB images (for
illustration ONLY). The signal reflected off the person occupies a small
region in the received RF signal. Further, there are many other reflections
from other objects.

Applying traditional unsupervised learning methods to RF data is not straight-
forward for the following reasons:

  * •

Traditional unsupervised methods either rely on strong augmentations, e.g.,
color jittering in contrastive learning, or require pretext tasks such as
colorizing grey-scale images or predicting image rotation. However, these RGB-
specific augmentations and tasks cannot be directly applied to RF signals. RF
data is quite different from RGB images. It is obtained by analyzing the
signal power reflected from different locations in space. There is no color
information in RF signals, and the signal is not invariant to rotation
transformation, making most of the existing unsupervised learning methods not
directly applicable to RF signals.

  * •

As shown in Figure 1, in RF signals, the information region that corresponds
to a person could be extremely small, occupying <1%. This is because the
majority of RF signals are reflected from objects in the environment.
Therefore, directly applying unsupervised learning methods on RF signals could
be strongly biased by background noises, and fail to learn useful information
about the person.

  * •

RF signals carry much information that is irrelevant to the person or task.
Not only human but almost all objects could reflect signals. In particular,
only a fraction of an RF signal traverses a wall, and the rest reflects off
walls and other objects in the environment. The percentage of the signal that
traverses a wall vs. the amount that reflects off it depends on the material
and surface of the wall, and is not easy to estimate. The same can be said
about almost all objects in the environment. Furthermore, it is not
immediately clear which part of the reflected signal corresponds to the person
or people in the scene. This makes it hard for unsupervised learning methods
to focus on representations that are useful for human sensing and avoid
irrelevant information.

In this paper, we introduce a new unsupervised learning framework, Trajectory-
Guided Unsupervised Learning (TGUL) to solve the above challenges brought in
by RF signals. We first introduce modified data augmentations and self-
supervised tasks that are suitable for RF signals. We then solve the problem
of sparse relevant information and eliminate the irrelevant information. To do
so, we leverage signal processing techniques that can be applied without any
supervision. Specifically we use a radar-based module to detect people and
track their trajectories. Then at any point in time, we zoom in on the region
of interest that contains the person, and eliminate signals reflected from
other objects. Unsupervised training loss is only added inside this region of
interest to avoid the model learning background noises.

We adapt TGUL to predictive learning and contrastive learning algorithms. We
evaluate and compare it with state-of-the-art supervised baselines for RF-
based sensing tasks, including 3D pose estimation, action recognition, and
person re-identification. Our results show that contrastive learning methods
could still be strongly biased to learning shortcut information in RF signals,
and discarding useful information about the person, while predictive learning
methods learn useful information and consistently improve the performance over
supervised baselines under the fine-tuning setting.

To summarize, this paper takes an important step towards extending
unsupervised representation learning to new modalities that are hard to
interpret by humans. Specifically it makes the following contributions:

  * •

The paper introduces trajectory-guided unsupervised learning (TGUL), a novel
unsupervised learning framework for RF signals. We show that TGUL is widely
applicable to a variety of RF-based human sensing tasks such as pose
estimation, action recognition, and person re-identification.

  * •

We demonstrate for the first time the possibility of boosting the performance
of radio-based human sensing tasks by leveraging unlabeled radio signals. For
example, with extra unlabeled RF data, the performance of RF-based pose
estimation could be improved by 11.3%. We further show that even without extra
unlabeled data, pre-training the network using TGUL can still improve the
performance by 5.7%.

  * •

The paper also shows that state of the art contrastive learning techniques
such as SimCLR, MoCo, and BYOL do not work well on radio signals. They tend to
learn shortcut information that are contrastive but do not help the downstream
tasks. The vulnerability of contrastive learning to shortcuts has been
observed in the context of RGB data, where color distribution can be a
shortcut that prevents contrastive learning from learning semantic information
from images [6]. In the context of RGB data, it is possible to design data
augmentations to break such shortcuts, e.g., color jittering. However, since
RF signals cannot be interpreted by humans, one cannot easily design data
augmentations that alter RF signals in a manner that eliminates the shortcut
without hampering the original task. As a result, contrastive learning does
not perform well on RF signals.

##  2 Related Work

RF-based Vision. There is a growing literature that uses radio signals to
“see” people’s poses and activities through walls and in dark settings.
Compared with RGB data, radio signals have several advantages: they can
traverse walls and occlusions; work in daytime and in darkness; and are not
human-interpretable and hence more privacy preserving. Thus, RF-based vision
has emerged as an attracting research direction that enables new applications
in health care and smart homes [3, 11, 27, 20, 44, 23, 33, 37, 24, 21, 4, 38,
40, 35, 31].

Previous RF-based vision works include human pose estimation, action
recognition, captioning, person re-identifications and so on [44, 46, 45, 37,
24, 9, 42, 12, 11, 20, 35]. However, past works rely on supervised learning.
Given the difficulty of annotating RF datasets with ground truth labels, past
work is limited to relatively small datasets.

Unsupervised Learning from RGB Data. Unsupervised learning methods used in
RGB-based vision can be divided into three categories: self-supervision using
pretext tasks, predictive approaches, and contrastive approaches. Early work
on unsupervised representation learning has focused on designing pretext tasks
and training the network to predict their pseudo labels. Such tasks include
solving jigsaw puzzles [26], colorize grey-scale images [41] or predicting
image rotation [13]. However, pretext tasks have to be handcrafted and are
highly dependent on the properties of RGB data, making them hard to be
generalized to other modalities.

Unsupervised learning based on predictive models including auto-regressive
(AR) and auto-encoding (AE) models show less dependence on RGB data. The
family of auto-encoders provides a popular framework for unsupervised
representation learning using a predictive loss [18, 30, 36]. It trains an
encoder to generate low-dimensional latent codes that could reconstruct the
entire high-dimensional inputs. There are many types of AEs, such as denoising
auto-encoders [36], which corrupt the input and let the latent codes
reconstruct it, and variational auto-encoders [30], which force the latent
codes to follow a prior distribution. However, relevant information in radio
signals is typically very sparse. Therefore, traditional AE models can easily
ignore the region in RF signal that is relevant to the person and their
action.

Recently, contrastive learning has become widely used for learning effective
representations without human supervision. The representations learned with
this approach generalize well to downstream tasks, and in some cases can
surpass the performance of supervised models [6, 7, 8, 16]. The core idea of
contrastive learning is to keep features from positive sample pairs close, and
features from negative samples far from each other. Today, multiple successful
contrastive learning frameworks exist, with small differences in how they pick
negative samples. They include SimCLR [6], the momentum-contrastive approach
(MoCo) [16], Contrastive-Multiview-Coding [32], and BYOL [14].

However, contrastive learning could suffer from the issue of “shortcut” –
i.e., it can learn easy features that are irrelevant to the downstream task,
and ignore relevant features [6, 34]. For example, in the case of images, the
color distribution can be a shortcut that prevents contrastive learning from
learning the semantic information in an image [6]. To avoid such shortcut,
contrastive learning methods use color jittering as a data augmentation. In
Section 6, we show that state-of-the-art contrastive learning methods can fail
to learn meaningful representations due to shortcuts in RF signals. However,
unlike shortcuts in RGB data, since RF signals cannot be interpreted by
humans, one cannot easily design data augmentations that alter the RF signal
in a manner that eliminates the shortcut without hampering the original task.

##  3 RF Signals Preliminary

Figure 2: Illustration of RF signals as a pair of horizontal and vertical
heatmaps after subtracting static objects, and an RGB image recorded at the
same time. Red color refers to high signal power, blue refers to low power.
Figure 3: RF-based human sensing framework. The RF heatmaps are first fed to a
Feature Network to extract spatio-temporal features, and ROI Align with the
labeled bounding box is used to crop out the regions with humans. Each
downstream task has a unique sub-network to extract task-specific features and
predictions.

RF-based human sensing relies on transmitting a low power radio signal and
receiving its reflections from the environment and nearby people [44, 46, 24,
9, 42, 12, 45, 1, 35, 33, 43, 47, 20, 11]. These technologies typically use a
radio device that combines FMCW (Frequency Modulated Continuous Wave) and
antenna arrays [1]. This allows the radio to operate like a consumer radar,
and separate RF signal reflections from different objects in space.
Specifically, FMCW separates RF reflections based on the distance of the
reflecting object, whereas antenna arrays separate reflections based on their
spatial direction.

As in past work, we consider a radio device with two antenna arrays: a
horizontal one and a vertical one. As a result, the RF signals can be
expressed as two heatmaps, as shown in Fig. 2. The horizontal heatmap is a
projection of the radio signals on a plane parallel to the ground, and the
vertical heatmap is a projection of the radio signals on a plane perpendicular
to the ground. We use the term RF frame to refer to a pair of horizontal and
vertical heatmaps.

##  4 RF-Based Human Sensing Framework

We choose the following three RF-based sensing tasks to validate the
performance of our unsupervised learning method:

  * •

(a) RF-Based 3D Pose Estimation [44, 46], which uses RF signals to infer the
3D locations of 14 keypoints on the human body: head, neck, shoulders, arms,
wrists, hips, knees, and feet.

  * •

(b) RF-Based action recognition [24], which analyzes radio signals to infer
human actions and interactions in the presence of occlusions and in poor
lighting.

  * •

(c) RF-Based Re-Id [12], which recognizes a person-of-interest across
different places and times by analyzing the radio signals that bounce off
their bodies.

In all these tasks, the neural network has the general structure in Figure 3.
The model first uses a spatio-temporal convolutional feature network
consisting of 9 residual blocks to extract features from the input RF frames.
It then crops out a bounding box around the person in the feature map.
Finally, the cropped features are fed to a task-specific sub-network to
generate frame-level features. The feature network is the same for all tasks,
but different tasks could have different sub-networks specifically designed
for that task.

##  5 Trajectory Guided Unsupervised Learning

In this section, we adapt contrastive and predictive unsupervised learning
algorithms to learn representations from unlabeled RF signals.

Figure 4: Predictive Learning adapted to RF signals. In the Signal Processing
Module (orange box), we use background subtraction to remove static objects
and moving detection to obtain bounding boxes for each person in the scene.
After this, RF frames are randomly masked and filled with zero. In the Neural
Network Module (blue box), a feature extraction network (green box) is applied
to extract features from RF inputs, and then an ROI Align module is applied
with the bounding box generated by the signal processing module to crop out
regions containing humans. A decoder network is then adopted to predict
original input ROIs from cropped features. The reconstruction loss between the
cropped ROI of the original heatmap and the predicted ROIs are used to train
the model.

###  5.1 Signal Processing for ROI Detection

Unlike standard RGB datasets, where the object of interest typically occupies
many of the pixels in an image, the RF signals reflected from a person only
occupy a small portion of an RF heatmap (e.g., < 1%). Therefore, we need to
track and zoom in on the person before we apply unsupervised learning methods
to RF signals, otherwise, the unsupervised learning method is likely to learn
information about the background instead of the person.

Traditional approaches for zooming in on individuals or objects in RGB images
typically rely on learning a bounding box of this object from ground truth
labels, which are not available in the context of unsupervised learning. Thus,
instead of using methods from computer vision, we leverage the properties of
radio signals. Specifically, there are mainly two challenges in detecting the
region of interest in the RF signal. The first is to be able to zoom in on the
radio signals reflected from a specific location or region in space
(presumably the location of the person) and ignore the rest of the RF signal.
To address this issue, we leverage the horizontal and vertical heatmaps, which
correspond to the projection of the radio signals on a horizontal plane and a
vertical plane. Thus, to zoom in on the signal reflected from a region in
space, we can simply crop out the projection of this region on the horizontal
and vertical heatmaps.

The second challenge is to determine which region in space contains the
person. In RGB data, detecting a person typically requires complex algorithms
or neural networks trained using a large dataset. In contrast, we use the fact
that FMCW radios can operate as a radar system that tracks and detect moving
objects. In most indoor scenarios, people are the only large moving objects.
Therefore, we can adapt radar detection algorithms to detect the person and
generate their trajectory as they move around. When the person becomes static,
the trajectory will stay at the location where the person stops moving. Then
it will start tracking the person again when he or she starts moving again. To
be more specific, we use a signal-processing technique called WiTrack [2] to
automatically track and generate the trajectory of the person. It first
subtracts from heatmap the median over a long period to remove the reflections
from static objects in the scene. Then it computes the difference between
consecutive frames to detect moving persons and generate their trajectories.
We then generate bounding boxes of size 1m$\times$1.5m for the person in each
frame based on their trajectory.

###  5.2 Predictive Unsupervised Learning from RF

We first consider learning unsupervised representations by trying to predict
how the human body changes the radio signals that bounce of it. Our predictive
learning framework for RF signals is illustrated in Figure 4. Given an input
sequence of RF heatmaps $x$ consisting of $T$ RF frames, $x_{1},\cdots,x_{T}$,
we first perform the aforementioned signal processing step to detect regions
of interest $b_{i}$ that contain people. We then randomly mask $t$ frames and
fill them with zeros to get the masked input $M(x)$. Then the masked input is
passed through an encoder network $E$ with parameter $\theta$, an ROI align
module $\mathbb{R}$ which crops out regions of interest in feature space using
$b_{i}$, and a decoder network $D$, with parameter $\delta$, to obtain the
reconstruction result $D_{\delta}(\mathbb{R}(E_{\theta}(M(x))_{i},b_{i}))$.
The reconstruction loss $\mathcal{L}_{r}$ is defined as the reconstruction
error between the cropped ROI from the original input and the reconstructed
one $D_{\delta}(\mathbb{R}(E_{\theta}(M(x))_{i},b_{i}))$:

$\mathcal{L}_{r}=\sum_{i=1}^{T}||D_{\delta}(\mathbb{R}(E_{\theta}(M(x))_{i},b_{i}))-\mathbb{R}(x_{i},b_{i})||_{2}.$

Instantiation. For the encoder, we use the same network architecture as prior
works [46, 11, 24], which is a spatio-temporal convolutional feature network
consisting of 9 residual blocks. For the decoder, we use a spatial de-
convolutional network with 3 residual blocks. For predictive learning, we
adapt two standard predictive methods: Auto-encoder [18] and Inpainting [29].
For Auto-encoder, there is no masking operation on the input, i.e., $M$ is an
identical mapping. For Inpainting, for each RF sequence consisting of $T=100$
RF frames, we randomly mask out five 5-frame segments within this sequence and
fill them with zeros.

Figure 5: Contrastive Learning on RF signals. In the Signal Processing Module
(orange box), we use background subtraction to remove static objects and
moving detection to obtain bounding boxes for each person in the scene. After
this, different transformation $\tau_{1}$ and $\tau_{2}$ to the bounding box
are applied to generate positive samples. Both branches share the same feature
extraction network (green box), and then use ROI Align with the given bounding
box to crop out regions containing humans. A non-linear projection head is
used to generate normalized features. Contrastive loss is used to train the
model.

###  5.3 Contrastive Learning from RF

We also consider adapting contrastive learning to RF signals. The contrastive
learning framework for RF signals is illustrated in the orange box in Figure
5. For each input RF heatmap sequence $x$ with $T$ frames
$x_{1},\cdots,x_{T}$, we first perform the same signal processing step as in
predictive learning to detect regions of interest and produce a bounding box
$b_{i}$ for each frame $x_{i}$. Then we generate a pair of positive samples by
performing two data augmentations $\tau_{1}$ and $\tau_{2}$ to the generated
bounding boxes from the signal processing step, resulting in augmented
bounding boxes $\tau_{1}(b_{i}),\tau_{2}(b_{i})$. Then we forward the inputs
signal with the augmented bounding boxes separately into the encoder $E$
parameterized by $\theta$, an ROI align module $\mathbb{R}$ and a multi-layer
nonlinear projection head $H$ parameterized by $h$ to get the latent
representations $z_{2i}$ and $z_{2i+1}$ for these two positive samples.
Moreover, since human actions are continuous, features within $S=10$ frames of
$x_{i}$ is also taken as positive pairs. We use the commonly used InfoNCE loss
[6] as the contrastive loss $\mathcal{L}_{c}$. Namely, for a batch of $N$ RF
segments containing $NT$ frames $x_{i}^{j},i=1,\cdots,T,j=1,\cdots,N$,

$\begin{split}\mathcal{L}_{c}=-\sum_{j=1}^{N}\sum_{i=1}^{T}\sum_{m=1}^{S}\log\frac{\exp\big{(}\text{sim}(z^{j}_{2i},z^{j}_{2i-S+2m})/t\big{)}}{\sum\limits_{l=1}^{N}\sum\limits_{k=1}^{2T}\mathbbm{1}_{k\neq
2i||l\neq j}\exp\big{(}\text{sim}(z^{j}_{2i},z^{l}_{k})/t\big{)}}\end{split}$

where $\text{sim}(u,v)=u^{T}v/(\|u\|_{2}\|v\|_{2})$ denotes the dot product
between the normalized $u$ and $v$ (i.e., cosine similarity),
$t\in\mathbb{R}^{+}$ is a scalar temperature parameter, and $z_{2i},z_{2i+1}$
are the encoded features of positive pairs generated from $x_{i}$, i.e.,
$z_{2i}=H_{h}(\mathbb{R}(E_{\theta}(x_{i}),\tau_{1}(b)))$ and
$z_{2i+1}=H_{h}(\mathbb{R}(E_{\theta}(x_{i}),\tau_{2}(b)))$.

Note that the augmentations $\tau_{1},\tau_{2}$ here are crop and resize
augmentations on the bounding boxes, instead of the whole input. This is
because due to the sparsity of RF signals, performing crop and resize on the
whole input may crop out the region of interest.

Instantiation For the encoder, we use the same network architecture as prior
works [46, 11, 24]. For the projection head, we use a spatial convolutional
network with one residual block. For negative pairs, we follow the original
negative sampling strategy of whatever contrastive learning method TGUL build
on. Since there is no prior work that applies existing contrastive learning
methods to RF data, we implement SimCLR [6], MoCo [16], CPC [17] and BYOL [14]
for RF data by ourselves. The SimCLR implementation is what we just described
in Figure 5. For MoCo, we use the same data augmentations as SimCLR, except
that the positive feature pairs are generated by two feature networks, a
normal one and a momentum one. The loss is the same as the one used in [16].
For BYOL, we use the same data augmentation operations as SimCLR, and follow
the implementation of [14] to generate predictions, projections, and loss. For
CPC, we follow the design in [17]. Specifically, we add a GRU after the RF
feature extractor. The output of the GRU at every frame is then used as the
context to predict the features in 1.5s in the future using the contrastive
loss.

Table 1: Evaluation of different models on different RF tasks under fixed
feature extractor setting and the relative improvements over network
initialized randomly. $\downarrow$ indicates the smaller the value, the better
the performance; $\uparrow$ indicates the larger the value, the better the
performance.

Tasks | 3D Pose Estimation | Action Recognition | Person Re-ID (Campus) | Person Re-ID (Home)
---|---|---|---|---
Metrics | Pose Err.↓ (mm) | mAP↑ | mAP↑ | CMC-1↑ | CMC-5↑ | mAP↑ | CMC-1↑ | CMC-5↑
$\theta=0.1$ | $\theta=0.5$
Random init | 60.1 | 60.5 | 53.3 | 28.1 | 43.8 | 68.8 | 30.1 | 54.2 | 74.6
SimCLR + TGUL | 80.5 | 4.2 | 0 | 29.8 | 44.1 | 67.5 | 31.2 | 55.1 | 73.8
MoCo + TGUL | 77.2 | 5.1 | 0.18 | 29.1 | 44.7 | 65.3 | 30.5 | 54.5 | 74.0
CPC + TGUL | 78.7 | 3.6 | 0 | 30.0 | 42.7 | 69.5 | 30.7 | 54.0 | 75.3
BYOL + TGUL | 79.3 | 4.7 | 0 | 29.5 | 44.4 | 66.7 | 30.7 | 54.6 | 73.5
Autoencoder | 59.4 | 62.3 | 54.2 | 29.0 | 44.5 | 67.0 | 31.1 | 55.5 | 75.5
Autoencoder + TGUL | 55.7 | 71.1 | 63.2 | 43.8 | 69.7 | 87.2 | 35.2 | 61.5 | 81.9
Inpainting | 58.0 | 63.9 | 55.4 | 30.2 | 48.1 | 70.5 | 32.8 | 57.7 | 76.5
Inpainting + TGUL | 51.1 | 72.3 | 65.5 | 49.8 | 73.1 | 90.5 | 38.5 | 64.2 | 84.7
Improvement | +15.0% | +19.5% | +22.9% | +77.2% | +66.9% | +31.5% | +27.9% | +18.5% | +13.5%

Table 2: Evaluation of different models on different RF tasks under fine-
tuning setting and the relative improvements over supervised training from
random initialized network. $\downarrow$ indicates the smaller the value, the
better the performance; $\uparrow$ indicates the larger the value, the better
the performance.

Tasks | 3D Pose Estimation | Action Recognition | Person Re-ID (Campus) | Person Re-ID (Home)
---|---|---|---|---
Metrics | Pose Err.↓ (mm) | mAP↑ | mAP↑ | CMC-1↑ | CMC-5↑ | mAP↑ | CMC-1↑ | CMC-5↑
$\theta=0.1$ | $\theta=0.5$
Supervised [24, 11] | 38.4 | 90.1 | 87.8 | 59.5 | 82.1 | 95.5 | 46.4 | 74.6 | 89.5
SimCLR + TGUL | 38.8 | 89.8 | 87.4 | 59.0 | 81.7 | 94.1 | 45.9 | 73.8 | 88.5
MoCo + TGUL | 38.3 | 89.7 | 87.2 | 59.3 | 82.0 | 94.5 | 46.4 | 74.3 | 89.7
CPC + TGUL | 38.6 | 89.9 | 87.5 | 59.4 | 81.5 | 94.0 | 46.0 | 74.5 | 89.1
BYOL + TGUL | 38.5 | 89.7 | 87.2 | 59.4 | 81.9 | 94.5 | 46.6 | 74.5 | 89.5
Autoencoder | 38.5 | 90.0 | 87.7 | 59.1 | 81.9 | 95.5 | 45.9 | 74.2 | 88.6
Autoencoder + TGUL | 37.5 | 91.2 | 87.9 | 59.7 | 82.8 | 95.5 | 46.8 | 74.6 | 89.8
Inpainting | 38.2 | 90.5 | 88.0 | 59.3 | 82.1 | 95.7 | 46.2 | 74.4 | 89.2
Inpainting + TGUL | 36.2 | 91.7 | 88.7 | 60.1 | 83.3 | 95.9 | 47.5 | 75.3 | 90.3
Improvement | +5.7% | +1.8% | +1.0% | +1.0% | +1.5% | +0.4% | +2.4% | +0.9% | +0.9%

##  6 Experiments

RF Datasets. We use two real-world RF datasets used in past work: RF-MMD [24]
and RRD [11]. RF-MMD is an RF-based 3D pose-estimation and action-recognition
dataset consisting of 25 hours of RF data with 30 volunteers performing 35
actions in 10 different indoor environments. RRD is an RF-based person re-
identification dataset. It contains 863 RF sequences from 100 different
identities at 5 different locations on a campus, and 6305 RF sequences from 38
different identities in 19 homes.

Evaluation Metrics. We use the performance metrics used by past work for each
task. Specifically, for 3D pose estimation, we use the average $l_{2}$
distance between 14 predicted keypoints and their ground-truth locations as
the evaluation metric. For action recognition, we use mean average precision
(mAP) at different intersection-over-union (IoU) thresholds $\theta$ to
evaluate the accuracy of the model in detecting and classifying an action
event. For person re-identification, we separate the dataset into query set
and gallery set. The query samples and gallery samples are then encoded to
feature vectors. We calculate the cosine distance between the features of each
query sample and each gallery sample, and rank the distance to get the top-N
closest gallery samples for each query sample. Based on the ranking results,
we compute the mean average precision (mAP) and the cumulative matching
characteristic (CMC) at rank-1 and rank-5.

Evaluation Settings. We evaluate the performance under both fixed feature
extractor setting and fine-tuning setting. In the fixed feature extractor
setting, we use the pre-trained RF feature extractor as initialization of the
RF feature extractor for each downstream task, and only those task-specific
sub-networks (Fig. 3) are trained during downstream task training. In the
fine-tuning setting, we fine-tune the whole model on the downstream task.

Training Details. For all approaches, we train the network for 50 epochs with
a batch size of 128, using the Adam optimizer [22] with 1e-3 learning rate and
1e-5 weight decay. We use PyTorch [28] for implementations.

###  6.1 Main Results

We evaluate our unsupervised learning method on multiple downstream RF-based
tasks, and report the results in Tables 2 and 2. Note that contrastive leaning
methods could not be directly applied to RF signals without TGUL because the
RGB-based augmentations is not directly applicable to RF signals. Table 2
shows the results of the fixed feature extractor setting. The table reveals
two findings. First, predictive pre-training using TGUL learns useful
unsupervised representations and delivers 15.0% relative improvement in 3D
pose estimation error, 19.5% relative improvement in action recognition mAP at
$\theta=0.1$, 77.2% relative improvement in person ReID mAP on RRD-Campus, and
27.9% relative improvement on RRD-Home. Further, inpainting, i.e., predicting
missing RF signals, is more powerful than autoencoding though both deliver
gains.

Second, the table shows that contrastive pre-training performs worse on all
the downstream tasks than predictive pre-training, and even worse than random
initialization for the 3D pose estimation and action recognition tasks. This
is because contrastive learning learns shortcut semantics irrelevant to the
tasks of interest. We further investigate this issue of shortcut in Section
6.2.

Table 2 shows the results of the fine-tuning setting. The table shows that
initializing supervised models with Inpainting+TGUL’s representations can
achieve 5.7% relative improvement in 3D pose estimation error, 1.8% relative
improvement in action recognition mAP at $\theta=0.1$, 1.0% relative
improvement in person ReID mAP on RRD-Campus, and 2.4% relative improvement on
RRD-Home. In contrast, initializing supervised models with contrastive
representations is not useful.

Tables 2 and 2 also demonstrate the effectiveness of TGUL. Directly applying
Autoencoder or Inpainting on RF signals only brings limited improvements over
randomly initialized network under fixed feature network setting, and cannot
improve the performance under fine-tuning setting. On the other hand,
Autoencoder and Inpainting with TGUL achieves much better performance under
both settings. This is because TGUL is effective in forcing the network to
focus on the person and pay less attention to other regions which contain
mostly noises, and thus achieves better performance than without this
guidance.

Overall, the results show that our framework is the first unsupervised method
to boost the performance of RF-based human sensing, bringing the gains of
unsupervised learning to this new data modality.

###  6.2 Feature Visualization

|  |
---|---|---
(a) Original Heatmap | (b) CL Features | (c) PL features
Figure 6: Contrastive learning (CL) can be biased to shortcut and discard
useful information in the input, while predictive learning (PL) can preserve
the information in the input.

As shown in Table 2 and 2, contrastive pre-training methods perform poorly and
even worse than random initialization of the feature network. This is because
contrastive learning is likely to exploit shortcut information (e.g., the
distance of the person to the RF device) and discard other useful information
for downstream tasks. In Figure 6, we visualize the features generated by
feature networks pre-trained using contrastive learning and predictive
learning. As shown in the figure, the features network pre-trained using
contrastive learning discard most of the information in the input. On the
other hand, the feature network pre-trained using predictive learning can
preserve and abstract the information about the person in the input and thus
improve the performance.

###  6.3 Ablation Studies

We perform ablation studies on our proposed unsupervised learning framework.
All ablation studies are performed on the RF-MMD [24] dataset with the
downstream task of pose estimation, where we report the average error across
all keypoints.

Masking Strategy: When applied to RF signals, the Inpainting task is performed
by masking some frames in a short window of RF signals and reconstructing them
from the remaining frames. We compare the performance of Inpainting with
different masking strategies. The input RF signal is a 100-frames RF sequence.
We compare 3 different masking strategies: center segment (25 frames in the
center of the input), random segments (five 5-frame segments at random
positions of input), and random (each frame is masked with 0.25 probability).
As shown in the results below, the random segments masking strategy performs
the best.

Table 3: Pose error (the lower the better) with different masking strategies. Mask Strategy | Pose Err.↓ (mm)
---|---
Center Segment | 36.8
Random Segments | 36.2
Random | 36.4

Size of Masked Segment: Next, we study the influence of the size of each
masked segment. Note that when the mask size equals 0, no frame is masked, and
the inpainting task becomes a normal auto-encoder. In all of these
experiments, we mask 5 segments and follow the strategy of randomly locating
the masked segments in the input. As shown in Table 4, the best performance is
for masking 5 consecutive frames, i.e., a segment of size 5.

Table 4: Pose error with different sizes of masked segments. Masking Size | 0 | 1 | 3 | 5 | 10
---|---|---|---|---|---
Pose Err.↓ (mm) | 37.5 | 37.0 | 36.3 | 36.2 | 36.7

Benefits of Unlabeled Data for RF-Based Sensing: By enabling RF-based tasks to
leverage unsupervised representation learning, our framework allows RF-based
sensing to benefit from a large amount of unlabeled RF data. To evaluate the
benefit of such unlabeled data, we simulate the scenario where only a handful
of labeled RF data is available and a large amount of RF data is unlabeled
data. Specifically, we randomly select 10% of the training set of RF-MMD to be
RF-MMD-S to serves as the small labeled dataset. We compare the performance of
Inpainting when it is pre-trained on RF-MMD-S and RF-MMD and finetuned on RF-
MMD-S.

Table 5: Performance of Inpainting on RF-MMD with a small amount of labeled data. The results in the table use RF-MMD-S for finetuning, which is a randomly selected 10% subset of RF-MMD which is used for unsupervised pre-training. The rest of the dataset is used without labels. The table shows that our framework can further improve the performance with more unlabeled data. Methods | Pose Err.↓ (mm)
---|---
Training from scratch (RF-MMD-S) | 48.7
Inpainting on RF-MMD-S+finetune | 46.1
Inpainting on RF-MMD+finetune | 43.2

As shown in Table 5, our framework can improve the performance of RF-based 3D
pose estimation by 5.3% without using any additional unlabeled data. This is
because our framework can learn a general representation of RF signals and
thus provide better generalization ability. Indeed, our framework improves the
pose error further by another 6.0% by leveraging the unlabeled data. This
demonstrates the potential of using our framework to leverage large-scale
unlabeled RF data to improve the performance of RF-based human sensing
methods.

##  7 Conclusion

We introduced TGUL, the first unsupervised learning framework for RF-based
human sensing tasks. We adapted state-of-the-art unsupervised learning methods
from RGB data to RF data by leveraging radar technology to focus unsupervised
learning on the person’s trajectory, and design data augmentations suitable
for RF signals. We showed that contrastive learning could be strongly biased
by shortcuts in RF data, making it a poor choice for this modality. We further
demonstrate the potential of unsupervised learning methods based on predictive
tasks for learning useful representations from RF data. Extensive empirical
results on multiple RF datasets and tasks show that TGUL could consistently
improve the performance of RF-based sensing models over supervised learning
baselines.

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# Measures of Fluctuations for a Liquid Near Critical Drying.

Mary K. Coe  Robert Evans  Nigel B. Wilding  H.H. Wills Physics Laboratory,
University of Bristol, Bristol BS8 1TL, U.K.

###### Abstract

We investigate density fluctuations in a liquid close to a solvophobic
substrate at which a surface critical drying transition occurs. Using
classical density functional theory, we determine three measures of the
spatial extent and strength of the fluctuations, i.e., the local
compressiblity $\chi_{\mu}(z)$, the local thermal susceptibility $\chi_{T}(z)$
and the reduced density $\chi_{*}(z)$; z is the distance from the substrate.
Whilst the first measure is frequently used, the second and third were
introduced very recently by Eckert et.al., Phys. Rev. Lett. 125, 268004
(2020). For state points in the critical drying regime, all three measures,
each scaled by its bulk value, exhibit very similar forms and the ratio of
$\chi_{T}(z)$ to $\chi_{\mu}(z)$, for fixed $z$ in the vapour-liquid
transition region, is constant. Using a scaling treatment of surface
thermodynamics we show that such behaviour is to be expected on general
grounds.

##  I Introduction

Recent studies have linked the emergence of vapour-like films near to
hydrophobic and solvophobic planar substrates in contact with liquids to an
underlying surface phase transition called drying [1, 2, 3]. Drying is the
analogue of the more common wetting where, for sufficiently attractive
substrates, a macroscopically thick film of liquid can intrude between the
bulk vapour and the substrate. Specifically, the drying transition that occurs
when the bulk liquid is at liquid-vapour coexistence and the strength of
attractive interactions between the substrate and fluid is lowered, leading to
a macroscopically thick intruding vapor film, has been shown to be critical in
almost all cases [3]. In addition, a continuous transition occurs for purely
repulsive, or very weakly attractive, substrates as the chemical potential,
$\mu$, of an oversaturated liquid is decreased towards the coexistence
chemical potential, $\mu_{co}$, e.g. [4, 5]. This latter transition is
referred to as complete drying.

Continuous transitions are, of course, associated with enhanced density
fluctuations. The recent work of Evans et. al. [1, 2] has related the enhanced
density fluctuations observed near solvophobic substrates to the critical
drying transition. Whilst earlier work on hydrophobic systems also observed an
enhancement of density fluctuations [6, 7, 8], no link was made to critical
drying. Many of the earlier studies quantified such fluctuations using the
probability distributions of particle occupancy in sub-volumes immediately
adjacent to hydrophobic and solvophobic substrates [7, 6]. However, these
measures provide scant information about the spatial extent of the
fluctuations, and how their magnitude depends on the strength of substrate-
liquid attraction and the deviation from bulk liquid-vapour coexistence, given
by $\delta\mu=\mu-\mu_{co}$. Moreover, these earlier studies do not address
the important issue of proximity to a critical drying transition.

Measures which incorporate spatial resolution have typically been termed local
compressibilities [8, 9], and have focused on derivatives of the density
profile, $\rho(\mathbf{r})$, of the inhomogeneous fluid. The most natural, and
perhaps most widely used, measure is the local compressibility adopted by
Stewart and Evans [9]

$\chi_{\mu}(\mathbf{r})=\left(\frac{\partial\rho(\mathbf{r})}{\partial\mu}\right)_{T}$
(1)

where the derivative is w.r.t. the chemical potential $\mu$ of the bulk
reservoir and the temperature $T$ is fixed. This measure has been utilised in
several previous studies of hydrophobic and solvophobic systems, and has been
shown to be a powerful indicator of the approach to a critical drying
transition [1, 2]. It provides a much sharper signature of the onset of drying
than the density profile itself, i.e. the growth in thickness of a drying film
of vapor is much slower than that of the maximum of $\chi_{\mu}(\mathbf{r})$
on approaching the transition. Note that in some earlier papers, for example
[5, 10, 11], $\chi_{\mu}(\mathbf{r})$ was termed the local susceptibility.

Here we enquire whether there are other measures that could provide additional
information about the nature of density fluctuations in the inhomogeneous
liquid close to critical drying. We were motivated by the recent work of
Eckert et. al. [12] who defined two new measures, the local thermal
susceptibility $\chi_{T}(\mathbf{r})$ and the reduced density
$\chi_{*}(\mathbf{r})$, in a similar manner to the local compressibility
$\chi_{\mu}(\mathbf{r})$. Specifically, they defined these measures as [12]

$\chi_{T}(\mathbf{r})=\left(\frac{\partial\rho(\mathbf{r})}{\partial
T}\right)_{\mu}$ (2)
$\chi_{*}(\mathbf{r})=\rho(\mathbf{r})-\mu\chi_{\mu}(\mathbf{r})-T\chi_{T}(\mathbf{r})$
(3)

Using Grand Canonical Monte Carlo simulations, the authors determined the
three quantities, which they term fluctuation profiles, for various model
fluids under a variety of confinements, and found that the profiles differed
considerably from one other, suggesting they reflect different aspects of the
density fluctuations.

At first glance it is not obvious what precisely these quantities measure.
Evans et.al. [2] showed that $\chi_{\mu}(\mathbf{r})$ is the correlator of the
total number operator $N$ with the local number density operator at position
$\mathbf{r}$. For a planar substrate $\chi_{\mu}(\mathbf{r})$ is an integral
of the density-density pair correlation function [9]. Eckert et.al. [12]
identified $\chi_{T}(\mathbf{r})$ as the correlator (covariance) of the total
entropy operator $S$ with the local number density operator and they showed
that $\chi_{*}(\mathbf{r})$ is the difference between the local density and
the correlator of the Hamiltonian operator $\mathcal{H}$ with the local number
density operator. Near a surface critical transition it is expected that all
three quantities should reflect the divergence of the density-density
correlation length measured parallel to the substrate. However, the precise
nature of the divergence of each quantity is not known and is investigated
here.

We use classical DFT to calculate $\chi_{\mu}(z)$ and the new measures
$\chi_{T}(z)$ and $\chi_{*}(z)$ for a truncated Lennard-Jones (LJ) fluid
adsorbed at a planar substrate that exerts an external substrate-fluid
potential decaying algebraically with the distance $z$ from the substrate, for
state points close to the critical drying transition. For this model system,
which is often employed in simulations, it is known that critical drying
occurs precisely when the strength of substrate-fluid attraction vanishes,
i.e. in the hard-wall limit [1, 2]. This allows us to make detailed
comparisons between theory and DFT results. We find that the three measures,
each normalized by its value in bulk, exhibit very similar forms in the near
critical drying regime. Moreover, at a given $T$, the ratio of $\chi_{T}(z)$
to $\chi_{\mu}(z)$, measured at fixed $z$ in the edge of the vapour film, is
found to be constant in the near critical regime. We explain this observation
using i) a scaling treatment of surface thermodynamics and ii) an effective
interface potential analysis. Remarkably the result we find from both
treatments not only accounts for our DFT results but turns out to be a close
analogue of the ratio in bulk where, considering the density of the bulk fluid
$\rho_{b}\equiv\rho_{b}(\mu,T)$, it follows that

$\chi_{T,b}=-\left(\frac{\partial\rho_{b}}{\partial\mu}\right)_{T}\left(\frac{\partial\mu}{\partial
T}\right)_{\rho_{b}}\equiv-\chi_{\mu,b}\left(\frac{\partial\mu}{\partial
T}\right)_{\rho_{b}}$ (4)

i.e., the ratio of $\chi_{T,b}$ to $\chi_{\mu,b}$ is the negative of the
temperature derivative of $\mu$ at fixed density. Note that
$\chi_{\mu,b}=\rho_{b}^{2}\kappa_{T}$, where $\kappa_{T}$ is the usual (bulk)
isothermal compressibility.

Our paper is organized as follows: in Sec. II we describe the model fluid and
substrate, and how these are treated within our microscopic DFT calculations.
Sec. III describes the underlying theory, i.e. our treatment of the surface
thermodynamics and the pertinent interface potential. We conclude in Sec. IV
with a discussion of our results and their general relevance for fluctuations
near interfacial phase transitions.

##  II DFT Calculations for a Model Fluid.

We consider the same model fluid as that treated in [2], i.e. a truncated LJ
fluid with particles of diameter $\sigma$ and well-depth $\varepsilon$. We use
Fundamental Measure Theory [13, 14] to treat repulsive interactions between
fluid particles, modelling these as hard-spheres and employing the original
Rosenfeld hard-sphere functional. Attraction is incorporated using the
standard DFT mean field theory treatment [15]. Specifically, the attractive
pair potential used within DFT is given by [2, 15]

$\phi_{att}(r)=\begin{cases}-\varepsilon&r<r_{min}\\\
4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]&r_{min}<r<r_{c}\\\
0&r>r_{c}\end{cases}$ (5)

where $r$ is the distance between the centres of two fluid particles,
$r_{min}=2^{1/6}\sigma$ is the minimum of the pair potential and
$r_{c}=2.5\sigma$ is the cut-off radius of interaction. Within this mean-field
approach the bulk critical temperature is
$k_{\mathrm{B}}T_{c}/\varepsilon=1.319442$. This model fluid is in contact
with an impenetrable smooth planar substrate composed of particles of diameter
$\sigma_{s}$ of homogeneous density $\rho_{s}$. The substrate exists in the
x-y plane and has its surface located at $z=0$. The external potential exerted
by the substrate on a fluid particle takes the form [2]

$V_{ext}(z)=\begin{cases}\infty&z<0\\\
\varepsilon_{sf}\left[\frac{2}{15}\left(\frac{\sigma_{s}}{z+z_{min}}\right)^{9}-\left(\frac{\sigma_{s}}{z+z_{min}}\right)^{3}\right]&z>0\end{cases}$
(6)

where $\varepsilon_{sf}=2\pi\rho_{s}\varepsilon_{s}\sigma_{s}^{3}/3$ is a
measure of the substrate-fluid attraction strength, $\varepsilon_{s}$ is the
well depth of the substrate particle-fluid particle LJ potential, and
$z_{min}=(2/5)^{1/6}\sigma_{s}$ is the location of the minimum of the
interaction potential. We set $\sigma_{s}$ = $\sigma$ and, for numerical
reasons, shift the minimum of the potential so that it occurs at the surface
of the planar substrate, see [2]. The DFT program used within this work can be
found at [16].

##  III Theory

###  III.1 Surface Thermodynamics and Scaling Argument

We consider a system consisting of a smooth planar substrate in contact with a
liquid which is at a state point close to its critical drying transition. Due
to the symmetry of the substrate, the density profile of such a system varies
only along the direction perpendicular to the substrate, $z$, and has the form
of a vapour film near to the substrate, with a smooth transition to the bulk
liquid near $z=\ell_{eq}$, as sketched in fig. 1.

Figure 1: Illustration of the density profile $\rho(z)$ in the near drying
situation. A macroscopic planar substrate is in contact with a bulk fluid in
its liquid phase. At equilibrium, a film of vapour of thickness $\ell_{eq}$
intrudes between the substrate and liquid. Its extent depends on the state
point, i.e. how close this is to bulk coexistence, measured by
$\beta\delta\mu$, and the strength of substrate-fluid attraction, measured by
$\varepsilon_{s}$.

We assume the density profile of such a system can be written as a smooth
function, $\rho(z)=S(z-l_{eq})$. Using the definitions given in equations
(1,2), it follows that

$\displaystyle\chi_{\mu}(\ell_{eq})$
$\displaystyle=-\rho^{\prime}(\ell_{eq})\left(\frac{\partial\ell_{eq}}{\partial\mu}\right)_{T}$
$\displaystyle\chi_{T}(\ell_{eq})$
$\displaystyle=-\rho^{\prime}(\ell_{eq})\left(\frac{\partial\ell_{eq}}{\partial
T}\right)_{\mu}$ (7)

where $\rho^{\prime}(\ell_{eq})$ is the spatial derivative of the density
profile evaluated at $\ell_{eq}$. Clearly

$\frac{\chi_{T}(\ell_{eq})}{\chi_{\mu}(\ell_{eq})}=\left(\frac{\partial\ell_{eq}}{\partial
T}\right)_{\mu}\left(\frac{\partial\ell_{eq}}{\partial\mu}\right)_{T}^{-1}$
(8)

$\ell_{eq}$ can be defined from a thermodynamic quantity, namely the excess
Gibbs adsorption, $\Gamma$, as $\ell_{eq}\equiv-\Gamma/A\Delta\rho$ [17],
where $A$ is the surface area of the substrate-fluid interface and
$\Delta\rho=(\rho_{l}-\rho_{v})$ is the difference between the coexisting
liquid and vapour densities of the bulk fluid. Eq. (8) can therefore be
rewritten as

$\frac{\chi_{T}(\ell_{eq})}{\chi_{\mu}(\ell_{eq})}=\left(\frac{\partial\Gamma}{\partial
T}\right)_{\mu}\left(\frac{\partial\Gamma}{\partial\mu}\right)_{T}^{-1}$ (9)

Noting that $\Gamma\equiv\Gamma(\delta\mu,T)$, and recalling
$\delta\mu=\mu-\mu_{co}$, the first partial derivative on the r.h.s. can be
written as

$\left(\frac{\partial\Gamma}{\partial
T}\right)_{\mu}=\left(\frac{\partial\Gamma}{\partial
T}\right)_{\delta\mu}-\left(\frac{\partial\Gamma}{\partial\delta\mu}\right)_{T}\left(\frac{\partial\mu_{co}}{\partial
T}\right)$ (10)

It is then important to ascertain which contribution diverges fastest and
therefore determines the scaling behaviour close to critical drying. From the
standard Gibbs’ adsorption equation $\Gamma$ can be written as [18]

$\frac{\Gamma}{A}=-\left(\frac{\partial\gamma_{sl}}{\partial\mu}\right)_{T}$
(11)

where $\gamma_{sl}$ is the substrate-liquid surface tension, defined as the
excess grand potential per unit area. Close to critical drying, the singular
part of the tension responsible for the divergence of $\Gamma$, can be written
as a scaling function, $\Sigma$, of the variable
$\delta\mu/\tilde{t}^{\Delta}$ [19, 4], where $\tilde{t}=(T-T_{D})/T_{D}$
measures the deviation from $T_{D}$, the drying temperature for a given choice
of fluid-fluid (ff) and substrate-fluid (sf) interaction potentials. $\Delta$
is the (surface) gap exponent, i.e., the analogue of the gap exponent that
enters the standard scaling function for the bulk free energy density. The
singular contribution to the tension is given by

$\gamma_{sing}\sim|\tilde{t}|^{2-\alpha_{s}}\Sigma\left(\frac{\delta\mu}{|\tilde{t}|^{\Delta}}\right)$
(12)

where $\alpha_{s}$ is the surface equivalent of the bulk heat capacity
critical exponent. Using eq. (11), $\Gamma_{sing}$ can therefore be written as

$|\Gamma_{sing}|\sim|\tilde{t}|^{2-\alpha_{s}-\Delta}\mathcal{L}\left(\frac{\delta\mu}{|\tilde{t}|^{\Delta}}\right)$
(13)

where $\mathcal{L}$ is the first derivative of $\Sigma$, and is itself a
scaling function. The temperature and chemical potential derivatives are then

$\displaystyle\left(\frac{\partial|\Gamma_{sing}|}{\partial
T}\right)_{\delta\mu}$
$\displaystyle\sim|\tilde{t}|^{1-\alpha_{s}-\Delta}\mathcal{L}\left(\frac{\delta\mu}{|\tilde{t}|^{\Delta}}\right)$
$\displaystyle+|\tilde{t}|^{1-\alpha_{s}-\Delta}\left(\frac{\delta\mu}{|\tilde{t}|^{\Delta}}\right)\mathcal{L}^{\prime}\left(\frac{\delta\mu}{|\tilde{t}|^{\Delta}}\right)$
$\displaystyle\left(\frac{\partial|\Gamma_{sing}|}{\partial\mu}\right)_{T}$
$\displaystyle\sim|\tilde{t}|^{2-\alpha_{s}-2\Delta}\mathcal{L}^{\prime}\left(\frac{\delta\mu}{|\tilde{t}|^{\Delta}}\right)$
(14)

where $\mathcal{L}^{\prime}$ is the first derivative of $\mathcal{L}$, and the
term
$(\delta\mu/|\tilde{t}|^{\Delta})\mathcal{L}^{\prime}(\delta\mu/|\tilde{t}|^{\Delta})$
can be recognised to also be a scaling function. Comparing the exponents, we
deduce that, for the second term in eq. (10) to diverge faster than the first,
$\Delta>1$. If this were to be the case, then eq. (9) would become

$\frac{\chi_{T}(\ell_{eq})}{\chi_{\mu}(\ell_{eq})}\sim-\frac{\partial\mu_{co}}{\partial
T}$ (15)

where it is understood that we are considering the ratio of the fastest
diverging contributions. The resulting ratio is simply minus the gradient of
the bulk coexistence curve at the given temperature. Note the striking
similarity to the corresponding ratio of bulk quantities, eq. (4).

So what is the value of the gap exponent $\Delta$ for a particular system? It
is well-known that $\Delta$ satisfies [19]

$\Delta=2-\alpha_{s}-\beta_{s}$ (16)

where $\beta_{s}$ is the surface critical exponent for the adsorption.
$\alpha_{s}$, $\beta_{s}$ and therefore $\Delta$ are dependent on the
respective ranges of ff and sf interactions [4, 19]. For systems where ff and
sf interactions are both long-ranged (LR), $\alpha_{s}=-1$ and $\beta_{s}=-1$
(a linear divergence of the adsorption)[4, 19] which leads to $\Delta=4$.
Interactions of this type best describe experimental systems [3]. Thus, the
latter are expected to obey eq. (15), implying that all three measures of
density fluctuations diverge in the same way on the approach to the critical
drying transition.

A more challenging case is that of short-ranged (SR) ff and long-ranged (LR)
sf interactions which is pertinent for many simulation studies. In this case,
critical drying is not temperature dependent, rather this transition occurs at
exactly $\varepsilon_{s}=0$ for all temperatures below the bulk critical
temperature [3]. $\varepsilon_{s}$ therefore acts as the measure of deviation
from the drying critical point and takes the role of $\tilde{t}$ in the
scaling arguments above. (We return to this important point below.) For the
case of SR ff LR sf interactions, $\alpha_{s}=1$ and $\beta_{s}=0$ (a
logarithmic divergence of the adsorption) [2] which leads to a gap exponent
$\Delta=1$. Therefore this is a borderline case for the applicability of eq.
(15). In the next subsection we argue that such systems do indeed obey eq.
(15).

###  III.2 Interface Potential Analysis

In order to ascertain whether the case of SR ff LR sf does obey eq. (15) it is
necessary to resort to a mesoscopic analysis. Following Evans et. al. [2], the
excess grand potential per unit surface area of a truncated LJ fluid in
contact with a planar wall can be written as

$\omega_{ex}(\ell)=\gamma_{sl}+\gamma_{lv}+\omega_{B}(\ell)+\delta\mu\Delta\rho\ell$
(17)

where $\Delta\rho$ and $\delta\mu$ are as defined earlier and
$\omega_{B}(\ell)$ is the binding potential, which is the free energy required
to bind the liquid-vapour interface to the planar substrate at a distance
$\ell$. For a system with SR ff LR sf interactions the binding potential takes
the form [2, 3]

$\omega_{B}(\ell)=a(T)\exp(-\ell/\xi_{b})+b(T)\ell^{-2}$ (18)

where $\xi_{b}$ is the correlation length of the bulk vapour (the phase that
potentially ’wets’ the substrate), $a(T)$ is a positive coefficient,
proportional to $\Delta\rho$, with the dimensions of energy per unit area, and
$b(T)$ is

$b(T)=-b_{o}\rho_{s}\varepsilon_{s}\sigma_{s}^{6}$ (19)

Here, $b_{o}=\pi\Delta\rho/3$, and $\rho_{s}$, $\varepsilon_{s}$ and
$\sigma_{s}$ are as discussed in Sec. II. The temperature dependence of both
coefficients in eq. (18) is determined by that of $\Delta\rho$.

Minimising eq. (17) w.r.t. $\ell$ yields the equation determining the
equilibrium film thickness:

$-\frac{\ell_{eq}}{\xi_{b}}=\ln\left(\frac{\xi_{b}}{a(T)}\right)+\ln\left(\delta\mu\Delta\rho-\frac{2b(T)}{\ell_{eq}^{3}}\right)$
(20)

Clearly critical drying at bulk coexistence, $\delta\mu=0$, occurs at
$\varepsilon_{s}=0$ for all temperatures $T$. This equation therefore predicts
$\ell_{eq}\sim-\ln\varepsilon_{s}+3\ln\ell_{eq}$, in agreement with Evans et.
al. [2]. Note that the film thickness diverges only in the limit
$\varepsilon_{s}\rightarrow 0$ implying that critical drying occurs for all
temperatures below the bulk critical temperature and that $\varepsilon_{s}$
takes on the role of $\tilde{t}$ within the scaling arguments, as outlined in
Sec.III.1. On the approach to complete drying at a hard wall, where
$\varepsilon_{s}=0$, we find $\ell_{eq}\sim-\ln\delta\mu$, in agreement with
the standard result for complete drying (or wetting) from off-coexistence with
SR forces, e.g. [5] and [4].

Substituting eq. (20) into eq. (7) yields

$\chi_{\mu}(\ell_{eq})=\xi\Delta\rho\rho^{\prime}(\ell_{eq})\left(\delta\mu\Delta\rho-\frac{2b}{\ell_{eq}^{3}}\left(1-\frac{3\xi}{\ell_{eq}}\right)\right)^{-1}$
(21)

and

$\displaystyle\chi_{T}(\ell_{eq})$
$\displaystyle=-\xi\Delta\rho\rho^{\prime}(\ell_{eq})\left(\delta\mu\Delta\rho-\frac{2b}{\ell_{eq}^{3}}\left(1-\frac{3\xi}{\ell_{eq}}\right)\right)^{-1}\frac{\partial\mu_{co}}{\partial
T}$
$\displaystyle=-\chi_{\mu}(\ell_{eq})\left(\frac{\partial\mu_{co}}{\partial
T}\right)$ (22)

In deriving eq. (22) we have neglected the temperature dependence of the bulk
correlation length $\xi_{b}$. Including this gives a further, more slowly
diverging (logarithmic) contribution.

Eq. (22) is in agreement with eq. (15); recall we address the ratio of fastest
diverging contributions. Hence, we have shown that the borderline case of SR
ff LR sf interactions should obey the same relation predicted in the previous
subsection that used surface thermodynamics and scaling. We deduce that for SR
ff LR sf interactions on the approach to critical drying both
$\chi_{\mu}(\ell_{eq})$ and $\chi_{T}(\ell_{eq})$ should diverge as
$\varepsilon_{s}^{-1}$. On the approach to complete drying we expect both
$\chi_{\mu}(\ell_{eq})$ and $\chi_{T}(\ell_{eq})$ to diverge as
$\sim\delta\mu^{-1}$.

##  IV DFT Results

$\chi_{\mu}(z)$ and $\chi_{T}(z)$ are calculated within DFT by performing
numerical derivatives of the density profile with respect to $\mu$ and $T$,
respectively. The bulk value $\chi_{\mu,b}$ is obtained by calculating the
isothermal compressibility as outlined in Sec. I, whilst the bulk value
$\chi_{T,b}$ is most easily obtained using a numerical derivative.
$\chi_{*}(z)$ and $\chi_{*,b}$ follow directly from eq. (3).

Figure 2: (a) Fluctuation profiles, normalized by their bulk values, for a
truncated LJ fluid in contact with a planar hard wall, $\varepsilon_{s}=0.0$.
The measure of density fluctuations is denoted as $\chi_{x}$, where
$x=\mu,T,*$. Raw fluctuation profiles are given in the inset, and have been
made dimensionless by multiplying by $\varepsilon\sigma^{3}$,
$\varepsilon\sigma^{3}/k_{\mathrm{B}}$ and $\sigma^{3}$ in the cases of
$\chi_{\mu}$, $\chi_{T}$ and $\chi_{*}$ respectively. (b) The corresponding
density profile, scaled by its bulk value. The state point for the bulk liquid
is $T=0.775T_{c}$ and $\beta\delta\mu=10^{-3}$.

Fig. 2(a) compares fluctuation profiles obtained using DFT for a planar hard
wall close to (complete) drying. The chemical potential deviation
$\beta\delta\mu=10^{-3}$ is chosen to mimic water at near ambient conditions.
Fig. 2(b) shows the density profile of the system for comparison. The
fluctuation profiles are scaled by their bulk values in the main plot of fig.
2(a) and unscaled in the inset. Comparing first the unscaled profiles, it is
evident that $\chi_{*}(z)$ is by far the largest in magnitude , whilst
$\chi_{\mu}(z)$ and $\chi_{T}(z)$ are similar in magnitude. $\chi_{\mu}(z)$
and $\chi_{T}(z)$ appear to mirror one another, with the latter taking
negative values and the former remaining positive. The differing sign is
easily explained by the behaviour of the density profile upon increasing $\mu$
and $T$. When $\mu$ is increased at fixed $T$ the system moves away from bulk
coexistence and hence the thickness of the vapour film, measured by
$\ell_{eq}$, decreases. The gradient of the density profile in eq. (7) is
positive and it follows that $\chi_{\mu}(z)$ is also positive. In contrast,
increasing $T$ at fixed $\mu$ moves the system closer to liquid-vapour
coexistence which increases $\ell_{eq}$ so that $\chi_{T}(z)$ is negative.
(Note that there is no a priori reason for the peak in the $\chi_{\mu}(z)$ and
the trough in $\chi_{T}(z)$ profiles to be perfectly aligned.) Observing the
signs of the individual terms in eq. (3) it is clear why in the inset of fig.
2(a), $\chi_{*}(z)$ appears to be the most pronounced measure of density
fluctuations near to drying. However, this is somewhat misleading. When the
fluctuation profiles are scaled by their bulk values, fig.2(a) shows that
$\chi_{\mu}(z)$ exhibits the most pronounced maximum. All three scaled
profiles are positive and their maxima lie very close together, at about
3.5$\sigma$, near where the gradient of the density profile is largest. On
decreasing $\beta\delta\mu$ the three (scaled) fluctuation profiles exhibit
the same shapes but the position of the maxima increases, consistent with the
increase in $\ell_{eq}$, and each peak height increases rapidly, consistent
with what we expect from Sec. III.2 on approaching complete drying. This
implies that $\chi_{\mu}(z)$ is the most sensitive measure of density
fluctuations in systems near to drying and dictates the behaviour of the other
two measures 111In the present analysis we have taken the thermal de Broglie
wavelength, $\Lambda$, to be a constant,independent of $T$. This is the usual
convention in simulation studies. In [12] the consequences of allowing
temperature dependence in $\Lambda$ are addressed. However, when comparing
diverging contributions, as we do here, the choice of convention does not
matter..

We can now examine the reliability of the prediction of eq. (15) by using DFT
to calculate $\chi_{T}(\ell_{eq})$ and $\chi_{\mu}(\ell_{eq})$ for various
$\beta\delta\mu$ and values of $\varepsilon_{sf}$, proportional to
$\varepsilon_{s}$. $\ell_{eq}$ is calculated from the DFT result for the
density profile using $\ell_{eq}\equiv-\Gamma/A\Delta\rho$, i.e. from the
measured Gibbs adsorption $\Gamma$. The pertinent ratio is to be compared to
$\partial\mu_{co}/\partial T$, calculated from the bulk coexistence curve. The
latter is found in the usual way by equating the pressure and chemical
potential of the phases. In the case of DFT employing the Rosenfeld
functional, the relevant bulk fluid is described by the Percus-Yevick hard
sphere model, supplemented by the interparticle attraction of eq. 5 treated
within mean field [15]. For the temperature we consider, $T=0.775T_{c}$,
$(\partial\mu_{co}/\partial T)\approx 0.834$, in reduced units. We choose to
define the relative error between the ratio of compressibilities and the
gradient of the coexistence curve as

$\delta_{e}=\frac{\left|\frac{\chi_{T}(\ell_{eq})}{\chi_{\mu}(\ell_{eq})}+\frac{\partial\mu_{co}}{\partial
T}\right|}{\frac{\partial\mu_{co}}{\partial T}}$ (23)

and plot results for a variety of systems near to critical drying in fig. 3.
As the limit of critical drying is approached, i.e.
$\beta\delta\mu=0,\varepsilon_{sf}=0$, $\delta_{e}$ approaches zero,
indicating that eq. (15) is obeyed. This is displayed very clearly for the two
smallest values of $\beta\delta\mu$. The anomalous point, very close to
critical drying, with $\beta\delta\mu=10^{-5}$ and $\varepsilon_{sf}=0$, is
related to the numerical difficulty in evaluating the ratio of local
compressibilities in this regime where density fluctuations are especially
strong.

Figure 3: Comparison of the relative error, $\delta_{e}$, between each side of
eq. (15) for systems close to critical drying. The dimensionless ratio in the
abscissa measures the strength of sf attraction. The temperature was fixed at
$T=0.775T_{c}$.

##  V Conclusions

We have shown that in the vicinity of a critical drying transition all three
measures of density fluctuations introduced by Eckert et.al. [12] diverge in
the same way, i.e. with the same critical exponent. This was demonstrated: i)
in Sec. III.1 by considering the divergence of the local compressibility and
thermal susceptibility at $\ell_{eq}$ within a general scaling treatment of
surface thermodynamics for a wide class of model systems and ii) in Sec. III.2
by considering the form of the same quantities using an interface potential
(binding potential) analysis for the borderline case of SR ff and LR sf
potentials. DFT results for a system consisting of a (SR) truncated LJ fluid
in contact with a planar wall with algebraically decaying (LR) sf attraction
were presented in Sec. IV. These confirm the binding potential predictions and
provide explicit results for $\chi_{\mu}(z)$, $\chi_{T}(z)$ and $\chi_{*}(z)$,
in the vicinity of the critical drying transition. Comparing these three
measures of fluctuations, Fig. 2 shows that $\chi_{\mu}(z)$ provides the
sharpest indicator of the critical drying transition, drives the form of the
other measures and therefore can be thought of as the most sensitive measure
of density fluctuations in this region.

Whilst the results presented in Sec. III.2 and IV have focused on systems with
SR ff LR sf interactions, which are most pertinent to computational studies,
we note that the general result of eq. (15) is expected to hold for any system
with a gap exponent $\Delta>1$. This includes systems with LR ff LR sf
interactions, which have $\Delta=4$ [19, 4] and which are pertinent to
experimental systems where dispersion forces always prevail at the longest
length scales.  It also includes systems with SR ff SR sf interactions, where
within mean-field, $\alpha_{s}=0$ and $\beta_{s}=0$, and hence $\Delta=2$ [4].
We note that $\chi_{\mu}(z)$ has been calculated exactly for critical wetting
in a two dimensional model using transfer matrix methods [21]. Extending the
analysis to $\chi_{T}(z)$ shows that all measures of density fluctuations
exhibit similar forms, however the local compressibility diverges faster than
the thermal susceptibility222We thank a referee for bringing this paper to our
attention, and for deriving an expression for $\chi_{T}(z)$.. Finally, we note
that it is possible to derive an expression similar to eq. (15) for fluids
adsorbed at smooth curved surfaces. In this case, eq. (15) acquires an
additional curvature dependent term which constrains further the divergence of
the local density fluctuations [23].

###### Acknowledgements.

We thank T. Eckert and M. Schmidt for forwarding details of their work in
advance of publication which motivated our present investigation. R.E.
acknowledges Leverhulme Trust Grant EM-2020-029\4. This work used the
facilities of the Advanced Computing Research Centre, University of Bristol.

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