# Vicious walks with long-range interactions

Igor Goncharenko, Ajay Gopinathan  School of Natural Sciences, University of
California, Merced, California, 95343, USA

###### Abstract

The asymptotic behaviour of the survival or reunion probability of vicious
walks with short-range interactions is generally well studied. In many
realistic processes, however, walks interact with a long ranged potential that
decays in $d$ dimensions with distance $r$ as $r^{-d-\sigma}$. We employ
methods of renormalized field theory to study the effect of such long range
interactions. We calculate, for the first time, the exponents describing the
decay of the survival probability for all values of parameters $\sigma$ and
$d$ to first order in the double expansion in $\varepsilon=2-d$ and
$\delta=2-d-\sigma$. We show that there are several regions in the $\sigma-d$
plane corresponding to different scalings for survival and reunion
probabilities. Furthermore, we calculate the leading logarithmic corrections
for the first time.

###### pacs:

64.60.ae, 64.60.F-, 05.40.Jc, 64.60.Ht

##  I Introduction

Systems consisting of diffusing particles or random walks interacting by means
of a long-range potential are non-equilibrium systems, which describe
different phenomena in physics, chemistry and biology. From a physical
perspective they are used to study metastable supercooled liquids Supercool ;
Dean , melting in type-II high-temperature superconductors Nelson , electron
transport in quasi-one-dimensional conductors Quasi1d  and carbon nanotubes
Nanotube . From a chemical viewpoint the interest in these systems lies in the
fact that some diffusion-controlled reactions processes rely on the diffusion
of long-range interacting particles which react after they are closer than an
effective capture distance. Some examples include radiolysis in liquids
ParkDeem , electronic energy transfer reactions Klafter  and a large variety
of chemical reactions in amorphous media RDreview . From a biological
viewpoint, the investigation of these systems is helpful in understanding the
dynamics of interacting populations in terms of predator-prey models Krap-
Redner ; Bray  and membrane inclusions with curvature-mediated interactions
Reynwar1 ; Reynwar2 .

Vicious walks (VW) are a class of non-intersecting random walks, where the
process is terminated upon the first encounter between walkers Fisher . The
fundamental physical quantity describing VW is the survival probability which
is defined as the probability that no pair of particles has collided up to
time $t$. Diffusing particles or walks that are not allowed to meet each other
but otherwise remain free, we call pure VW. The behavior of pure VW is
generally well-known. The survival probability for such a system has been
computed in the framework of renormalization group theory in arbitrary spatial
dimensions up to two-loop order Cardy ; Bhat1 ; Bhat2 . These approximations
have been confirmed by exact results available in one dimension from the
solution of the boundary problem of the Fokker-Plank equation Krap-Redner ;
Bray , using matrix model formalism Katori  and Bethe ansatz technique Derrida
. On the other hand the effect of long range interactions has been extensively
investigated in many-body problems. It has been shown that the existence of
long-range disorder leads to a rich phase diagram with interesting crossover
effects Halp ; Bla ; Prud . If the potential is Coulomb-like ($\sim
r^{-1-\sigma}$) then systems in one dimension behave similar to a one-
dimensional version of a Wigner crystal Wigncrist  for $\sigma<0$ and similar
to a Luttinger liquid for $\sigma\geq 0$ Mor-Zab . If the potential is
logarithmic then in the long-time limit the dynamics of particles are
described by non-intersecting paths Hinrichsen ; Katori . The generalization
of VW that includes the effect of long range interactions has not attracted
much attention in the literature. Up to our knowledge there was one attempt to
study long-range VW Bhat3 . Here the authors considered the case of a long-
range potential decaying as $gr^{-\sigma-d}$, where $g$ is a coupling
constant. It was shown by applying the Wilson momentum shell renormalization
group that only one of the critical exponents characterize long-range VW. For
a specific value of $\sigma$ ($\sigma=2-d$) they show that the exponent
$\gamma$, which determines the decay of the asymptotic survival probability
with time, is given by the expression:

$\gamma=\frac{p(p-1)}{4}u_{1},$ (1)

where $p$ the number of VW in the system,
$u_{1}=(\varepsilon/2+[(\varepsilon/2)^{2}+g]^{1/2})$ and $\varepsilon=2-d$.
There are limitations to the above approach. First, it is restricted to a
single form of the potential ($\sim r^{-2}$) and systems such as membrane
inclusions and chemical reactions have different power-law potentials. Second,
it considers identical walkers but one would like to have results if the
diffusion constant of all walkers are different. Finally it is not convenient
to compute higher-loop corrections using the Wilson formalism.

In this paper we reconsider the problem of long-range VW using methods of
Callan-Symanzyk renormalized field theory in conjunction with an expansion in
$\varepsilon=2-d$ and $\delta=2-d-\sigma$. We note that it is more convenient
to compute logarithmic and higher loop corrections by using this method. We
derive the asymptotics of the survival and reunion probability for all values
of the parameters $(\sigma,d)$ for the first time.

Table 1: One-loop survival probability of $p$ sets of particles with $n_{j}$ particles in each set large-time asymptotic at different regions of the $\sigma-d$ plane. We refer to Figure 3 for specific value of $\sigma$ and $d$ in each region. Region | Survival probability |
---|---|---
I | $t^{-(d-2)/2}+t^{-(d+\sigma-2)/2}$ |
II | $t^{-\frac{1}{2}\sum_{ij}n_{i}n_{j}\varepsilon}$ |
III | $t^{-\frac{u_{1}}{2}\sum_{ij}n_{i}n_{j}(1+\delta/2\log t)}$ |
IV | $t^{-(d-2)/2}$ |
V, $d=2$ | $t^{-\frac{\sqrt{g_{0}}}{2}\sum_{ij}n_{i}n_{j}(1+\delta/2\log t)}$ |
VI, $\sigma=2-d$ |  $t^{-\frac{u_{1}}{2}\sum_{ij}n_{i}n_{j}}$ 111$u_{1}$ is defined by the formula (1). |

In this paper we will show that there are several regions in $\sigma-d$ plane
in which we have different behavior of the critical exponent. Our results are
summarized in Table I. We note that results on the line $\sigma+d=2$ have been
obtained before Bhat3 . Regions I and IV correspond to Gaussian or mean-field
behavior (see Figure 3). In region II we found that the system reproduces pure
VW. Logarithmic corrections in region III and at the short-range upper
critical dimension $d=2$ have been obtained as series expansion in
$\delta=2-\sigma-d$.

The remainder of this paper is organized as follows: Section II reviews the
field theoretic formulation of long range VW and describes Feynman rules and
dimensionalities of various quantities. In section III we derive the value of
all fixed points and study their stability. Section IV presents results for
the critical exponents and logarithmic corrections of various dynamical
observables. Section V contains our concluding remarks. In Appendix A we give
the details of the computation of some integrals that appear in Section III.

##  II Modelling VW with long-range interations

As the starting point of the description of our model we consider $p$ sets of
diffusing particles or random walks with $n_{i}$ particles in each set
$i=1\dots p$, with a pairwise intraset interaction which includes a local or
short-range part and a non-local or long-range tail. The local part determines
the vicious nature of walks: if two walks belonging to the different sets are
brought close to each other, both are annihilated. Walks belonging to the same
set are supposed to be independent. At $t=0$ all particles start in the
vicinity of the origin. We are interested in the survival and reunion
probabilities of walks at time $t>0$.

A continuum description of a system of $N$ Brownian particles $X_{i}$ with
two-body interactions is simplified by the coarse-graining procedure in which
a large number of microscopic degrees of freedom are averaged out. Their
influence is simply modelled as a Gaussian noise-term in the Langevin
equations. A convenient starting point for the description of the stochastic
dynamics is the path-integral formalism. Then the system under consideration
is modeled by the classical action

$S=\int\limits_{0}^{+\infty}dt\left(\sum\limits_{i=1}^{N}\dot{X}_{i}^{2}/(2D_{i})+\sum\limits_{i<j}V(X_{i}-X_{j})\right)$
(2)

where $t$ is (imaginary-)time, $X_{i}(t)$ is the $d$-dimensional vector
denoting the position of $i$th particle at time $t$. $D_{i}$ is an $i$th
particle diffusion coefficient. The path-integral representation of the
probability density function for the particle displacements from their
original positions is given by the functional ${\cal Z}=\int{\cal
D}X\exp[-S]$. The survival probability is defined as the expectation value

$P(t)=\langle\prod_{i,j}[1-\delta(X_{i}(t)-X_{j}(t))]\rangle$ (3)

with respect to the functional ${\cal Z}$. It is computed in the framework of
usual perturbation theory and will be a sum of integrals over internal degrees
of freedom. It is more convenient to perform these integrations in Fourier
space. To do this we would need the Fourier transform of the interaction
potential $V(r)$. We note that it is comprised of a short-range part of the
form $V_{0}(r)=\lambda\delta(r)$ and a long-range part which decays with the
distance $r$ as a power law, $V_{l}(r)=gr^{-d-\sigma}$. The Fourier transform
of the latter is divergent if $\sigma\geq 0$. We introduce the cut-off
parameter $a$ to regularize the singularity
$V_{l}(r)=g(r^{2}+a^{2})^{-(d+\sigma)/2}$. Fourier transformation of this
function is given by the expression

$V_{l}(q)=g\frac{\pi^{d/2}2^{\sigma}}{\Gamma(\frac{d+\sigma}{2})}(q/a)^{\sigma/2}K_{\sigma/2}(aq),$
(4)

where $K_{\sigma}$ is the modified Bessel-function with index $\sigma$. Small
$a$ expansion of (4) at leading order yields

$V_{l}(q)\sim g\begin{cases}q^{\sigma},&\mbox{if}\quad\sigma\neq 0\\\
\log(aq),&\mbox{if }\quad\sigma=0\\\ \end{cases}$ (5)

where we used the property $K_{-\sigma}(x)=K_{\sigma}(x)$ of the Bessel
function. The non-universal coefficient coming from the Taylor expansion can
be absorbed by the appropriate renormalization of the constant $g$. Special
cases when $\sigma$ is even gives logarithmic behavior. Effectively it does
not change our results. So we focus on the typical term $q^{\sigma}$.

The second quantized version of the action (2) can be constructed using
standard methods Doi ; Peliti . The generalization of the action to the long-
range interacting case is also known Mahan ; Fetter . The result is

$S(\phi_{i},\phi^{\dagger}_{i})=\int
dtd^{d}x\\{\sum_{i}[\phi^{\dagger}_{i}\partial_{t}\phi_{i}+D_{i}\nabla\phi^{\dagger}_{i}\nabla\phi_{i}]\\}+\int
dtd^{d}xd^{d}y\sum\limits_{i<j}\phi^{\dagger}_{i}(t,x)\phi_{i}(t,x)V_{ij}(x-y)\phi^{\dagger}_{j}(t,y)\phi_{j}(t,y)\;.$
(6)

The first term describes the evolution of free random walks with diffusion
constants $D_{i}$. The potential is

$V_{ij}(x-y)=\lambda_{ij}\delta(x-y)+g_{ij}V(x-y),$ (7)

and we refer to $\lambda_{ij},g_{ij}$ as short-range and long-range coupling
constants respectively.

A dynamic response functional associated with the action (6) is

$\mathcal{Z}=\int\mathcal{D}\phi\mathcal{D}\phi^{\dagger}e^{-S(\phi_{i},\phi^{\dagger}_{i})}$
(8)

where $\phi_{i}\left(x,t\right)$ is the complex scalar field. After the
quantization we may treat $\phi^{\dagger}_{i}\left(x,t\right)$ as the creation
operator which creates a particle of sort $i$ at point $x$ at time $t$. Having
the dynamic response functional, correlation functions can be computed as
functional averages (path integrals) of monomials of $\phi$ and
$\phi^{\dagger}$ with the weight
$\exp\left\\{-S(\phi,\phi^{\dagger})\right\\}$.

Figure 1:  Feynman rules for the theory (6). Notice that both $\lambda$ and
$g$ vertices appear with differenet $i$ and $j$ indices and that $g$ has
momentum dependence. Figure 2:  One-loop Feynman diagrams contributing to
$\lambda_{Rij}$

As a first step towards the renormalization group analysis of this model, we
discuss the dimensions of various quantities in (6) expressed in terms of
momentum:

$[t]=p^{-2}\quad[\phi]=p^{d}\quad[\lambda]=p^{2-d}\quad[g]=p^{2-d-\sigma}.$
(9)

The naive dimension of the coupling constant $g$ allows us to identify the
upper critical dimension $d_{c}(\sigma)=2-d-\sigma$. For $\sigma>0$, the
short-range term naively dominates the long-range term and we expect to have
the behavior of the system similar to the case of pure VW. We will reserve the
symbol $\varepsilon$ ($\varepsilon=2-d$) to denote deviations from the short-
range critical dimension $d_{c}=2$, and $\delta$ ($\delta=2-d-\sigma$) for the
deviations from the long-range critical dimension $d_{c}(\sigma)$. If
$\sigma=0$ then the critical dimension of the long-range part coincides with
the short-range part and we have the non-trivial correction to the asymptotic
behavior due to long-range interactions. This boundary separates mean-field or
Gaussian behavior from long-range behavior. For $\sigma<0$ the long-range term
dominates the short-range term and we expect to have non-trivial corrections
to the behavior of the system.

Now we consider diagrammatic representation elements of model (6). In zero-
loop approximation the vertex 4-point function takes a simpler form after
Laplace-Fourier transformation:

$\Gamma^{(2,2)}_{ij}(s,p)=V_{ij}(p_{1}+p_{2})\delta(\sum\limits_{k}p_{k}).$
(10)

The same transformation applied to the bare propagator yields:

$\Gamma^{(1,1)}_{j}(s,p)=(s+D_{i}p^{2})^{-1}$ (11)

We note that there are no vertices in (6) that produce diagrams which dress
the propagator, implying there is no field renormalization. As a consequence
the bare propagator (11) is the full propagator for the theory. Feynman rules
are summarized in Figure 1. There are two vertices in the theory: one is a
short-range $\lambda$-vertex and another is a long-range momentum dependent
$g$-vertex. Each external line of the vertex corresponds to a functionally
independent field. The propagator is formed by contracting appropriate lines
from different vertices. We recall the propagator is the correlation function
of $\phi_{i}$ and $\phi^{\dagger}_{i}$ fields only.

Physical observables are computed with the help of correlation functions. The
probability that $p$ sets of particles with $n_{i}$ particles in each set
start at the proximity of the origin and finish at $x_{i,\alpha_{i}}$ ($i$
index enumerates different sets and $\alpha_{i}$ index enumerates particles in
set $i$) without intersecting each other can be obtained by generalizing eqn
(3). In the field theoretical formulation, this probability becomes the
following correlation function:

$G(t)=\int\prod_{i=1}^{p}\prod_{\alpha_{i}=1}^{n_{i}}d^{d}x_{i,\alpha_{i}}\langle\phi_{i}(t,x_{i,\alpha_{i}})(\phi^{\dagger}_{i}(0,0))^{n_{i}}\rangle,$
(12)

In the Feynman representation it is the vertex with $2N$ ($N=\sum_{j}n_{j}$)
external lines. In the first order of the perturbation theory one needs to
contract these lines with corresponding lines of the vertices in Figure 1.
Since there are many independent fields in the correlation function (12) this
operation can be done in many ways. It yields a combinatorial factor,
$n_{i}n_{j}$, in front of each diagram, which is the number of ways of
constructing a loop from the $n_{i}$ lines of type $i$ and $n_{j}$ lines of
type $j$ on the one hand and one line of type $i$ and one line of type $j$ on
the other hand. From the next section we will see that the survival
probability scales as $G(t)\sim t^{-\gamma}$, where $\gamma$ is the critical
exponent. If all walks are free, $\gamma=0$. In the presence of interactions
we expect $\gamma$ to be a universal quantity that does not depend on the
intensity of the short-range interaction $\lambda_{ij}$. It is convenient to
introduce the so called truncated correlation function which is obtained from
(12) by factoring out external lines:

$\Gamma(t)=G(t)/(\Gamma^{(1,1)})^{2N}$ (13)

Another physical observable, the reunion probability, is defined as the
probability that $p$ sets of particles with $n_{i}$ particles in each set
start at the proximity of the origin and without colliding into each other
finish at the proximity of some point at time $t$:

$R(t)=\int
d^{d}x\prod_{i=1}^{p}\langle\phi_{i}(t,x)^{n_{i}}(\phi^{\dagger}_{i}(0,0))^{n_{i}}\rangle,$
(14)

In the Feynman representation it is depicted as the watermelon diagram with
$2N$ stripes. We note that if the theory is free this expression is the
product of free propagators and at the large-time limit the return probability
scales as $R_{\cal O}(t)\sim t^{-(N-1)d/2}$. If interactions are taken into
account it becomes $R(t)\sim t^{-(N-1)d/2-2\gamma}$, where $\gamma$ is
survival probability exponent. The reason that it enters with the factor 2 is
the following. If we cut a watermelon diagram of the reunion probability
correlation function in the middle then it produces two vertex diagrams with
$2N$ external lines of the survival probability correlation function. As a
result the reunion probability is the product of two survival probabilities.
It remains true in all orders of perturbation theory. For a rigorous proof we
refer to Bhat2 .

##  III The Renormalization of observables

Figure 3:  The critical behavior of vicious walks with long-range interactions
in the different regions of the $(\sigma,d)$ plane. Region I and IV correspond
to the mean field short-range behavior, in region II will be critical short-
range behavior, region III is the long-range behavior. The lines $d=2$ and
$\sigma+d=2$ represent regions V and VI respectively.

While computing correlation functions like (12) perturbatively one faces
divergent integrals when $d=d_{c}$. The convenient scheme developed for
dealing with these divergences follows Callan-Symanzik renormalization-group
analysis Zinn ; Amit . Within this scheme we start with the bare correlation
function $G(t;\lambda,g)$, where $\lambda=\\{\lambda_{ij}\\}$, and
$g=\\{g_{ij}\\}$ denote the set of bare short-range and long-range coupling
constants. In the renormalized theory it becomes
$G_{R}(t;\lambda_{R},g_{R},\mu)$. From dimensional analysis it follows that

$G_{R}(t;\lambda_{R},g_{R},\mu)=G_{R}(t\mu;\lambda_{R},g_{R}),$ (15)

where $\mu$ is the renormalization scale. The scale invariance leads to the
expression

$G_{R}(t;\lambda_{R},g_{R},\mu)=Z(\lambda_{R},g_{R},\mu)G(t;\lambda,g).$ (16)

Here functions $Z$ are chosen in such a way that
$G_{R}(t,\lambda_{R},g_{R},l)$ remains finite when the cut-off is removed at
each order in a series expansion of $\lambda_{R}$, $g_{R}$, $\varepsilon$ and
$\delta$. From the fact that $G(t,\lambda,g)$ does not depend on the
renormalization scale $\mu$ we get the Callan-Symanzik equation

$\left(\mu\frac{\partial}{\partial\mu}+\beta_{g}\frac{\partial}{\partial
g}+\beta_{u}\frac{\partial}{\partial u}-\gamma\right)G_{R}=0,$ (17)

where the $\beta$-functions are defined by

$\beta_{\lambda}(\lambda_{R},g_{R})=\mu\frac{\partial}{\partial\mu}\lambda_{R}\qquad\beta_{g}(\lambda_{R},g_{R})=\mu\frac{\partial}{\partial\mu}g_{R}$
(18)

and the function $\gamma$ by

$\gamma(\lambda_{R},g_{R})=\mu\frac{\partial}{\partial\mu}\ln Z.$ (19)

The renormalization group functions are understood as the expansion in double
series of coupling constants $\lambda$ and $g$ and deviations from the
critical dimension $\varepsilon$ and $\delta$. We take
$\delta=O(\varepsilon)$. The coefficient $Z(\lambda_{R},g_{R},\mu)$ is fixed
by the normalization conditions. It is more convenient to impose these
conditions on the Laplace transform of the truncated correlation function
(13). One sets the following condition then

$\Gamma_{R}(\mu)=1,$ (20)

when $s=\mu$. We note that the same multiplicative renormalization factor $Z$
yields $\Gamma$ finite. From this fact one can infer that

$\Gamma(\mu;\lambda,g)=Z(\mu;\lambda,g)^{-1}.$ (21)

If we express unrenormalized couplings in terms of renormalized ones (21) we
will obtain the equation for finding $Z$ explicitly.

The equation (17) can be solved by the method of characteristics. Within this
method we let couplings depend on the scale which is parametrized by
$\mu(x)=x\mu$. Here $x$ is introduced as a parametrization variable of the RG
flow and is not to be confused with position. Henceforth $x$ will refer to
this parametrization variable. We introduce running couplings
$\bar{\lambda}(x)$ and $\bar{g}(x)$. They satisfy the equations

$x\frac{d}{dx}\bar{g}(x)=\beta_{g}(\bar{\lambda}(x),\bar{g}(x))\quad
x\frac{d}{dx}\bar{\lambda}(x)=\beta_{\lambda}(\bar{\lambda}(x),\bar{g}(x)).$
(22)

The renormalized value should be defined by the initial conditions
$\bar{\lambda}(1)=\lambda_{R}$ and $\bar{g}(1)=g_{R}$. the solution of the
equation is then

$G_{R}(t)=e^{\int\limits_{1}^{\mu
t}\gamma(\bar{\lambda}(x),\bar{g}(x))dx/x}G_{R}(\mu^{-1};\bar{\lambda}(\mu
t),\bar{g}(\mu t),\mu)$ (23)

Next we calculate the first-order contribution to the renormalized vertices.
The $\lambda$-vertex is renormalized by the set of diagrams that are shown in
Figure 2. We notice that there are no diagrams producing the momentum
dependent $g$-vertex in the theory (6). This statement is the corollary of the
fact that only independent fields of power one enter into the expression of
the vertex and there are no higher powers of fields. Also we keep in mind that
the renormalized couplings are defined by the value of the vertex function
taken at zero external momenta. It produces the following expression:

$\begin{cases}\lambda_{Rij}&=\lambda_{ij}-\frac{1}{2}(\lambda_{ij}^{2}I_{1}+2\lambda_{ij}g_{ij}I_{2}+g_{ij}^{2}I_{3})\\\
g_{Rij}&=g_{ij}\\\ \end{cases}$ (24)

where $I_{k}=I_{k}(\sigma;D_{i},D_{j})$ are one-loop integrals corresponding
to the diagrams $a$, $b$, $c$ in the Figure 2 respectively. Using the Feynman
rules we can explicitly write them down:

$I_{k}=\int\frac{d^{d}q}{(2\pi)^{d}}\frac{q^{(k-1)\sigma}}{2s+(D_{i}+D_{j})q^{2}},\quad
k=1,2,3.$ (25)

We will use dimensional regularization procedure to compute these integrals.
The details of the computation are summarized in Appendix A. We note that
integrals will diverge logarithmically at different values of the spatial
dimension $d$. For this reason it leads to different critical behavior in
different regions of the $\sigma-d$ plane (see Figure 3). These regions
correspond to four possibilities for $\varepsilon=2-d$ and $\delta=2-d-\sigma$
to be positive or negative. Only if $\delta=O(\varepsilon)$ or, in other
words, if both $\varepsilon$ and $\delta$ are infinitesimally small but the
ratio $\varepsilon/\delta$ is finite we expect non-zero fixed points of the
renormalization group flow. Similar approximation have been used before Halp
but for different models with long-range disorder. It allows us to follow the
standard procedure of deriving the $\beta$-functions which consists of two
steps.

First, we express unrenormalized couplings in terms of the renormalized. For
the short-range coupling constant $\lambda$ it can be done by solving the
quadratic equation in (24). Expanding the square root and keeping terms up to
the second order we infer that

$\begin{cases}\lambda_{ij}&=\lambda_{Rij}+\frac{1}{2}(\lambda_{Rij}^{2}\frac{a_{d}}{\varepsilon}+2\lambda_{Rij}g_{Rij}\frac{b_{d}}{\delta}+g_{Rij}^{2}\frac{c_{d}}{2\delta-\varepsilon})\\\
g_{ij}&=g_{Rij}\\\ \end{cases}$ (26)

where $a_{d}$, $b_{d}$ and $c_{d}$coefficients have been found explicitly in
Appendix A. Now we introduce dimensionless renormalized couplings

$\bar{g}_{Rij}=a_{d}(2s)^{-\delta/2}\quad\bar{\lambda}_{Rij}=b_{d}(2s)^{-\varepsilon/2}.$
(27)

An important observation is that $c_{d}a_{d}=b_{d}^{2}$ which can be verified
by explicit substitution (see Appendix A). Multiplying the first and second
equation in (26) by the factors $a_{d}$ and $b_{d}$ respectively, and using
redefinitions (27) we can condense all pre-factors in the right hand side of
the equations into the dimensionless constants.

Second, we differentiate equations (26) with respect to the scaling parameter
$\mu$. Using definitions (18) and the fact that bare couplings do not depend
on the scale, we derive

$\begin{cases}\beta_{\lambda,ij}&=-\varepsilon\bar{\lambda}_{Rij}+(\bar{\lambda}_{Rij}+\bar{g}_{Rij})^{2}\\\
\beta_{g,ij}&=-\delta\bar{g}_{Rij}\\\ \end{cases}$ (28)

where the right hand side is understood as the leading contribution to the
$\beta$-functions from the double expansions in $\lambda,g$ and
$\varepsilon,\delta$. From (28) we see that it is convenient to introduce new
coupling constants $u_{Rij}=\bar{\lambda}_{Rij}+\bar{g}_{Rij}$. After this
step the renormalization group equations read

$\begin{cases}\beta_{u,ij}&=-\varepsilon u_{Rij}+u_{Rij}^{2}-g_{Rij}\\\
\beta_{g,ij}&=-\delta g_{Rij}\\\ \end{cases}$ (29)

We note that in the last equations $g$ coupling constant has been redefined
$\sigma\bar{g}_{Rij}\to g_{Rij}$.

Fixed points are zeros of the $\beta$-functions. If $\delta\neq 0$ then the
last equation in (29) is zero only when $g_{*}=0$. Then the first equation has
two solutions $u=0$ and $u=\varepsilon$. If $\delta=0$ then $g$ plays the role
of a parameter and the fixed points are determined by the roots of the
quadratic equation

$0=-\varepsilon u+u^{2}-g$ (30)

which are real if $g\geq-(\varepsilon/2)^{2}$ and we find

$u_{1,2}=\varepsilon/2\pm\sqrt{(\varepsilon/2)^{2}+g}.$ (31)

All fixed points are listed in the Table II. The stability of these fixed
points is determined by the matrix of partial derivatives

$\beta_{*}=-\left(\begin{array}[]{cc}\partial\beta_{u}/\partial
u&\partial\beta_{u}/\partial g\\\ \partial\beta_{g}/\partial
u&\partial\beta_{g}/\partial g\end{array}\right)_{u=u_{*},g=g_{*}}$ (32)

Eigenvalues are listed in the Table 2. The Gaussian fixed point is stable in
all directions for $\varepsilon<0$ and $\delta<0$ which corresponds to region
I in Figure 3. In this region we find both short-range(pure VW) and long-range
mean-field behavior depending on the sign of $\sigma$. On the contrary, for
$\varepsilon>0$ and $\delta>0$ we find that the Gaussian fixed point is
unstable(irrelevant) in all directions and the short-range (pure VW) fixed
point is stable(relevant) only in $u$-direction. It means that long-range
interactions will play a leading role. This region corresponds to region III
in Figure 3. Next for $\varepsilon>0$ and $\delta<0$ we find that the short-
range (pure VW) fixed point is stable in all directions. It means that the
system is insensitive to the long-range tail. This region corresponds to
region II in Figure 3. Finally for $\varepsilon<0$ and $\delta>0$ we find that
the short-range (pure VW) fixed point is unstable in all directions and the
system will be described by mean-field at long time.

Table 2:  Fixed points for flow equations (29) and the corresponding eigenvalues $(\lambda_{1},\lambda_{2})$ of the stability matrix (32). We note that $u_{1}$ and $u_{2}$ are values of the  Fixed point | $(u_{*},g_{*})$ | $(\lambda_{1},\lambda_{2})$
---|---|---
Gaussian | $(0,0)$ | $(\varepsilon,\delta)$
Pure VW | $(\varepsilon,0)$ | $(-\varepsilon,\delta)$
LR stable | $(u_{1},0)$ | $(-\sqrt{\varepsilon^{2}-4g},0)$
LR unstable | $(u_{2},0)$ | $(\sqrt{\varepsilon^{2}-4g},0)$

##  IV Calculation of critical exponents and discussion

Here we describe our method of computing critical exponents. It is based on
the formula (21) from the previous section. First, we obtain the leading
divergent part of the correlation function. The renormalized correlation
function depends on the scale $\mu$ but it appears in all formulas in
combination with time: $\mu t$. Second, since we have found the bare coupling
constant as a function of renormalized (dressed) couplings we express
correlation function in terms of dressed couplings. Finally using the
normalization condition (20) and the definition (19) we differentiate $Z$ with
respect to $\mu\partial/\partial\mu$ to obtain the exponent $\gamma$. The
poles should cancel after this operation.

In section 2 it was explained that the truncated correlation function in the
one-loop approximation is given by the formula

$\Gamma(t;\lambda,g)=1-\sum_{i,j}n_{i}n_{j}\left(\lambda_{ij}I_{1}+g_{ij}I_{2}\right).$
(33)

Here integrals are the same as in (25).

We start our analysis with the region I. Notice that truncated correlation
function $\Gamma(t)$ and survival probability $G(t)$ have similar large time
behavior. We use large momentum cut-off to compute integrals $I_{1}$ and
$I_{2}$ as in formula (64) in Appendix A. The renormalization of coupling
constants is trivial in this case. Therefore the leading contribution to the
survival probability is given by

$G(t)\sim t^{(2-d)/2}+g_{0}t^{(2-d-\sigma)/2},$ (34)

where $g_{0}$ is non-universal coefficient and we will not need its exact
value. We notice that if $\sigma>0$ the second term will decay faster than the
first term and in the long-time limit it will produce the same behavior as
mean-field pure VW. On the other hand if $\sigma<0$ the first term will decay
faster and long-range interactions will play a leading role. Many authors
observed similar behavior in various systems with long-range defects Halp ;
Bla ; Prud . Intuitively if potential falls fast with distance than the system
effectively represent system with short-range potential where particle
interact when they are close to each other.

Region IV exhibits similar behavior. Now the integral $I_{2}$ is computed with
the help of the dimensional regularization (58) and the integral $I_{1}$
remains the same. From the fact (15) one can infer that the survival
probability scales as

$G(t)\sim t^{(2-d)/2}.$ (35)

Short-range behavior dominates because the running coupling constant will flow
towards the Gaussian fixed point at long time limit which is the only stable
fixed in this region. This result is exact regardless the number of loops one
takes into account.

In Region II the computation is as follows.

$\ln Z=\sum
n_{i}n_{j}\left(\lambda_{ij}\frac{a_{d}}{\varepsilon}+g_{ij}t^{(2-d-\sigma)/2}\right),$
(36)

so plugging the result from (59) to (36) we obtain at the fixed point
$(\lambda_{*}=\varepsilon,g=0)$

$\gamma=-\frac{1}{2}\sum n_{i}n_{j}\varepsilon$ (37)

And we reproduce the pure VW behavior. This result is the reflection of the
fact that the renormalization-group trajectories run away to stable pure VW
fixed point. It is with agreement with the results obtained by Katori in
Katori  for $d=1$, and the logarithmic intraset particle interactions. The
irrelevance of the long-range interaction in lower dimensions is a typical
phenomenon observed in a various out of equilibrium interacting particle
systems.

We now consider regions III, V and VI. Integrals in (33) are computed via
dimensional regularization. Taking the inverse of (33) and then logarithm one
can obtain at the leading order:

$\log Z=\sum
n_{i}n_{j}\left(\lambda_{ij}\frac{a_{d}}{\varepsilon}+g_{ij}\frac{b_{d}}{\delta}\right)$
(38)

where $a_{d}$ and $b_{d}$ are defined in Appendix A in (59) and (60). We note
that after taking the derivative the poles in (38) will cancel in the limit of
$\delta=O(\varepsilon)$. Also one recalls the expansion (26) and the
redefinitions in (27). Using (19) we show that the expression for the function
$\gamma$ which determines critical exponent takes the form

$\gamma=-\frac{1}{2}\sum_{ij}n_{i}n_{j}u_{R}$ (39)

Evaluated at the stable fixed point
$(u_{1}=\varepsilon/2+\sqrt{(\varepsilon/2)^{2}+g}$ it gives the following
result:

$\gamma=-\frac{1}{2}\sum_{ij}n_{i}n_{j}u_{1},$ (40)

and the survival probability scales as $G(t)\sim t^{\gamma}$.

We will now find the logarithmic corrections to this scaling law. The running
coupling constant can be found from the flow equation (29):
$\bar{g}(x)=e^{-\delta x}g$. In the case $\delta,\varepsilon=0$ (the
intersection of regions V and VI) the flow equation for $\bar{u}(x)$ is

$x\frac{d\bar{u}(x)}{dx}=-\bar{u}^{2}(x)+g$ (41)

and the solution is

$\bar{u}(x)=\sqrt{g}\tanh(\sqrt{g}\log
x+\phi_{0})\sim\sqrt{g}\tanh(\sqrt{g}\log x),$ (42)

where $\phi_{0}$ is the initial condition and we do not need its exact form.
After plugging this expression into the (23) we infer

$\int\limits_{1}^{\mu
t}\gamma(\bar{u},\bar{g})\frac{dx}{x}\sim\log(\cosh(\sqrt{g}\log\mu t))$ (43)

Thus the survival probability is

$G(t)\sim\cosh(\sqrt{g}\log t)^{-\frac{1}{2}\sum n_{i}n_{j}}$ (44)

In the limit of large time $\cosh(\sqrt{g}\log t)\sim t^{\sqrt{g}}$ implying
$gamma=-\frac{1}{2}\sum_{ij}n_{i}n_{j}\sqrt{g}$ which is consistent with
equation (40). For negative coupling constant $g<0$ the solution in (42)
becomes

$\bar{u}(x)\sim-\sqrt{|g|}\tan(\sqrt{|g|}\log x)$ (45)

The integral (43) is divergent if $t>\exp(\pi/2\sqrt{|g|})$ which leads to the
result that the survival probability is zero beyond this time. For smaller
times one has $G(t)\sim\cos(\sqrt{|g|}\log t)^{-\frac{1}{2}\sum n_{i}n_{j}}$.
Thus, upto one-loop order approximation, It implies that if walks are
attracted to each other then all of them will annihilate at some finite time.
This might be a signature of faster than power law decay and we expect to have
corrections to this behavior at higher loop approximation.

Next we consider the case when $\varepsilon=0$ and $\delta\neq 0$ but $\delta$
remains small i.e. region V. The flow equation for the $\bar{u}(x)$ is

$x\frac{d\bar{u}(x)}{dx}=-\bar{u}^{2}(x)+gx^{-\delta}$ (46)

and the solution can be found by the method of perturbation. Up to the first
order

$\bar{u}(x)=\sqrt{g}\tanh(\sqrt{g}\log
x)+\delta\sqrt{g}\log(x)\tanh(\sqrt{g}\log x)$ (47)

After plugging this expression into eqn (23) we infer

$\int\limits_{1}^{\mu
t}\gamma(\bar{u},\bar{g})\frac{dx}{x}\sim-\frac{1}{2}\sum
n_{i}n_{j}\left(\log(t^{\sqrt{g}})+\frac{1}{2}\delta\sqrt{g}\log^{2}(t)\right)$
(48)

Therefore we have the correction to the survival probability in the form

$G\sim t^{-\frac{1}{2}\sum n_{i}n_{j}\sqrt{g}(1+\delta/2(\log t))}$ (49)

Now we extend our analysis to the case when $\varepsilon>0$, corresponding to
regions III and VI. The evolution of the coupling constant is

$x\frac{d}{dx}\bar{u}(x)=\varepsilon\bar{u}-\bar{u}^{2}+gx^{\delta}$ (50)

We choose the ansatz in the form $\bar{u}(x)=u_{0}(x)+\delta v(x)$. For
$\delta=0$ (i.e. region VI) the equation for $u_{0}(x)$ reads

$x\frac{d}{dx}u_{0}(x)=\varepsilon u_{0}-u_{0}^{2}+g$ (51)

and we reproduce the result (40). We now extend to the case where
$\varepsilon,\delta>0$ (region III). Here we will need the exact solution to
(51) to find the corrections:

$u_{0}(x)=\frac{Cx^{u_{1}-u_{2}}u_{1}+u_{2}}{1+Cx^{u_{1}-u_{2}}},$ (52)

where $C=(u_{R}-u_{2})/(u_{1}-u_{R})$. The logarithmic correction follows from
the form of the perturbation. The equation for $v(x)$ is

$x\frac{d}{dx}v(x)=\varepsilon v-2u_{0}v-g\log x$ (53)

The solution can be found explicitly as a combination of hypergeometric
functions. In the most interesting case, $\varepsilon=1$ ($d=1$) the
hypergeometric functions are degenerate and become linear functions.
Corrections to the integral then read

$\int\limits_{1}^{\mu t}\gamma dx/x\sim\frac{1}{2}\delta
u_{1}\log^{2}(t)+\log(t)(t)^{u_{1}-u_{2}})$ (54)

In the limit of large time only the first term contributes to the exponent and
the survival probability scales as

$G\sim t^{-\frac{1}{2}\sum n_{i}n_{j}u_{1}(1+\delta/2\log t)}$ (55)

##  V Conclusion

In summary, we studied long-range vicious walks using the methods of Callan-
Symanzik renormalized field theory. Our work confirms the previously known RG
fixed point structure including their stability regions. We calculated the
critical exponents for all values of $\sigma$ and $d$ to first order in
$\varepsilon$ expansion and to all orders in $\delta$ expansion, which have
hitherto been known only for $d+\sigma=2$. Our results indicate that,
depending on the exact values of $d$ and $\sigma$, the system can be dominated
by either short range (pure VW) or long range behaviors. In addition, we
calculated the leading logarithmic corrections for several dynamical
observables that are typically measured in simulations.

We hope that our work stimulates further interest in long-range vicious walks.
It would be interesting to see further simulation results for the critical
exponents for $d>1$ and for logarithmic corrections. Also, it would be
interesting to have analytical and numerical results for other universal
quantities such as scaling functions and amplitudes.

##  VI Acknowledgments

AG would like to acknowledge UC Merced start-up funds and a James S. McDonnell
Foundation Award for Studying Complex Systems.

## Appendix A

Effective four-point function (one-particle irreducible, 1PI) that appeared in
(24) is composed of usual short-range and new momentum dependent vertices.
This gives rise to integrals (25). The first integral $\mu=1$ has been
evaluated in Cardy  by using alpha representation
$1/(q^{2}+s)=\int_{0}^{+\infty}d\alpha e^{i(q^{2}+s)\alpha}$ and the result is

$I_{1}=K_{d}(2s)^{-\varepsilon/2}\Gamma(\varepsilon/2).$ (56)

We notice that since there is no angular dependence one can perform $d-1$
integrations and one will be left with one dimensional integral. To compute
this integral we use the formula GR :

$\int\limits_{0}^{+\infty}dx\frac{x^{\nu-1}}{P+Qx^{2}}=\frac{1}{2P}\left(\frac{P}{Q}\right)^{\nu/2}\Gamma\left(\frac{\nu}{2}\right)\Gamma\left(1-\frac{\nu}{2}\right)$
(57)

We see that in our case $P=s$, $Q=(D_{i}+D_{j})$ and $\nu=d+(\mu-1)\sigma$.
This immediately gives the result:

$\displaystyle I_{\mu}=$
$\displaystyle\frac{K_{d}}{2}\left(\frac{1}{(D_{i}+D_{j})}\right)^{\frac{d+(\mu-1)\sigma}{2}}s^{\frac{d+(\mu-1)\sigma}{2}-1}\times$
$\displaystyle\times\Gamma\left(\frac{d+(\mu-1)\sigma}{2}\right)\Gamma\left(1-\frac{d+(\mu-1)\sigma}{2}\right)\,,$
(58)

where $K_{d}=2^{d-1}\pi^{-d/2}\Gamma^{-1}(d/2)$ is the surface area of
$d$-dimensional unit sphere.

It is convenient to define

$a_{d}=\frac{K_{d}}{2}\left(\frac{2}{(D_{i}+D_{j})}\right)^{d/2}(2s)^{-\varepsilon/2}$
(59)
$b_{d}=\frac{K_{d}}{2}\left(\frac{2}{(D_{i}+D_{j})}\right)^{(d+\sigma)/2}(2s)^{-\delta/2}$
(60)
$c_{d}=\frac{K_{d}}{2}\left(\frac{2}{(D_{i}+D_{j})}\right)^{(d+2\sigma)/2}(2s)^{-(2\delta-\varepsilon)/2}$
(61)

So integral $I_{\mu}$ in the limit of $\delta=O(\varepsilon)$ can be written
as:

$I_{1}=\frac{a_{d}}{\varepsilon},\quad I_{2}=\frac{b_{d}}{\delta},\quad
I_{3}=\frac{c_{d}}{2\delta-\varepsilon}.$ (62)

We used an expansion $\Gamma(\varepsilon/2)\sim 2/\varepsilon$ for small
$\varepsilon$. An important property of coefficients (59) - (61) is that

$c_{d}a_{d}=b^{2}_{d},$ (63)

which can be verified by direct substitution.

Now we compute mean field integrals:

$I_{\mu}=\int d^{d}qdtq^{d+\sigma}\exp(-t(D_{i}+D_{j})q^{2})\sim
t^{-(d+\sigma-2)/2},$ (64)

where we assumed that the large momentum cut-off is imposed and corresponding
coupling constants have been renormalized. The non-universal coefficient is
not important.

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# RADA: Robust Adversarial Data Augmentation for Camera Localization in
Challenging Weather

Jialu Wang1, Muhamad Risqi U. Saputra2, Chris Xiaoxuan Lu3, Niki Trigoni1, and
Andrew Markham1 1Jialu Wang, Niki Trigoni, and Andrew Markham are with the
Department of Computer Science, University of Oxford, UK {jialu.wang,
niki.trigoni<EMAIL_ADDRESS>Risqi U. Saputra is with the
Data Science Department at Monash University Indonesia
<EMAIL_ADDRESS>Xiaoxuan Lu is with the School of Informatics
at the University of Edinburgh, UK<EMAIL_ADDRESS>
###### Abstract

Camera localization is a fundamental and crucial problem for many robotic
applications. In recent years, using deep-learning for camera-based
localization has become a popular research direction. However, they lack
robustness to large domain shifts, which can be caused by seasonal or
illumination changes between training and testing data sets. Data augmentation
is an attractive approach to tackle this problem, as it does not require
additional data to be provided. However, existing augmentation methods blindly
perturb all pixels and therefore cannot achieve satisfactory performance. To
overcome this issue, we proposed RADA, a system whose aim is to concentrate on
perturbing the geometrically informative parts of the image. As a result, it
learns to generate minimal image perturbations that are still capable of
perplexing the network. We show that when these examples are utilized as
augmentation, it greatly improves robustness. We show that our method
outperforms previous augmentation techniques and achieves up to two times
higher accuracy than the SOTA localization models (e.g., AtLoc and MapNet)
when tested on ‘unseen’ challenging weather conditions.

Index terms–Data augmentation, camera Localization, deep learning to
automation, adversarial training, domain shift.

##  I Introduction

Figure 1:  Upper: Traditional training method for deep camera localization
models. Lower: Our proposed training method with RADA.

Camera localization refers to the problem of recovering the 3D position and
orientation of a camera given an image or a set of images as the input. This
problem is fundamental and crucial for many robotic applications such as
autonomous driving, domestic robot navigation, augmented reality, and so on.
Camera localization system was initially developed based on hand-crafted
features. However, the extracted features were vulnerable to cross-weather,
such as lighting and saturation [29]. Recently, deep-learning based models
were proposed to alleviate those problems by automatically extracting
informative features from images [1, 2, 3]. However, they remain less robust
towards real-world scene variations, especially when the model is tested on
unseen scenarios with a domain shift, such as a change in ambient weather,
impacting appearance and illumination [28].

One possible solution to improve the robustness of existing deep learning-
based localization models is to use basic data augmentation to diversify the
training data distribution. For example, [1, 3] shifted the RGB pixel values,
[17] used Gaussian noise, and [27] employed blur, HSV shift, and other
perturbations to simulate cross weather interference. However, these domain
adaptation techniques are data agnostic. They perturb all image pixels in a
uniform fashion, corrupting important information for camera localization.

In recent studies [18, 19, 20], adversarial training is considered as one of
the most promising ways to improve robustness especially for classification
problems [6, 21], [24], [25, 26]. Nevertheless, it has not been used in camera
localization under domain shift. [28] used standard data augmentation methods
(e.g. Gaussian blur) to improve the robustness of steering angle predictions
but then they utilize ‘adversarial training’ to fine-tune the model’s hyper-
parameters (e.g., variance of Gaussian). Recently, [27] evaluated the
performance of various adversarial attacks on autonomous driving data,
demonstrating the brittleness of many approaches. Inspired by this paper, we
take a step further, and explore if these attacks can be used in camera
localization so as to achieve greater robustness for cross-weather adaptation.
Nevertheless, our experiments in Section IV-D indicates that directly adopting
the existing adversarial training methods blindly perturb all pixels and yield
unsatisfactory performance.

In this paper, we propose a novel adversarial data augmentation system for
robust camera localization, dubbed as RADA. Instead of corrupting the entire
image uniformly, RADA concentrates on perturbing only the geometrically
informative parts of the image, especially those extensively used by camera
localization such as trees, building landmarks, etc. In that sense, RADA
generates perturbation that can greatly reduce the performance of localization
models while at the same time minimally altering the original image. We
demonstrate how this seemingly paradoxical minimal perturbation actually
yields more robust and noise-resistant performance in reality when faced with
cross-domain interference and unseen weather conditions. In summary, our key
contributions are given as follows:

  * •

We propose RADA, a Robust Adversarial Data Augmentation system for camera
localization. RADA focuses on perturbing the geometrically informative part of
images such that it can significantly improve the robustness of the
localization models. To the best of our knowledge, this is the first work on
adversarial learning for camera localization.

  * •

We demonstrate the effectiveness of RADA by applying it on two state-of-the-
art localization models, namely AtLoc [3] and MapNet [2], such that their
accuracy can be improved up to more than two fold, especially when they were
tested on unseen weathers.

  * •

We perform extensive experiments on the public Oxford RobotCar dataset,
including analysing the trade-off between the quality of the training data and
the incremental gain of RADA.

##  II Related Work

###  II-A Data Augmentation Approaches

Data augmentation is a technique to increase the scale and the complexity of
training dataset in order to improve the model robustness [9]. This is
achieved by introducing artificially generated data samples without changing
their corresponding labels, and without requiring data from the target domain
to be provided. [12] divides the existing computational visual data
augmentation techniques into two categories: basic image manipulation and
deep-learning approaches.

#### Basic image manipulation

These are a group of approaches that either expand the data set in a
relatively random and blind manners, such as by employing color space or
geometric transformations, or give a strong assumption on the noise
distribution such as by using Gaussian noise. S. Brahmbhatt et al. [1]
simulated the interference caused by the change of time and weather in their
MapNet paper by adjusting the brightness of images (via ColorJitter toolbox).
Compared to learning-based augmentation approaches, improvement in system
robustness is limited.

#### Deep-learning approaches

Generative Adversarial Networks (GAN) and adversarial training are two
different deep-learning based data-augmentation approaches. GAN approaches
[13] utilize a generator network to attempt to fool the classifier in the
training phase. The original dataset is then artificially augmented by the
generator via the learned data distribution. [6] showed the two methods are
closely related and can even be combined to achieve complementary advantages.

###  II-B Adversarial Training

Adversarial training is a deep-learning based data augmentation technique that
directly augments the data by maximizing the loss function of the classifier
[4, 5]. It was originally proposed by [14] based on the principle of gradient
ascent. They multiply the sign of the gradient by a very small step
$\varepsilon$ to obtain the perturbation, and named this attack FGSM. [23]
proposed FGM which replaces the sign of the gradient with the unit vector of
gradient direction to avoid the fact that different pixels are perturbed by
the same step size. PGD [6] optimized the perturbation mechanism with an extra
constraint through multiple iterations. Subsequently, EAT [22] and ALP [21]
improved training efficiency by increasing sample diversity and using ‘pairing
loss’ respectively.

Figure 2: The comparison of perturbation results between Gaussian, FGSM [23],
and RADA (ours). $x$, $\Delta x$, and $x^{\prime}$ are the original image, the
generated perturbations, and the perturbed image respectively. The histogram
of perturbations are obtained by equally dividing $\Delta x^{\prime}$ into 9
sub-squares and computing the frequency (note that pixels without interference
are omitted. The Gaussian perturbed outputs were clipped within [0,1]. ). Both
Gaussian and FGSM histograms indicate that their perturbations are distributed
uniformly in all regions. In contrast, the perturbations generated by RADA are
concentrated in the regions which contain discriminative geometrical
structures that are important for localization (e.g., trees, buildings). By
training the localization model on $x^{\prime}$, the network does not overfit
to weather-specific structure and thus the robustness to cross weather
variations will be significantly improved.

##  III Methods

###  III-A Background

The mathematical expressions for generating the (augmented) adversarial
samples is illustrated as follows. Let $\theta$ be the model $g_{\theta}$’s
weights, $L$ is the loss function between the labels $y$ and the predictions
$g_{\theta}(x)$, $\widetilde{x}$ is the generated adversarial sample, and the
small constant $\varepsilon$ describes the step size of the perturbation. The
problem of generating adversarial samples boils down to finding a set of
perturbed input images $\widetilde{x}$, which makes the loss function $L$ as
large as possible as follows:

$\widetilde{x}=\mathop{argmax}\;L(g_{\theta}(x),y)$ (1)

Eq. (1) can be simplified by solving the following first order Taylor
expansion:

$\widetilde{x}=\mathop{argmax}\limits_{\widetilde{x}:||\widetilde{x}-x||_{norm}\leq\varepsilon}{(\bigtriangledown_{x}L(g_{\theta}(x),y))^{T}(\widetilde{x}-x)}$
(2)

When $norm=\infty$, the perturbation $r_{adv}$ and the adversarial sample
$\widetilde{x}$ then can be obtained by:

$\widetilde{x}=x+r_{adv}=x+\varepsilon\;\mathop{sign}(\bigtriangledown_{x}L(g_{\theta}(x),y))$
(3)

for  _each epoch_ do

1\. Use the range of $x$ to calculate the perturbation threshold $\eta_{th}$
(Eq. (5)) ;

2\. Freeze the model parameter $\theta$, obtain $L_{\theta}(p,p^{*})$ (Eq. (6)
and (7)), and calculate the gradient $g$ (Eq. (8));

3\. Calculate $r_{adv}$ by using $\mathop{g}$ (Eq. (9)). If any perturbation
value exceeds the threshold $\eta_{th}$, assign $\eta_{th}$ to that
perturbation;

4\. Calculate adversarial sample $x_{adv}$ using $r_{adv}$ (Eq. (10));

5\. If any pixel value in $x_{adv}$ exceeds the range $[0,255]$, assign the
nearest boundary (0 or 255) to that pixel;

6\. Use $x_{adv}$ to train the model and update the weights $\theta$;

end for

Algorithm 1 RADA Algorithm

###  III-B Proposed Perturbation Mechanism

The perturbation $r_{adv}$ in Eq. (3) can be redefined into Eq. (4) where
$pow=0$ as follows:

$r_{adv}=\varepsilon\times sign(g)\times g^{pow},\qquad pow=0$ (4)

As described, existing methods either take $pow=0$ (e.g., FGSM) or $pow=1$
(e.g., FGM, PGD, etc.). However, our experiments show that directly using this
traditional adversarial training approach cannot satisfactorily improve the
robustness (see Sec. IV-D). This is because all pixels will receive the
maximum perturbation along the gradient direction which corrupts all pixel
values, whether they are key for localization or not, generating a scattering
of confounding dots (see Fig. 2). On the contrary, our aim is to concentrate
on perturbing only the geometrically informative parts of the image. As a
result, the method should be able to learn to generate minimal perturbation
whilst greatly reducing the localization performance. Then, when the models
are retrained using these perturbed images in addition to the original
training set, the robustness will improve significantly. To this end, the
following changes have been made to the traditional methods:

(1) Adversarial perturbation. In our approach, we take $pow\textgreater 1$ and
a relatively larger $\varepsilon$ in Eq. (4) which enables RADA to not only
take the direction of the gradient, but to also learn how sensitive the
network is to a particular pixel, and then focuses the perturbation efforts
towards that pixel. As a result, it can concentrate on perturbing small
geometrically informative regions of an image as seen in Fig. 2.

(2) Threshold: perturbation constraint mechanism. Before each batch of
training, we calculate a perturbation threshold from the pixel range so as to
limit the size of the perturbation. This design enables the perturbation to be
adaptive to the input dataset. As shown in Eq. (5) below, $\eta_{th}$ is the
threshold used to limit the perturbation and $x_{max}$ and $x_{min}$ are the
maximum and minimum pixel values of each batch. The constant $\eta$ is the
number of thresholds, which is positively correlated with the accuracy.

$\eta_{th}=(x_{max}-x_{min}){\div}\eta$ (5)

(3) Clipping: Adversarial sample constraint mechanism. To avoid invalid
values, we further clip the value of the perturbed pixels into the range of
[0,255]. Fig.4 (d) shows that this restriction can effectively reduce the
confounding pixels.

(4) Untargeted perturbation. Unlike the targeted perturbation approaches [12]
(e.g., by turning a sunny image into a rainy image such that the method is
robust to the rainy image) which typically fail to adapt to ‘unseen’
conditions (e.g., snowy conditions), our approach uses untargeted perturbation
[27]. Therefore, it can adapt to a wide variety of unseen weather conditions
without specifying the target domain.

###  III-C Proposed Adversarial Training for Camera Localization

A deep camera localization model is trained to predict the 6-DoF camera poses
${p=(t,w)}$ by inputting an RGB image $x$ and learning the model weights
$\theta$ through back-propagation algorithm. Given the estimated camera pose
$p$ and its corresponding ground truth $p^{*}$, the loss function
$L_{\theta}(p,p^{*})$ is defined as follows:

$L_{\theta}(p,p^{*})=\sum_{i=1}^{|D|}h(p_{i},p_{i}^{*})+\alpha\sum_{i,j=1,i\not=j}^{|D|}h(v_{i,j},v_{i,j}^{*})$
(6)
$h(p,p^{*})=||t-t^{*}||_{1}e^{-\beta}+\beta+||w-w^{*}||_{1}e^{-\gamma}+\gamma$
(7)

where $i$,$j$ refers to the indexes of two adjacent images,
$v_{i,j}=(t_{i}-t_{j},w_{i}-w_{j})$ is the relative pose between $p_{i}$ and
$p_{j}$, and $h\\{.\\}$ is the difference between $p$ and $p^{*}$. $\beta$ and
$\gamma$ are trainable parameters used to balance the loss between translation
$t$ and rotation $w$. Let $L_{\theta}^{r}(p,p^{*})$ be the loss function in
Eq. (6) after freezing the weights $\theta$, then the gradient $g$ is defined
as the derivative w.r.t. $x$ as follows:

$g=\bigtriangledown_{x}L_{\theta}^{r}(p,p^{*})$ (8)

Based on the gradient ascent principle [14], the perturbation $r_{adv}$ is
equal to the gradient direction multiplied by a constant step $\varepsilon$.
By plugging in Eq. (8) into Eq. (4), then we obtain our perturbation for
camera localization as follows:

$r_{adv}=\varepsilon\times
sign(\bigtriangledown_{x}L_{\theta}^{r}(p,p^{*}))\times(\bigtriangledown_{x}L_{\theta}^{r}(p,p^{*}))^{pow},pow>1$
(9)

The generated adversarial sample $x_{adv}$ can be defined as:

$x_{adv}=x+r_{adv}$ (10)

Finally, the overall workflow of generating RADA adversarial samples is shown
in Algorithm 1. The RADA system pipeline is illustrated in Fig. 1.

TABLE I: RobotCar Dataset Selection Sequence | Time | Tag | Model
---|---|---|---
loop | 2014-06-26-09-24-58 | overcast | Training
loop | 2014-06-26-08-53-56 | overcast | Training
loop | 2014-06-23-15-36-04 | sunny | Testing
loop | 2014-06-24-14-09-07 | over-exposure | Testing
fullA | 2014-11-28-12-07-13 | overcast | Training
fullA | 2014-12-02-15-30-08 | overcast | Training
fullA | 2014-11-25-09-18-32 | rain | Testing
fullB | 2015-02-13-09-16-26 | overcast | Training
fullB | 2015-02-03-08-45-10 | snow | Testing

##  IV EXPERIMENTS

###  IV-A Localization model and Evaluation metrics

In order to demonstrate the efficacy of our adversarial data augmentation, we
use RADA to enhance two state-of-the-art camera localization models, namely
MapNet [2] and AtLoc [3].

#### MapNet

MapNet is a DNN for localization. MapNet is a recent deep learning based
camera localization model that can automatically extract features and directly
recover the absolute camera pose from input images. It achieves a relative
higher robustness to illumination changes with an additional geometric
constraint between pairs of frames.

#### Atloc

AtLoc [3] is a SOTA camera localization model, which shows high robustness to
dynamic objects and changing illumination by introducing a self-attention
mechanism.

For a fair comparison, in this paper, we used the same evaluation strategy for
both MapNet and AtLoc, i.e. mean translation error (m) and mean rotation error
(degree).

###  IV-B Dataset

Oxford RobotCar dataset [16] was collected from a 10km route in an autonomous
Nissan LEAF car through central Oxford over a year. The collected data
contains most of weather conditions, including rainy, snowy, over-exposure,
and nighttime, which makes it particularly challenging for camera localization
problem. Details about selected sequences can be found in Table. I. For each
sequence, we adopted input images recorded by the stereo centre camera
sequence 01 with a resolution of 1280 x 960. The interpolations of INS data
were labelled as the corresponding ground truth.

TABLE II: Comparing RADA with SOTA methods using MapNet and AtLoc Testing Sequence | AtLoc [3] | AtLoc with Gaussian | AtLoc with FGSM | AtLoc with RADA (ours)
---|---|---|---|---
sunny | 8.86m, $4.67^{\circ}$ | 10.93m, $5.97^{\circ}$ | 8.11m, $3.60^{\circ}$ |  6.93m, $\mathbf{3.40^{\circ}}$
over-exposure | 22.17m, $17.72^{\circ}$ |  8.80m, $9.38^{\circ}$ | 9.54m, $8.76^{\circ}$ | 9.23m, $\textbf{8.84}^{\circ}$
rain | 8.99m, $2.15^{\circ}$ | 12.40m, $2.96^{\circ}$ | 12.90m, $2.74^{\circ}$ | 6.87m, $\textbf{1.90}^{\circ}$
snow | 37.99m, $8.18^{\circ}$ | 29.21m, $10.82^{\circ}$ | 24.56m, $11.12^{\circ}$ |  13.07m, $\textbf{8.15}^{\circ}$
average | 19.50m, $8.18^{\circ}$ | 15.34m, $7.28^{\circ}$ | 13.78m, $6.55^{\circ}$ |  9.02m, $\textbf{5.57}^{\circ}$
Testing Sequence | MapNet [2] | MapNet with Gaussian | MapNet with FGSM | MapNet with RADA (ours)
sunny |  9.84m, $\textbf{3.96}^{\circ}$ | 69.48m, $50.22^{\circ}$ | 16.89m, $11.65^{\circ}$ | 17.26m, $6.82^{\circ}$
over-exposure | 18.49m, $12.88^{\circ}$ | 49.80m, $39.33^{\circ}$ | 19.74m, 8.27∘ |  16.89m, $11.65^{\circ}$
rain | 15.00m, $3.72^{\circ}$ | 16.11m, $3.60^{\circ}$ | 13.39m, $3.60^{\circ}$ |  12.10m, 2.56∘
snow | 29.16m, $9.06^{\circ}$ | 36.22m, $12.04^{\circ}$ | 23.18m, $8.63^{\circ}$ |  12.91m, $\textbf{2.65}^{\circ}$
average | 18.12m, $7.41^{\circ}$ | 42.90m, $26.30^{\circ}$ | 18.3m, $8.04^{\circ}$ |  14.79m, $\textbf{5.92}^{\circ}$
Figure 3: Trajectories on rainy day (upper) and snowy day (lower) test set.The
ground truth (black line), predicted trajectories (red line) and starting
point (start) are visualized above.

###  IV-C Implementation Details

For all RADA adversarial perturbations, we set the power of the gradient
$pow=1.5$, the step size of the perturbation $\varepsilon$ = 158 (Eq. (9)),
and the number of threshold $\eta$ = 10 (Eq. (5)). For all the FGSM
perturbations, we set $\varepsilon$ = 0.3, which is a typical value for this
method. For the Gaussian perturbation, we set a commonly used distribution
whose mean and variance are 0 and 0.05 respectively. All models were trained
until convergence. For a fair evaluation, all challenging test weathers are
hidden/excluded from the training data.

###  IV-D Overall Performance

For all evaluations, we trained both AtLoc and MapNet on good weather dataset
(overcast) by using different data augmentation methods, and then tested them
in good weather (sunny) as well as ’unseen’ challenging weathers (over-
exposure, rainy, and snowy days). For non-augmented system, we trained basic
AtLoc and MapNet by utilizing the original, unmodified dataset. For naive
augmentation system, we add Gaussian noise to the training dataset. For
conventional deep learning-based augmentation system, we employ FGSM [14] to
perturb the dataset. Apart from the basic AtLoc and MapNet, all models were
trained by using both the original and the perturbed data. Fig. 3 visualizes
some of the predicted trajectory for AtLoc.

Table II shows the numerical evaluation results in terms of mean translation
error (m) and mean rotation error (degree). All competing systems work
properly when they were tested on the good weather data (sunny), but they
suffer from accuracy degradation on other challenging conditions. Unmodified
AtLoc and MapNet degrade significantly when they were tested on rainy, snowy
and over-exposure data. Sometimes, Gaussian and FGSM can improve the network’s
robustness, but most of the time they confuse the network since the generated
perturbation is either random or too noisy such that it corrupts critical
pixel information used for camera localization. In contrast, AtLoc and MapNet
modified with RADA outperform the other techniques on most of good and
challenging weather conditions. This owes to its ability to focus the
perturbation on geometrically important features for localization such that 1)
the perturbed image resembles the original image, keeping important
information for localization, and 2) the perturbation forces the network to
not overfit to any weather-specific structures, increasing the robustness to
cross weather variations as described in Fig. 2. On average, RADA yields
$36.7\%$ and $26.5\%$ smaller translation and rotation error respectively for
both AtLoc and MapNet.

TABLE III: Analysis on the Incremental Gain of RADA % rainy data in training set | Test on rainy data | Test on sunny data
---|---|---
base AtLoc | RADA | base AtLoc | RADA
0% | 38.04m, $46.6^{\circ}$ | 34.57m, $55.17^{\circ}$ | 3.93m, $7.14^{\circ}$ | 5.88m, $8.12^{\circ}$
20% | 28.28m, $24.44^{\circ}$ | 16.29m, $5.29^{\circ}$ | 17.67m, $10.61^{\circ}$ | 10.74m, $4.61^{\circ}$
40% | 24.1m, $29.87^{\circ}$ | 10.97m, $7.55^{\circ}$ | 21.07m,,$40.36^{\circ}$ | 13.78m, $31.47^{\circ}$

###  IV-E Perturbation Histogram

To compare the perturbation results from different methods, the perturbation
histograms are plotted as seen in Fig. 2. As shown in Fig. 2, the perturbation
of Gaussian is normally distributed on the whole image, although it is heavily
clipped in the brighter areas of the image. FGSM perturbation follows binary
distribution in all regions as it perturbs every pixel with the same step size
regardless the gradient scale. In general, both FGSM and Gaussian
perturbations are evenly distributed on the whole image, generating a lot of
noises and confounding pixels. On the contrary, RADA acts in a very different
manner. As shown in Fig. 2, RADA produces very wide and large perturbations
over trees and buildings, but relatively less perturbations over sky and road.
This is because RADA does not only take the direction of the gradient, but
also learns how sensitive the network is to a particular pixel and then it
focuses the perturbation efforts towards that pixel. In particular, RADA
targets its perturbation on small geometrically informative regions rather
than polluting the whole image. In that sense, RADA learns that road itself or
the sky, which exists in most of the scenes, are not spatially discriminative
features for camera localization. However other features like trees and
buildings are important to estimate camera location and thus it concentrates
its perturbation on these parts. As a result, RADA learns to generate
efficient perturbations to the images, where these minimal perturbations are
sufficient to perplex the network and can be used to improve the robustness to
cross-weather variations.

Figure 4: The perturbed images generated by RADA after turning off some
building blocks. (a) Original image, (b) Complete version of RADA, (c) RADA
without clipping mechanism, (d) RADA without threshold, and (e) RADA without
threshold and clipping. It can be clearly seen that our proposed threshold and
clipping mechanism can effectively avoid polluting crucial localization
information or generating lots of confounding dots. TABLE IV: Ablation study
of RADA on RobotCar (Basic AtLoc denotes the original AtLoc Model without
RADA)

| Over-exposure | Rainy
---|---|---
Basic AtLoc | 22.17, $17.72^{\circ}$ | 8.99, $2.15^{\circ}$
Complete RADA (ours) |  9.23m, $\textbf{8.84}^{\circ}$ |  6.87m, $\textbf{1.90}^{\circ}$
No clipping RADA (ours) | 12.39, $10.98^{\circ}$ | 10.96, $2.69^{\circ}$
No threshold RADA (ours) | 11.99, $10.79^{\circ}$ | 8.82, $2.05^{\circ}$
no threshold, no clipping RADA (ours) | 12.66, $12.75^{\circ}$ | 17.28, $2.95^{\circ}$

###  IV-F Ablation Study

We also conducted ablation study under different weather conditions using
AtLoc model. In Table IV, basic AtLoc is compared with a version trained with
complete RADA, a clipping version of RADA trained without thresholding, a
thresholding version trained without clipping, and a no-thresholding-no-
clipping version of RADA. All models were trained in good condition and then
tested in different ‘unseen’ weather conditions. The rest settings are kept
the same for fair comparison. The complete RADA achieved the best performance
among all. This comparison indicates that all incomplete RADA versions are
less accurate than the complete RADA, showing that the thresholding and
clipping mechanism can produce effective constraints (see Fig. 4).

###  IV-G Analysis on the Incremental Gain of RADA

We further investigate what the relative gains from RADA are when supplied
with a small proportion of data from the target domain e.g. a ratio of
80%overcast to 20% rainy. We tried different mixtures of dataset by
incorporating 0 to 40% of rainy day in the overcast training set. We then
tested the model on two sets, one on the good (overcast) weather and another
on the challenging rainy data. For each mixture of training data, we compute
the mean error of basic AtLoc and the one trained with RADA. As shown in Table
III, if we increase the percentage of rainy data from 0 to $40\%$ in the
training set, RADA yields $68.3\%$ and $86.4\%$ smaller translation and
rotation error respectively, especially when it is tested on rainy data. In
summary, by adding a small portion of data from the target domain, it can
significantly improve the accuracy of RADA.

##  V CONCLUSIONS

We propose RADA, an adversarial data augmentation system for camera
localization that can concentrate on perturbing the geometrically informative
parts of the image. RADA is a simple plugin that can be used on virtually any
deep-learning based camera localization network to generate adversarial
training examples. RADA learns to generate minimal perturbations that can
perplex the localization network and largely improve robustness. By using
RADA, we demonstrated the possibility of using adversarial training to solve
domain shift problem in camera localization. Through further experiments, we
have also explored the trade-off between the quality of training dataset and
the incremental benefit of RADA. For future work, an auto-encoder could be
used to generate advanced adversarial perturbation in which the feature
distribution is expected to be closer to the one generated from the real
domain-shift problems. This might improve further the performance of RADA in a
more challenging situation such as nighttime.

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