# Study of invariance of nonextensive statistics under the uniform energy
spectrum translation

A.S. Parvan

###### Abstract

The general formalisms of the $q$-dual statistics, the Boltzmann-Gibbs
statistics, and three versions of the Tsallis statistics known as Tsallis-1,
Tsallis-2, and Tsallis-3 statistics have been considered in the canonical
ensemble. We have rigorously proved that the probability distribution of the
Tsallis-1 statistics is invariant under the uniform energy spectrum
translation at a fixed temperature. This invariance demonstrates that the
formalism of the Tsallis-1 statistics is consistent with the fundamentals of
the equilibrium statistical mechanics. The same results we have obtained for
the probability distributions of the Tsallis-3 statistics, Boltzmann-Gibbs
statistics, and $q$-dual statistics. However, we have found that the
probability distribution of the Tsallis-2 statistics, the expectation values
of which are not consistent with the normalization condition of probabilities,
is indeed not invariant under the overall shift in energy as expected.

###### keywords:

Nonextensive statistics , Tsallis statistics , $q$-dual statistics

organization=Bogoliubov Laboratory of Theoretical Physics, Joint Institute for
Nuclear Research,city=Dubna, postcode=141980, country=Russia
organization=Department of Theoretical Physics, Horia Hulubei National
Institute of Physics and Nuclear Engineering,city=Bucharest-Magurele,
country=Romania

##  1 Introduction

In modern physics, the microscopic foundation of the statistical mechanics’
paradigm based on the nonextensive entropy measure [1] is a primordial task.
It is well-known that one of the fundamental aspects of the theory of the
usual Boltzmann-Gibbs statistical mechanics is the invariance of the
probability distribution under the uniform translation of energy spectrum.
This invariance is a consequence of the first law of equilibrium
thermodynamics, which in the case of a cyclic process, may be written in the
form of the integral of energy change [2]:

$\oint dU=0.$ (1)

In thermodynamics, the internal energy $U$ is a state function and the
infinitesimal change $dU$ of the internal energy associated with any
infinitesimal transformation of state is a total differential [3]. The total
change in energy $U$ is independent of the transformation path. It depends
only on the initial and final states. Thus, in thermodynamics, the internal
energy $U$ can be only defined up to an arbitrary additive constant $U_{0}$
[2]:

$U=U(T,V,N)+U_{0}.$ (2)

Then, the invariance of the probability distribution under the uniform energy
spectrum translation may be considered as a statement of the first law of
thermodynamics (1) and additivity (2).

Nowadays, there are at least three versions of the Tsallis statistics [1, 4]
based on the same generalized entropy [1, 5, 6, 7] (see more concrete
explanations in ref. [8]) that differ from each other only in the definition
of the expectation values of operators. The first variant of the Tsallis
statistics [1, 4, 9], also called the Tsallis-1 statistics, is defined by the
standard expectation values of operators as in the Boltzmann-Gibbs statistics.
Such expectation values are consistent with the normalization condition of
probabilities in total agreement with the statistical mechanics and theory of
probability requirements. The second variant of the Tsallis statistics (the
Tsallis-2 statistics) [1, 4, 10] is based on the generalized expectation
values of operators, which are not consistent with the normalization condition
of probabilities. Such unconventional expectation values lead to an
inconsistent connection between the statistical mechanics, probability theory,
and theory of equilibrium thermodynamics. In the third variant of the Tsallis
statistics [4], also called the Tsallis-3 statistics, the normalized
generalized expectation values of operators are used. However, in contrast
with the Tsallis-2 statistics, the normalized generalized expectation values
of the Tsallis-3 statistics are consistent with the normalization condition of
probabilities of microstates.

The present paper is motivated by the uncertainties dominant in high-energy
physics concerning the correct variant of the Tsallis statistics. Generally,
in high-energy physics, the phenomenological Tsallis distribution [11, 12, 13,
14] and different Tsallis-like distributions [15, 16, 17] of the power-law
form inspired by the Tsallis statistics [1] are used to describe the
experimental data for the transverse momentum distribution of particles
produced in proton-proton and heavy-ion collisions at the LHC and RHIC
energies [18, 19, 20, 21, 22, 23, 24, 25, 26]. These distributions are the
single-particle distribution functions, which have difficulties of the
fundamental character in the Tsallis statistics (see the detailed proof in
refs. [27, 28, 29]). Nevertheless, such difficulties for the phenomenological
Tsallis distribution are eliminable in the framework of the $q$-dual
statistics introduced in ref. [30]. In high-energy physics, the consistent
single-particle distribution functions were found only for the Tsallis-1
statistics [27, 28]. However, in the high-energy physics community, only the
Tsallis-3 statistics is most frequently considered correct since it is
hypothesized that only the probability distribution of the Tsallis-3
statistics is invariant under the uniform energy spectrum translation (see,
for example, the reference [31]). In this paper, we are going to show that the
Tsallis-1 statistics is also invariant under the uniform energy spectrum
translation.

The conviction that only the Tsallis-3 statistics is invariant under the
uniform energy spectrum translation has its origins in Tsallis’ works. In ref.
[8], it was stated that the probability distribution of the Tsallis-1
statistics is not invariant under the overall shift in energy at fixed
Lagrange multiplier $\beta$ associated with the energy, i.e., the
thermodynamical quantities as functions of $\beta$ depend on the choice of the
origin of energies. In refs. [4, 8], the same statement was given for the
Tsallis-2 statistics. However, in the same refs. [4, 8], it was shown that the
probability distribution of the Tsallis-3 statistics is invariant under the
overall shift in energy at fixed Lagrange multiplier $\beta$. Based on these
conclusions, in refs. [8, 31], it is stated that of all three formulations of
the Tsallis statistics only the Tsallis-3 statistics is consistent with the
fundamentals of the equilibrium statistical mechanics. However, this statement
contradicts the results of the earlier work [32] in which it was demonstrated
that the probability distribution of the Tsallis-1 statistics is also
invariant under the uniform energy spectrum translation.

In the present paper, in comparison with ref. [32], we introduce a more
general and explicit method to demonstrate the invariance of the probability
distributions under the uniform energy spectrum translation. Using this new
method, we strongly prove that the probability distribution of the Tsallis-1
statistics is invariant under the overall shift in energy at a fixed
temperature. This important property is found in the framework of the complete
formalism of the Tsallis-1 statistics obtained in ref. [9] and in present
paper. We also prove the invariance of the probability distributions of the
Tsallis-3 statistics, the $q$-dual statistics, and Boltzmann-Gibbs statistics.
However, we demonstrate that the probability distribution of the Tsallis-2
statistics is not invariant under the energy shift, even in terms of
temperature.

The paper is organized as follows. In Secs. 2 – 6 we discuss the general
formalisms of the Boltzmann-Gibbs statistics, Tsallis-1 statistics, Tsallis-2
statistics, Tsallis-3 statistics, and $q$-dual statistics, respectively, and
study their invariance under the energy shift. The discussions and summary are
given in Sec. 7.

##  2 Boltzmann-Gibbs statistics

The Boltzmann-Gibbs statistics is defined by the Boltzmann-Gibbs entropy with
the probabilities $p_{i}$ of the microstates of the system normalized to unity

$\displaystyle S=$ $\displaystyle-\sum_{i}p_{i}\ln p_{i},$ (3) $\displaystyle
1=$ $\displaystyle\sum\limits_{i}p_{i}$ (4)

and by the standard expectation values of an operator $A$

$\langle A\rangle=\sum\limits_{i}p_{i}A_{i},$ (5)

where $A_{i}=\langle i|A|i\rangle$ is the value of operator $A$ in the $i$-th
microscopic state of the system. Here and throughout the paper we use the
system of natural units $\hbar=c=k_{B}=1$.

Let us consider the canonical ensemble. The thermodynamic potential of the
canonical ensemble, the free energy $F$, is the Legendre transform of the
fundamental thermodynamic potential, the energy $E$:

$\displaystyle F=$ $\displaystyle E-TS=\sum\limits_{i}p_{i}(E_{i}+T\ln
p_{i}),$ (6) $\displaystyle E=$ $\displaystyle\sum_{i}p_{i}E_{i},$ (7)

where $E$ is the mean energy and $E_{i}$ is the energy of the $i$-th
microscopic state of the system.

The unknown probabilities $\\{p_{i}\\}$ are found from the first and second
laws of thermodynamics (the principle of maximum entropy). This means that in
the canonical ensemble the set of equilibrium probabilities $\\{p_{i}\\}$ can
be derived from the constrained local extremum of the thermodynamic potential
(6) by the method of the Lagrange multipliers (see, for example, Refs. [33,
34, 35]):

$\displaystyle\Phi=$ $\displaystyle F-\lambda\varphi,$ (8)
$\displaystyle\varphi=$ $\displaystyle\sum\limits_{i}p_{i}-1=0,$ (9)
$\displaystyle\frac{\partial\Phi}{\partial p_{i}}=$ $\displaystyle 0,$ (10)

where $\lambda$ is an arbitrary real constant.

Substituting Eq. (6) into Eqs. (8)–(10), we obtain

$p_{i}=e^{\frac{\Lambda-E_{i}}{T}}.$ (11)

where $\Lambda\equiv\lambda-T$ and $\partial E_{i}/\partial p_{i}=0$.
Substituting Eq. (11) into Eq. (6) and using Eq. (4), we obtain

$F=\Lambda=\lambda-T$ (12)

and [36]

$\displaystyle p_{i}=$ $\displaystyle
e^{\frac{F-E_{i}}{T}}=e^{-S+\frac{E-E_{i}}{T}}\equiv\frac{1}{Z}e^{-\frac{E_{i}}{T}},$
(13) $\displaystyle Z=$ $\displaystyle\sum\limits_{i}e^{-\frac{E_{i}}{T}},$
(14) $\displaystyle F=$ $\displaystyle-T\ln Z,$ (15)

where $Z$ is the partition function.

Let us prove that the probability distribution (13) is invariant under an
overall shift in energy of microstates

$E^{\prime}_{i}=E_{i}+E_{0},$ (16)

where $E_{0}$ is the same constant for all microstates of the system. Let us
suppose that the probability (13) is invariant under the transformation (16),
i.e.

$p^{\prime}_{i}=p_{i}.$ (17)

Then the entropy (3), the energy (7) and the free energy (6) under the energy
shift (16) are transformed as

$\displaystyle S^{\prime}=$ $\displaystyle-\sum_{i}p^{\prime}_{i}\ln
p^{\prime}_{i}=-\sum_{i}p_{i}\ln p_{i}=S,$ (18) $\displaystyle E^{\prime}=$
$\displaystyle\sum_{i}p^{\prime}_{i}E^{\prime}_{i}=\sum_{i}p_{i}(E_{i}+E_{0})=E+E_{0},$
(19) $\displaystyle F^{\prime}=$
$\displaystyle\sum\limits_{i}p^{\prime}_{i}(E^{\prime}_{i}+T\ln
p^{\prime}_{i})=\sum\limits_{i}p_{i}(E_{i}+E_{0}+T\ln p_{i})=F+E_{0}.$ (20)

Using Eqs. (13), (16) and (18)–(20), we obtain

$p^{\prime}_{i}=e^{\frac{F^{\prime}-E^{\prime}_{i}}{T}}=e^{-S^{\prime}+\frac{E^{\prime}-E^{\prime}_{i}}{T}}=e^{\frac{F-E_{i}}{T}}=e^{-S+\frac{E-E_{i}}{T}}=p_{i}.$
(21)

Equation (21) is equivalent to Eq. (17). Thus we have proved that the
probability distribution $p_{i}$ of the Boltzmann-Gibbs statistics is
invariant under an overall shift in energy. Note that the invariance of the
distribution (13) can be obtained explicitly from the last equality of Eq.
(13) and Eq. (14) without introducing the assumption (17).

##  3 Tsallis-1 statistics

The Tsallis-$1$ statistics [1] is defined by the generalized entropy with the
probabilities normalized to unity [1, 4]

$\displaystyle S=$ $\displaystyle\sum\limits_{i}\frac{p_{i}^{q}-p_{i}}{1-q},$
(22) $\displaystyle 1=$ $\displaystyle\sum\limits_{i}p_{i}$ (23)

and by the standard expectation values

$\langle A\rangle=\sum\limits_{i}p_{i}A_{i},$ (24)

where $q\in\mathbb{R}$ is a real parameter taking values $0<q<\infty$. In the
Gibbs limit $q\to 1$, the entropy (22) recovers the Boltzmann-Gibbs entropy
(3) and the Tsallis-1 statistics is reduced to the usual Boltzmann-Gibbs
statistics (see Sect. 2).

Let us consider the canonical ensemble. The thermodynamic potential $F$ of the
canonical ensemble can be written as

$\displaystyle F=$ $\displaystyle
E-TS=\sum\limits_{i}p_{i}\left[E_{i}+T\frac{p_{i}^{q-1}-1}{q-1}\right],$ (25)
$\displaystyle E=$ $\displaystyle\sum_{i}p_{i}E_{i},$ (26)

where $E$ is the mean energy and $E_{i}$ is the energy of the $i$-th
microscopic state of the system.

Substituting Eq. (25) into Eqs. (8)–(10) and using Eq. (23), we obtain the
equilibrium probabilities of microstates of the system [9]

$p_{i}=\left[1+\frac{q-1}{q}\frac{\Lambda-E_{i}}{T}\right]^{\frac{1}{q-1}}$
(27)

and

$\sum\limits_{i}\left[1+\frac{q-1}{q}\frac{\Lambda-
E_{i}}{T}\right]^{\frac{1}{q-1}}=1,$ (28)

where $\Lambda\equiv\lambda-T$ and $\partial E_{i}/\partial p_{i}=0$. In the
Gibbs limit $q\to 1$, the probability distribution (27) recovers the
Boltzmann-Gibbs probability distribution (11). The expectation values of an
operator $A$ takes the form [9]

$\langle A\rangle=\sum\limits_{i}A_{i}\left[1+\frac{q-1}{q}\frac{\Lambda-
E_{i}}{T}\right]^{\frac{1}{q-1}},$ (29)

where the norm function $\Lambda$ is the solution of Eq. (28).

Substituting Eq. (27) into Eq. (25) and using Eqs. (23) and (26), we obtain
[9]

$F=E-TS=\frac{1}{q}[\Lambda+(q-1)E]$ (30)

and [9]

$\Lambda=E-qTS.$ (31)

Substituting Eq. (31) into Eq. (27), we get (cf. with Eq. (13))

$p_{i}=\left[1-(q-1)S+\frac{q-1}{q}\frac{E-E_{i}}{T}\right]^{\frac{1}{q-1}}.$
(32)

The formula (32) was also obtained in ref. [32] with $\sum_{i}p_{i}^{q}$
instead of $1-(q-1)S$. From the definition of entropy (22) we have
$\sum_{i}p_{i}^{q}=1-(q-1)S$. In the Gibbs limit $q\to 1$, the probability
distribution (32) of the Tsallis-1 statistics recovers the Boltzmann-Gibbs
probability distribution (13). Comparing Eqs. (32) and (13), we can see that
the relation between the energies of microstates, the mean energy and the
entropy of system in the probability distribution (32) of the Tsallis-1
statistics is consistent with the same relation in the probability
distribution (13) of the Boltzmann-Gibbs statistics. Thus the probability
distribution of the Tsallis-1 statistics has the correct structure from the
point of view of the fundamentals of the statistical mechanics. It should be
stressed that the probability distribution in terms of the Lagrange multiplier
$\beta$ obtained in refs. [1, 4, 8] on the base of the Jaynes principle is
precisely reduced to the probability distribution (27) (as well as (32)) after
excluding the intermediate Lagrange multiplier $\beta$, which is a method
dependent. See the proof in the A.

Let us demonstrate that the probability distribution of the Tsallis-1
statistics is invariant under the overall energy shift (16),
$E^{\prime}_{i}=E_{i}+E_{0}$. Assuming that

$p^{\prime}_{i}=p_{i},$ (33)

we obtain

$\displaystyle S^{\prime}=$
$\displaystyle\sum\limits_{i}\frac{{p^{\prime}}_{i}^{q}-{p^{\prime}}_{i}}{1-q}=\sum\limits_{i}\frac{p_{i}^{q}-p_{i}}{1-q}=S,$
(34) $\displaystyle E^{\prime}=$
$\displaystyle\sum\limits_{i}p^{\prime}_{i}E^{\prime}_{i}=\sum\limits_{i}p_{i}(E_{i}+E_{0})=E+E_{0},$
(35) $\displaystyle\Lambda^{\prime}=$ $\displaystyle
E^{\prime}-qTS^{\prime}=E+E_{0}-qTS=\Lambda+E_{0}$ (36)

and

$F^{\prime}=\sum\limits_{i}p^{\prime}_{i}\left[E^{\prime}_{i}+T\frac{{p^{\prime}}_{i}^{q-1}-1}{q-1}\right]=\sum\limits_{i}p_{i}\left[E_{i}+E_{0}+T\frac{p_{i}^{q-1}-1}{q-1}\right]=F+E_{0}.$
(37)

Using Eqs. (16), (27), (32) and (34)–(36), we have (cf. with Eq. (21))

$p^{\prime}_{i}=\left[1+\frac{q-1}{q}\frac{\Lambda^{\prime}-E^{\prime}_{i}}{T}\right]^{\frac{1}{q-1}}=\left[1+\frac{q-1}{q}\frac{\Lambda-
E_{i}}{T}\right]^{\frac{1}{q-1}}=p_{i}$ (38)

and

$p^{\prime}_{i}=\left[1-(q-1)S^{\prime}+\frac{q-1}{q}\frac{E^{\prime}-E^{\prime}_{i}}{T}\right]^{\frac{1}{q-1}}=\left[1-(q-1)S+\frac{q-1}{q}\frac{E-E_{i}}{T}\right]^{\frac{1}{q-1}}=p_{i}.$
(39)

Equations (38) and (39) exactly coincide with Eq. (33). Thus we have proved
that the probability distribution $p_{i}$ of the Tsallis-1 statistics is
indeed invariant under an overall shift in energy.

##  4 Tsallis-2 statistics

The Tsallis-2 statistics is defined by the generalized entropy (22) with
probabilities $\\{p_{i}\\}$ of the microstates of system normalized to unity
(23) and by the generalized expectation values [1, 4, 10]

$\langle A\rangle=\sum\limits_{i}p_{i}^{q}A_{i},$ (40)

where $q\in\mathbb{R}$ is a real parameter taking values $0<q<\infty$. This
choice of the expectation values is inconsistent because it has the feature
$\langle 1\rangle\neq 1$.

The thermodynamic potential of the canonical ensemble for the Tsallis-$2$
statistics can be written as

$\displaystyle F=$ $\displaystyle
E-TS=\sum\limits_{i}p_{i}^{q}\left[E_{i}+T\frac{p_{i}^{1-q}-1}{1-q}\right],$
(41) $\displaystyle E=$ $\displaystyle\sum_{i}p_{i}^{q}E_{i},$ (42)

where $E$ is the mean energy of the system.

Substituting Eq. (41) into Eqs. (8)–(10) and using Eq. (23), we obtain the
equilibrium probabilities of microstates of the system [4]

$\displaystyle p_{i}=$
$\displaystyle\frac{1}{Z}\left[1-(1-q)\frac{E_{i}}{T}\right]^{\frac{1}{1-q}},$
(43) $\displaystyle Z=$
$\displaystyle\sum\limits_{i}\left[1-(1-q)\frac{E_{i}}{T}\right]^{\frac{1}{1-q}},$
(44)

where

$Z\equiv\left[1-(1-q)\frac{\Lambda}{T}\right]^{\frac{1}{1-q}}$ (45)

and $q\Lambda\equiv\lambda-T$. Then the expectation values (40) for the
Tsallis-$2$ statistics in the canonical ensemble can be rewritten in the
general form as [4]

$\langle
A\rangle=\frac{1}{Z^{q}}\sum\limits_{i}A_{i}\left[1-(1-q)\frac{E_{i}}{T}\right]^{\frac{q}{1-q}},$
(46)

where the partition function $Z$ is calculated by Eq. (44).

Substituting Eq. (43) into Eq. (41) and using Eqs. (45) and (22), and
introducing the function $\chi\equiv\sum_{i}p_{i}^{q}=1+(1-q)S$, we obtain

$F=E-TS=\Lambda=-T\frac{Z^{1-q}-1}{1-q}.$ (47)

Substituting Eq. (47) into Eq. (43), we have (cf. with Eqs. (13) and (32))

$p_{i}=\left[\frac{1-(1-q)\frac{E_{i}}{T}}{1-(1-q)\frac{F}{T}}\right]^{\frac{1}{1-q}}=\left[\frac{1-(1-q)\frac{E_{i}}{T}}{1+(1-q)S-(1-q)\frac{E}{T}}\right]^{\frac{1}{1-q}}.$
(48)

Comparing Eqs. (48) and (13), we observe that the relation between the
energies of microstates, the mean energy and the entropy of system in the
probability distribution of the Tsallis-2 statistics is incompatible with the
same relation for the probability distribution (13) of the Boltzmann-Gibbs
statistics. Thus the structure of the probability distribution of the
Tsallis-2 statistics does not reproduce the structure of the probability
distribution of the Boltzmann-Gibbs statistics. This discrepancy is an
additional deficiency of the Tsallis-2 statistics.

Let us verify the invariance of the probability distribution of the Tsallis-2
statistics under the overall energy shift (16), $E^{\prime}_{i}=E_{i}+E_{0}$.
Assuming that

$p^{\prime}_{i}=p_{i},$ (49)

we obtain $S^{\prime}=S$ (see Eq. (34)) and

$\displaystyle E^{\prime}=$
$\displaystyle\sum\limits_{i}{p^{\prime}}_{i}^{q}E^{\prime}_{i}=\sum\limits_{i}p_{i}^{q}(E_{i}+E_{0})=E+E_{0}\chi,$
(50) $\displaystyle F^{\prime}=$
$\displaystyle\sum\limits_{i}{p^{\prime}}_{i}^{q}\left[E^{\prime}_{i}+T\frac{{p^{\prime}}_{i}^{1-q}-1}{1-q}\right]=\sum\limits_{i}p_{i}^{q}\left[E_{i}+E_{0}+T\frac{p_{i}^{1-q}-1}{1-q}\right]=F+E_{0}\chi.$
(51)

Using Eqs. (16), (48) and (34), (50), (51), we get (cf. with Eq. (21))

$p^{\prime}_{i}=\left[\frac{1-(1-q)\frac{E^{\prime}_{i}}{T}}{1-(1-q)\frac{F^{\prime}}{T}}\right]^{\frac{1}{1-q}}=\left[\frac{1-(1-q)\frac{E_{i}+E_{0}}{T}}{1-(1-q)\frac{F+E_{0}\chi}{T}}\right]^{\frac{1}{1-q}}\neq
p_{i}$ (52)

and

$p^{\prime}_{i}=\left[\frac{1-(1-q)\frac{E^{\prime}_{i}}{T}}{1+(1-q)S^{\prime}-(1-q)\frac{E^{\prime}}{T}}\right]^{\frac{1}{1-q}}=\left[\frac{1-(1-q)\frac{E_{i}+E_{0}}{T}}{1+(1-q)S-(1-q)\frac{E+E_{0}\chi}{T}}\right]^{\frac{1}{1-q}}\neq
p_{i}.$ (53)

Equations (52) and (53) are not equivalent to Eq. (49). Thus the probability
distribution $p_{i}$ of the Tsallis-2 statistics is not invariant under an
overall shift in energy.

##  5 Tsallis-3 statistics

The Tsallis-3 statistics is defined by the generalized entropy (22) with
probabilities $\\{p_{i}\\}$ of the microstates of system normalized to unity
(23) and by the normalized generalized expectation values [4]

$\langle
A\rangle=\frac{\sum\limits_{i}p_{i}^{q}A_{i}}{\sum\limits_{i}p_{i}^{q}},$ (54)

where $q\in\mathbb{R}$ is a real parameter taking values $0<q<\infty$. In
contrast with the Tsallis-2 statistics, this choice of the expectation values
is consistent because it has the feature $\langle 1\rangle=1$.

Let us consider the canonical ensemble. The thermodynamic potential of the
canonical ensemble for the Tsallis-$3$ statistics can be written as

$\displaystyle F=$ $\displaystyle
E-TS=\frac{1}{\chi}\sum\limits_{i}p_{i}^{q}\left[E_{i}+T\chi\frac{p_{i}^{1-q}-1}{1-q}\right],$
(55) $\displaystyle E=$ $\displaystyle\frac{1}{\chi}\sum_{i}p_{i}^{q}E_{i},$
(56)

where $\chi\equiv\sum_{i}p_{i}^{q}=1+(1-q)S$ and $E$ is the mean energy of the
system.

Substituting Eq. (55) into Eqs. (8)–(10) and using Eq. (23), we obtain the
equilibrium probabilities of microstates of the system [4]

$\displaystyle p_{i}=$
$\displaystyle\frac{1}{Z}\left[1+(1-q)\frac{E-E_{i}}{T\chi}\right]^{\frac{1}{1-q}},$
(57) $\displaystyle Z=$
$\displaystyle\sum\limits_{i}\left[1+(1-q)\frac{E-E_{i}}{T\chi}\right]^{\frac{1}{1-q}},$
(58)

where

$\displaystyle Z\equiv$
$\displaystyle\left[1-(1-q)\frac{\Lambda}{T}\right]^{\frac{1}{1-q}},$ (59)
$\displaystyle\chi=$
$\displaystyle\frac{1}{Z^{q}}\sum\limits_{i}\left[1+(1-q)\frac{E-E_{i}}{T\chi}\right]^{\frac{q}{1-q}}$
(60)

and $q\Lambda\equiv\lambda-T$. Rewriting Eq. (57) into the form

$p_{i}Z^{1-q}=p_{i}^{q}\left[1+(1-q)\frac{E-E_{i}}{T\chi}\right]$ (61)

and taking the summation over the index $i$, we obtain [4]

$Z^{1-q}=\chi.$ (62)

Substituting Eq. (62) into Eqs. (56) – (58), (60) and combining Eqs. (58) and
(60), we obtain the probability distribution [4]

$p_{i}=\frac{1}{Z}\left[1+(1-q)\frac{E-E_{i}}{TZ^{1-q}}\right]^{\frac{1}{1-q}}$
(63)

and two norm equations

$\sum\limits_{i}\left[1+(1-q)\frac{E-E_{i}}{TZ^{1-q}}\right]^{\frac{1}{1-q}}=\sum\limits_{i}\left[1+(1-q)\frac{E-E_{i}}{TZ^{1-q}}\right]^{\frac{q}{1-q}}$
(64)

and

$E=\frac{1}{Z}\sum\limits_{i}E_{i}\left[1+(1-q)\frac{E-E_{i}}{TZ^{1-q}}\right]^{\frac{q}{1-q}}$
(65)

for two unknown norm functions $E$ and $Z$. Then the expectation values (54)
for the Tsallis-$3$ statistics in the canonical ensemble can be rewritten in
the general form as [4]

$\langle
A\rangle=\frac{1}{Z}\sum\limits_{i}A_{i}\left[1+(1-q)\frac{E-E_{i}}{TZ^{1-q}}\right]^{\frac{q}{1-q}},$
(66)

where the norm functions $Z$ and $E$ are calculated from the system of two
equations (64) and (65). Note that Eq. (62) (as well as Eq. (64)) represents
the equation of self-consistency of the Tsallis-3 statistics.

Substituting Eq. (63) into Eqs. (22) and (55), and using Eq. (62), we obtain
[4]

$\displaystyle S=$ $\displaystyle\frac{Z^{1-q}-1}{1-q},$ (67) $\displaystyle
F=$ $\displaystyle E-TS=E-T\frac{Z^{1-q}-1}{1-q}.$ (68)

Substituting Eq. (67) into Eq. (63), we have (cf. with Eqs. (13) and (32))

$p_{i}=\left[\frac{1}{1+(1-q)S}+(1-q)\frac{E-E_{i}}{T(1+(1-q)S)^{2}}\right]^{\frac{1}{1-q}}.$
(69)

Comparing Eqs. (69) and (13), we observe that the relation between the
energies of microstates, the mean energy and the entropy of system in the
probability distribution (69) of the Tsallis-3 statistics is not compatible
with the same relation for the probability distribution (13) of the Boltzmann-
Gibbs statistics. Thus the probability distribution of the Tsallis-3
statistics does not reproduce the structure of the probability distribution of
the Boltzmann-Gibbs statistics, which is correct from the point of view of the
fundamentals of the statistical mechanics. This discrepancy is a significant
deficiency of the Tsallis-3 statistics.

Let us prove that the probability distribution of the Tsallis-3 statistics is
invariant under the overall energy shift (16), $E^{\prime}_{i}=E_{i}+E_{0}$.
Assuming that

$p^{\prime}_{i}=p_{i},$ (70)

we obtain $S^{\prime}=S$ (see Eq. (34)) and

$\displaystyle\chi^{\prime}=$
$\displaystyle\sum\limits_{i}{p^{\prime}}_{i}^{q}=\sum\limits_{i}p_{i}^{q}=\chi,$
(71) $\displaystyle Z^{\prime}=$
$\displaystyle\left(\chi^{\prime}\right)^{\frac{1}{1-q}}=\left(\chi\right)^{\frac{1}{1-q}}=Z,$
(72) $\displaystyle E^{\prime}=$
$\displaystyle\frac{1}{\chi^{\prime}}\sum\limits_{i}{p^{\prime}}_{i}^{q}E^{\prime}_{i}=\frac{1}{\chi}\sum\limits_{i}p_{i}^{q}(E_{i}+E_{0})=E+E_{0}$
(73)

and

$F^{\prime}=E^{\prime}-TS^{\prime}=E+E_{0}-TS=F+E_{0}.$ (74)

Using Eqs. (16), (63), (72) and (73), we have (cf. with Eq. (21))

$p^{\prime}_{i}=\frac{1}{Z^{\prime}}\left[1+(1-q)\frac{E^{\prime}-E^{\prime}_{i}}{T{Z^{\prime}}^{1-q}}\right]^{\frac{1}{1-q}}=\frac{1}{Z}\left[1+(1-q)\frac{E-E_{i}}{TZ^{1-q}}\right]^{\frac{1}{1-q}}=p_{i}.$
(75)

Equation (75) coincides with Eq. (70). Thus we have proved that the
probability distribution $p_{i}$ of the Tsallis-3 statistics is invariant
under an overall shift in energy.

##  6 $q$-dual statistics

The $q$-dual statistics is defined by the generalized entropy obtained from
the entropy (22) by the multiplicative transformation $q\to 1/q$ with the
probabilities normalized to unity (see the reference [30])

$\displaystyle S=$
$\displaystyle\frac{q}{q-1}\sum\limits_{i}\left(p_{i}^{1/q}-p_{i}\right),$
(76) $\displaystyle 1=$ $\displaystyle\sum\limits_{i}p_{i}$ (77)

and by the standard expectation values

$\langle A\rangle=\sum\limits_{i}p_{i}A_{i},$ (78)

where $q\in\mathbb{R}$ is a real parameter taking values $0<q<\infty$. In the
Gibbs limit $q\to 1$, the entropy (76) recovers the Boltzmann-Gibbs entropy
(3) and the $q$-dual statistics is reduced to the usual Boltzmann-Gibbs
statistics (see Sect. 2). Note that the $q$-dual statistics under the
transformation $q\to 1/q$ is reduced to the Tsallis-1 statistics (see Sect.
3). It should be stressed that the $q$-dual statistics was especially
introduced in physics in ref. [30] in order to justify the phenomenological
Tsallis transverse momentum distribution [11, 12], which is largely used in
the high energy physics.

Let us consider the canonical ensemble. The thermodynamic potential $F$ of the
canonical ensemble can be written as

$\displaystyle F=$ $\displaystyle
E-TS=\sum\limits_{i}p_{i}\left[E_{i}+Tq\frac{p_{i}^{\frac{1-q}{q}}-1}{1-q}\right],$
(79) $\displaystyle E=$ $\displaystyle\sum_{i}p_{i}E_{i},$ (80)

where $E$ is the mean energy and $E_{i}$ is the energy of the $i$-th
microscopic state of the system.

Substituting Eq. (79) into Eqs. (8)–(10) and using Eq. (77), we obtain the
equilibrium probabilities of microstates of the system

$p_{i}=\left[1+(1-q)\frac{\Lambda-E_{i}}{T}\right]^{\frac{q}{1-q}}$ (81)

and

$\sum\limits_{i}\left[1+(1-q)\frac{\Lambda-
E_{i}}{T}\right]^{\frac{q}{1-q}}=1,$ (82)

where $\Lambda\equiv\lambda-T$ and $\partial E_{i}/\partial p_{i}=0$. In the
Gibbs limit $q\to 1$, the probability distribution (81) recovers the
Boltzmann-Gibbs probability distribution (11). The expectation values of an
operator $A$ takes the form

$\langle A\rangle=\sum\limits_{i}A_{i}\left[1+(1-q)\frac{\Lambda-
E_{i}}{T}\right]^{\frac{q}{1-q}},$ (83)

where the norm function $\Lambda$ is the solution of Eq. (82).

Substituting Eq. (81) into Eq. (79) and using Eqs. (77) and (80), we obtain

$F=E-TS=q\left[\Lambda+\frac{1-q}{q}E\right]$ (84)

and

$\Lambda=E-\frac{1}{q}TS.$ (85)

Substituting Eq. (85) into Eq. (81), we have (cf. with Eqs. (13) and (32))

$p_{i}=\left[1-\frac{1-q}{q}S+(1-q)\frac{E-E_{i}}{T}\right]^{\frac{q}{1-q}}.$
(86)

In the Gibbs limit $q\to 1$, the probability distribution (86) of the $q$-dual
statistics recovers the Boltzmann-Gibbs probability distribution (13).
Comparing Eqs. (86) and (13), we observe that the relation between the
energies of microstates, the mean energy and the entropy of system in the
probability distribution (86) of the $q$-dual statistics is compatible with
the same relation for the probability distribution (13) of the Boltzmann-Gibbs
statistics. Thus the probability distribution of the $q$-dual statistics has
the correct structure.

Let us prove that the probability distribution of the $q$-dual statistics is
invariant under the overall energy shift (16), $E^{\prime}_{i}=E_{i}+E_{0}$.
Assuming that

$p^{\prime}_{i}=p_{i},$ (87)

we obtain

$\displaystyle S^{\prime}=$
$\displaystyle\frac{q}{q-1}\sum\limits_{i}\left({p^{\prime}}_{i}^{1/q}-p^{\prime}_{i}\right)=\frac{q}{q-1}\sum\limits_{i}\left(p_{i}^{1/q}-p_{i}\right)=S,$
(88) $\displaystyle E^{\prime}=$
$\displaystyle\sum\limits_{i}p^{\prime}_{i}E^{\prime}_{i}=\sum\limits_{i}p_{i}(E_{i}+E_{0})=E+E_{0},$
(89) $\displaystyle\Lambda^{\prime}=$ $\displaystyle
E^{\prime}-\frac{1}{q}TS^{\prime}=E+E_{0}-\frac{1}{q}TS=\Lambda+E_{0}$ (90)

and

$F^{\prime}=\sum\limits_{i}p^{\prime}_{i}\left[E^{\prime}_{i}+Tq\frac{{p^{\prime}}_{i}^{\frac{1-q}{q}}-1}{1-q}\right]=\sum\limits_{i}p_{i}\left[E_{i}+E_{0}+Tq\frac{p_{i}^{\frac{1-q}{q}}-1}{1-q}\right]=F+E_{0}.$
(91)

Using Eqs. (16), (81), (86) and (88)–(90), we obtain (cf. with Eq. (21))

$p^{\prime}_{i}=\left[1+(1-q)\frac{\Lambda^{\prime}-E^{\prime}_{i}}{T}\right]^{\frac{q}{1-q}}=\left[1+(1-q)\frac{\Lambda-
E_{i}}{T}\right]^{\frac{q}{1-q}}=p_{i}$ (92)

and

$p^{\prime}_{i}=\left[1-\frac{1-q}{q}S^{\prime}+(1-q)\frac{E^{\prime}-E^{\prime}_{i}}{T}\right]^{\frac{q}{1-q}}=\left[1-\frac{1-q}{q}S+(1-q)\frac{E-E_{i}}{T}\right]^{\frac{q}{1-q}}=p_{i}.$
(93)

Equations (92) and (93) exactly coincide with Eq. (87). Thus we have proved
that the probability distribution $p_{i}$ of the $q$-dual statistics is indeed
invariant under an overall shift in energy.

##  7 Conclusions

We have considered the general formalisms of the Boltzmann-Gibbs statistics,
the $q$-dual statistics, the Tsallis-1 statistics, the Tsallis-2 statistics,
and the Tsallis-3 statistics in the canonical ensemble. The equilibrium
probability distributions have been analytically derived from a constrained
local extremum of the thermodynamic potential. The canonical ensemble for the
$q$-dual statistics has been studied for the first time. We have found that
the probability distributions of all these statistics may be written in
different representations. For the probability distributions of the Tsallis-1
statistics, the Tsallis-3 statistics, the $q$-dual statistics, and Boltzmann-
Gibbs statistics, we have obtained a representation in which the probability
distribution is a function of the difference between the mean energy and the
energy of the microstate. However, for the Tsallis-2 statistics, such a
representation does not exist. We have found that the probability
distributions of the Tsallis-1 and $q$-dual statistics have the correct
structure from the point of view of the fundamentals of the statistical
mechanics since the relation between the energies of microstates, the mean
energy, and the entropy of the system in these probability distributions is
consistent with the same relation in the probability distribution of the
Boltzmann-Gibbs statistics. However, the probability distributions of the
Tsallis-2 and Tsallis-3 statistics do not reproduce this structure of the
probability distribution of the Boltzmann-Gibbs statistics, which is correct
from the point of view of the fundamentals of the statistical mechanics. Thus
this discrepancy is a significant deficiency of the Tsallis-2 and Tsallis-3
statistics.

We have found that the probability distribution of the Tsallis-1 statistics in
terms of the Lagrange multiplier $\beta$ obtained in the Tsallis’ original
paper on the base of the Jaynes principle is precisely reduced to the
probability distribution obtained in the present paper and the previous paper
of the same author. It has been obtained after excluding the intermediate
Lagrange multiplier $\beta$ and representing the final results in terms of the
thermodynamic temperature of the system. Thus we have proved the equivalence
of the Jaynes principle with the method of extremization of the thermodynamic
potential. This fact demonstrates that the final results for the probability
distribution of the Tsallis-1 statistics are independent of the method of
calculation.

We have introduced a new general and explicit method to demonstrate the
invariance of the probability distributions under the uniform energy spectrum
translation. This method does not depend on the factorization property of the
exponential function. Using this method, we have studied the invariance of the
probability distributions of the Boltzmann-Gibbs statistics, the $q$-dual
statistics, and three versions of the Tsallis statistics under an overall
shift in energy at a fixed temperature. We have demonstrated that the
probability distribution of the Tsallis-1 statistics is invariant under the
uniform energy spectrum translation at a fixed temperature. We also prove the
invariance of the probability distributions of the Tsallis-3 statistics, the
$q$-dual statistics, and Boltzmann-Gibbs statistics. However, we have found
that the probability distribution of the Tsallis-2 statistics, the expectation
values of which are not consistent with the normalization condition of
probabilities, is not invariant under the overall shift in energy. Thus this
fact presents a severe deficiency for the Tsallis-2 statistics as it violates
the fundamentals of physics.

Acknowledgments: This work was supported in part by the joint research project
of JINR and IFIN-HH.

##  Appendix A Probability distribution of Tsallis-1 statistics from the
Jaynes principle

Let us show that the probability distribution obtained in the original paper
of C. Tsallis [1] in terms of the Lagrange multiplier $\beta$ associated with
the energy represents an intermediate result and in the terms of temperature
$T$ it is reduced to the probability distribution obtained in ref. [9] and
present paper.

In ref. [1], the Tsallis-1 statistics is defined by the generalized entropy
obtained from Eq. (22) with the probabilities normalized to unity

$\displaystyle S=$ $\displaystyle\frac{1-\sum\limits_{i}p_{i}^{q}}{q-1},$ (94)
$\displaystyle 1=$ $\displaystyle\sum\limits_{i}p_{i}$ (95)

and by the standard expectation values

$\langle A\rangle=\sum\limits_{i}p_{i}A_{i},$ (96)

where $q\in\mathbb{R}$ is a real parameter taking values $0<q<\infty$.

In ref. [1], the Lagrange function of canonical ensemble is defined as

$\Phi=S+\alpha\sum\limits_{i}p_{i}-\alpha\beta(q-1)\sum\limits_{i}p_{i}E_{i},$
(97)

where $\alpha$ and $\beta$ are the Lagrange multipliers. Imposing
$\partial\Phi/\partial p_{i}=0$, one obtains the equilibrium probabilities of
microstates of the system

$\displaystyle p_{i}=$
$\displaystyle\frac{1}{Z}\left[1-\beta(q-1)E_{i}\right]^{\frac{1}{q-1}},$ (98)
$\displaystyle Z=$
$\displaystyle\sum\limits_{i}\left[1-\beta(q-1)E_{i}\right]^{\frac{1}{q-1}},$
(99)

where $Z=(\alpha(q-1)/q)^{-1/(q-1)}$.

Substituting Eq. (98) into Eq. (94), we have

$\displaystyle S=$
$\displaystyle\frac{1}{q-1}+\frac{Z^{1-q}}{1-q}\left[1-\beta(q-1)E\right],$
(100) $\displaystyle E=$ $\displaystyle\sum\limits_{i}p_{i}E_{i},$ (101)

where $E$ is the mean energy. Differentiating $S$, $p_{i}$ and $E$ with
respect to $\beta$, we obtain

$\displaystyle\frac{\partial S}{\partial\beta}=$ $\displaystyle
Z^{1-q}\left\\{\frac{\partial\ln
Z}{\partial\beta}[1-\beta(q-1)E]+E+\beta\frac{\partial
E}{\partial\beta}\right\\},$ (102) $\displaystyle\frac{\partial
p_{i}}{\partial\beta}=$ $\displaystyle-\frac{\partial\ln
Z}{\partial\beta}p_{i}-\frac{1}{Z}E_{i}\left[1-\beta(q-1)E_{i}\right]^{\frac{1}{q-1}-1},$
(103) $\displaystyle\frac{\partial E}{\partial\beta}=$
$\displaystyle\sum\limits_{i}\frac{\partial p_{i}}{\partial\beta}E_{i},$ (104)

where $\partial E_{i}/\partial\beta=0$. Multiplying Eq. (103) by the factor
$[1-\beta(q-1)E_{i}]$ and taking the summation over $i$, and using Eqs. (95),
(101), (104) and the equation $\sum\limits_{i}\partial p_{i}/\partial\beta=0$,
we obtain

$-\frac{\partial\ln Z}{\partial\beta}=\frac{E-\beta(q-1)\frac{\partial
E}{\partial\beta}}{1-\beta(q-1)E}$ (105)

and

$\frac{\partial S}{\partial\beta}=q\beta Z^{1-q}\frac{\partial
E}{\partial\beta}.$ (106)

The temperature in the statistical mechanics is defined from the fundamental
thermodynamic potential $E$ as

$T\equiv\frac{\partial E}{\partial S}=\frac{\partial E/\partial\beta}{\partial
S/\partial\beta}.$ (107)

Substituting Eq. (106) into Eq. (107), we obtain

$\beta=\frac{Z^{q-1}}{qT}.$ (108)

Thus the Lagrange multiplier $\beta$ is a function of the thermodynamic
temperature $T$.

Substituting Eq. (108) into Eqs. (98)–(100), we have

$\displaystyle p_{i}=$
$\displaystyle\frac{1}{Z}\left[1-\frac{q-1}{q}\frac{E_{i}}{TZ^{1-q}}\right]^{\frac{1}{q-1}},$
(109) $\displaystyle Z=$
$\displaystyle\sum\limits_{i}\left[1-\frac{q-1}{q}\frac{E_{i}}{TZ^{1-q}}\right]^{\frac{1}{q-1}}$
(110)

and

$\displaystyle S=$
$\displaystyle\frac{1-Z^{1-q}}{q-1}+\frac{1}{q}\frac{E}{T},$ (111)
$\displaystyle F=$ $\displaystyle
E-TS=-T\frac{1-Z^{1-q}}{q-1}+\frac{q-1}{q}E.$ (112)

Using Eqs. (109), (111), we obtain exactly Eq. (32) of the Tsallis-1
statistics:

$p_{i}=\left[1-(q-1)S+\frac{q-1}{q}\frac{E-E_{i}}{T}\right]^{\frac{1}{q-1}}.$
(113)

Let us introduce a new function $\Lambda$ instead of $Z$ in the form

$Z^{1-q}\equiv 1+\frac{q-1}{q}\frac{\Lambda}{T},$ (114)

Substituting Eq. (114) into Eqs. (109), (110), we obtain exactly Eqs. (27) and
(28) of the Tsallis-1 statistics:

$p_{i}=\left[1+\frac{q-1}{q}\frac{\Lambda-E_{i}}{T}\right]^{\frac{1}{q-1}}$
(115)

and

$\sum\limits_{i}\left[1+\frac{q-1}{q}\frac{\Lambda-
E_{i}}{T}\right]^{\frac{1}{q-1}}=1.$ (116)

Substituting Eq. (114) into Eqs. (111), (112), we obtain exactly Eqs. (30) and
(31) of the Tsallis-1 statistics:

$F=E-TS=\frac{1}{q}[\Lambda+(q-1)E]$ (117)

and

$\Lambda=E-qTS.$ (118)

Thus the formalism of the Tsallis-1 statistics given in refs. [1, 4] is
reduced to the formalism obtained in ref. [9] and present paper after
excluding the intermediate Lagrange multiplier $\beta$, which is a method
dependent. Note that this proof in a concise form can be found also in ref.
[9].

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# Prompt-IML: Image Manipulation Localization with pre-trained foundation
models through prompt tuning

###### Abstract

Deceptive images can be shared in seconds with social networking services,
posing substantial risks. Tampering traces, such as boundary artifacts and
high-frequency information, have been significantly emphasized by massive
networks in the Image Manipulation Localization (IML) field. However, they are
prone to image post-processing operations, which limit the generalization and
robustness of existing methods. We present a novel Prompt-IML framework. We
observe that humans tend to discern the authenticity of an image based on both
semantic and high-frequency information, inspired by which, the proposed
framework leverages rich semantic knowledge from pre-trained visual foundation
models to assist IML. We are the first to design a framework that utilizes
visual foundation models specially for the IML task. Moreover, we design a
Feature Alignment and Fusion module to align and fuse features of semantic
features with high-frequency features, which aims at locating tampered regions
from multiple perspectives. Experimental results demonstrate that our model
can achieve better performance on eight typical fake image datasets and
outstanding robustness.

Index Terms—  manipulation localization, prompt tuning, feature fusion,
attention mechanism

##  1 Introduction

The commonly encountered image processing techniques are copy-move, splicing,
and inpainting, all of which have the capability to alter the original
semantic content of images. Meanwhile, the rapid advancement of image editing
tools has substantially reduced the difficulty and cost associated with
producing deceptive images. Consequently, there is a pressing need to
accurately locate the manipulated regions in deceptive images.

The Image Manipulation Localization (IML) task is to granularly locate
tampered regions within images. With the advancement of deep learning,
researchers are attempting to establish massive manipulation localization
networks [7]. Many existing methods revolve around specific tampering traces,
seeking the optimal feature representation through carefully designed network
architectures [2, 3, 1]. However, tampering traces are prone to image post-
processing operations [31, 8]. It’s a contributing factor to the limited
robustness and generalization of the aforementioned methods.

Fig. 1: Prompt-IML incorporates semantic information with high-frequency
information to improve the performance. Semantic information may notice
specific objects (circled red region) to assist the manipulation localization.

Moreover, we notice that existing methods ignore a key element of images to
achieve generalizability and robustness, that is the semantic information [8].
Humans naturally observe the coherence of the semantic information within the
picture to identify fake images. Semantic information plays a principal part
in many computer vision tasks [24]. We believe that it is not exceptional in
IML. Compared with features related to tampering traces, semantic features are
more robust to image post-processing. Therefore, employing the semantic
features of images as another supplement for judgment will assist the task of
IML. However, training a network with rich semantic knowledge using limited
available datasets is challenging [25]. Besides, typical methods often utilize
high-frequency features from images to identify manipulations, which brings
new challenges of aligning semantic features with them [26].

Fig. 2: Architecture overview. The Channel, Spatial, Deformable represents the
procedure of Eq. 4, Eq. 5, and Eq. 7.

To overcome the aforementioned challenges, we exploit pre-trained visual
foundation models to acquire semantic features of images through prompt
tuning. Fig. 1 exhibits the difference between the proposed method and typical
methods. We propose prompt-IML that actively utilizes semantic information
along with high-frequency information for manipulation localization.
Specifically, We use BayarConv to extract the high-frequency features of
images and feed them into subsequent networks for further processing.
Simultaneously, we raise a semantic feature extraction network, which adheres
to the architectural design of visual foundation models and is initialized
with pre-trained weights. During training, we freeze it and attach several
learnable prompt embeddings to image token sequences to adjust the semantic
features. Then, we facilitate interaction between semantic features and high-
frequency features through a designed Feature Alignment and Fusion (FAF)
module, which involves multiple attention mechanisms to enhance features and
locate tampered regions from multiple perspectives.

Our main contributions are summarized as follows:

  * •

We are the first to design a framework that utilizes visual foundation models
specially for the IML task. Incorporating semantic information with high-
frequency information for discernment aligns more with the logic circuit of
human judgment regarding image veracity.

  * •

We propose an FAF module, that enables adapting visual foundation models to
IML tasks through prompt tuning. The proposed FAF module involves multiple
attention mechanisms to align and fuse semantic features with high-frequency
features.

  * •

Experiments on eight datasets demonstrate the generalizability of the proposed
framework. Extensive experiments prove the robustness of the framework against
image post-processing operations.

##  2 Related Works

Image Manipulation Localization. With the advancement of deep learning,
researchers have embarked on efforts to establish end-to-end manipulation
localization networks. MVSSNet++[8] integrates multi-scale features, contour
features, and high-frequency features of images for feature extraction and
utilizes spatial-channel attention for enhanced feature fusion. PSCC-Net[7]
proposes a progressive spatial-channel attention module, utilizing multi-scale
features and dense cross-connections to generate tampering masks of various
granularity. These works involve the meticulous design of network
architectures to acquire more optimal feature representations regarding
tampering traces. Although these methods have achieved decent performance in
the IML task, the choice of feature representation for tampering traces still
significantly impacts the model’s generalization and robustness.

Tuning Visual Foundation Models. Compared to fine-tuning, prompt tuning is an
efficient, low-cost way of adapting an AI foundation model to new downstream
tasks without retraining the model and updating its weights. This technique
was first used in NLP, and VPT[27] is an efficient way to adapt it for the
visual domain. Recently, EVP[10] achieved granular manipulated region
localization by adjusting the embedding representation of images and
incorporating high-frequency information. They attempt to adapt the pre-
trained model through prompt tuning to various downstream tasks, including
IML. However, due to the simple feature fusion design, their model performs
poorly.

##  3 Proposed Method

###  3.1 Approach Overview

Fig. 2 illustrates the architecture of the proposed prompt-IML. The complete
pipeline consists of two phases, i.e., Feature Extraction Network (FEN) and
Manipulation Localization Network (MLN). The FEN comprises two parallel
branches: one extracts semantic features, and the other focuses on extracting
high-frequency features. Given the differences between them, we employ a
carefully designed FAF module to fuse features. This module primarily utilizes
various attention mechanisms to facilitate interaction between the features.
The multi-scale features outputted during the FEN stage are ultimately fed
into the MLN. The MLN aggregates feature information through layer-wise up-
sampling and outputs the final prediction results.

###  3.2 Feature Extraction Network

Dual-Branch Architecture. We use two branches to extract multiple features of
images, and both share the same structure based on Swin-Transformer [28]. The
semantic branch is initialized with pre-trained weights and remains untrained
during training to preserve the optimal semantic representations.
Specifically, let $\mathbf{X}\in\mathbb{R}^{H\times W\times C}$ be the input
image. For the semantic branch, we get the input
$\mathbf{F}^{0}_{sem}\in\mathbb{R}^{(H^{\prime}\times W^{\prime})\times
C^{\prime}}$ through partitioning the image into specified-sized patches:

$\mathbf{F}^{0}_{sem}=\operatorname{Norm}(\operatorname{Conv}(\mathbf{X}))+\mathbf{F}_{PE},$
(1)

where $H^{\prime}\times W^{\prime}$ represents the number of the patches,
$\mathbf{F}_{PE}$ is a learnable positional embedding. For the high-frequency
branch, we employ a set of BayarConv with kernels varying sizes, which prevent
information loss caused by a fixed-size receptive field, to get the input
$\mathbf{F}^{0}_{\textit{hfq}}\in\mathbb{R}^{(H^{\prime}\times
W^{\prime})\times C^{\prime}}$:

$\mathbf{F}^{0}_{\textit{hfq}}=\operatorname{Concat}(\\{\operatorname{BayarConv_{i\times
i}(\mathbf{X})},\ i=3,5,7\\}),$ (2)

where $i$ symbolizes the kernel size.

To comprehensively consider both global and local information and mitigate
information loss[8], multi-scale features
$\mathbf{F}^{i}_{sem},\mathbf{F}^{i}_{\textit{sem}}\in\mathbb{R}^{(H_{i}\times
W_{i})\times C_{i}},i=1,...,4$ are generated for the subsequent procedure in
each branch. Specifically, each branch comprises four layers, namely sem layer
and hfq layer, each of which consists of several blocks. The forward
propagation process in each block can be described below:

$\begin{split}\mathbf{z}^{i,j}&=\operatorname{SelfAttn}(\operatorname{LN}(\mathbf{F}^{i,j-1}))+\mathbf{F}^{i,j-1},\\\
\mathbf{F}^{i,j}&=\operatorname{MLP}(\operatorname{LN}(\mathbf{z}^{i,j}))+\mathbf{z}^{i,j},\end{split}$
(3)

where $j=1,...,N_{i},\mathbf{F}^{i,0}=\mathbf{F}^{i}$,
$\operatorname{LN}(\cdot)$ denotes layer normalization, and $\mathbf{F}^{i,j}$
denotes the output of the $i$-th layer and $j$-th block.

Fig. 3: Manipulation localization results on images originating from multiple
datasets. The 3-rd column represents the results of our method, while columns
4 to 10 depict the results of another six SOTA methods.

Feature Alignment and Fusion Module. Attention mechanisms are widely used to
enhance features in IML task [MVSS]. We propose a FAF module for better
alignment and fusion of multiple features, which consists of channel
attention, spatial attention, and deformable attention. First, we employ
average pooling operation to reduce features, denoted by overline. Then, they
are concatenated on dimension $C_{i}$, which is denoted by $[\cdot]$, and fed
into an MLP to generate corresponding channel-attention vectors
$\mathbf{W}^{C}_{s^{i}},\mathbf{W}^{C}_{h^{i}}\in\mathbb{R}^{C_{i}}$, the
above procedure can be formulated as:

$\begin{split}\mathbf{W}^{C}_{s^{i}},\mathbf{W}^{C}_{h^{i}}&=\operatorname{ChannelAttn}(\mathbf{F}^{i}_{sem},\mathbf{F}^{i}_{\textit{hfq}})\\\
&=\operatorname{Split}\left(\operatorname{MLP}\left(\left[{\overline{\mathbf{F}^{i}_{sem}}},{\overline{\mathbf{F}^{i}_{\textit{hfq}}}}\right]\right)\right),\end{split}$
(4)

where $\operatorname{Split}$ is the reverse operation of
$\operatorname{Concat}$. To obtain the spatial attention vector, we use two
1×1 convolutions with an intermediate ReLU layer, denoted by $g(\cdot)$, to
aggregate spatial information, spatial-attention vectors
$\mathbf{W}^{S}_{s^{i}},\mathbf{W}^{S}_{h^{i}}\in\mathbb{R}^{H_{i}\times
W_{i}}$ can be obtained:

$\begin{split}\mathbf{W}^{S}_{s^{i}},\mathbf{W}^{S}_{h^{i}}&=\operatorname{SpatialAttn}\left(\mathbf{F}^{i}_{sem},\mathbf{F}^{i}_{\textit{hfq}}\right)\\\
&=\operatorname{Split}\left(\operatorname{Conv}\left({g}\left(\operatorname{Conv}\left(\left[\mathbf{F}^{i}_{sem},\mathbf{F}^{i}_{\textit{hfq}}\right]\right)\right)\right)\right).\end{split}$
(5)

Finally, we align branch features by applying attention vectors crosswise,
which gives out the input of the next layer by residual adding, for
$i=1,2,3,4$:

$\begin{split}&\mathbf{F}^{C}_{s^{i}}=\mathbf{W}^{C}_{s^{i}}\odot\mathbf{F}^{i}_{sem},\quad\mathbf{F}^{S}_{s^{i}}=\mathbf{W}^{S}_{s^{i}}\odot\mathbf{F}^{i}_{sem},\\\
&\mathbf{F}^{C}_{h^{i}}=\mathbf{W}^{C}_{h^{i}}\odot\mathbf{F}^{i}_{\textit{hfq}},\quad\mathbf{F}^{S}_{h^{i}}=\mathbf{W}^{S}_{h^{i}}\odot\mathbf{F}^{i}_{\textit{hfq}},\\\
&\mathbf{F}^{i}_{sem}:=\mathbf{F}^{i}_{sem}+\mathbf{F}^{C}_{h^{i}}+\mathbf{F}^{S}_{h^{i}},\\\
&\mathbf{F}^{i}_{\textit{hfq}}:=\mathbf{F}^{i}_{\textit{hfq}}+\mathbf{F}^{C}_{s^{i}}+\mathbf{F}^{S}_{s^{i}}.\end{split}$
(6)

Then, we fuse semantic feature $\mathbf{F}^{i}_{sem}$ and high-frequency
feature $\mathbf{F}^{i}_{\textit{hfq}}$ to get the input $\mathbf{F}_{d}^{i}$
of the MLN. Tampering operations affect a certain number of pixels rather than
a single pixel, restricting the attention range is more advantageous in
suppressing sporadic positive responses to the features. Therefore, we utilize
deformable attention[29] for enhancement. The fusion process can be described
by the following equations:

$\begin{split}&\mathbf{attn}_{s}=\operatorname{DfA}(\operatorname{query}=\mathbf{F}^{i}_{{sem}},\operatorname{value}=\mathbf{F}^{i}_{\textit{hfq}}),\\\
&\mathbf{attn}_{h}=\operatorname{DFA}(\operatorname{query}=\mathbf{F}^{i}_{\textit{hfq}},\operatorname{value}=\mathbf{F}^{i+1}_{{sem}}),\\\
&\mathbf{F}_{d}^{i}=\gamma_{1}*(\mathbf{F}^{i}_{{sem}}+\mathbf{attn}_{s})+\gamma_{2}*(\mathbf{F}^{i}_{\textit{hfq}}+\mathbf{attn}_{h}),\end{split}$
(7)

where $\gamma_{1}$, $\gamma_{2}$ are learnable parameters, and DFA means
Deformable Attention.

###  3.3 Manipulation Localization Network

MLN adopt the architecture of Mask2Former[24], which involves two parts: the
Pixel Decoder and the Transformer Decoder. The Pixel Decoder is primarily
responsible for progressively upsampling features from low resolution to high
resolution. The Transformer Decoder utilizes a single query embedding and
multi-scale features as inputs. The use of multi-scale features is
advantageous for locating small tampered regions, while query embeddings,
combined with Masked-Attention, help restrict Cross-Attention to the tampered
regions for extracting tampering-related features.

###  3.4 Prompt Tuning Method

We leverage rich semantic features from pre-trained visual foundation models
through prompt tuning. For each basic block, we concatenate unique prompt
embeddings and image tokens as input:

$\displaystyle\mathbf{F}^{i,j}=\operatorname{{Block}^{i,j}_{sem}}\left(\left[\mathbf{P}^{i,j},\mathbf{F}^{i,j-1}\right]\right),i=1,2,...,N_{i},$
(8)

where $N_{i}$ symbolizes the total number of blocks in $i$-th layer. Assume
input with batch size of $B$, where $\mathbf{P}^{i,j}\in\mathbb{R}^{B\times
n_{p}\times C_{i}}$. We expand $\mathbf{P}^{i,j}$ after partitioning to alter
dimensions to $\mathbb{R}^{(B*n_{w})\times n_{p}\times C}$, ensuring that each
window contains exactly $n_{p}$ prompt embeddings for self-attention
computation. After merging windows, we average on $n_{w}$ groups of prompt
tokens to reshape back as $\mathbf{P}^{i,j}\in\mathbb{R}^{B\times n_{p}\times
C}$.

Fig. 4: Robustness evaluation against 6 different perturbations. Test dataset is CASIA1, and AUC is the evaluation metric. The x-axis symbolizes the perturbation severity level from 0 to 9 with 0 being no perturbation. Table 1: The comparison of image manipulation localization performance (F1 score with fixed threshold: 0.5). The best performance in each column are bolded and the second best underlined. Method | CASIA1 | NIST16 | COVER | IMD20 | Columbia | DEF-12K | In-the-Wild | Korus | Average
---|---|---|---|---|---|---|---|---|---
FCN [11] | 0.441 | 0.167 | 0.199 | 0.210 | 0.223 | 0.130 | 0.192 | 0.122 | 0.211
DeepLabv3 [12] | 0.429 | 0.237 | 0.151 | 0.216 | 0.442 | 0.068 | 0.220 | 0.120 | 0.235
MFCN [1] | 0.346 | 0.243 | 0.148 | 0.170 | 0.184 | 0.067 | 0.161 | 0.118 | 0.180
RRU-Net [2] | 0.291 | 0.200 | 0.078 | 0.159 | 0.264 | 0.033 | 0.178 | 0.097 | 0.163
HPFCN [4] | 0.173 | 0.172 | 0.104 | 0.111 | 0.115 | 0.038 | 0.125 | 0.097 | 0.117
MantraNet [3] | 0.187 | 0.158 | 0.236 | 0.164 | 0.452 | 0.067 | 0.314 | 0.110 | 0.211
H-LSTM [5] | 0.156 | 0.357 | 0.163 | 0.202 | 0.149 | 0.059 | 0.173 | 0.143 | 0.175
SPAN [6] | 0.143 | 0.211 | 0.144 | 0.145 | 0.503 | 0.036 | 0.196 | 0.086 | 0.183
PSCC [7] | 0.335 | 0.173 | 0.220 | 0.197 | 0.503 | 0.072 | 0.303 | 0.114 | 0.240
EVP [10] | 0.483 | 0.210 | 0.114 | 0.233 | 0.277 | 0.090 | 0.231 | 0.113 | 0.219
CAT-Net [9] | 0.237 | 0.102 | 0.210 | 0.257 | 0.206 | 0.206 | 0.217 | 0.085 | 0.190
MVSS-Net++ [8] | 0.513 | 0.304 | 0.482 | 0.270 | 0.660 | 0.095 | 0.295 | 0.102 | 0.340
PIM [13] | 0.566 | 0.280 | 0.251 | 0.419 | 0.680 | 0.167 | 0.418 | 0.234 | 0.377
Ours | 0.581 | 0.343 | 0.414 | 0.423 | 0.801 | 0.194 | 0.414 | 0.266 | 0.430

##  4 Experiment

###  4.1 Experimental Setup

Datasets. During the training phase, we utilize the CASIA2[14] and synthetic
datasets used by PSCC-Net[7] as the training set. To achieve a balance between
training effectiveness and efficiency, we randomly sample 30,000 images from
the synthetic dataset for training. During the testing phase, we employ eight
common datasets to assess our model’s performance: CASIA1[15], COVER[17],
IMD20[22], NIST16[18], Columbia[16], DEFACTO-12K[21], In-the-Wild[20], and
Korus[19].

Implementation Details. We train our model on two RTX 3060 GPUs with a batch
size of 14. We employ weighted cross-entropy loss as the objective function.
Input images are resized to $512\times 512$. We utilize the AdamW optimizer
with $\beta_{1}=0.9$, $\beta_{2}=0.999$, weight decay 0.05, and cosine
annealing warm restarts strategy. The maximum learning rate is set to 1e-4,
and the minimum learning rate is 1e-6. The model is trained for 80 epochs,
including 5 warm-up epochs.

###  4.2 Comparisons

We compare our method with 13 state-of-the-art models to comprehensively
assess the model’s performance. We follow the metrics of previous work [13]
for evaluation, the experimental results are shown in Table 1, in which the
best results are bolded and the second-best results are underlined. The
proposed model places either 1st or 2nd on all datasets, demonstrating its
effectiveness and generalization ability. It is worth noting that H-LSTM
achieves favorable performance on the NIST16 dataset, primarily owing to the
specially fine-tuning. MVSSNet++ is a meticulously designed network that fully
leverages boundary artifacts and high-frequency information from forged
images. However, by integrating the semantic and high-frequency information,
we achieve a 9% F1-score improvement in average. EVP fails to achieve
satisfactory generalization performance, possibly due to its less effective
fusion strategy. The manipulation localization results of various methods are
illustrated in Fig. 3, in which our approach transcends limitations associated
with specific types of datasets, demonstrating its efficacy in effectively
addressing a wide array of tampering methods.

Table 2: Image Manipulation Localization Performance(F1 score with fixed threshold: 0.5) Setting | Sem | HP | F.Align | F.Fuse | F1-score
---|---|---|---|---|---
1 | ✓ | - | - | - | 0.481
2 | - | ✓ | - | - | 0.392
3 | ✓ | ✓ | - | - | 0.505
4 | ✓ | ✓ | ✓ | - | 0.555
5 | ✓ | ✓ | - | ✓ | 0.517
6 | ✓ | ✓ | ✓ | ✓ | 0.581

###  4.3 Robustness Test

In the real-world scenario, manipulated images may suffer from various post-
processing techniques, leading to the fading or disappearance of tampering
traces, which significantly compromises the model’s performance. We follow the
setup introduced by [13], introducing six common perturbations to mimic post-
processing effects of brightness, contrast, darkening, dithering, pink noise
and JPEG2000 compression. We evaluate the robustness of each method on the
CASIA1 dataset, with pixel-level localization AUC scores presented in Fig. 4.
The results demonstrate the necessity of incorporating semantic information,
as the aligned semantic features supplement the high-frequency feature well,
which contributes in the robustness of the proposed method.

###  4.4 Ablation Study

To assess the effectiveness of the modules we design, we conduct comprehensive
ablation experiments. Table 2 presents the specific experimental settings and
corresponding F1 scores testing on the CASIA1 dataset. Experiment 1 utilize
only the semantic branch through prompt tuning. Experiment 2, on the other
hand, solely employ a high-frequency branch trained from scratch. The results
demonstrate that either semantic or high-frequency information is vital in the
IML task. Furthermore, we investigate the effectiveness of the designed
alignment and fusion method via experiments 3 to 5. We ablate the deformable
attention in fusion by substituting it with element-wise addition. The results
exhibit the effectiveness of the designed multiple attention mechanisms.

##  5 Conclusions

We present Prompt-IML, which introduces semantic information of pre-trained
visual foundation models into IML tasks. The semantic information is leveraged
through prompt tuning and fused with high-frequency information of images.
Experimental results on typical IML datasets demonstrate the effectiveness of
the proposed method.

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