# The finitistic dimension of a triangulated category

Henning Krause  Fakultät für Mathematik
Universität Bielefeld
D-33501 Bielefeld
Germany<EMAIL_ADDRESS>
###### Abstract.

The finitistic dimension of a triangulated category is introduced. For the
category of perfect complexes over a ring it is shown that this dimension is
finite if and only if the small finitistic dimension of the ring is finite.

###### Key words and phrases:

Finitistic dimension, triangulated category

###### 2020 Mathematics Subject Classification:

18G80 (primary), 16E10 (secondary)

The finitistic dimension of a ring is a homological invariant which is
conjectured to be finite for any finite dimensional algebra over a field [2].
Despite many efforts the conjecture remains open. So it seems appropriate to
take a step back in order to understand this homological invariant beyond the
specific context of modules over a finite dimensional algebra. This is
precisely a situation where triangulated categories turn out to be useful,
because they enable the transfer of concepts.

Let us consider the category of perfect complexes over a noetherian scheme. A
recent result of Neeman characterises the existence of bounded t-structures
[11], confirming a conjecture of Antieau, Gepner, and Heller [1]. This
requires the scheme to be of finite dimension, and an analysis of the proof
reveals that there is an analogue of this theorem in a non-commutative
setting. It turns out that the condition on the scheme to be of finite
dimension translates into the finiteness of the finitistic dimension. Thus a
unified proof requires a definition of finitistic dimension for triangulated
categories, and this is precisely what this paper offers. That there is such a
concept is not surprising, because it is well known that the finiteness of the
finitistic dimension of a ring is a derived invariant, even though the integer
values of this dimension in a derived equivalence class may differ [13].

The purpose of this note is to describe the finiteness of the finitistic
dimension of a ring as a property of the triangulated category of perfect
complexes. To this end we introduce for any triangulated category its
finitistic dimension. This is in the spirit of Rouquier’s dimension [15]; for
a wide class of rings its finiteness characterises regularity. Note that we
focus on the small finitistic dimension. For the big finitistic dimension and
a characterisation of its finiteness via the unbounded derived category we
refer to recent work of Rickard [14].

* * *

Let $\mathcal{T}$ be a triangulated category with suspension
$\Sigma\colon\mathcal{T}\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\mathcal{T}$.
From [3, 15] we recall the following definition. For an object $X$ in
$\mathcal{T}$ and $n\geq 0$ set

$\operatorname{thick}^{n}(X):=\begin{cases}\operatorname{add}\varnothing&n=0,\\\
\operatorname{add}\\{\Sigma^{i}X\mid i\in\mathbb{Z}\\}&n=1,\\\
\operatorname{add}\\{\operatorname{cone}\phi\mid\phi\in\operatorname{Hom}(\operatorname{thick}^{1}(X),\operatorname{thick}^{n-1}(X))&n>1,\end{cases}$

where $\operatorname{add}\mathcal{X}$ denotes the smallest full subcategory
containing $\mathcal{X}$ that is closed under finite direct sums and direct
summands. Also, set

$\operatorname{thick}(X):=\bigcup_{n\geq 0}\operatorname{thick}^{n}(X).$

The triangulated category $\mathcal{T}$ is _finitely generated_ when
$\mathcal{T}=\operatorname{thick}(X)$ for some object $X$, and $\mathcal{T}$
is _strongly finitely generated_ if its _dimension_

$\operatorname{dim}\mathcal{T}:=\inf\\{n\geq 0\mid\exists
X\in\mathcal{T}\colon\operatorname{thick}^{n+1}(X)=\mathcal{T}\\}$

is finite.

We continue with some further notation. For a pair of objects $X,Y$ in
$\mathcal{T}$ and $n\geq 0$ we write $h(X,Y)\leq n$ when for any pair
$i,j\in\mathbb{Z}$

$\operatorname{Hom}(X,\Sigma^{i}Y)\neq
0\neq\operatorname{Hom}(X,\Sigma^{j}Y)\;\;\implies\;\;|i-j|<n.$

We set

$\hom^{n}(X):=\\{Y\in\mathcal{T}\mid h(X,Y)\leq n\\}.$

The _amplitude_ of $X$ is

$\operatorname{amp}(X):=\sup\\{|n|\geq
0\mid\operatorname{Hom}(X,\Sigma^{n}X)\neq 0\\}.$

We record some elementary properties of $\hom^{n}(X)$.

######  Lemma 1.

For $X\in\mathcal{T}$ and $n\geq 0$ we have

  1. (1)

$\hom^{0}(X)=\\{Y\in\mathcal{T}\mid\operatorname{Hom}(X,\Sigma^{i}Y)=0\text{
for all }i\in\mathbb{Z}\\}$,

  2. (2)

$\hom^{1}(X)=\\{Y\in\mathcal{T}\mid\operatorname{Hom}(X,\Sigma^{i}Y)\neq
0\text{ for at most one }i\in\mathbb{Z}\\}$,

  3. (3)

$\hom^{n}(X)=\hom^{n}(\Sigma^{i}X)$ for each $i\in\mathbb{Z}$,

  4. (4)

$\hom^{n}(X)=\hom^{n+|i|}(X\oplus\Sigma^{i}X)$ for each $i\in\mathbb{Z}$,

  5. (5)

$\hom^{n}(X)\subseteq\hom^{n}(Y)$ for each $Y\in\operatorname{add}X$,

  6. (6)

$\hom^{n}(X)\subseteq\hom^{n}(Y)$ for each exact triangle $Y^{\prime}\to Y\to
Y^{\prime\prime}\to\Sigma Y^{\prime}$ with
$Y^{\prime},Y^{\prime\prime}\in\operatorname{add}X$.∎

######  Lemma 2.

Let $Y\in\operatorname{thick}(X)$. For each $p\geq 0$ there are inclusions

$\operatorname{thick}^{p}(Y)\subseteq\operatorname{thick}^{q}(X)\quad\text{and}\quad\hom^{p}(X)\subseteq\hom^{q}(Y)\quad\text{for}\quad
q\gg 0.$

###### Proof.

Let $Y\in\operatorname{thick}^{n}(X)$. Then
$\operatorname{thick}^{p}(Y)\subseteq\operatorname{thick}^{p+n}(X)$. The
second inclusion follows from the previous lemma by an induction on the number
$n$ of steps needed to build $Y$ from $X$. ∎

The following definition provides a finiteness condition for triangulated
categories which lies in between ‘finitely generated’ and ‘strongly finitely
generated’.

Recall from [12] that an object $X$ is _homologically finite_ if it satisfies
$h(X,Y)<\infty$ for each object $Y$.

######  Definition 3.

An object $X$ of a triangulated category $\mathcal{T}$ is a _finitistic
generator_ of $\mathcal{T}$ if $X$ is homologically finite and

$\hom^{p}(X)\subseteq\operatorname{thick}^{p}(X)\quad\text{for all}\quad p\geq
0.$

The _finitistic dimension_ 111While the term ‘finitistic dimension’ goes back
to Bass [2], the term ‘finitistic’ is often attributed to Hilbert’s Programme.
Hilbert intended to formalise mathematics based on the use of finite methods.
This was meant as a contribution towards the foundations of mathematics, but
his syzygy theorem (about invariants, and in modern language about modules of
finite projective dimension) may well be considered as an early instance for
this line of thought. In fact, Hilbert mentions ‘the use or the knowledge of a
syzygy’ as a model for the notion of simplicity when he formulated his
unpublished 24th problem that asks for ‘criteria of simplicity, or proof of
the greatest simplicity of certain proofs’. For Hilbert’s 24th problem we
refer to [9, 17], and for an excellent exposition of Hilbert’s finitism, see
[16]. of $\mathcal{T}$ is

$\operatorname{fin{.}dim}\mathcal{T}:=\inf\\{\operatorname{amp}(X)\mid X\text{
is a finitistic generator of }\mathcal{T}\\}.$

As usual, $\operatorname{fin{.}dim}\mathcal{T}=\infty$ if no finitistic
generator exists.

######  Remark 4.

(1) If an object $X$ is homologically finite, then $\mathcal{T}=\bigcup_{p\geq
0}\hom^{p}(X)$. Thus a finitistic generator $X$ of $\mathcal{T}$ satisfies
$\mathcal{T}=\operatorname{thick}(X)$.

(2) If $\mathcal{T}$ admits a finitistic generator, then all objects in
$\mathcal{T}$ are homologically finite.

(3) If $\operatorname{thick}^{n+1}(X)=\mathcal{T}$ for some $X\in\mathcal{T}$,
then $\hom^{p}(Y)\subseteq\operatorname{thick}^{p}(Y)$ for
$Y=X\oplus\Sigma^{n}X$ and all $p\geq 0$. Thus strong finite generation
implies that there is a finitistic generator, provided that all objects are
homologically finite.

Let $A$ be a ring and let $\mathcal{P}(A)$ denote the class of $A$-modules $M$
that admit a finite resolution

$0\longrightarrow P_{n}\longrightarrow\cdots\longrightarrow
P_{1}\longrightarrow P_{0}\longrightarrow M\longrightarrow 0$

with all $P_{i}$ finitely generated projective. We denote

$\operatorname{fin{.}dim}A:=\sup\\{\operatorname{proj{.}dim}M\mid
M\in\mathcal{P}(A)\\}$

the _small finitistic dimension_ of $A$; this is a slight variation of the
original definition in [2] which takes the supremum of
$\operatorname{proj{.}dim}M$ where $M$ runs through all finitely generated
$A$-modules having finite projective dimension. Our definition seems natural
as the modules in $\mathcal{P}(A)$ are precisely the ones which become compact
(or perfect) when viewed as an object in the derived category of $A$. Note
that both definitions coincide when the ring $A$ is noetherian.

We are now in the position to characterise $\operatorname{fin{.}dim}A<\infty$
in terms of the category of perfect complexes. Let $\operatorname{Perf}(A)$
denote the category of perfect complexes over $A$.

######  Theorem 5.

For a ring $A$ we have

$\operatorname{fin{.}dim}A<\infty\;\;\iff\;\;\operatorname{fin{.}dim}\operatorname{Perf}(A)<\infty.$

More precisely, let $X$ be a finitistic generator of $\operatorname{Perf}(A)$.
Then there are $p,q\geq 0$ such that

$\hom^{1}(A)\subseteq\hom^{p}(X)\quad\text{and}\quad\operatorname{thick}^{1}(X)\subseteq\operatorname{thick}^{q}(A),$

which implies

$\operatorname{fin{.}dim}\operatorname{Perf}(A)\leq\operatorname{fin{.}dim}A<p+q.$

We will use the following well known fact, which is a consequence of the ghost
lemma (cf. [8, Lemma 2.4]).

######  Lemma 6.

For $M\in\mathcal{P}(A)$ and $n>0$ we have

$\operatorname{proj{.}dim}M<n\;\;\iff\;\;M\in\operatorname{thick}^{n}(A)\subseteq\mathrm{perf}(A).\qed$

###### Proof of Theorem 5.

The crucial observation is that taking the cohomology of a complex identifies
the objects in $\hom^{1}(A)$ with the modules in $\mathcal{P}(A)$.

Suppose first that $d=\operatorname{fin{.}dim}A<\infty$. We show by induction
on $p\geq 0$ that $\hom^{p}(A)\subseteq\operatorname{thick}^{p+d}(A)$. The
case $p=0$ is clear and we may assume $p>0$. Let $X\in\hom^{p}(A)$ and suppose
its cohomology

$H^{n}(X)=\operatorname{Hom}(A,\Sigma^{n}X)$

is concentrated in degrees $0,\ldots,p-1$. Then we may assume $X^{n}=0$ for
$n\geq p$. For $p=1$ the complex $X$ identifies with
$H^{0}(X)\in\mathcal{P}(A)$ and then Lemma 6 implies
$X\in\operatorname{thick}^{d+1}(A)$. For $p>1$ we may write $X$ as an
extension of the truncation

$X^{\prime}\colon\quad\cdots\longrightarrow X^{p-4}\longrightarrow
X^{p-3}\longrightarrow X^{p-2}\longrightarrow 0\longrightarrow\cdots$

and a complex concentrated in degree $p-1$. We have

$X^{\prime}\in\hom^{p-1}(A)\subseteq\operatorname{thick}^{p-1+d}(A)$

and therefore $X\in\operatorname{thick}^{p+d}(A)$. Now set
$A^{\prime}=A\oplus\Sigma^{d}A$. Then we have

$\hom^{p+d}(A^{\prime})=\hom^{p}(A)\subseteq\operatorname{thick}^{p+d}(A)=\operatorname{thick}^{p+d}(A^{\prime})$

for $p\geq 0$ and $\hom^{d}(A^{\prime})=0$, using Lemma 1. Thus $A^{\prime}$
is a finitistic generator and
$\operatorname{fin{.}dim}\operatorname{Perf}(A)\leq\operatorname{fin{.}dim}A$
follows since $\operatorname{amp}(A^{\prime})=d$.

Suppose now that $X$ is a finitistic generator of $\operatorname{Perf}(A)$.
Using Lemma 2 we have

$\hom^{1}(A)\subseteq\hom^{p}(X)\subseteq\operatorname{thick}^{p}(X)\subseteq\operatorname{thick}^{n}(A)$

for some $n,p\geq 0$. Thus $\operatorname{fin{.}dim}A<n$ by Lemma 6. ∎

######  Remark 7.

There are examples of rings such that $\operatorname{fin{.}dim}A=\infty$ and
$\operatorname{fin{.}dim}A^{\mathrm{op}}=0$; cf. [7]. We have
$\operatorname{Perf}(A^{\mathrm{op}})\simeq\operatorname{Perf}(A)^{\mathrm{op}}$,
and therefore finiteness of finitistic dimension is not a symmetric notion for
triangulated categories.

The following remark collects some known facts about the existence of strong
and finitistic generators of $\operatorname{Perf}(A)$. We give references, and
full proofs are provided in the appendix.

######  Remark 8.

Let $A$ be a noetherian ring. Strong finite generation of
$\operatorname{Perf}(A)$ holds if and only if the global dimension is finite.
Clearly, this implies that $A$ is _regular_ , so each finitely generated
$A$-module has finite projective dimension [15, Proposition 7.25]. In general
(and despite such claims in the literature), the converse is not true even
when $A$ is commutative, because there are regular rings having infinite
finitistic dimension [10, Appendix, Example 1], so

$A\text{
regular}\quad\;\not\\!\\!\\!\\!\implies\quad\operatorname{gl{.}dim}A<\infty.$

However when $A$ is in addition semilocal, then $A$ is regular if and only if
$A$ has finite global dimension [18, Corollary 2]. This provides many examples
where $\operatorname{Perf}(A)$ has a finitistic generator but no strong
generator, because one may take for $A$ any finite dimensional algebra over a
field that has infinite global dimension and finite finitistic dimension
(keeping in mind the question/conjecture from [2] that for finite dimensional
algebras the finitistic dimension is always finite).

######  Example 9.

Let $\mathcal{T}$ be an idempotent complete algebraic triangulated category.
If $\operatorname{fin{.}dim}\mathcal{T}=0$, then there exists a ring $A$ with
$\operatorname{fin{.}dim}A=0$ and a triangle equivalence
$\mathcal{T}\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{Perf}(A)$.

###### Proof.

A generator $X$ of $\mathcal{T}$ with $\operatorname{amp}(X)=0$ is a tilting
object. Thus for $A=\operatorname{End}(X)$ the functor
$\operatorname{RHom}(X,-)$ provides an equivalence
$\mathcal{T}\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{Perf}(A)$;
see [6, Proposition 9.1.20]. Moreover,
$\hom^{1}(A)\subseteq\operatorname{thick}^{1}(A)$ implies
$\operatorname{fin{.}dim}A=0$ by Lemma 6. ∎

## Appendix: Strong finite generation

In this appendix we record with proofs some facts that are well known, but
only to those knowing them well. Let $\mathcal{A}$ be an abelian category with
a projective generator $P$. Thus any object $X\in\mathcal{A}$ admits an
epimorphism $P^{n}\to X$ for some positive integer $n$. Let $\mathcal{P}$
denote the full exact subcategory of projective objects in $\mathcal{A}$. We
consider the derived categories of bounded complexes and the inclusion
$\mathcal{P}\to\mathcal{A}$ induces a triangle equivalence

$\mathbf{D}^{\mathrm{b}}(\mathcal{P})\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{thick}(P)\subseteq\mathbf{D}^{\mathrm{b}}(\mathcal{A})$

which we view as an identification.

######  Proposition 10.

The following holds for
$\mathbf{D}^{\mathrm{b}}(\mathcal{P})\subseteq\mathbf{D}^{\mathrm{b}}(\mathcal{A})$.

  1. (1)

$\mathbf{D}^{\mathrm{b}}(\mathcal{P})=\mathbf{D}^{\mathrm{b}}(\mathcal{A})$
$\iff$
$\\{X\in\mathcal{A}\mid\operatorname{proj{.}dim}X<\infty\\}=\mathcal{A}$.

  2. (2)

$\operatorname{dim}\mathbf{D}^{\mathrm{b}}(\mathcal{P})<\infty$ $\iff$
$\operatorname{gl{.}dim}\mathcal{A}<\infty$.

###### Proof.

(1) The equality
$\mathbf{D}^{\mathrm{b}}(\mathcal{P})=\mathbf{D}^{\mathrm{b}}(\mathcal{A})$
holds if and only if each object from $\mathcal{A}$, viewed as a complex
concentrated in degree zero, belongs to
$\mathbf{D}^{\mathrm{b}}(\mathcal{P})$. But an object from $\mathcal{A}$
belongs to $\mathbf{D}^{\mathrm{b}}(\mathcal{P})$ precisely when it has finite
projective dimension.

(2) Suppose first that $\operatorname{gl{.}dim}\mathcal{A}<n$. Then
$\operatorname{thick}^{n}(P)=\mathbf{D}^{\mathrm{b}}(\mathcal{A})$ by [4,
Theorem 8.3] or [8, Lemma 2.5]. For the converse suppose that
$\operatorname{thick}(P)=\operatorname{thick}^{n}(P)$ for some integer $n$. We
claim that $\operatorname{gl{.}dim}\mathcal{A}<n$. This follows from Lemma 6
once we know that each object $X\in\mathcal{A}$ has finite projective
dimension. To see this choose a projective resolution $Q\to X$ and for $1\leq
i\leq n$ let $\phi_{i}$ denote the following chain map between appropriate
truncations of $Q$:

${\cdots}$${0}$${0}$${Q^{-n-i}}$${\cdots}$${Q^{-i}}$${Q^{-i+1}}$${0}$${\cdots}$${\cdots}$${0}$${Q^{-n-i-1}}$${Q^{-n-i}}$${\cdots}$${Q^{-i}}$${0}$${0}$${\cdots}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$

It is clear that $H^{*}(\phi_{i})=0$ for all $1\leq i\leq n$. Thus
$\operatorname{Hom}(-,\phi_{i})$ vanishes on $\operatorname{thick}^{1}(P)$,
and therefore $\operatorname{Hom}(-,\phi)$ vanishes on
$\operatorname{thick}^{n}(P)$ for $\phi=\phi_{n}\circ\cdots\circ\phi_{1}$.
This follows from the ghost lemma and implies that $\phi$ is null-homotopic
since $\operatorname{thick}(P)=\operatorname{thick}^{n}(P)$; see [15, Lemma
4.11] or [5, Theorem 1]. Then it follows that the kernel of the differential
$Q^{-n}\to Q^{-n+1}$ is projective. Thus $X$ has finite projective dimension.
∎

One may ask for general criteria on $\mathcal{A}$ such that the conditions in
(1) and (2) are equivalent. To indicate some connection we include the
following.

######  Lemma 11.

Suppose that every non-zero object in $\mathcal{A}$ has a projective cover and
a maximal subobject. Then we have

$\operatorname{gl{.}dim}\mathcal{A}<\infty\quad\iff\quad\\{X\in\mathcal{A}\mid\operatorname{inj{.}dim}X<\infty\\}=\mathcal{A}.$

###### Proof.

One direction is clear and requires no assumption on $\mathcal{A}$. For the
converse it is convenient to identify $\mathcal{A}$ with the category of
finitely presented modules over $\operatorname{End}(P)$ via the functor
$X\mapsto\operatorname{Hom}(P,X)$. The assumption on $\mathcal{A}$ implies
that this ring is semiperfect, and therefore $S=P/\operatorname{rad}P$ is
semisimple. We claim that $X\in\mathcal{A}$ is projective if and only if
$\operatorname{Ext}^{1}(X,S)=0$. To see this choose a projective cover

$\xi\colon\quad 0\longrightarrow\Omega(X)\longrightarrow P(X)\longrightarrow
X\longrightarrow 0.$

If $\Omega(X)\neq 0$, then there is a non-zero morphism $\pi\colon\Omega(X)\to
S$ such that $\pi\circ\xi\neq 0$ in $\operatorname{Ext}^{1}(X,S)$ since
$\Omega(X)\subseteq\operatorname{rad}P(X)$. From this characterisation of
projectivity it follows that $\operatorname{gl{.}dim}\mathcal{A}$ is bounded
by $\operatorname{inj{.}dim}S$. ∎

### Acknowledgement

I am grateful to Janina Letz for useful comments and for sharing her
computations of the finitistic dimension in some interesting examples. This
work was supported by the Deutsche Forschungsgemeinschaft (SFB-TRR 358/1 2023
- 491392403).

## References

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11institutetext: Lehigh University, Computer Science and Engineering,
Bethlehem, PA, USA
11email<EMAIL_ADDRESS>
# Adversarial Regression Learning
for Bone Age Estimation

Youshan Zhang  Brian D. Davison

###### Abstract

Estimation of bone age from hand radiographs is essential to determine
skeletal age in diagnosing endocrine disorders and depicting the growth status
of children. However, existing automatic methods only apply their models to
test images without considering the discrepancy between training samples and
test samples, which will lead to a lower generalization ability. In this
paper, we propose an adversarial regression learning network ($ARLNet$) for
bone age estimation. Specifically, we first extract bone features from a fine-
tuned Inception V3 neural network and propose regression percentage loss for
training. To reduce the discrepancy between training and test data, we then
propose adversarial regression loss and feature reconstruction loss to
guarantee the transition from training data to test data and vice versa,
preserving invariant features from both training and test data. Experimental
results show that the proposed model outperforms state-of-the-art methods.

###### Keywords:

Adversarial learning Dataset shift Bone age estimation.

##  1 Introduction

Skeletal age estimation from hand radiology images is extensively used in
endocrinological disease diagnosis, judgment of children’s growth, and genetic
disorder diagnoses [1]. Bone age basically reflects the appearance of the
hand. As children grow, their bones become longer and change from cartilage to
proper bones, such that we can estimate how old a child is based on the
average age of children with similar bone images.

Bone age assessment starts with taking X-ray images of children’s hands. For
decades, the assessment of bone age was usually based on manual visual
assessment of bone development of palms and wrists [2, 3]. Hence, it is
tedious and error prone. Therefore, it is necessary to develop an accurate
automatic bone age algorithm. Recently, deep neural networks have become
widely used in the medical field. Automatic bone age estimation using deep
learning methods is much faster than manual labeling, while accuracy exceeds
conventional methods [4]. Most existing deep learning based methods take
advantage of well-trained models on ImageNet datasets such as VGG16 [5],
ResNet50 [6], or Inception V3 [7] for feature extraction, and add a regression
layer to output bone age.

Although many methods achieve effective results in bone age estimation, they
still suffer from one challenge: models are only optimized with training data,
while the differences between training and test samples are omitted. Such
models hence have a lower generalization ability to test samples if they are
different from training data. To address this challenge, we aggregate three
different loss functions in one framework: regression percentage error loss,
adversarial regression loss, and feature reconstruction loss.

The contributions of this paper are three-fold:

  1. 1.

To reduce the data discrepancy, we first extract features from a fine-tuned
Inception V3 neural network and propose a regression percentage error loss,
which addresses both individual prediction error and mean prediction error;

  2. 2.

To the best of our knowledge, we are the first to propose an adversarial
regression loss to reduce the data difference during the training;

  3. 3.

The proposed feature reconstruction loss is able to maintain the feature
consistency information of training and test samples.

Extensive experiments are conducted on a large-scale Bone-Age dataset and
achieve superior results over state-of-the-art methods. Our architecture is
also successfully applied to another regression task: face age estimation.

##  2 Related work

In recent decades, bone age estimation has changed from manual assessment to
automatic estimation algorithms. Deep learning methods have proved to be
better than traditional machine learning methods in estimating in bone age.

Halabi et al. [8] reported the best five models for bone age estimation of the
Radiological Society of North America (RSNA) challenge. The best model
utilized the Inception V3 architecture concatenated with gender information.
Data augmentation was leveraged to prevent overfitting and improve
performance. The best performance achieves an error of 4.2 months (mean
absolute difference, or MAD) in the challenge. Iglovikov et al. [9] first used
a dilated U-net to segment hand, and then they removed the background and
normalized images. They finally calculated the affine transformations to
register images. The bone age assessment model consists of a VGG-style
regression model and classification model. The classification model aimed to
output the bone age class, while the regression model output the bone age. The
model achieved 4.20 months MAD. Chen et al. [10] proposed an attention-guided
approach to automatically localize the discriminative regions to estimate bone
age. They then aggregated different regions for bone age assessment. Their
model achieved 4.7 months MAD. Similarly, Escoba et al. [1] developed an
approach to focus on local analysis of anatomical ROIs. They presented the
additional bounding boxes and ROIs annotations during training and proposed a
hand detection and hand pose estimation model to extract local information for
bone age estimation. Pan et al. [11] combined four different less-correlated
and relatively high-performing models, and they found that creating a ensemble
model based on several weak models convincingly outperformed single-model
prediction for bone age assessment.

(a) Sample training images

(b) Sample test images (can be quite faint)

Figure 1: Some bone radiology images from training and test data (differences
between the training and test samples cause the difficulty of transfer
learning).

Transfer learning applies existing knowledge to a new domain. Although many
methods take advantage of pre-trained ImageNet models to estimate the bone
age, they do not consider the differences between training and testing data,
which can sometimes be quite obvious, as shown in Fig. 1. The discrepancy of
data (named data shift/bias) will cause a vulnerability in the trained model,
which causes poorer generalization on test data [12, 13]. Transductive
transfer learning uses both labeled training samples and unlabeled test
samples to train the model and then uses a trained model to infer the label of
the unlabeled test set. Hence, the data bias is mitigated via such a paradigm,
which can improve the performance of the test data [14]. In this paper, we
propose adversarial regression learning to estimate bone age and
simultaneously reduce the data shift between training and test datasets.

##  3 Method

###  3.1 Motivation

As shown in Fig. 1, we observe differences between training and test data.
Unlike previous work that only optimizes the model based on training data, we
utilize the idea of transductive learning—we train the neural network using
both the labeled training and unlabeled test data, which reduces the
discrepancy between them. We adopt learning theory from [15] that the test
risk can be minimized via bounding the training risk and discrepancy between
them as follows.

Theorem 1 Let $h$ be a hypothesis, $\epsilon_{tr}(h)$ and $\epsilon_{te}(h)$
represents the training and test risk (or error), respectively.

$\displaystyle\epsilon_{te}(h)\leq\epsilon_{tr}(h)+d_{\mathcal{H}}(\mathcal{D}_{tr},\mathcal{D}_{te})+C$
(1)

where $d_{\mathcal{H}}(\mathcal{D}_{tr},\mathcal{D}_{te})$ is the
$\mathcal{H}$-divergence of training and test data, $C$ is the adaptability to
quantify the error in an ideal hypothesis space of training and test data,
which should be a sufficiently small constant.

###  3.2 Problem

Bone age estimation is a regression problem. Given training data
$\mathcal{X}_{tr}$ (including bone radiology image and gender information)
with its labels $\mathcal{Y}_{tr}\in(0,\mathbb{R}^{+})^{N_{tr}}$ and test data
$\mathcal{X}_{te}$ without its labels
($\mathcal{Y}_{te}\in(0,\mathbb{R}^{+})^{N_{te}}$ for evaluation only), the
goal in bone age estimation is to learn a regressor $\mathcal{R}$ to minimize
the test data risk and reduce the discrepancy between training and test data.

For most existing models, in the absence of data shift, regression models
simply learn a regressor $\mathcal{R}$ that performs the task on training
data, and minimizes the loss function in Eq. 2:

$\mathcal{L}_{tr}(\mathcal{X}_{tr},\mathcal{Y}_{tr})=\mathbb{E}[\ell(\mathcal{R}(\mathcal{X}_{tr}),\mathcal{Y}_{tr})],$
(2)

where $\mathbb{E}[\cdot]$ is the expectation and $\ell$ can be any appropriate
loss function. Eq. 2 only minimizes the training risk $\epsilon_{tr}(h)$.

We define $\ell$ as the regression percentage error loss,
$\ell=\mathcal{L_{P}}=\mathcal{L_{M}}+\alpha\mathcal{L_{D}}$, where
$\mathcal{L_{M}}$ is the mean absolute percentage error loss,
$\mathcal{L_{D}}$ is the proposed absolute mean discrepancy loss and $\alpha$
is the balance factor between two loss functions. Specifically,
$\mathcal{L_{M}}=\frac{1}{N_{tr}}\sum_{i=1}^{N_{tr}}|\frac{\mathcal{Y}_{tr_{i}}-\mathcal{Y}_{tr_{i}}^{\prime}}{\mathcal{Y}_{tr_{i}}}|,\
\ \
\mathcal{L_{D}}=|\frac{\overline{\mathcal{Y}_{tr}}-\overline{\mathcal{Y}_{tr}^{\prime}}}{\overline{\mathcal{Y}_{tr}}}|$,
where $\overline{\mathcal{Y}_{tr}}$ and $\overline{\mathcal{Y}_{tr}^{\prime}}$
denote the mean actual and predicted bone age in training dataset. For
training data, such a loss $\ell$ can capture both individual and mean
percentage error.

In Eq. 2, the loss of training data did not reduce the data shift issue.
Therefore, previous works have lower generalization to the test data. To
mitigate the effects of data shift, we follow previous adversarial learning
[16, 17] to map samples across different domains to a common space, and then
invariant features are maintained. Hence, our model can learn on the training
data, while retaining high generalization ability on test data.

Figure 2: The architecture of our proposed adversarial regression learning
network ($ARLNet$). Training and test features are first extracted via $G$.
Then, $G(\mathcal{X}_{tr})$ and $G(\mathcal{X}_{te})$ are fed into shared
layers. The concatenate layer combines features from preceding layers and the
gender. The weights of shared layers are updated by labeled training and
unlabeled test data. $\Phi_{tr\shortrightarrow te}$ and
$\Phi_{te\shortrightarrow tr}$ is the reconstruction from training data to
test data and from test data to training data using hidden reconstruction
layers (the detailed reconstruction layers are in Fig. 3). $\mathcal{L_{P}}$:
regression percentage error loss, $\mathcal{L_{AR}}$: adversarial regression
loss, $\mathcal{L}_{Recon}$: feature reconstruction loss, $\mathcal{R}$:
regressor and $\mathcal{D_{R}}$: data regressor

.

Figure 3: Feature reconstruction layers ($\mathcal{L}_{Recon}$). The shared
layers are optimized by both $G(\mathcal{X}_{tr})$ and $G(\mathcal{X}_{te})$.
For $G(\mathcal{X}_{tr})$, the output from reconstruction layers should be
closer to $G(\mathcal{X}_{tr})$, but it is under the conversion of
$\Phi_{te\shortrightarrow tr}$ since the shared layers are also optimized by
$G(\mathcal{X}_{te})$, it is hence from $te$ to $tr$. Similarly, the output of
reconstruction layer for $G(\mathcal{X}_{te})$ is under the conversion of
$\Phi_{tr\shortrightarrow te}$.

In our framework, we employ a two-stage data discrepancy reduction procedure.
We first extract bone features from any pre-trained or fine-tuned network. The
distribution of training and test data are initially aligned, which reduced
the discrepancy between them. We then use adversarial regression learning to
find invariant features to further reduce the differences between training and
test data. $d_{\mathcal{H}}(\mathcal{D}_{tr},\mathcal{D}_{te})\approx
Dist(\mathcal{X}_{tr},\mathcal{X}_{te})+Dist(G(\mathcal{X}_{tr}),G(\mathcal{X}_{te})),$
specifically,

$\displaystyle
Dist(\mathcal{X}_{tr},\mathcal{X}_{te})=||\frac{1}{N_{tr}}\sum_{i=1}^{N_{tr}}G(\mathcal{X}_{tr}^{i})-\frac{1}{N_{te}}\sum_{j=1}^{N_{te}}G(\mathcal{X}_{te}^{j})||_{\mathcal{H}}$
$\displaystyle
Dist(G(\mathcal{X}_{tr}),G(\mathcal{X}_{te}))=||\frac{1}{N_{tr}}\sum_{i=1}^{N_{tr}}\Phi(G(\mathcal{X}_{tr}^{i}))-\frac{1}{N_{te}}\sum_{j=1}^{N_{te}}\Phi(G(\mathcal{X}_{te}^{j}))||_{\mathcal{H}}$

where $G$ is feature extractor from any pre-trained or fine-tuned neural
network, and $\Phi$ is feature extractor from adversarial regression learning
in Sec. 3.3. $\mathcal{H}$ is the Reproducing Kernel Hilbert Space (RKHS)
space.

###  3.3 Adversarial regression learning

Adversarial learning is widely used to mitigate data shift issues [16, 17]. It
minimizes the domain discrepancy by a feature extractor and a domain
discriminator. The domain discriminator aims to distinguish the source domain
from the target domain, while the feature extractor aims to learn domain-
invariant representations to fool the domain discriminator. Given data
representations from feature extractor $G$, we can learn a discriminator $D$,
which can distinguish the two domains using the following binary cross-entropy
loss function:

$\displaystyle\mathcal{L_{A}}(G(\mathcal{X}_{tr}),$ $\displaystyle
G(\mathcal{X}_{te}))=-\mathbb{E}_{x_{tr}\sim
G(\mathcal{X}_{tr})}[\text{log}D(x_{tr})]-\mathbb{E}_{x_{te}\sim
G(\mathcal{X}_{te})}[\text{log}(1-D(x_{te}))]$ (3)

However, adversarial learning is typically applied in classification problems.
The binary cross-entropy in Eq. 3 is also frequently used in improving the
accuracy of classification problems. The cross-entropy loss is not proper to
indicate the data difference in a regression problem. Therefore, we need a
regression loss function to distinguish the training and test datasets, and we
adopt the adversarial learning for the regression problem as follows.

$\displaystyle\mathcal{L_{AR}}(\Phi_{tr\shortrightarrow
te},G(\mathcal{X}_{tr}),G(\mathcal{X}_{te}))=\mathbb{E}_{x\sim(\mathcal{X}_{tr}\cup\mathcal{X}_{te})}[\ell(\mathcal{D_{R}}(x),\mathcal{Y}_{\mathcal{DL}})]$
(4)
$\displaystyle=\frac{1}{n_{tr}+n_{te}}\sum_{k=1}^{n_{tr}+n_{te}}|\frac{\mathcal{Y}^{k}_{\mathcal{{DL}}}-\mathcal{Y}^{k^{\prime}}_{\mathcal{{DL}}}}{\mathcal{Y}^{k}_{\mathcal{{DL}}}+\epsilon}|+|\frac{\overline{\mathcal{\mathcal{Y}_{\mathcal{{DL}}}}}-\overline{\mathcal{Y}^{{}^{\prime}}_{\mathcal{DL}}}}{\overline{\mathcal{Y}_{\mathcal{{DL}}}+\epsilon}}|$

where $\mathcal{L_{AR}}$ is adversarial regression loss,
$\Phi_{tr\shortrightarrow te}$ is the mapping from training to test data;
$\mathcal{D_{R}}$ is the adapted data regressor, and
$\mathcal{X}_{tr}\cup\mathcal{X}_{te}$ contains all training and test samples;
$\mathcal{Y}_{\mathcal{{DL}}}$ is the data label in Fig. 2, specifically, $0$
is the label for the training data and $1$ is the label for the test data;
$\mathcal{Y}^{k^{\prime}}_{\mathcal{{DL}}}$ is the prediction of data label;
$\overline{\mathcal{Y}_{\mathcal{DL}}}$ and
$\overline{\mathcal{Y}^{{}^{\prime}}_{\mathcal{DL}}}$ denote the mean actual
and predicted values of training or test dataset and $\epsilon$ is a small
number (1e-9) to prevent division by 0. $\mathcal{L_{AR}}$ can measure how
well-matched the training and test data are from both individual and overall
differences. A perfect regression model would have an adversarial regression
loss of 0.

To this end, the data regressor $\mathcal{D_{R}}$ learns the data discrepancy
by maximizing adversarial regression loss $\mathcal{L_{AR}}$ with the fixed
$\Phi$, and the feature extractor $\Phi$ aims to learn domain-invariant
representations via minimizing $\mathcal{L_{AR}}$ with the optimal regressor
$\mathcal{R}$. Eq. LABEL:eq:regre guarantees that $tr\shortrightarrow te$,
that is, given training samples, it will learn a map from training to test
samples, while minimizing the
$\mathcal{L}_{tr}(\mathcal{X}_{tr},\mathcal{Y}_{tr})$ in Eq. 2. However, Eq.
LABEL:eq:regre only guarantees training data close to test data, and it does
not ensure that $\Phi_{te\shortrightarrow tr}$ maintains the features of the
training samples. We hence introduce another mapping from test data to
training data $\Phi_{te\shortrightarrow tr}$ and train it with the same
adversarial regression loss as in $\Phi_{tr\shortrightarrow te}$ as shown in
Eq. 5. The only difference is that the $1$ is the new data label for training
data, and $0$ is the new data label for test data.

$\displaystyle\mathcal{L_{AR}}(\Phi_{te\shortrightarrow
tr},G(\mathcal{X}_{tr}),G(\mathcal{X}_{te}))$ (5)

Therefore, we define the adversarial regression loss as:

$\displaystyle\mathcal{L_{AR}}(G(\mathcal{X}_{tr}),G(\mathcal{X}_{te}))$
$\displaystyle=\mathcal{L_{AR}}(\Phi_{tr\shortrightarrow
te},G(\mathcal{X}_{tr}),G(\mathcal{X}_{te}))$ (6)
$\displaystyle+\mathcal{L_{AR}}(\Phi_{te\shortrightarrow
tr},G(\mathcal{X}_{tr}),G(\mathcal{X}_{te}))$

To encourage the training and test information to be preserved during the
adversarial regression learning, we propose a feature reconstruction loss in
our model. Details of the feature reconstruction layers are shown in Fig. 3;
the reconstruction layers are right behind the shared layers, and it aims to
reconstruct extracted features and maintain feature consistency during the
conversion process without losing features. The feature reconstruction loss is
defined as:

$\displaystyle\mathcal{L}$ $\displaystyle{}_{Recon}(\Phi_{tr\shortrightarrow
te},\Phi_{te\shortrightarrow tr},G(\mathcal{X}_{tr}),G(\mathcal{X}_{te}))$ (7)
$\displaystyle=\mathbb{E}_{x_{tr}\sim
G(\mathcal{X}_{tr})}[\ell^{\prime}(\Phi_{te\shortrightarrow
tr}(\Phi_{tr\shortrightarrow te}(x_{tr}))-x_{tr})]$
$\displaystyle+\mathbb{E}_{x_{te}\sim
G(\mathcal{X}_{te})}[\ell^{\prime}(\Phi_{tr\shortrightarrow
te}(\Phi_{te\shortrightarrow tr}(x_{te}))-x_{te})]$

where $\ell^{\prime}$ is the mean square error loss, which calculates the
difference between true features and reconstructed features.

###  3.4 Overall objective

We combine the three loss functions to formalize our objective function:

$\displaystyle\mathcal{L}(\mathcal{X}_{tr},\mathcal{X}_{te},\mathcal{Y}_{tr},\Phi_{tr\shortrightarrow
te},\Phi_{te\shortrightarrow tr})$
$\displaystyle=\mathcal{L}_{tr}(\mathcal{X}_{tr},\mathcal{Y}_{tr})+\beta\mathcal{L_{AR}}(G(\mathcal{X}_{tr}),G(\mathcal{X}_{te}))$
(8) $\displaystyle+\gamma\mathcal{L}_{Recon}(\Phi_{tr\shortrightarrow
te},\Phi_{te\shortrightarrow tr},G(\mathcal{X}_{tr}),G(\mathcal{X}_{te})),$

where $\beta$ and $\gamma$ are trade-off parameters. Our model ultimately
solves the following optimization problem. It minimizes the difference during
the transition from the training to test data and from test to training data.
Meanwhile, it maximizes the identification ability of training or test
data.111Source code is available at
https://github.com/YoushanZhang/Adversarial-Regression-Learning-for-Bone-Age-
Estimation.

$\displaystyle\mathop{\operatorname*{arg\,min}}\min_{\begin{subarray}{c}\Phi_{tr\shortrightarrow
te}\\\ \Phi_{te\shortrightarrow
tr}\end{subarray}}\max_{\mathcal{D}_{tr},\mathcal{D}_{te}}$
$\displaystyle\mathcal{L}(\mathcal{X}_{tr},\mathcal{X}_{te},\mathcal{Y}_{tr},\Phi_{tr\shortrightarrow
te},\Phi_{te\shortrightarrow tr})$

##  4 Experiments

We primarily validate our methods using the Bone-Age dataset. To demonstrate
the generalizability of the architecture of $ARLNet$, we also evaluate our
approach on the task of predicting age using two face datasets.

###  4.1 Datasets

#### Bone-Age

includes data from the 2017 Pediatric Bone Age Challenge, which is organized
by the Radiological Society of North America (RSNA). The statistics of the
Bone-Age dataset are shown in Tab. 1 (Note that the test data is clearly
different from the training data.) It also includes gender information
associated with the bone age.

#### Face-Age

is from the UTKFace dataset, which contains 9780 images with ages from 1 to
116. It also includes gender information with its associated images (more
details can be found in Zhang et al. [18]). We also consider age regression in
this dataset. The statistics of Face-Age dataset are shown in Tab. 2.

Table 1: Statistics of Bone-Age dataset Bone-Age | # Total | # Male | # Female | Bone ages
---|---|---|---|---
Training set | 12611 | 6833 | 5778 | 10.8 $\pm$ 3.5
Validation set | 1425 | 773 | 652 | 10.8 $\pm$ 3.5
Test set | 200 | 100 | 100 | 8.8 $\pm$ 3.6
Table 2: Statistics of Face-Age dataset Face-Age | # Total | # Male | # Female | Face ages
---|---|---|---|---
Training set | 4890 | 2183 | 2707 | 29.43 $\pm$ 24.79
Test set | 4890 | 2189 | 2701 | 29.41 $\pm$ 24.76

#### MORPH II

is from MORPH database, which contains more than 55,000 face images of 13,000
individuals aged from 16 to 77 years. It includes images with detailed age,
gender, and many ethnicities). We followed the training/testing settings of
Shen et al. [19] in the experiments, which selects 5,492 images of Caucasians.
The final performance is reported using five-fold cross-validation.

###  4.2 Implementation details

In the Bone-Age dataset, the features are extracted from an Inception V3
neural network through the last fully connected layer. We attach a regression
layer to output the bone age, and gender is also another input for the model.
One represented feature vector has the size of $1\times 1000$ and is
corresponding to one bone image. For the Bone-Age dataset, the represented
feature vectors for training, validation, and test data have a size of
$11611\times 1000$, $1425\times 1000$, and $200\times 1000$, respectively.
Similarly, we extract features from the last fully connected layer in a pre-
trained Inception V3 neural network for the Face-Age dataset. The size of
feature vectors of training and test data is $4890\times 1000$. For MORPH II,
the size of feature vectors of training and test data are $4394\times 1000$,
and $1098\times 1000$, respectively. We then train the model based on these
extracted feature vectors. The parameters of $ARLNet$ are first tuned based on
the performance of the validation dataset of the Bone-Age dataset. We then
applied these parameters to test data of Bone-Age, Face-Age and MORPH II
datasets.

The numbers of units of the dense layer in shared layers are 512, 8, and 1,
while the numbers of units in the reconstruction layers are opposite (1, 8,
and 512). The dropout rate is 0.5. Our implementation is based on Keras and
the parameters settings are $\alpha=10$, $\beta=\gamma=0.5$, learning rate:
$\eta=0.0001$, batch size = 128, number of iterations is 300 and the optimizer
is Adam. We first train the regression model using the percentage error loss
on the labeled training data. Next, we perform adversarial regression learning
using the extracted features and feature reconstruction loss, which yields the
learned parameters for the feature transformations $\Phi_{tr\shortrightarrow
te}$ and $\Phi_{te\shortrightarrow tr}$. We also compare our results with 15
state-of-the-art methods (including both traditional methods and deep neural
networks). All re-implemented methods are marked in bold in Tables 4 and 4.

###  4.3 Evaluation

We use mean absolute error to evaluate our model:
$\text{MAE}=\frac{1}{N}\sum\nolimits_{i=1}^{N}(|y_{i}-\hat{y_{i}}|),$ where
$y$ is the provided age, and $\hat{y}$ is the predicted age.

#### Bone-Age.

The comparison of MAE is listed in Tab. 4. We find that $ARLNet$ has the
lowest MAE versus all other approaches (23% reduction from the best baseline).
Specifically, online software has the worst performance among all methods. It
is a pruned version of Inception V3 [8], which excludes gender information,
and the difference in performance between the two models is significant (more
than 65% reduction). The MAE of Inception V3 is 4.20 months, while MAE of the
pruned version is 12.35 months. This demonstrates that gender is important in
predicting bone age. The performance of online software could be regarded as a
lower bound. SVR is a traditional method, and its performance relies on the
extracted features; it also does not consider the discrepancy between training
and test data. GSM has better results than the SVR model since it samples more
data between training and test data, which reduces the discrepancy between
them. The performance of pre-trained models (VGG16 and Xception) also leads to
higher MAE values. The is because the pre-trained model is trained based on
the ImageNet dataset, while it does not contain information from bone images.
In addition, we observe that two similar methods, DANN and ADDA, also have
high MAE values. There are two possible reasons. First, the feature extraction
layers of DANN and ADDA models are too shallow to extract detailed information
for radiology bone images. Second, the adversarial learning of these two
models only considers the transition from training data to test data.

Table 3: Results comparison of Bone-Age dataset Methods | MAE
---|---
Online software [20] | 12.35
SVR [21] | 10.70
VGG16 [5] | 7.57
GSM [14] | 6.06
DANN [16] | 5.26
ADDA [17] | 4.91
Xception [22] | 4.39
ResNet50 [4] | 6.00
U-Net-VGG [9] | 4.97
Ice Module [8] | 4.40
Inception V3 [8] | 4.20
Ensembles Models [11] | 3.93
$ARLNet$ | 3.01

Table 4: MAE comparison on Face-Age and MORPH II datasets  Methods | Face-Age | MORPH II
---|---|---
VGG16 [5] | 10.40 | 5.37
VGG19 [5] | 9.64 | 4.93
SVR [21] | 9.10 | 5.77
ResNet50 [6] | 8.64 | 4.02
Xception [22] | 8.49 | 3.88
Inception V3 [22] | 8.36 | 3.65
GSM [14] | 8.06 | 3.35
DANN [16] | 6.26 | 3.01
ADDA [17] | 5.91 | 2.73
ARN [23] | - | 3.00
DRFs [19] | - | 2.91
DCNN [24] | - | 2.75
$ARLNet$ | 4.50 | 2.28

#### Face-Age.

The comparison of MAE is listed in Tab. 4. $ARLNet$ again has the lowest MAE
value, outperforming all others (24% reduction than the best baseline). We
notice that results from Inception V3 are close to our model since our model
is based on extracted features from the pre-trained Inception V3. However, our
model has lower MAE than Inception V3 model, which demonstrates that the
adversarial regression learning is useful in the regression problem. We find
that the SVR model is better than some neural networks (VGG16 and Googlenet)
since SVR uses the features from the Inception V3 model. SVR normally has a
higher error than the fine-tuned IncetionV3 network. In addition, the
performance of two domain adaptation methods (DANN and ADDA) have results
close to that of our model. The underlying reason is that face images are
easier to find data discrepancy and extract features since images are RGB
images, while radiology bone images are significantly different from RGB
images, it is difficult for these two methods to extract fine-grained bone
features.

#### MORPH II.

Tab. 4 also compares performance between $ARLNet$ and state-of-the-art
methods. ARLNet achieves the best performance, reducing the error rate by
17.1% compared to the best baseline model (DCNN), suggesting that adversarial
regression learning is effective in reducing the discrepancy between training
and test data, and achieves lower MAE.

##  5 Discussion

There are three reasons that our model outperforms state-of-the-art methods.
First, the proposed percentage loss $\ell$ is able to consider both the
individual and mean percentage error. Second, we propose adversarial
regression loss, which can maintain transition from training data to test data
and transition from test data to training data. Third, we propose feature
reconstruction loss, which further guarantees the consistency of training and
test samples.

Table 5: Ablation experiments of effects of different loss functions of Bone-Age dataset Methods |  MAE
---|---
$ARLNet$$-\mathcal{L}_{D}-\mathcal{L}_{AR}-\mathcal{L}_{Recon}$ |  4.52
$ARLNet$$-\mathcal{L}_{M}-\mathcal{L}_{AR}-\mathcal{L}_{Recon}$ |  4.38
$ARLNet$$-\mathcal{L}_{D}-\mathcal{L}_{Recon}$ |  4.17
$ARLNet$$-\mathcal{L}_{M}-\mathcal{L}_{Recon}$ |  4.03
$ARLNet-\mathcal{L}_{AR}-\mathcal{L}_{Recon}$ |  3.91
$ARLNet-\mathcal{L}_{AR}$ |  3.45
$ARLNet-\mathcal{L}_{Recon}$ |  3.26
$ARLNet$ |  3.01

#### Ablation study.

To better demonstrate the performance of our model, we report the effects of
different loss functions on classification accuracy in Tab. 5
($\mathcal{L_{D}}$: absolute mean discrepancy loss, $\mathcal{L_{M}}$: mean
absolute percentage error loss, and $\mathcal{L_{AR}}$: adversarial regression
loss and $\mathcal{L}_{Recon}$: feature reconstruction loss).
“$ARLNet-\mathcal{L}_{D}-\mathcal{L}_{AR}-\mathcal{L}_{Recon}$” is implemented
without absolute mean discrepancy loss, adversarial regression loss, and
feature reconstruction loss. It is a simple model, which only considers the
labeled training data using mean absolute percentage error loss.
“$ARLNet-\mathcal{L}_{AR}-\mathcal{L}_{Recon}$” reports results without
performing the additional adversarial regression loss and feature
reconstruction loss. “$ARLNet-\mathcal{L}_{Recon}$” trains the model with
percentage error loss and adversarial regression loss without the feature
reconstruction loss. We observe that with the increasing of the number of loss
functions, the robustness of our model keeps improving. Therefore, we can
conclude that all these different loss functions are important in maximizing
regression performance.

#### What can we learn from ARLNet?

As shown in the ablation study, we know the effects of different loss
functions. The adversarial regression loss has a dominant effect on the final
results. Differing from traditional machine learning that only optimizes
models using training data, our $ARLNet$ considers transductive learning [14],
and reduces the discrepancy between training and test data. It will be useful
to include the test data during the training without any labels. Although
there are no labels for the test data, the discrepancy between training
samples and test samples is minimized, the test risk is thus reduced if there
is a difference between training and test data. Therefore, the performance of
the model will be improved if we feed the test data during the training
processes with ARL.

##  6 Conclusion

We have presented an adversarial regression learning network ($ARLNet$) for
bone age estimation that reduces the discrepancy between training and test
data. In particular, we consider the problem from the traditional training
protocol to adversarial regression learning. The adversarial regression
learning consists of adversarial regression and feature reconstruction losses.
The adversarial regression loss can push the prototype of bone ages computed
in either training or test data close in the embedding space, and maintain
invariant representations across two datasets. In addition, the proposed
feature reconstruction loss further guarantees the structure and content from
training and test data, and it will take the decision of regressor into
account to align feature distribution, which leads to domain-invariant
representations. Our approach provides a more than 20% error reduction over
the state of the art in two age regression tasks.

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