# Fair Training of Decision Tree Classifiers

Francesco Ranzato1, Caterina Urban2, Marco Zanella1

###### Abstract

We study the problem of formally verifying individual fairness of decision
tree ensembles, as well as training tree models which maximize both accuracy
and individual fairness. In our approach, fairness verification and fairness-
aware training both rely on a notion of stability of a classification model,
which is a variant of standard robustness under input perturbations used in
adversarial machine learning. Our verification and training methods leverage
abstract interpretation, a well established technique for static program
analysis which is able to automatically infer assertions about stability
properties of decision trees. By relying on a tool for adversarial training of
decision trees, our fairness-aware learning method has been implemented and
experimentally evaluated on the reference datasets used to assess fairness
properties. The experimental results show that our approach is able to train
tree models exhibiting a high degree of individual fairness w.r.t. the natural
state-of-the-art CART trees and random forests. Moreover, as a by-product,
these fair decision trees turn out to be significantly compact, thus enhancing
the interpretability of their fairness properties.

##  1 Introduction

The widespread adoption of data-driven automated decision-making software with
far-reaching societal impact, e.g., for credit scoring (Khandani, Kim, and Lo
2010), recidivism prediction (Chouldechova 2017), or hiring tasks (Schumann et
al. 2020), raises concerns on the fairness properties of these tools (Barocas
and Selbst 2016). Several fairness verification and bias mitigation approaches
for machine learning (ML) systems have been proposed in recent years, e.g.
(Aghaei, Azizi, and Vayanos 2019; Grari et al. 2020; Roh et al. 2020; Ruoss et
al. 2020; Urban et al. 2020; Yurochkin, Bower, and Sun 2020; Zafar et al.
2017) among the others. However, most works focus on neural networks (Roh et
al. 2020; Ruoss et al. 2020; Urban et al. 2020; Yurochkin, Bower, and Sun
2020) or on group-based notions of fairness (Grari et al. 2020; Zafar et al.
2017), e.g., demographic parity (Dwork et al. 2012) or equalized odds (Hardt,
Price, and Srebro 2016). These notions of group-based fairness require some
form of statistical parity (e.g. between positive outcomes) for members of
different protected groups (e.g. gender or race). On the other hand, they do
not provide guarantees for individuals or other subgroups. By contrast, in
this paper we focus on _individual fairness_ (Dwork et al. 2012), intuitively
meaning that similar individuals in the population receive similar outcomes,
and on decision tree ensembles (Breiman 2001; Friedman 2001), which are
commonly used for tabular datasets since they are easily interpretable ML
models with high accuracy rates.

#### Contributions.

We propose an approach for verifying individual fairness of decision tree
ensembles, as well as training tree models which maximize both accuracy and
fairness. The approach is based on _abstract interpretation_ (Cousot and
Cousot 1977; Rival and Yi 2020), a well known static program analysis
technique, and builds upon a framework for training robust decision tree
ensembles called Meta-Silvae (Ranzato and Zanella 2020b), which in turn
leverages a verification tool for robustness properties of decision trees
(Ranzato and Zanella 2020a). Our approach is fully parametric on a given
underlying abstract domain representing input space regions containing similar
individuals. We instantiate it with a product of two abstract domains: (a) the
well-known abstract domain of hyper-rectangles (or boxes) (Cousot and Cousot
1977), that represents exactly the standard notion of similarity between
individuals based on the $\ell_{\infty}$ distance metric, and does not lose
precision for the univariate split decision rules of type $x_{i}\leq k$; and
(b) a specific relational abstract domain which accounts for one-hot encoded
categorical features.

Our Fairness-Aware Tree Training method, called FATT, is designed as an
extension of Meta-Silvae (Ranzato and Zanella 2020b), a learning methodology
for ensembles of decision trees based on a genetic algorithm which is able to
train a decision tree for maximizing both its accuracy and its robustness to
adversarial perturbations. We demonstrate the effectiveness of FATT in
training accurate and fair models on the standard datasets used in the
literature on fairness. Overall, the experimental results show that our fair-
trained models are on average between $35\%$ and $45\%$ more fair than
naturally trained decision tree ensembles at an average cost of $-3.6\%$ of
accuracy. Moreover, it turns out that our tree models are orders of magnitude
more compact and thus more interpretable. Finally, we show how our models can
be used as “hints” for setting the size and shape hyper-parameters (i.e.,
maximum depth and minimum number of samples per leaf) when training standard
decision tree models. As a result, this hint-based strategy is capable to
output models that are about $20\%$ more fair and just about $1\%$ less
accurate than standard models.

#### Related Work.

The most related work to ours is by Aghaei et al. (Aghaei, Azizi, and Vayanos
2019), Raff et al. (Raff, Sylvester, and Mills 2018) and Ruoss et al. (Ruoss
et al. 2020).

By relying on the mixed-integer optimization learning approach by Bertsimas
and Dunn (Bertsimas and Dunn 2017), Aghaei et al. (Aghaei, Azizi, and Vayanos
2019) put forward a framework for training fair decision trees for
classification and regression. The experimental evaluation shows that this
approach mitigates unfairness as modeled by their notions of disparate impact
and disparate treatment at the cost of a significantly higher training
computational cost. Their notion of disparate treatment is distance-based and
thus akin to individual fairness with respect to the nearest individuals _in a
given dataset_ (e.g., the $k$-nearest individuals). In contrast, we consider
individual fairness with respect to the nearest individuals _in the input
space_ (thus, also individuals that are not necessarily part of a given
dataset).

Raff et al. (Raff, Sylvester, and Mills 2018) propose a regularization-based
approach for training fair decision trees as well as fair random forests. They
consider both group fairness as well as individual fairness with respect to
the $k$-nearest individuals in a given dataset, similarly to Aghaei et al.
(Aghaei, Azizi, and Vayanos 2019). In their experiments they use a subset of
the datasets that we consider in our evaluation (i.e., the Adult, German, and
Health datasets). Our fair models have higher accuracy than theirs (i.e.,
between $2\%$ and $5.5\%$) for all but one of these datasets (i.e., the Health
dataset). Interestingly, their models (in particular those with worse accuracy
than ours) often have accuracy on par with a constant classifier due to the
highly unbalanced label distribution of the datasets (cf. Table 1).

Finally, Ruoss et al. (Ruoss et al. 2020) have proposed an approach for
learning individually fair data representations and training neural networks
(rather than decision tree ensembles as we do) that satisfy individual
fairness with respect to a given similarity notion. We use the same notions of
similarity in our experiments (cf. Section 6.1).

##  2 Background

Given an input space $X\subseteq\mathbb{R}^{d}$ of numerical vectors and a
finite set of labels $\mathcal{L}=\\{y_{1},\ldots,y_{m}\\}$, a classifier is a
function $C\colon X\rightarrow\wp_{+}(\mathcal{L})$, where
$\wp_{+}(\mathcal{L})$ is the set of nonempty subsets of $\mathcal{L}$, which
associates at least one label to every input in $X$. A training algorithm
takes as input a dataset $D\subseteq X\times\mathcal{L}$ and outputs a
classifier $C\colon X\rightarrow\wp_{+}(\mathcal{L})$ which optimizes some
objective function, such as the Gini index or the information gain for
decision trees.

Categorical features can be converted into numerical ones through one-hot
encoding, where a single feature with $k$ possible distinct categories
$\\{c_{1},...,c_{k}\\}$ is replaced by $k$ new binary features with values in
$\\{0,1\\}$. Then, each value $c_{j}$ of the original categorical feature is
represented by a bit-value assignment to the new $k$ binary features in which
the $j$-th feature is set to $1$ (and the remaining $k-1$ binary features are
set to $0$).

Classifiers can be evaluated and compared through several metrics. Accuracy on
a test set is a basic metric: given a ground truth test set $T\subseteq
X\times\mathcal{L}$, the accuracy of $C$ on $T$ is
$\mathit{acc}_{T}(C)\triangleq|\\{(\boldsymbol{x},y)\in
T~{}|~{}C(\boldsymbol{x})=\\{y\\}\\}|/|T|$. However, according to a growing
belief (Goodfellow, McDaniel, and Papernot 2018), accuracy is not enough in
machine learning, since robustness to adversarial inputs of a ML classifier
may significantly affect its safety and generalization properties (Carlini and
Wagner 2017; Goodfellow, McDaniel, and Papernot 2018). Given an input
perturbation modeled by a function $P\colon X\rightarrow\wp(X)$, a classifier
$C:X\rightarrow\wp_{+}(\mathcal{L})$ is _stable_ (Ranzato and Zanella 2020a)
on the perturbation $P(\boldsymbol{x})$ of $\boldsymbol{x}\in X$ when $C$
consistently assigns the same label(s) to every attack ranging in
$P(\boldsymbol{x})$, i.e.,

$\operatorname*{stable}(C,\boldsymbol{x},P)\stackrel{{\scriptstyle{\mbox{$\triangle$}}}}{{\Leftrightarrow}}\forall\boldsymbol{x}^{\prime}\in
P(\boldsymbol{x})\colon C(\boldsymbol{x}^{\prime})=C(\boldsymbol{x}).$

When the sample $\boldsymbol{x}\in X$ has a ground truth label
$y_{\boldsymbol{x}}\in\mathcal{L}$, robustness of $C$ on $\boldsymbol{x}$
boils down to stability $\operatorname*{stable}(C,\boldsymbol{x},P)$ together
with correct classification $C(\boldsymbol{x})=\\{y_{\boldsymbol{x}}\\}$.

We consider standard classification decision trees commonly referred to as
CARTs (Classification And Regression Trees) (Breiman et al. 1984). A decision
tree $t\colon X\rightarrow\wp_{+}(\mathcal{L})$ is defined inductively. A base
tree $t$ is a single leaf $\lambda$ storing a frequency distribution of labels
for the samples of the training dataset, hence
$\lambda\in[0,1]^{|\mathcal{L}|}$, or, equivalently,
$\lambda\colon\mathcal{L}\rightarrow[0,1]$. Some algorithmic rule converts
this frequency distribution into a set of labels, typically as
$\operatorname*{arg\,max}_{y\in\mathcal{L}}\lambda(y)$. A composite tree $t$
is $\gamma(\operatorname*{\mathit{split}},t_{l},t_{r})$, where
$\operatorname*{\mathit{split}}\colon
X\rightarrow\\{\mathbf{tt},\mathbf{ff}\\}$ is a Boolean split criterion for
the internal parent node of its left and right subtrees $t_{l}$ and $t_{r}$;
thus, for all $\boldsymbol{x}\in X$,
$t(\boldsymbol{x})\triangleq\textbf{if}~{}\operatorname*{\mathit{split}}(\boldsymbol{x})~{}\textbf{then}~{}t_{l}(\boldsymbol{x})~{}\textbf{else}~{}t_{r}(\boldsymbol{x})$.
Although split rules can be of any type, most decision trees employ univariate
hard splits of type
$\operatorname*{\mathit{split}}(\boldsymbol{x})\triangleq\boldsymbol{x}_{i}\leq
k$ for some feature $i\in[1,d]$ and threshold $k\in\mathbb{R}$.

Tree ensembles, also known as forests, are sets of decision trees which
together contribute to formulate a unique classification output. Training
algorithms as well as methods for computing the final output label(s) vary
among different tree ensemble models. Random forests (RFs) (Breiman 2001) are
a major instance of tree ensemble where each tree of the ensemble is trained
independently from the other trees on a random subset of the features.
Gradient boosted decision trees (GBDT) (Friedman 2001) represent a different
training algorithm where an ensemble of trees is incrementally build by
training each new tree on the basis of the data samples which are mis-
classified by the previous trees. For RFs, the final classification output is
typically obtained through a voting mechanism (e.g., majority voting), while
GBDTs are usually trained for binary classification problems and use some
binary reduction scheme, such as one-vs-all or one-vs-one, for multi-class
classification.

##  3 Individual Fairness

Dwork et al. (Dwork et al. 2012) define _individual fairness_ as “the
principle that two individuals who are similar with respect to a particular
task should be classified similarly”. They formalize this notion as a
Lipschitz condition of the classifier, which requires that any two individuals
$\boldsymbol{x},\boldsymbol{y}\in X$ whose distance is
$\delta(\boldsymbol{x},\boldsymbol{y})\in[0,1]$ map to distributions
$D_{\boldsymbol{x}}$ and $D_{\boldsymbol{y}}$, respectively, such that the
statistical distance between $D_{\boldsymbol{x}}$ and $D_{\boldsymbol{y}}$ is
at most $\delta(\boldsymbol{x},\boldsymbol{y})$. The intuition is that the
output distributions for $\boldsymbol{x}$ and $\boldsymbol{y}$ are
indistinguishable up to their distance
$\delta(\boldsymbol{x},\boldsymbol{y})$. The distance metric $\delta\colon
X\times X\rightarrow\mathbb{R}_{\geq 0}$ is problem specific and satisfies the
basic axioms
$\delta(\boldsymbol{x},\boldsymbol{y})=\delta(\boldsymbol{y},\boldsymbol{x})$
and $\delta(\boldsymbol{x},\boldsymbol{x})=0$.

By following Dwork et al’s standard definition (Dwork et al. 2012), we
consider a classifier $C\colon X\rightarrow\wp_{+}(\mathcal{L})$ to be fair
when $C$ outputs the same set of labels for every pair of individuals
$\boldsymbol{x},\boldsymbol{y}\in X$ which satisfy a similarity relation
$S\subseteq X\times X$. Thus, $S$ can be derived from a distance $\delta$ as
$(\boldsymbol{x},\boldsymbol{y})\in
S\stackrel{{\scriptstyle{\mbox{$\triangle$}}}}{{\Leftrightarrow}}\delta(\boldsymbol{x},\boldsymbol{y})\leq\epsilon$,
where $\epsilon\in\mathbb{R}$ is a threshold of similarity. In order to
estimate a fairness metric for a classifier $C$, we count how often $C$ is
fair on sets of similar individuals ranging into a test set $T\subseteq
X\times\mathcal{L}$:

$\textstyle{\operatorname*{\mathit{fair}}_{T}(C)}\triangleq\displaystyle\frac{|\\{(\boldsymbol{x},y)\in
T~{}|~{}\operatorname*{fair}(C,\boldsymbol{x},S)\\}|}{|T|}$ (1)

where $\operatorname*{fair}(C,\boldsymbol{x},S)$ is defined as follows:

######  Definition 3.1 (Individual Fairness).

A classifier $C\colon X\rightarrow\wp_{+}(\mathcal{L})$ is _fair_ on an input
sample $\boldsymbol{x}\in X$ with respect to a similarity relation $S\subseteq
X\times X$, denoted by $\operatorname*{fair}(C,\boldsymbol{x},S)$, when
$\forall\boldsymbol{x}^{\prime}\in
X\colon(\boldsymbol{x},\boldsymbol{x}^{\prime})\in S\Rightarrow
C(\boldsymbol{x}^{\prime})=C(\boldsymbol{x})$. ∎

Hence, fairness for a similarity relation $S$ boils down to stability on the
perturbation $P_{S}(\boldsymbol{x})\triangleq\\{\boldsymbol{x}^{\prime}\in
X~{}|~{}(\boldsymbol{x},\boldsymbol{x}^{\prime})\in S\\}$, namely, for all
$\boldsymbol{x}\in X$,

$\operatorname*{fair}(C,\boldsymbol{x},S)\;\Leftrightarrow\;\operatorname*{stable}(C,\boldsymbol{x},P_{S})$
(2)

Let us remark that fairness is orthogonal to accuracy since it does not depend
on the correctness of the label assigned by the classifier, so that that
training algorithms that maximize accuracy-based metrics do not necessarily
achieve fair models. Thus, this is also the case of a natural learning
algorithm for CART trees and RFs, that locally optimizes split criteria by
measuring entropy or Gini impurity, which are both indicators of the correct
classification of training data.

It is also worth observing that fairness is monotonic with respect to the
similarity relation, meaning that

$\operatorname*{fair}(C,\boldsymbol{x},S)\wedge S^{\prime}\subseteq
S\;\Rightarrow\;\operatorname*{fair}(C,\boldsymbol{x},S^{\prime})$ (3)

We will exploit this monotonicity property, since this implies that, on one
hand, fair classification is preserved for smaller similarity relations and,
on the other hand, fairness verification and fair training is more challenging
for larger similarity relations.

##  4 Verifying Fairness

As individual fairness is equivalent to stability, individual fairness of
decision trees can be verified by Silva (Ranzato and Zanella 2020a), an
abstract interpretation-based algorithm for checking stability properties of
decision tree ensembles.

###  4.1 Verification by Silva

Silva performs a static analysis of an ensemble of decision trees in a so-
called abstract domain $A$ that approximates properties of real vectors,
meaning that each abstract value $a\in A$ represents a set of real vectors
$\gamma(a)\in\wp(\mathbb{R}^{d})$. Silva approximates an input region
$P(\boldsymbol{x})\in\wp(\mathbb{R}^{d})$ for an input vector
$\boldsymbol{x}\in\mathbb{R}^{d}$ by an abstract value $a\in A$ such that
$P(\boldsymbol{x})\subseteq\gamma(a)$ and for each decision tree $t$, it
computes an over-approximation of the set of leaves of $t$ that can be reached
from some vector in $\gamma(a)$. This is computed by collecting the
constraints of split nodes for each root-leaf path, so that each leaf
$\lambda$ of $t$ stores the minimum set of constraints $C_{\lambda}$ which
makes $\lambda$ reachable from the root of $t$. It is then checked if this set
of constraints $C_{\lambda}$ can be satisfied by the input abstract value
$a\in A$: this check is denoted by $a\models^{?}C_{\lambda}$ and its
_soundness_ requirement means that if some input sample
$\boldsymbol{z}\in\gamma(a)$ may reach the leaf $\lambda$ then $a\models
C_{\lambda}$ must necessarily hold. When $a\models C_{\lambda}$ holds the leaf
$\lambda$ is marked as reachable from $a$. For example, if
$C_{\lambda}=\\{x_{1}\leq 2,\neg(x_{1}\leq-1),x_{2}\leq-1\\}$ then an abstract
value such as $\langle{x_{1}\leq 0,x_{2}\leq 0}\rangle$ satisfies
$C_{\lambda}$ while a relational abstract value such as $x_{1}+x_{2}=4$ does
not. This over-approximation of the set of leaves of $t$ reachable from $a$
allows us to compute a set of labels, denoted by
$t^{A}(a)\in\wp_{+}(\mathcal{L})$ which is an over-approximation of the set of
labels assigned by $t$ to all the input vectors ranging in $\gamma(a)$, i.e.,
$\cup_{\boldsymbol{z}\in\gamma(a)}t(\boldsymbol{z})\subseteq t^{A}(a)$ holds.
Thus, if $P(\boldsymbol{x})\subseteq\gamma(a)$ and
$t^{A}(a)=t(\boldsymbol{x})$ then $t$ is stable on $P(\boldsymbol{x})$.

For standard classification trees with hard univariate splits of type
$x_{i}\leq k$, we will use the well-known hyper-rectangle abstract domain
$\operatorname*{HR}$ whose abstract values for vectors
$\boldsymbol{x}\in\mathbb{R}^{d}$ are of type

$h=\langle{\boldsymbol{x}_{i}\in[l_{1},u_{1}],\dots,\boldsymbol{x}_{d}\in[l_{d},u_{d}]}\rangle\in\textstyle\operatorname*{HR}_{d}$

where lower and upper bounds $l,u\in\mathbb{R}\cup\\{-\infty,+\infty\\}$ with
$l\leq u$ (more on this abstract domain can be found in (Rival and Yi 2020)).
Thus, $\gamma(h)=\\{\boldsymbol{x}\in\mathbb{R}^{d}\mid\forall
i.\,l_{i}\leq\boldsymbol{x}_{i}\leq u_{i}\\}$. The hyper-rectangle abstract
domain guarantees that for each leaf constraint $C_{\lambda}$ and
$h\in\operatorname*{HR}$, the check $h\models^{?}C_{\lambda}$ is (sound and)
_complete_ , meaning that $h\models C_{\lambda}$ holds iff there exists some
input sample in $\gamma(h)$ reaching $\lambda$. This completeness property
therefore entails that the set of labels $t^{\operatorname*{HR}}(h)$ computed
by this analysis coincides exactly with the set of classification labels
computed by $t$ for all the samples in $\gamma(h)$, so that for the
$\ell_{\infty}$-based perturbation such that
$P_{\infty}(\boldsymbol{x})=\gamma(h)$ then it turns out that $t$ is stable on
$P_{\infty}(\boldsymbol{x})$ iff $t^{\operatorname*{HR}}(h)=t(\boldsymbol{x})$
holds.

In order to analyse a forest $F$ of trees, Silva reduces the whole forest to a
single tree $t_{F}$, by stacking every tree $t\in F$ on top of each other,
i.e., each leaf becomes the root of the next tree in $F$, where the ordering
of this unfolding operation does not matter. Then, each leaf $\lambda$ of this
huge single tree $t_{F}$ collects all the constraints of the leaves in the
path from the root of $t_{F}$ to $\lambda$. Since this stacked tree $t_{F}$
suffers from a combinatorial explosion of the number of leaves, Silva deploys
a number of optimisation strategies for its analysis. Basically, Silva
exploits a best-first search algorithm to look for a pair of input samples in
$\gamma(a)$ which are differently labeled, hence showing instability. If one
such instability counterexample can be found then instability is proved and
the analysis terminates, otherwise stability is proved. Also, Silva allows to
set a safe timeout which, when met, stops the analysis and outputs the current
sound over-approximation of labels.

###  4.2 Verification with One-Hot Enconding

As described above, the soundness of Silva guarantees that no true reachable
leaf is missed by this static analysis. Moreover, when the input region
$P(\boldsymbol{x})$ is defined by the $\ell_{\infty}$ norm and the static
analysis is performed using the abstract domain of hyper-rectangles
$\operatorname*{HR}$, Silva is also complete, meaning that no false positive
(i.e., a false reachable leaf) can occur. However, that this is not true
anymore when dealing with classification problems involving some categorical
features.

[$\textit{color}_{\textit{white}}\leq 0.5$ [$\\{\ell_{1}\\}$, name = n1]
[$\\{\ell_{2}\\}$] ] at (n1) [left=6ex,above=1ex]$t_{1}$;
[$\textit{color}_{\textit{black}}\leq 0.5$ [$\\{\ell_{2}\\}$]
[$\\{\ell_{1}\\}$, name = n2] ] at (n2) [right=6ex,above=1ex]$t_{2}$;

The diagram above depicts a toy forest $F$ consisting of two trees $t_{1}$ and
$t_{2}$, where left/right branches are followed when the split condition is
false/true. Here, a categorical feature
$\textit{color}\in\\{\textit{white},\textit{black}\\}$ is one-hot encoded by
$\textit{color}_{\textit{white}},\textit{color}_{\textit{black}}\in\\{0,1\\}$.
Since colors are mutually exclusive, every white individual in the input
space, i.e.
$\textit{color}_{\textit{white}}=1,\textit{color}_{\textit{black}}=0$, will be
labeled as $\ell_{1}$ by both trees. However, by running the stability
analysis on the hyper-rectangle
$\langle\textit{color}_{\textit{white}}\in[0,1],\textit{color}_{\textit{black}}\in[0,1]\rangle$,
Silva would mark the classifier as unstable because there exists a sample in
$[0,1]^{2}$ whose output is
$\\{\ell_{1},\ell_{2}\\}\neq\\{\ell_{1}\\}=F(\textit{color}_{\textit{white}}=1,\textit{color}_{\textit{black}}=0)$.
This is due to the point $(0,0)\in[0,1]^{2}$ which is a feasible input sample
for the analysis, although it does not represent any actual individual in the
input space. In fact, $t_{1}(0,0)=\\{\ell_{2}\\}$ and
$t_{2}(0,0)=\\{\ell_{1}\\}$, so that by a majority voting
$F(0,0)=\\{\ell_{1},\ell_{2}\\}$, thus making $F$ unstable (i.e., unfair) on
$(1,0)$ (i.e., on white individuals).

To overcome this issue, we instantiate Silva to an abstract domain which is
designed as a reduced product (more details on reduced products can be found
in (Rival and Yi 2020)) with a relational abstract domain keeping track of the
relationships among the multiple binary features introduced by one-hot
encoding a categorical feature. More formally, this relational domain
maintains the following two additional constraints on the $k$ features
$x^{c}_{1},...,x^{c}_{k}$ introduced by one-hot encoding a categorical
variable $x^{c}$ with $k$ distinct values:

  1. 1.

the possible values for each $x^{c}_{i}$ are restricted to $\\{0,1\\}$;

  2. 2.

the sum of all $x^{c}_{i}$ must satisfy $\sum_{i=1}^{k}x^{c}_{i}=1$.

Hence, these conditions guarantee that any abstract value for
$x^{c}_{1},...,x^{c}_{k}$ represents precisely one possible category for
$x^{c}$. This abstract domain for a categorical variable $x$ with $k$ distinct
values is denoted by $\operatorname*{OH}_{k}(x)$. In the example above, any
hyper-rectangle
$\langle\textit{color}_{\textit{white}}\in[0,1],\textit{color}_{\textit{black}}\in[0,1]\rangle$
is reduced by $\operatorname*{OH}_{2}(\textit{color})$, so that just two
different values
$\langle\textit{color}_{\textit{white}}=0,\textit{color}_{\textit{black}}=1\rangle$
and
$\langle\textit{color}_{\textit{white}}=1,\textit{color}_{\textit{black}}=0\rangle$
are allowed.

Summing up, the generic abstract value of the reduced hyper-rectangle domain
computed by the analyzer Silva for data vectors consisting of $d$ numerical
variables $x^{j}\in\mathbb{R}$ and $m$ categorical variables $c^{j}$ with
$k_{j}\in\mathbb{N}$ distinct values is:

$\textstyle\langle{x^{j}\in[l_{j},u_{j}]}\rangle_{j=1}^{d}\times\langle{c^{j}_{i}\in\\{0,1\\}\mid\sum_{i=1}^{k_{j}}c^{j}_{i}=1}\rangle_{j=1}^{m}$

where $l_{j},u_{j}\in\mathbb{R}\cup\\{-\infty,+\infty\\}$ and $l_{j}\leq
u_{j}$.

##  5 FATT: Fairness-Aware Training of Trees

Several algorithms for training robust decision trees and ensembles have been
put forward (Andriushchenko and Hein 2019; Calzavara, Lucchese, and Tolomei
2019; Calzavara et al. 2020; Chen et al. 2019; Kantchelian, Tygar, and Joseph
2016; Ranzato and Zanella 2020b). These algorithms encode the robustness of a
tree classifier as a loss function which is minimized either by either exact
methods such as MILP or by suboptimal heuristics such as genetic algorithms.

The robust learning algorithm of (Ranzato and Zanella 2020b), called Meta-
Silvae, aims at maximizing a tunable weighted linear combination of accuracy
and stability metrics. Meta-Silvae relies on a genetic algorithm for evolving
a population of trees which are ranked by their accuracy and stability, where
tree stability is computed by resorting to the verifier Silva (Ranzato and
Zanella 2020a). At the end of this genetic evolution, Meta-Silvae returns the
best tree(s). It turns out that Meta-Silvae typically outputs compact models
which are easily interpretable and often achieve accurate and stable models
already with a single decision tree rather than a forest. By exploiting the
equivalence (2) between individual fairness and stability and the
instantiation of the verifier Silva to the product abstract domain tailored
for one-hot encoding, we use Meta-Silvae as a learning algorithm for decision
trees, called FATT, that enhances their individual fairness.

While standard learning algorithms for tree ensembles require tuning some
hyper-parameters, such as maximum depth of trees, minimum amount of
information on leaves and maximum number of trees, Meta-Silvae is able to
infer them automatically, so that the traditional tuning process is not
needed. Instead, some standard parameters are required by the underlying
genetic algorithm, notably, the size of the evolving population, the maximum
number of evolving iterations, the crossover and mutation functions (Holland
1984; Srinivas and Patnaik 1994). Moreover, we need to specify the objective
function of FATT that, for learning fair decision trees, is given by a
weighted sum of the accuracy and individual fairness scores over the training
set. It is worth remarking that, given an objective function, the genetic
algorithm of FATT converges to an optimal (or suboptimal) solution regardless
of the chosen parameters, which just affect the rate of convergence and
therefore should be chosen for tuning its speed.

Crossover and mutation functions are two main distinctive features of the
genetic algorithm of Meta-Silvae. The crossover function of Meta-Silvae
combines two parent trees $t_{1}$ and $t_{2}$ by randomly substituting a
subtree of $t_{1}$ with a subtree of $t_{2}$. Also, Meta-Silvae supports two
types of mutation strategies: grow-only, which only allows trees to grow, and
grow-and-prune, which also allows pruning the mutated trees. Finally, let us
point out that Meta-Silvae allows to set the basic parameters used by generic
algorithms: population size, selection function, number of iterations. In our
instantiation of Meta-Silvae to fair learning: the population size is kept
fixed to $32$, as the experimental evaluation showed that this provides an
effective balance between achieved fairness and training time; the standard
roulette wheel algorithm is employed as selection function; the number of
iterations is typically dataset-specific.

##  6 Experimental Evaluation

We consider the main standard datasets used in the fairness literature and we
preprocess them by following the steps of Ruoss et al. (Ruoss et al. 2020,
Section 5) for their experiments on individual fairness for deep neural
networks: (1) standardize numerical attributes to zero mean and unit variance;
(2) one-hot encoding of all categorical features; (3) drop rows/columns
containing missing values; and (4) split into train and test set. These
datasets concern binary classification tasks, although our fair learning
naturally extends to multiclass classification with no specific effort. We
will make all the code, datasets and preprocessing pipelines of FATT publicly
available upon publication of this work.

Adult.

The Adult income dataset (Dua and Graff 2017) is extracted from the 1994 US
Census database. Every sample assigns a yearly income (below or above $50K) to
an individual based on personal attributes such as gender, race, and
occupation.

Compas.

The COMPAS dataset contains data collected on the use of the COMPAS risk
assessment tool in Broward County, Florida (Angwin et al. 2016). Each sample
predicts the risk of recidivism for individuals based on personal attributes
and criminal history.

Crime.

The Communities and Crime dataset (Dua and Graff 2017) contains socio-
economic, law enforcement, and crime data for communities within the US. Each
sample indicates whether a community is above or below the median number of
violent crimes per population.

German.

The German Credit dataset (Dua and Graff 2017) contains samples assigning a
good or bad credit score to individuals.

Health.

The heritage Health dataset (https://www.kaggle.com/c/hhp) contains physician
records and insurance claims. Each sample predicts the ten-year mortality
(above or below the median Charlson index) for a patient.

|  | Training Set | Test Set
---|---|---|---
dataset | #features | size | positive | size | positive
adult | 103 | 30162 | 24.9% | 15060 | 24.6%
compas | 371 | 4222 | 53.3% | 1056 | 55.6%
crime | 147 | 1595 | 50.0% | 399 | 49.6%
german | 56 | 800 | 69.8% | 200 | 71.0%
health | 110 | 174732 | 68.1% | 43683 | 68.0%

Table 1: Overview of Datasets.

Table 1 displays size and distribution of positive samples for these datasets.
As noticed by (Ruoss et al. 2020), some datasets exhibit a highly unbalanced
label distribution. For example, for the adult dataset, the constant
classifier $C(\boldsymbol{x})=1$ would achieve $75.4\%$ test set accuracy and
$100\%$ individual fairness with respect to any similarity relation. Hence, we
follow (Ruoss et al. 2020) and we will evaluate and report the balanced
accuracy of our FATT classifiers, i.e.,

$0.5\,(\frac{\textit{truePositive}}{\textit{truePositive}+\textit{falseNegative}}+\frac{\textit{trueNegative}}{\textit{trueNegative}+\textit{falsePositive}})$.

###  6.1 Similarity Relations

We consider four different types of similarity relations, as described by
Ruoss et al. (Ruoss et al. 2020, Section 5.1). In the following, let
$I\subseteq\mathbb{N}$ denote the set of indexes of features of an individual
after one-hot encoding.

noise:

Two individuals $\boldsymbol{x},\boldsymbol{y}\in X$ are similar when a subset
of their (standardized) numerical features indexed by a given subset
$I^{\prime}\subseteq I$ differs less than a given threshold $\tau\geq 0$,
while all the other features are unchanged:
$(\boldsymbol{x},\boldsymbol{y})\in S_{\textit{noise}}$ iff
$|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}|\leq\tau$ for all $i\in
I^{\prime}$, and ${\boldsymbol{x}}_{i}={\boldsymbol{y}}_{i}$ for all $i\in
I\smallsetminus I^{\prime}$. For our experiments, we consider $\epsilon=0.3$
in the standardized input space, e.g., for adult two individuals are similar
if their age difference is at most 3.95 years.

cat:

Two individuals are similar if they are identical except for one or more
categorical sensitive attributes indexed by $I^{\prime}\subseteq I$:
$(\boldsymbol{x},\boldsymbol{y})\in S_{\textit{cat}}$ iff
${\boldsymbol{x}}_{i}={\boldsymbol{y}}_{i}$ for all $i\in I\smallsetminus
I^{\prime}$. For adult and german, we select the gender attribute. For compas,
we identify race as sensitive attribute. For crime, we consider two
individuals similar regardless of their state. Lastly, for health, neither
gender nor age group should affect the final prediction.

noise-cat:

Given noise and categorical similarity relations $S_{\textit{noise}}$ and
$S_{\textit{cat}}$, their union $S_{\textit{noise-cat}}\triangleq
S_{\textit{noise}}\cup S_{\textit{cat}}$ models a relation where two
individuals are similar when some of their numerical attributes differ up to a
given threshold while the other attributes are equal except some categorical
features.

conditional-attribute:

Here, similarity is a disjunction of two mutually exclusive cases. Consider a
numerical attribute $\boldsymbol{x}_{i}$, a threshold $\tau\geq 0$ and two
noise similarities $S_{n_{1}},S_{n_{2}}$. Two individuals are defined to be
similar if their $i$-th attributes are similar for $S_{n_{1}}$ and are bounded
by $\tau$ or these attributes are above $\tau$ and similar for $S_{n_{2}}$:
$S_{\textit{cond}}\triangleq\\{(\boldsymbol{x},\boldsymbol{y})\in
S_{n_{1}}~{}|~{}{\boldsymbol{x}}_{i}\leq\tau,\,{\boldsymbol{y}}_{i}\leq\tau\\}\cup\\{(\boldsymbol{x},\boldsymbol{y})\in
S_{n_{2}}~{}|~{}{\boldsymbol{x}}_{i}>\tau,\,{\boldsymbol{y}}_{i}>\tau\\}$. For
adult, we consider the median age as threshold $\tau=37$, and two noise
similarities based on age with thresholds $0.2$ and $0.4$, which correspond to
age differences of $2.63$ and $5.26$ years respectively. For german, we also
consider the median age $\tau=33$ and the same noise similarities on age, that
correspond to age differences of $0.24$ and $0.47$ years.

Note that our approach is not limited to supporting these similarity
relations. Further domain-specific similarities can be defined and handled by
our approach by instantiating the underlying verifier Silva with an
appropriately over-approximating abstract domain to retain soundness.
Moreover, if the similarity relation can be precisely represented in the
chosen abstract domain, we also retain completeness.

###  6.2 Setup

Our experimental evaluation compares CART trees and Random Forests with our
FATT tree models. CARTs and RFs are trained by scikit-learn. We first run a
preliminary phase for tuning the hyper-parameters for CARTs and RFs. In
particular, we considered both entropy and Gini index as split criteria, and
we checked maximum tree depths ranging from $5$ to $100$ with step $10$. For
RFs, we scanned the maximum number of trees ($5$ to $100$, step $10$). Cross
validation inferred the optimal hyper-parameters, where the datasets have been
split in $80\%$ training and $20\%$ validation sets. The hyper-parameters of
FATT (i.e, weights of accuracy and fairness in the objective function, type of
mutation, selection function, number of iterations) by assessing convergence
speed, maximum fitness value and variance among fitness in the population
during the training phase. FATT trained single decision trees rather than
forests, thus providing more compact and interpretable models. It turned out
that accuracy and fairness of single FATT trees are already competitive, where
individual fairness may exceed $85\%$ for the most challenging similarities.
We therefore concluded that ensembles of FATT trees do not introduce
statistically significant benefits over single decision trees. Since FATT
trees are stochastic by relying on random seeds, each experimental test has
been repeated 1000 times and the results refer to their median value.

###  6.3 Results

| Acc. % | Bal.Acc. % | Individual Fairness $\operatorname*{\mathit{fair}}_{T}$ %
---|---|---|---
| cat | noise | noise-cat
Dataset | RF | FATT | RF | FATT | RF | FATT | RF | FATT | RF | FATT
adult | 82.76 | 80.84 | 70.29 | 61.86 | 91.71 | 100.00 | 85.44 | 95.21 | 77.50 | 95.21
compas | 66.57 | 64.11 | 66.24 | 63.83 | 48.01 | 100.00 | 35.51 | 85.98 | 30.87 | 85.98
crime | 80.95 | 79.45 | 80.98 | 79.43 | 86.22 | 100.00 | 31.83 | 75.19 | 32.08 | 75.19
german | 76.50 | 72.00 | 63.62 | 52.54 | 91.50 | 100.00 | 92.00 | 99.50 | 90.00 | 99.50
health | 85.29 | 77.87 | 83.27 | 73.59 | 7.84 | 99.99 | 47.66 | 97.04 | 2.91 | 97.03
Average | 78.41 | 74.85 | 72.88 | 66.25 | 65.06 | 100.00 | 58.49 | 90.58 | 46.67 | 90.58

Table 2: RF and FATT comparison.

Table 2 shows a comparison between RFs and FATTs. We show accuracy, balanced
accuracy and individual fairness with respect to the noise, cat, and noise-cat
similarity relations as computed on the test sets $T$. As expected, FATT trees
are slightly less accurate than RFs — $~{}3.6\%$ on average, which also
reflects to balanced accuracy — but outperform them in every fairness test. On
average, the fairness increment ranges between $+35\%$ to $+45\%$ among
different similarity relations. Table 3 shows the comparison for the
conditional-attribute similarity, which applies to adult and german datasets
only. Here, the average fairness increase of FATT models is $+8.5\%$.

| Individual Fairness $\operatorname*{\mathit{fair}}_{T}$ %
---|---
Dataset | RF | FATT
adult | 84.75 | 94.12
german | 91.50 | 99.50

Table 3: Comparison for conditional-attribute.

Figure 1: Accuracy (top) and Fairness (bottom).

Fig. 1 shows the distribution of accuracy and individual fairness for FATT
trees over 1000 runs of the FATT learning algorithm. This plot is for fairness
with respect to noise-cat similarity, as this is the most challenging relation
to train for (this is a consequence of (3)). We can observe a stable behaviour
for accuracy, with $\approx 50\%$ of the observations laying within one
percentile from the median. The results for fairness are analogous, although
for compas we report a higher variance of the distribution, where the lowest
observed fairness percentage is $\approx 10\%$ higher than the corresponding
one for RFs. We claim that this may depend by the high number of features in
the dataset, which makes fair training a challenging task.

| Model size | Avg. verification time per sample (ms)
---|---|---
| cat | noise | noise-cat
Dataset | RF | FATT | RF | FATT | RF | FATT | RF | FATT
adult | 1427 | 43 | 0.03 | 0.02 | 0.03 | 0.02 | 0.03 | 0.02
compas | 147219 | 75 | 0.36 | 0.07 | 0.47 | 0.07 | 0.61 | 0.07
crime | 14148 | 11 | 0.12 | 0.07 | 2025.13 | 0.07 | 2028.47 | 0.07
german | 5743 | 2 | 0.06 | 0.03 | 0.06 | 0.02 | 0.07 | 0.03
health | 2558676 | 84 | 1.40 | 0.06 | 0.91 | 0.05 | 3.10 | 0.06

Table 4: Model sizes and verification times.

Table 4 compares the size of RF and FATT models, defined as total number of
leaves. It turns out that FATT tree models are orders of magnitude smaller
and, thus, more interpretable than RFs (while having comparable accuracy and
significantly enhanced fairness). Let us also remark that the average
verification time per sample for our FATT models is always less than $0.01$
milliseconds.

| FATT | Natural CART | Hinted CART
---|---|---|---
Dataset | Acc. % | Fair. % | Size | Acc. % | Fair. % | Size | Acc. % | Fair. % | Size
adult | 80.84 | 95.21 | 43 | 85.32 | 77.56 | 270 | 84.77 | 87.46 | 47
compas | 64.11 | 85.98 | 75 | 65.91 | 22.25 | 56 | 65.91 | 22.25 | 56
crime | 79.45 | 75.19 | 11 | 77.69 | 24.31 | 48 | 77.44 | 60.65 | 8
german | 72.00 | 99.50 | 2 | 75.50 | 57.50 | 115 | 73.50 | 86.00 | 4
health | 77.87 | 97.03 | 84 | 83.85 | 79.98 | 2371 | 82.25 | 93.64 | 100
Average | 74.85 | 90.58 | 43 | 77.65 | 52.32 | 572 | 76.77 | 70.00 | 43

Table 5: Decision trees comparison.

Finally, in Table 5 compares FATT models with natural CART trees in terms of
accuracy, size, and fairness with respect to the noise-cat similarity. While
CARTs are approximately $3\%$ more accurate than FATT on average, they are
roughly half less fair and more than ten times larger.

It is well known that decision trees often overfit (Bramer 2007) due to their
high number of leaves, thus yielding unstable/unfair models. Post-training
techniques such as tree pruning are often used to mitigate overfitting (Kearns
and Mansour 1998), although they are deployed when a tree has been already
fully trained and thus often pruning is poorly beneficial. As a byproduct of
our approach, we trained a set of natural CART trees, denoted by Hint in Table
5, which exploits hyper-parameters as “hinted” by FATT training. In
particular, in this “hinted” learning of CART trees, the maximum tree depth
and the minimum number of samples per leaf are obtained as tree depth and
minimum number of samples of our best FATT models. Interestingly, the results
in Table 5 show that these “hinted” decision trees have roughly the same size
of our FATT trees, are approximately $20\%$ more fair than natural CART trees
and just $1\%$ less accurate. Overall, it turns out that the general
performance of these “hinted” CARTs is halfway between natural CARTs and
FATTs, both in term of accuracy and fairness, while having the same
compactness of FATT models.

##  7 Conclusion

We believe that this work contributes to push forward the use of formal
verification methods in decision tree learning, in particular a very well
known program analysis technique such as abstract interpretation is proved to
be successful for training and verifying decision tree classifiers which are
both accurate and fair, improve on state-of-the-art CART and random forest
models, while being much more compact and interpretable. We also showed how
information from our FATT trees can be exploited to tune the natural training
process of decision trees. As future work we plan to extend further our
fairness analysis by considering alternative fairness definitions, such as
group or statistical fairness.

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# A note on Samelson product in $Sp(n)$

Sajjad Mohammadi

###### Abstract

Let $m$ and $n$ be two positive integers such that $m<n$. Let $Q_{n-m+1}$ be
the symplectic quasi-projective space of rank $n-m+1$. In this article, we
will study the order of the Samelson product $S^{4m-1}\wedge
Q_{n-m+1}\rightarrow Sp(n)$.

Keywords: Samelson product; Symplectic group.

$2010$ Mathematics Subject Classification: $55Q15$; $55P10$.

##  1 Introduction

For two maps $f_{1}\colon A_{1}\rightarrow X$ and $f_{2}\colon
A_{2}\rightarrow X$ into a homotopy associative $H$-space $X$ with inverse,
the Samelson product of $f_{1}$ and $f_{2}$ is defined as the following
composition

$A_{1}\wedge A_{2}\overset{f_{1}\wedge f_{2}}{\longrightarrow}X\wedge
X\overset{c}{\longrightarrow}X$

and is denoted by $\langle f_{1},f_{2}\rangle$, where the map $c$ is the
reduced commutator map. Let $G$ be a simple compact connected Lie group.
Samelson products play a crucial role in the homotopy theory and have many
applications in the homotopy classification of gauge groups of principal
$G$-bundles, the study of $H$-spaces, homotopy commutativity, homotopy
normality, and the study of self groups of Lie groups. In recent years,
Samelson products have been studied extensively for classical Lie groups and
valuable results have been obtained. For example, see [2], [4], [5], [8], [9],
[10] and [12].

Let $m$ and $n$ be two positive integers such that $m<n$. Let $Q_{n-m+1}$ be
the symplectic quasi-projective space of rank $n-m+1$, we denote the inclusion
$Q_{n-m+1}\rightarrow Sp(n)$ by $\epsilon_{m,n}$. Also, let
$\varepsilon_{m,n}\colon S^{4m-1}\rightarrow Sp(n)$ represents the generator
of $\pi_{4m-1}(Sp(n))$. In this article, we will study the order of the
Samelson product $\langle\varepsilon_{m,n},\epsilon_{m,n}\rangle\colon
S^{4m-1}\wedge Q_{n-m+1}\rightarrow Sp(n)$. We will prove the following
theorem.

######  Theorem 1.1.

The order of the Samelson product
$\langle\varepsilon_{m,n},\epsilon_{m,n}\rangle\colon S^{4m-1}\wedge
Q_{n-m+1}\rightarrow Sp(n)$ is equal to

$\left\\{\begin{array}[]{ll}\frac{(2n+1)!}{(2n-2m+1)!}&\textrm{if $m$ is even
},\\\ \\\ \frac{2(2n+1)!}{(2n-2m+1)!}&\textrm{if $m$ is odd }.\\\
\end{array}\right.$

The theorem recovers the known case in [7] when $m=1$.

In Section 2, in cases where $m$ is an even integer and $m$ is an odd integer,
we will study the group $[\Sigma^{4m-1}Q_{n-m+1},Sp(n)]$. Then in Section 3,
regarding the Samelson product
$\langle\varepsilon_{m,n},\epsilon_{m,n}\rangle$ as an element of
$[\Sigma^{4m-1}Q_{n-m+1},Sp(n)]$, we will study the order of the Samelson
product $\langle\varepsilon_{m,n},\epsilon_{m,n}\rangle$.

##  2 The group $[\Sigma^{4m-1}Q_{n-m+1},Sp(n)]$

Our main goal in this section is to study the Samelson product
$\langle\varepsilon_{m,n},\epsilon_{m,n}\rangle$, where the map
$\epsilon_{m,n}\colon Q_{n-m+1}\rightarrow Sp(n)$ is the inclusion map. We
denote $Sp(\infty)/Sp(n)$ by $X_{n}$ and $[X,Sp(n)]$ by $Sp_{n}(X)$. Let
$Q_{n}$ be the symplectic quasi projective space of rank $n$ defined in [6].
This space has the following cellular structure

$Q_{n}=S^{3}\cup e^{7}\cup e^{11}\cup\cdots\cup e^{4n-1}.$

Put $X=S^{4m-1}\wedge Q_{n-m+1}$. Note that $X$ has the following cellular
structure

$X\simeq S^{4m+2}\cup e^{4m+6}\cup e^{4m+10}\cup\cdots\cup e^{4n+2}.$

Recall that as an algebra

$\displaystyle
H^{*}(Sp(n);\mathbb{Z})=\bigwedge(y_{3},y_{7},\cdots,y_{4n-1}),$
$\displaystyle H^{*}(Sp(\infty);\mathbb{Z})=\bigwedge(y_{3},y_{7},\cdots),$
$\displaystyle H^{*}(BSp(\infty);\mathbb{Z})=\mathbb{Z}[q_{1},q_{2},\cdots],$

where $y_{4i-1}=\sigma q_{i}$, $\sigma$ is the cohomology suspension and
$q_{i}$ is the $i-$th symplectic Pontrjagin class. Consider the projection map
$\pi:Sp(\infty)\rightarrow X_{n}$, also an algebra we have

$\displaystyle
H^{*}(X_{n};\mathbb{Z})=\bigwedge(\bar{y}_{4n+3},\bar{y}_{4n+7},\cdots),$
$\displaystyle H^{*}(\Omega
X_{n};\mathbb{Z})=\mathbb{Z}\\{b_{4n+2},b_{4n+6},\cdots,b_{8n+2}\\}\quad(*\leq
8n+2),$

where $\pi^{*}(\bar{y}_{4i+3})=y_{4i+3}$ and
$b_{4n+4j-2}=\sigma(\bar{y}_{4n+4j-1})$. Consider the following fibre sequence

$\displaystyle\Omega Sp(\infty)\overset{\Omega\pi}{\longrightarrow}\Omega
X_{n}\overset{\delta}{\longrightarrow}Sp(n)\overset{j}{\longrightarrow}Sp(\infty)\overset{\pi}{\longrightarrow}X_{n}.$
(2.1)

By applying the functor $[X,\quad]$ to fibration $(2.1)$, we get the following
exact sequence

$[X,\Omega Sp(\infty)]\overset{(\Omega\pi)_{*}}{\longrightarrow}[X,\Omega
X_{n}]\overset{\delta_{*}}{\longrightarrow}Sp_{n}(X)\overset{j_{*}}{\longrightarrow}[X,Sp(\infty)]\overset{\pi_{*}}{\longrightarrow}[X,X_{n}].\quad(\star)$

Note that $X_{n}$ has the cellular structure as following

$X_{n}\simeq S^{4n+3}\cup e^{4n+7}\cup e^{4n+11}\cup\cdots.$

Also, we have

$\Omega X_{n}\simeq S^{4n+2}\cup e^{4n+6}\cup e^{4n+10}\cup\cdots.$

According to the $CW$-structure of $X_{n}$, we have the following isomorphisms

$\pi_{i}(X_{n})=0\quad(for\hskip 5.69054pti\leq
4n+2),\qquad\pi_{4n+3}(X_{n})\cong\mathbb{Z}.$

Observe that $[X,Sp(\infty)]\cong[\Sigma
X,BSp(\infty)]\cong{\widetilde{KSp}}^{-1}(X)$. Apply ${\widetilde{KSp}}^{-1}$
to the cofibration sequence $S^{4m-1}\wedge Q_{n-m}\rightarrow X\rightarrow
S^{4n+2}$. We know that ${\widetilde{KSp}}^{-1}(S^{4n+2})=0$, for every $n\geq
1$, so by use of induction, we can conclude that
${\widetilde{KSp}}^{-1}(X)=0$. On the other hand, we know that $\Omega X_{n}$
is $(4n+1)$-connected and $H^{4n+2}(\Omega X_{n})\cong\mathbb{Z}$ which is
generated by $b_{4n+2}=\sigma(\bar{y}_{4n+3})$. The map $b_{4n+2}\colon\Omega
X_{n}\rightarrow K(\mathbb{Z},4n+2)$ is a loop map and is a
$(4n+3)$-equivalence. Since dim$X\leq 4n+2$, the map
$(b_{4n+2})_{*}\colon[X,\Omega X_{n}]\rightarrow H^{4n+2}(X)$ is an
isomorphism of groups. Thus we rewrite the exact sequence $(\star)$ as the
following exact sequence

$\displaystyle{\widetilde{KSp}}^{-2}(X)\overset{\psi}{\longrightarrow}H^{4n+2}(X)\rightarrow
Sp_{n}(X)\rightarrow 0,$

where we use the isomorphism
${\widetilde{KSp}}^{-i}(X)\cong[\Sigma^{i}X,BSp(\infty)]$. So we have the
exact sequence

$0\rightarrow Coker\psi\overset{\iota}{\longrightarrow}Sp_{n}(X)\rightarrow
0.$

Therefore we get the following lemma.

######  Lemma 2.1.

$Sp_{n}(X)\cong Coker\psi$. $\quad\Box$

In what follows, we will calculate the image of $\psi$.

Let $Y$ be a $CW$-complex with dim $Y\leq 4n+2$, we denote $[Y,U(2n+1)]$ by
$U_{2n+1}(Y)$. By Theorem $1.1$ in [3], there is an exact sequence

$\tilde{K}^{-2}(Y)\overset{\varphi}{\longrightarrow}H^{4n+2}(Y)\rightarrow
U_{2n+1}(Y)\rightarrow\tilde{K}^{-1}(Y)\rightarrow 0,$

for any $f\in\tilde{K}^{-2}(Y)$ the map $\varphi$ is defined as follows

$\varphi(f)=(2n+1)!ch_{2n+1}(f),$

where $ch_{i}$ denotes the $2i$-dimensional part of the Chern character. Also,
we use the isomorphism $\tilde{K}^{-i}(Y)\cong[\Sigma^{i}Y,BU(\infty)]$. In
this paper, we denote both the canonical inclusion $Sp(n)\hookrightarrow
U(2n)$ and the induced map
${\widetilde{KSp}}^{*}(X)\rightarrow{\tilde{K}}^{*}(X)$ by $c^{\prime}$. By
Theorem $1.3$ in [11], there is a commutative diagram

${{\widetilde{KSp}}^{-2}(X)}$${H^{4n+2}(X)}$${{\tilde{K}}^{-2}(X)}$${H^{4n+2}(X).}$$\psi$$c^{\prime}$$(-1)^{n+1}$$\varphi$
(2.2)

Therefore to calculate the image of $\psi$, we first calculate the image of
$\varphi$. We denote the free abelian group with a basis $e_{1},e_{2},\cdots$,
by $\mathbb{Z}\\{e_{1},e_{2},\cdots\\}$ and the direct sum of $k$ copies of
$\mathbb{Z}$ by $\mathbb{Z}^{k}$. We need the following lemmas.

######  Lemma 2.2.

The group ${\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{n-m})$ is isomorphic to the
direct sum of a finite number of groups $\mathbb{Z}_{n}$.

###### Proof.

To prove this lemma we will use induction. Note that we have the following
isomorphisms

$\displaystyle{\widetilde{KSp}}^{-2}(S^{4i+2})\cong\mathbb{Z},\quad$
$\displaystyle{\widetilde{KSp}}^{-2}(S^{4n+1})\cong 0,$
$\displaystyle{\widetilde{KSp}}^{-2}(S^{4n})\cong\left\\{\begin{array}[]{ll}\mathbb{Z}/2\mathbb{Z}&\textrm{if
$n$ is even },\\\ \\\ 0&\textrm{if $n$ is odd},\\\ \end{array}\right.\quad$
$\displaystyle{\widetilde{KSp}}^{-2}(S^{4n-1})\cong\left\\{\begin{array}[]{ll}0&\textrm{if
$n$ is even },\\\ \\\ \mathbb{Z}/2\mathbb{Z}&\textrm{if $n$ is odd}.\\\
\end{array}\right.$

Consider the homotopy cofibration sequence

$S^{4m+6}\overset{\Sigma^{4m}v^{\prime}}{\longrightarrow}S^{4m+3}\rightarrow\Sigma^{4m}Q_{2}\rightarrow
S^{4m+7}\overset{\Sigma^{4m+1}v^{\prime}}{\longrightarrow}S^{4m+4},$

where $v^{\prime}$ is a generator of $\pi_{6}(S^{3})\cong\mathbb{Z}_{12}$. By
applying ${\widetilde{KSp}}^{-2}$, we get the following exact sequence

$\displaystyle{\widetilde{KSp}}^{-2}(S^{4m+4})\overset{(\Sigma^{4m+1}v^{\prime})^{*}}{\longrightarrow}{\widetilde{KSp}}^{-2}(S^{4m+7})\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{2})$
$\displaystyle\rightarrow{\widetilde{KSp}}^{-2}(S^{4m+3})$
$\displaystyle\rightarrow{\widetilde{KSp}}^{-2}(S^{4m+6}).$

Now, in cases where $m$ is even and $m$ is odd, we obtain the following exact
sequences

$\displaystyle
0\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{2})\rightarrow\mathbb{Z}_{2}\rightarrow\mathbb{Z},$
$\displaystyle\mathbb{Z}_{2}\overset{(\Sigma^{4m+1}v^{\prime})^{*}}{\longrightarrow}\mathbb{Z}_{2}\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{2})\rightarrow
0,$

respectively. Since $\mathbb{Z}$ is torsion-free, any map
$\mathbb{Z}_{2}\rightarrow\mathbb{Z}$ is zero map. When $m$ is even then by
exactness we obtain the group ${\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{2})$ is
isomorphic to $\mathbb{Z}_{2}$. Let $m$ be odd. Since $Sp(\infty)$ is homotopy
euivalent to $\Omega^{4}O(\infty)$, we can determine the map
${\widetilde{KSp}}^{-2}(S^{4m+4})\overset{(\Sigma^{4m+1}v^{\prime})^{*}}{\longrightarrow}{\widetilde{KSp}}^{-2}(S^{4m+7})$
by
$(\Sigma^{4m+1}v^{\prime})^{*}\colon\pi_{4m+9}(SO(\infty))\rightarrow\pi_{4m+12}(SO(\infty))$.
Let $t_{1}$ be a generator of $\pi_{4m+9}(SO(\infty))\cong\mathbb{Z}_{2}$.
Then $(\Sigma^{4m+1}v^{\prime})^{*}(t_{1})$ is the composite
$S^{4m+12}\overset{\Sigma^{4m+6}v^{\prime}}{\longrightarrow}S^{4m+9}\overset{t_{1}}{\longrightarrow}SO(\infty)$.
Note that $t_{1}\circ\Sigma^{4m+6}v^{\prime}\simeq t_{1}\circ 2v_{4m+9}\simeq
2t_{1}\circ v_{4m+9}$ is null homotopic, where by [13] we have
$\Sigma^{4m+6}v^{\prime}\simeq 2v_{4m+9}$, stably. Thus the map
$(\Sigma^{4m+1}v^{\prime})^{*}$ is the zero map. Therefore we can conclude
that the group ${\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{2})$ is also isomorphic
to $\mathbb{Z}_{2}$.

Now, we study the case $Q_{3}$. By applying ${\widetilde{KSp}}^{-2}$ to the
homotopy cofibration sequence

$S^{4m+10}\overset{\theta}{\longrightarrow}\Sigma^{4m}Q_{2}\rightarrow\Sigma^{4m}Q_{3}\rightarrow
S^{4m+11}\overset{\Sigma\theta}{\longrightarrow}\Sigma^{4m+1}Q_{2},$

we get the following exact sequence

$\displaystyle{\widetilde{KSp}}^{-2}(\Sigma^{4m+1}Q_{2})\overset{(\Sigma\theta)^{*}}{\longrightarrow}{\widetilde{KSp}}^{-2}(S^{4m+11})\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{3})$
$\displaystyle\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{2})$
$\displaystyle\overset{\theta^{*}}{\longrightarrow}{\widetilde{KSp}}^{-2}(S^{4m+10}).$

Now, in cases where $m$ is even and $m$ is odd, we obtain the following exact
sequences

$\displaystyle\mathbb{Z}_{2}\overset{(\Sigma\theta)^{*}}{\rightarrow}\mathbb{Z}_{2}\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{3})\rightarrow\mathbb{Z}_{2}\overset{\theta^{*}}{\rightarrow}\mathbb{Z},$
$\displaystyle
0\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{3})\rightarrow\mathbb{Z}_{2}\overset{\theta^{*}}{\rightarrow}\mathbb{Z},$

respectively. When $m$ is odd then by exactness we obtain the group
${\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{3})$ is isomorphic to $\mathbb{Z}_{2}$.
Let $m$ be even. We show that the map
${\widetilde{KSp}}^{-2}(\Sigma^{4m+1}Q_{2})\overset{(\Sigma\theta)^{*}}{\longrightarrow}{\widetilde{KSp}}^{-2}(S^{4m+11})$
is trivial. Let $t_{1}$ be a generator of
${\widetilde{KSp}}^{-2}(\Sigma^{4m+1}Q_{2})\cong[\Sigma^{4m+1}Q_{2},\Omega
Sp(\infty)]$. Then we know that $(\Sigma\theta)^{*}(t_{1})$ is the composite
$S^{4m+11}\overset{\Sigma\theta}{\longrightarrow}\Sigma^{4m+1}Q_{2}\overset{t_{1}}{\longrightarrow}\Omega
Sp(\infty)$, where the map $t_{1}$ comes from the composition
$\Sigma^{4m+1}Q_{2}\overset{q}{\rightarrow}S^{4m+8}\overset{s}{\rightarrow}\Omega
Sp(\infty)$, where $q$ is the pinch map to the top cell and $s$ generates
${\widetilde{KSp}}^{-2}(S^{4m+8})\cong\mathbb{Z}_{2}$. Consider the following
commutative diagram

${S^{4m+11}}$${\Sigma^{4m+1}Q_{2}}$${\Omega Sp(\infty)}$${S^{4m+8}}$${\Omega
Sp(\infty)}$$\Sigma\theta$$\Sigma^{4m+5}\nu^{\prime}$$s\circ q$$q$$||$$s$

where the square clearly commutes and since $\Sigma^{4m+5}\nu^{\prime}$
generates $\pi_{4m+11}(S^{4m+8})\cong\mathbb{Z}_{24}$, the triangle is also
commutes. We wish to show that the upper direction around the diagram is
trivial. We can determine the map
${\widetilde{KSp}}^{-2}(S^{4m+8})\rightarrow{\widetilde{KSp}}^{-2}(S^{4m+11})$
by $\pi_{4m+13}(SO(\infty))\rightarrow\pi_{4m+16}(SO(\infty))$ that sends the
generator $s$ to the composite
$S^{4m+16}\overset{\Sigma^{4m+10}v^{\prime}}{\longrightarrow}S^{4m+13}\overset{s}{\longrightarrow}SO(\infty)$.
Note that $s\circ\Sigma^{4m+10}v^{\prime}\simeq s\circ 2v_{4m+13}\simeq
2s\circ v_{4m+13}$ is null homotopic. Thus the map $(\Sigma\theta)^{*}$ is the
zero map. Therefore we can conclude that the group
${\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{3})$ is also isomorphic to
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$.

Consider the cofibration sequence

$S^{4n-2}\overset{\theta}{\longrightarrow}\Sigma^{4m}Q_{n-m-1}\rightarrow\Sigma^{4m}Q_{n-m}\rightarrow
S^{4n-1}\overset{\Sigma\theta}{\longrightarrow}\Sigma^{4m+1}Q_{n-m-1}.$

This sequence induces an exact sequence

$\displaystyle{\widetilde{KSp}}^{-2}(\Sigma^{4m+1}Q_{n-m-1})\overset{(\Sigma\theta)^{*}}{\longrightarrow}{\widetilde{KSp}}^{-2}(S^{4n-1})\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{n-m})$
$\displaystyle\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{n-m-1})$
$\displaystyle\overset{\theta^{*}}{\longrightarrow}{\widetilde{KSp}}^{-2}(S^{4n-2}).$

When $n$ is even then we get the following exact sequence

$0\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{n-m})\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{n-m-1})\overset{\theta^{*}}{\longrightarrow}\mathbb{Z},$

where by the assumption of induction, the group
${\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{n-m-1})$ is isomorphic to the direct sum
of a finite number of groups $\mathbb{Z}_{n}$. Therefore by exactness we
obtain the group ${\widetilde{KSp}}^{-2}(S^{4m}\wedge Q_{n-m})$ is also
isomorphic to the direct sum of a finite number of groups $\mathbb{Z}_{n}$.
When $n$ is odd then we get the following exact sequence

$\displaystyle{\widetilde{KSp}}^{-2}(\Sigma^{4m+1}Q_{n-m-1})\overset{(\Sigma\theta)^{*}}{\longrightarrow}\mathbb{Z}_{2}\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{n-m})\rightarrow{\widetilde{KSp}}^{-2}(\Sigma^{4m}Q_{n-m-1})\overset{\theta^{*}}{\longrightarrow}\mathbb{Z}.$

We show that the map
${\widetilde{KSp}}^{-2}(\Sigma^{4m+1}Q_{n-m-1})\overset{(\Sigma\theta)^{*}}{\longrightarrow}{\widetilde{KSp}}^{-2}(S^{4n-1})$
is trivial. Similar to the discussion of case $Q_{3}$, let $t_{1}$ be a
generator of
${\widetilde{KSp}}^{-2}(\Sigma^{4m+1}Q_{n-m-1})\cong[\Sigma^{4m+1}Q_{n-m-1},\Omega
Sp(\infty)]$. Then we know that $(\Sigma\theta)^{*}(t_{1})$ is the composite
$S^{4n-1}\overset{\Sigma\theta}{\longrightarrow}\Sigma^{4m+1}Q_{n-m-1}\overset{t_{1}}{\longrightarrow}\Omega
Sp(\infty)$, where the map $t_{1}$ comes from the composition
$\Sigma^{4m+1}Q_{n-m-1}\overset{q}{\rightarrow}S^{4n-4}\overset{s}{\rightarrow}\Omega
Sp(\infty)$, where $q$ is the pinch map to the top cell and $s$ generates
${\widetilde{KSp}}^{-2}(S^{4n-4})\cong\mathbb{Z}_{2}$. Consider the following
commutative diagram

${S^{4n-1}}$${\Sigma^{4m+1}Q_{n-m-1}}$${\Omega
Sp(\infty)}$${S^{4n-4}}$${\Omega
Sp(\infty)}$$\Sigma\theta$$\Sigma^{4n-7}\nu^{\prime}$$s\circ q$$q$$||$$s$

where $\Sigma^{4n-7}\nu^{\prime}$ generates
$\pi_{4n-1}(S^{4n-4})\cong\mathbb{Z}_{24}$. We can determine the map
${\widetilde{KSp}}^{-2}(S^{4n-4})\rightarrow{\widetilde{KSp}}^{-2}(S^{4n-1})$
by $\pi_{4n+1}(SO(\infty))\rightarrow\pi_{4n+4}(SO(\infty))$ that sends the
generator $s$ to the composite
$S^{4n+4}\overset{\Sigma^{4n-2}v^{\prime}}{\longrightarrow}S^{4n+1}\overset{s}{\longrightarrow}SO(\infty)$.
Note that $s\circ\Sigma^{4n-2}v^{\prime}\simeq s\circ 2v_{4n+1}\simeq 2s\circ
v_{4n+1}$ is null homotopic. Thus the map $(\Sigma\theta)^{*}$ is the zero
map. Thus by exactness and also the assumption of induction, we conclude that
the group ${\widetilde{KSp}}^{-2}(S^{4m}\wedge Q_{n-m})$ is also isomorphic to
the direct sum of a finite number of groups $\mathbb{Z}_{n}$. ∎

######  Lemma 2.3.

The following hold:
$(a)$: there is an isomorphism
${\widetilde{KSp}}^{-2}(X)\rightarrow{\widetilde{KSp}}^{-2}(S^{4m+2})\oplus\cdots\oplus{\widetilde{KSp}}^{-2}(S^{4n+2})\cong\mathbb{Z}^{n-m+1}$
such that
${\widetilde{KSp}}^{-2}(X)=\mathbb{Z}\\{\xi_{1},\xi_{2},\cdots,\xi_{n-m+1}\\}$,
where $\xi_{1}\in{\widetilde{KSp}}^{-2}(S^{4m+2})$,
$\xi_{2}\in{\widetilde{KSp}}^{-2}(S^{4m+6})$, $\cdots$, and also
$\xi_{n-m+1}\in{\widetilde{KSp}}^{-2}(S^{4n+2})$,
$(b)$ there is an isomorphism
${\tilde{K}}^{-2}(X)\rightarrow{\tilde{K}}^{-2}(S^{4m+2})\oplus\cdots\oplus{\tilde{K}}^{-2}(S^{4n+2})\cong\mathbb{Z}^{n-m+1}$
such that
${\tilde{K}}^{-2}(X)=\mathbb{Z}\\{\xi^{\prime}_{1},\xi^{\prime}_{2},\cdots,\xi^{\prime}_{n-m+1}\\}$,
where $\xi^{\prime}_{1}\in{\tilde{K}}^{-2}(S^{4m+2})$,
$\xi^{\prime}_{2}\in{\tilde{K}}^{-2}(S^{4m+6})$, $\cdots$, and also
$\xi^{\prime}_{n-m+1}\in{\tilde{K}}^{-2}(S^{4n+2})$,
$(c):$ if $m$ is even integer then

$c^{\prime}(\xi_{1})=\xi^{\prime}_{1},\quad
c^{\prime}(\xi_{2})=2\xi^{\prime}_{2},\quad\cdots,\quad
c^{\prime}(\xi_{n-m+1})=\left\\{\begin{array}[]{ll}\xi^{\prime}_{n-m+1}&\textrm{$n$
is even },\\\ \\\ 2\xi^{\prime}_{n-m+1}&\textrm{$n$ is odd },\\\
\end{array}\right.$

and if $m$ is odd integer then

$c^{\prime}(\xi_{1})=2\xi^{\prime}_{1},\quad
c^{\prime}(\xi_{2})=\xi^{\prime}_{2},\quad\cdots,\quad
c^{\prime}(\xi_{n-m+1})=\left\\{\begin{array}[]{ll}\xi^{\prime}_{n-m+1}&\textrm{$n$
is even },\\\ \\\ 2\xi^{\prime}_{n-m+1}&\textrm{$n$ is odd }.\\\
\end{array}\right.$

###### Proof.

First, we consider the case $Q_{2}$. Put
$L_{1}={\widetilde{KSp}}^{-2}(S^{4m+3})$ and
$L_{2}={\widetilde{KSp}}^{-2}(S^{4m+5})$. The cofibration sequence

$S^{4m+2}\rightarrow S^{4m-1}\wedge Q_{2}\rightarrow S^{4m+6}$

induces the following commutative diagram of exact sequences

${L_{1}}$${{\widetilde{KSp}}^{-2}(S^{4m+6})}$${{\widetilde{KSp}}^{-2}(S^{4m-1}\wedge
Q_{2})}$${{\widetilde{KSp}}^{-2}(S^{{4m+2}})}$${L_{2}}$${0}$${{\tilde{K}}^{-2}(S^{4m+6})}$${{\tilde{K}}^{-2}(S^{4m-1}\wedge
Q_{2})}$${{\tilde{K}}^{-2}(S^{{4m+2}})}$${0,}$$c^{\prime}=\sigma_{j}$$c^{\prime}$$c^{\prime}=\sigma_{i}$

where $\sigma_{i}=1,\sigma_{j}=2$ when $m$ is even and
$\sigma_{i}=2,\sigma_{j}=1$ when $m$ is odd, (for example, see [7]). Note that
there is an isomorphism ${\tilde{K}}^{-2}(S^{2i})\cong\mathbb{Z}$. We know
that $L_{1}$ is isomorphic to $\mathbb{Z}_{2}$ if $m$ is even and is zero if
$m$ is odd. Also, we have that $L_{2}$ is zero. So when $m$ is even then we
get the following exact sequence

$\mathbb{Z}_{2}\rightarrow\mathbb{Z}\rightarrow{\widetilde{KSp}}^{-2}(S^{4m-1}\wedge
Q_{2})\rightarrow\mathbb{Z}\rightarrow 0.$

Thus by the exactness we get ${\widetilde{KSp}}^{-2}(S^{4m-1}\wedge Q_{2})$ is
isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. When $m$ is odd then we get the
following exact sequence

$0\rightarrow\mathbb{Z}\rightarrow{\widetilde{KSp}}^{-2}(S^{4m-1}\wedge
Q_{2})\rightarrow\mathbb{Z}\rightarrow 0.$

Thus this exact sequence splits and we get
${\widetilde{KSp}}^{-2}(S^{4m-1}\wedge Q_{2})$ is isomorphic to
$\mathbb{Z}\oplus\mathbb{Z}$. Note that in both cases
${\tilde{K}}^{-2}(S^{4m-1}\wedge Q_{2})$ is a free abelian group isomorphic to
$\mathbb{Z}\oplus\mathbb{Z}$. Therefore we have

${\widetilde{KSp}}^{-2}(S^{4m-1}\wedge
Q_{2})=\mathbb{Z}\\{\xi_{1},\xi_{2}\\},\quad{\tilde{K}}^{-2}(S^{4m-1}\wedge
Q_{2})=\mathbb{Z}\\{\xi^{\prime}_{1},\xi^{\prime}_{2}\\},$

where $\xi_{1}\in{\widetilde{KSp}}^{-2}(S^{4m+2})$,
$\xi_{2}\in{\widetilde{KSp}}^{-2}(S^{4m+6})$,
$\xi^{\prime}_{1}\in{\tilde{K}}^{-2}(S^{4m+2})$ and
$\xi^{\prime}_{2}\in{\tilde{K}}^{-2}(S^{4m+6})$.

Put $L_{3}={\widetilde{KSp}}^{-2}(S^{4m}\wedge Q_{n-m})$. There is a
cofibration sequence

$S^{4m-1}\wedge Q_{n-m}\rightarrow X\rightarrow S^{4n+2}.$

This sequence induces the following commutative diagram of exact sequences

${L_{3}}$${{\widetilde{KSp}}^{-2}(S^{4n+2})}$${{\widetilde{KSp}}^{-2}(X)}$${{\widetilde{KSp}}^{-2}(S^{4m-1}\wedge
Q_{n-m})}$${0}$${0}$${{\tilde{K}}^{-2}(S^{4n+2})}$${{\tilde{K}}^{-2}(X)}$${{\tilde{K}}^{-2}(S^{4m-1}\wedge
Q_{n-m})}$${0,\quad(\ast)}$$c^{\prime}=\sigma_{k}$$c^{\prime}$$c^{\prime}$

where $\sigma_{k}=1$ when $n$ is even and $\sigma_{k}=2$ when $n$ is odd. Now,
since by Lemma 2.2 the group $L_{3}$ is isomorphic to the direct sum of a
finite number of groups $\mathbb{Z}_{n}$ we can rewrite the upper row of the
exact sequence in diagram $(\ast)$ as the following exact sequence

$\displaystyle
0\rightarrow{\widetilde{KSp}}^{-2}(S^{4n+2})\rightarrow{\widetilde{KSp}}^{-2}(X)\rightarrow{\widetilde{KSp}}^{-2}(S^{4m-1}\wedge
Q_{n-m})\rightarrow 0.$

Therefore the upper and the down rows of the exact sequences in diagram
$(\ast)$ splits. Thus by induction, we can conclude that
${\widetilde{KSp}}^{-2}(X)$ and ${\tilde{K}}^{-2}(X)$ are free abelian groups
that are isomorphic to $\mathbb{Z}^{n-m+1}$ with their basis
$\xi_{1},\xi_{2},\cdots,\xi_{n-m+1}$ and
$\xi^{\prime}_{1},\xi^{\prime}_{2},\cdots,\xi^{\prime}_{n-m+1}$, respectively,
where $\xi_{n-m+1}\in{\widetilde{KSp}}^{-2}(S^{4n+2})$ and
$\xi^{\prime}_{n-m+1}\in{\tilde{K}}^{-2}(S^{4n+2})$. Now according to the
definition of the maps $c^{\prime}$, we can choose $\xi_{1}$,
$\xi^{\prime}_{1}$, $\xi_{2}$, $\xi^{\prime}_{2}$, $\cdots$, $\xi_{n-m+1}$ and
$\xi^{\prime}_{n-m+1}$ such that if $m$ is even integer then we have

$c^{\prime}(\xi_{1})=\xi^{\prime}_{1},\quad
c^{\prime}(\xi_{2})=2\xi^{\prime}_{2},\quad\cdots,\quad
c^{\prime}(\xi_{n-m+1})=\left\\{\begin{array}[]{ll}\xi^{\prime}_{n-m+1}&\textrm{$n$
is even },\\\ \\\ 2\xi^{\prime}_{n-m+1}&\textrm{$n$ is odd },\\\
\end{array}\right.$

and if $m$ is odd integer then we have

$c^{\prime}(\xi_{1})=2\xi^{\prime}_{1},\quad
c^{\prime}(\xi_{2})=\xi^{\prime}_{2},\quad\cdots,\quad
c^{\prime}(\xi_{n-m+1})=\left\\{\begin{array}[]{ll}\xi^{\prime}_{n-m+1}&\textrm{$n$
is even },\\\ \\\ 2\xi^{\prime}_{n-m+1}&\textrm{$n$ is odd }.\\\
\end{array}\right.$

∎

Consider the map $c^{\prime}\colon Sp(n-m+1)\rightarrow SU(2n-2m+2)$. By
restriction of the map $c^{\prime}$ to their quasi-projective spaces, we get
to map ${\bar{c}}^{\prime}\colon
Q_{n-m+1}\rightarrow\Sigma\mathbb{C}P^{2n-2m+1}$. The cohomologies of
$Q_{n-m+1}$ and $\Sigma\mathbb{C}P^{2n-2m+1}$ are given by

$\displaystyle
H^{*}(Q_{n-m+1})=\mathbb{Z}\\{{\bar{y}}_{3},{\bar{y}}_{7},\cdots,{\bar{y}}_{4n-4m+3}\\},$
$\displaystyle
H^{*}(\Sigma\mathbb{C}P^{2n-2m+1})=\mathbb{Z}\\{{\bar{x}}_{3},{\bar{x}}_{5},{\bar{x}}_{7},\cdots,{\bar{x}}_{4n-4m+3}\\},$

such that ${\bar{c}}^{\prime}({\bar{x}}_{3})={\bar{y}}_{3}$,
${\bar{c}}^{\prime}({\bar{x}}_{5})=0$,
${\bar{c}}^{\prime}({\bar{x}}_{7})={\bar{y}}_{7}$, $\cdots$, and
${\bar{c}}^{\prime}({\bar{x}}_{4n-4m+3})={\bar{y}}_{4n-4m+3}$. We denote a
generator of $\tilde{K}(S^{2n})\cong\mathbb{Z}$ by $\zeta_{n}$. Recall that

$H^{*}(\mathbb{C}P^{2n-2m+1})=\mathbb{Z}[t]/(t^{2n-2m+2}),\qquad
K(\mathbb{C}P^{2n-2m+1})=\mathbb{Z}[x]/(x^{2n-2m+2}),$

where $|t|=2$. Note that
${\tilde{K}}^{-2}(\Sigma^{4m}\mathbb{C}P^{2n-2m+1})\cong{\tilde{K}}^{0}(\Sigma^{4m+2}\mathbb{C}P^{2n-2m+1})$
is a free abelian group generated by $\zeta_{2m+1}\otimes x$,
$\zeta_{2m+1}\otimes x^{2}$, $\cdots$, $\zeta_{2m+1}\otimes x^{2n-2m+1}$, with
the following Chern characters

$\displaystyle ch_{2n+1}(\zeta_{2m}\otimes
x)=ch_{2m}(\zeta_{2m})ch_{2n-2m+1}(x)=\frac{1}{(2n-2m+1)!}\sigma^{4m}t^{2n-2m+1},$
$\displaystyle ch_{2n+1}(\zeta_{2m}\otimes
x^{2})=ch_{2m}(\zeta_{2m})ch_{2n-2m+1}(x^{2})=A\sigma^{4m},$ $\displaystyle
ch_{2n+1}(\zeta_{2m}\otimes
x^{3})=ch_{2m}(\zeta_{2m})ch_{2n-2m+1}(x^{3})=B\sigma^{4m},$
$\displaystyle\vdots$ $\displaystyle ch_{2n+1}(\zeta_{2m}\otimes
x^{2n-2m+1})=ch_{2m}(\zeta_{2m})ch_{2n-2m+1}(x^{2n-2m+1})=C\sigma^{4m},$

where $A$, $B$ and $C$ are equal to

$A=ch_{2n-2m+1}(x^{2})=\sum_{\begin{subarray}{c}i_{1}+i_{2}=2n-2m+1,\\\ 1\leq
i_{1}\leq
n-m\end{subarray}}ch_{i_{1}}xch_{i_{2}}x=\sum_{k=1}^{n-m}\frac{1}{k!(2n-2m+1-k)!}t^{2n-2m+1},$
$\displaystyle B=ch_{2n-2m+1}(x^{3})=$ $\displaystyle
ch_{1}x\sum_{\begin{subarray}{c}i_{1}+i_{2}=2n-2m,\\\ 1\leq i_{1}\leq
i_{2}\end{subarray}}ch_{i_{1}}xch_{i_{2}}x+ch_{2}x\sum_{\begin{subarray}{c}i_{1}+i_{2}=2n-2m-1,\\\
2\leq i_{1}\leq i_{2}\end{subarray}}ch_{i_{1}}xch_{i_{2}}x+\cdots$
$\displaystyle+ch_{n-m-1}x\sum_{\begin{subarray}{c}i_{1}+i_{2}=n-m+2,\\\
n-m-1\leq i_{1}\leq
i_{2}\end{subarray}}ch_{i_{1}}xch_{i_{2}}x+\sum_{\begin{subarray}{c}i_{1}+i_{2}=2n-2m+1,\\\
i_{1},i_{2}\geq 1\end{subarray}}ch_{i_{1}}xch_{i_{2}}x^{2},$

and

$\displaystyle C=ch_{2n-2m+1}(x^{2n-2m+1})=$ $\displaystyle
ch_{1}x\sum_{\begin{subarray}{c}i_{1}+\cdots+i_{2n-2m}=2n-2m,\\\ 1\leq
i_{1}\leq i_{2}\leq\cdots\leq
i_{2n-2m}\end{subarray}}ch_{i_{1}}x^{i_{1}}\cdots ch_{i_{2n-2m}}x^{i_{2n-2m}}$
$\displaystyle+ch_{2}x^{2}\sum_{\begin{subarray}{c}i_{1}+\cdots+i_{n-m-1}=2n-2m-1,\\\
2\leq i_{1}\leq i_{2}\leq\cdots\leq
i_{n-m-1}\end{subarray}}ch_{i_{1}}x^{i_{1}}\cdots ch_{i_{n-m-1}}x^{i_{n-m-1}}$
$\displaystyle+ch_{3}x^{3}\sum_{\begin{subarray}{c}i_{1}+\cdots+i_{n-m-2}=2n-2m-2,\\\
3\leq i_{1}\leq i_{2}\leq\cdots\leq
i_{n-m-2}\end{subarray}}ch_{i_{1}}x^{i_{1}}\cdots
ch_{i_{n-m-2}}x^{i_{n-m-2}}+\cdots$
$\displaystyle+ch_{n-m}x^{n-m}\sum_{\begin{subarray}{c}i_{1}=n-m+1,\\\
i_{1}\geq n-m\end{subarray}}ch_{i_{1}}x^{i_{1}}.$

Consider the map
${\bar{c}}^{\prime}\colon{\tilde{K}}^{-2}(\Sigma^{4m}\mathbb{C}P^{2n-2m+1})\rightarrow{\tilde{K}}^{-2}(S^{4m-1}\wedge
Q_{n-m+1})$, we can put $\xi^{\prime}_{1}$, $\xi^{\prime}_{2}$, $\cdots$ and
$\xi^{\prime}_{n-m+1}$ such that
$\xi^{\prime}_{1}={\bar{c}}^{\prime}(\zeta_{2m}\otimes x)$,
$\xi^{\prime}_{2}={\bar{c}}^{\prime}(\zeta_{2m}\otimes x^{3})$, $\cdots$, and
$\xi^{\prime}_{n-m+1}={\bar{c}}^{\prime}(\zeta_{2m}\otimes x^{2n-2m+1})$. We
have the following proposition.

######  Proposition 2.4.

The image of $\psi$ is generated by

$\left\\{\begin{array}[]{ll}\frac{(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}&\textrm{if
$m$ is even },\\\ \\\
\frac{2(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}&\textrm{if
$m$ is odd }.\\\ \end{array}\right.$

###### Proof.

Consider the following commutative diagram

${{\tilde{K}}^{-2}(\Sigma^{4m}\mathbb{C}P^{2n-2m+1})}$${H^{4n+2}(\Sigma^{4m}\mathbb{C}P^{2n-2m+1})}$${{\tilde{K}}^{-2}(S^{4m-1}\wedge
Q_{n-m+1})}$${H^{4n+2}(S^{4m-1}\wedge
Q_{n-m+1}),}$$\varphi^{\prime}$${\bar{c}}^{\prime}$$\cong$$\varphi$ (2.3)

where the map $\varphi^{\prime}$ is defined similarly to the map $\varphi$. By
definition of the map $\varphi^{\prime}$, we have

$\displaystyle\varphi^{\prime}(\zeta_{2m}\otimes
x)=(2n+1)!ch_{2n+1}(\zeta_{2m}\otimes
x)=\frac{(2n+1)!}{(2n-2m+1)!}\sigma^{4m}t^{2n-2m+1},$
$\displaystyle\varphi^{\prime}(\zeta_{2m}\otimes
x^{2})=(2n+1)!ch_{2n+1}(\zeta_{2m}\otimes x^{2})=A(2n+1)!\sigma^{4m},$
$\displaystyle\varphi^{\prime}(\zeta_{2m}\otimes
x^{3})=(2n+1)!ch_{2n+1}(\zeta_{2m}\otimes x^{3})=B(2n+1)!\sigma^{4m},$
$\displaystyle\vdots$ $\displaystyle\varphi^{\prime}(\zeta_{2m}\otimes
x^{2n-2m+1})=(2n+1)!ch_{2n+1}(\zeta_{2m}\otimes
x^{2n-2m+1})=C(2n+1)!\sigma^{4m}.$

Therefore according to the commutativity of diagram $(2.3)$, we have

$\displaystyle\varphi(\xi^{\prime}_{1})=\varphi^{\prime}(\zeta_{2m}\otimes
x)=\frac{(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3},$
$\displaystyle\varphi(\xi^{\prime}_{2})=\varphi^{\prime}(\zeta_{2m}\otimes
x^{3})=B^{\prime}(2n+1)!\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3},$
$\displaystyle\vdots$
$\displaystyle\varphi(\xi^{\prime}_{n-m+1})=\varphi^{\prime}(\zeta_{2m}\otimes
x^{2n-2m+1})=C^{\prime}(2n+1)!\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3},$

where $B=B^{\prime}t^{2n-2m+1}$ and $C=C^{\prime}t^{2n-2m+1}$. Thus by the
commutativity of diagram $(2.2)$, we get

$\psi(\xi_{1})=\varphi(c^{\prime}(\xi_{1}))=\pm\left\\{\begin{array}[]{ll}\frac{(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}&\textrm{if
$m$ is even },\\\ \\\
\frac{2(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}&\textrm{if
$m$ is odd },\\\ \end{array}\right.$
$\psi(\xi_{2})=\varphi(c^{\prime}(\xi_{2}))=\pm\left\\{\begin{array}[]{ll}2B^{\prime}(2n+1)!\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}&\textrm{if
$m$ is even },\\\ \\\
B^{\prime}(2n+1)!\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}&\textrm{if $m$ is odd
},\\\ \end{array}\right.$

$\qquad\vdots$

$\psi(\xi_{n-m+1})=\varphi(c^{\prime}(\xi_{n-m+1}))=\left\\{\begin{array}[]{ll}-(2n+1)!C^{\prime}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}&\textrm{if
$n$ is even },\\\ \\\
2(2n+1)!C^{\prime}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}&\textrm{if $n$ is odd
}.\\\ \end{array}\right.$

Since the coefficients $B^{\prime},\cdots,C^{\prime}$ are divisible by
$\frac{1}{(2n-2m+1)!}$, the image of $\psi$ is included in the submodule
generated by $\frac{(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}$
if $m$ is even and is included in the submodule generated by
$\frac{2(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}$ if $m$ is
odd. Thus we can conclude that

$Im\psi\cong\left\\{\begin{array}[]{ll}\mathbb{Z}\\{\frac{(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}\\}&\textrm{if
$m$ is even },\\\ \\\
\mathbb{Z}\\{\frac{2(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}\\}&\textrm{if
$m$ is odd }.\\\ \end{array}\right.$

∎

Therefore by Lemma 2.1 and Proposition 2.4, we get the following theorem.

######  Theorem 2.5.

There is an isomorphism

$[X,Sp(n)]\cong\left\\{\begin{array}[]{ll}\mathbb{Z}_{\frac{(2n+1)!}{(2n-2m+1)!}}&\textrm{if
$m$ is even },\\\ \\\ \mathbb{Z}_{\frac{2(2n+1)!}{(2n-2m+1)!}}&\textrm{if $m$
is odd }.\quad\Box\\\ \end{array}\right.$

##  3 Proof of Theorem 1.1

In this section, we will prove Theorem 1.1. We recall that $X=S^{4m-1}\wedge
Q_{n-m+1}$. In the following theorem that was proved in [11], we obtain the
commutators in the group $[X,Sp(n)]$.

######  Theorem 3.1.

Let $X$ be a $CW$-complex with dim $X\leq 4n+2$. For $\alpha,\beta\in
Sp_{n}(X)$, the commutator $[\alpha,\beta]$ in $Sp_{n}(X)$ is given as

$[\alpha,\beta]=\iota([\sum_{i+j=n+1}\alpha^{*}(y_{4i-1})\beta^{*}(y_{4j-1})]).\quad\Box$

Put $X^{\prime}=S^{4m-1}\times Q_{n-m+1}$. Consider the following compositions

$S^{4m-1}\times
Q_{n-m+1}\overset{p_{1}}{\longrightarrow}S^{4m-1}\overset{\varepsilon_{m,n}}{\longrightarrow}Sp(n),$
$S^{4m-1}\times
Q_{n-m+1}\overset{p_{2}}{\longrightarrow}Q_{n-m+1}\overset{\epsilon_{m,n}}{\longrightarrow}Sp(n),$

where the maps $p_{1}$ and $p_{2}$ are the first and the second projections.
Put $\varepsilon_{m,n}\circ p_{1}=\alpha$ and $\epsilon_{m,n}\circ
p_{2}=\beta$. The commutator $[\alpha,\beta]\in[X^{\prime},Sp(n)]$ is the
following composition

$X^{\prime}\overset{\bar{\Delta}}{\longrightarrow}X^{\prime}\wedge
X^{\prime}\overset{\alpha\wedge\beta}{\longrightarrow}Sp(n)\wedge
Sp(n)\overset{\gamma}{\longrightarrow}Sp(n),$

where $\bar{\Delta}$ and $\gamma$ are the reduced diagonal map and the
commutator map. Consider the following diagram

${X^{\prime}}$${X^{\prime}\times X^{\prime}}$${X^{\prime}}$${X}$${Sp(n)\wedge
Sp(n)}$${Sp(n)}$${X^{\prime}\wedge X^{\prime}}$$\Delta$$p_{1}\times
p_{2}$$q$$\varepsilon_{m,n}\wedge\epsilon_{m,n}$$\gamma$$\bar{\Delta}$$\quad\alpha\wedge\beta$

where $\Delta$ is the diagonal map and $q:X^{\prime}\rightarrow X$ is the
quotient map. Since the composition

$X^{\prime}=S^{4m-1}\times
Q_{n-m+1}\overset{\Delta}{\longrightarrow}X^{\prime}\times
X^{\prime}\overset{p_{1}\times p_{2}}{\longrightarrow}S^{4m-1}\times
Q_{n-m+1}=X^{\prime}$

is $1$, the Samelson product
$\langle\varepsilon_{m,n},\epsilon_{m,n}\rangle=\gamma\circ(\varepsilon_{m,n}\wedge\epsilon_{m,n})$
is equal to the commutator $[\alpha,\beta]$ in the group $[X,Sp(n)]$. So by
Theorem 3.1, we have

$[\alpha,\beta]=\iota([{\varepsilon_{m,n}}^{*}(y_{4m-1})\otimes{\epsilon_{m,n}}^{*}(y_{4n-4m+3})])=\iota([\sigma^{4m-1}\otimes\bar{y}_{4n-4m+3}]).$

Therefore by Theorem 2.5, when $m$ is even then the order of the Samelson
product $\langle\varepsilon_{m,n},\epsilon_{m,n}\rangle$ is equal to
$\frac{(2n+1)!}{(2n-2m+1)!}$ and when $m$ is odd then the order of the
Samelson product $\langle\varepsilon_{m,n},\epsilon_{m,n}\rangle$ is equal to
$\frac{2(2n+1)!}{(2n-2m+1)!}$. $\quad\Box$

## References

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