# Exploiting Multi-Label Correlation in Label Distribution Learning

Zhiqiang Kou12, Jing Wang12, Yuheng Jia12 and Xin Geng1
Corresponding author.Corresponding author.

###### Abstract

Label Distribution Learning (LDL) is a novel machine learning paradigm that
assigns label distribution to each instance. Many LDL methods proposed to
leverage label correlation in the learning process to solve the exponential-
sized output space; among these, many exploited the low-rank structure of
label distribution to capture label correlation. However, recent studies
disclosed that label distribution matrices are typically full-rank, posing
challenges to those works exploiting low-rank label correlation. Note that
multi-label is generally low-rank; low-rank label correlation is widely
adopted in multi-label learning (MLL) literature. Inspired by that, we
introduce an auxiliary MLL process in LDL and capture low-rank label
correlation on that MLL rather than LDL. In such a way, low-rank label
correlation is appropriately exploited in our LDL methods. We conduct
comprehensive experiments and demonstrate that our methods are superior to
existing LDL methods. Besides, the ablation studies justify the advantages of
exploiting low-rank label correlation in the auxiliary MLL.

## Introduction

Label distribution learning (LDL) (Geng 2016) is a novel learning paradigm
that provides fine-grained label information for each instance. Unlike
traditional learning paradigms, LDL introduces the label description degree
(Geng 2016) that is a real-value and quantify the relevance of one labels to a
specific example. The label description degrees of all labels form a label
distribution, which provides a comprehensive representation of labeling
information. Fig.1 showcases an image from a natural-scene (Geng and Luo 2014)
dataset. The average ratings are rescaled to form a label distribution
$\\{0.25,0.4,0.25,0.1\\}$, effectively capturing the varying degrees of
importance assigned to labels. LDL utilizes label description degrees to
effectively solve label ambiguity (Gao et al. 2017).

Figure 1: An image from a natural-scene dataset (Geng and Luo 2014) with a
label distribution.

LDL has an overwhelming output space, which exponentially grows with the
number of potential labels (Wang and Geng 2021a). To tackle this issue, label
correlation has emerged as a promising solution. Many LDL works have proposed
to leverage label correlation in the learning processes. To name a few, (Xu
and Zhou 2017) captured the low-rank structure of label distribution matrix to
incorporate label correlation. (Jia et al. 2019) leveraged the low-rank
assumption to capture label correlations shared by different groups of
samples. They effectively captured label correlations in a local context.
Additionally, (Ren et al. 2019) utilized a low-rank matrix to capture the
global label correlation and further updated it based on different clusters to
also explore local label correlations. Label correlation helps significantly
improve the performance of these LDL methods. Notice that the above mentioned
LDL methods assumed that label distribution has low-rank structure and relied
it to exploit label correlation. However, (Wang and Geng 2021a) has
demonstrated that label distribution matrices are usually full-rank, which
challenges the suitability of these approaches with low-rank assumption. So,
it is possible to exploit the low-rank label correlation in LDL more
efficiently?

Our work is mainly inspired by two observations. The first one is that low-
rank has been extensively applied to multi-label learning (MLL) to capture
label correlation (Xu, Tao, and Xu 2016)(Jing et al. 2015)(Liu et al. 2021)(Wu
et al. 2020)(Xu, Tao, and Xu 2016)(Yu et al. 2018). The second one is that
label distribution has rich supervision information and implicitly contains
multi-label information. To see that, for the example in Fig. 1 we can observe
that the given label distribution contains implicit multi-labels of {tree,
mountain, water}. Our basic idea is to introduce an auxiliary MLL process in
LDL and exploit the low-rank label correlation on the MLL part, for example,
by assuming the MLL matrix is low-rank. That is, the low-rank assumption is
added on the MLL part instead of the LDL part.

Following the strategy, we propose two novel LDL methods TLRLDL and TKLRLDL to
exploit low-rank label correlation. First, we propose two methods to generate
multi-label from label distribution. The first one utilizes a threshold to
separate label distribution into multi-label, and the second one selects the
top-$k$ labels having the largest label description degrees as the positive
labels. Next, we learn label distribution and the generated multi-label
simultaneously, and capture the low-rank label correlation in the MLL process.
We conduct expensive experiments to justify that the proposed methods
outperform existing LDL approaches. Besides, the ablation studies validate the
advantages of exploiting the low-rank label correlation in the MLL process. To
sum up, our major contributions are as follows:

  * •

As far as we know, this is the first work to introduce an auxiliary MLL
process in LDL and exploit MLL label correlation for LDL.

  * •

We exploit label correlation in LDL by capturing the low-rank structure in an
auxiliary MLL process, which is more reasonable than directly exploiting the
low-rank label correlation of LDL.

  * •

We conduct extensive experiments to validate the advantages of our methods
over existing LDL algorithms and the superiority of exploiting the low-rank
label correlation in the auxiliary MLL process.

## Related Work

### Lable Distribution Learning

As a novel learning paradigm, LDL introduces label distribution to accurately
capture the degree of labeling for each instance. This unique characteristic
has sparked significant research interest in the field of LDL. In this
section, we provide a brief overview of the existing studies in LDL.

The existing LDL (Le et al. 2023)(Tan et al. 2023) algorithms can be broadly
classified into three categories. The first category involves transforming the
LDL problem into a single-label learning problem by assigning weights to the
training samples. Representative algorithms in this category include PT-SVM
and PT-Bayes, which utilize SVM and Bayes classifiers to solve the transformed
weighted single-label learning problem. The second category focuses on
adapting traditional machine learning algorithms to handle the LDL problem.
For instance, the K-nearest neighbors (KNN) classifier identifies the top k
neighbors of an instance and predicts the label distribution by averaging the
labels of these neighbors. Back-Propagation (BP) neural networks directly
optimize the descriptive degree of the final prediction through the BP
algorithm. The third category comprises specialized algorithms such as IIS-LDL
and BFGS-LDL. These algorithms formulate LDL as a regression problem and
employ improved iterative scaling and quasi-Newton methods, respectively, to
solve the regression problem efficiently. However, these LDL algorithms do not
take label correlation into account.

### Label Correlation in LDL

In recent years, researchers have recognized the challenge of the vast output
space of LDL (Gao et al. 2017)(Wang and Geng 2019)(Shen et al. 2017)(Wang and
Geng 2021b)(Ren and Geng 2017) and have developed various approaches to
address this issue. These methods can be categorized into three main types: 1)
global label correlation, 2) local label correlation, and 3) both global and
local label correlations. In the first category, (Zhou, Xue, and Geng 2015)
introduced a weighted Jeffrey’s divergence (Cha 2007) to capture label
correlation by assigning weights based on the Pearson correlation coefficient.
(Xu and Zhou 2017) incorporated the low-rank structure of label distribution
by applying trace-norm regularization. In the second category, (Jia et al.
2019) utilized a local low-rank structure to implicitly capture the local
label correlations. In the third category, (Ren et al. 2019) introduced LDL-
LCLR, a method that leverages both global and local label correlations. It
utilizes a low-rank matrix to capture global label correlation and updates the
matrix based on different clusters to explore local label correlation.

However, many of these works rely on the assumption of low-rank to exploit
label correlation. As reported by (Wang and Geng 2021a), label distribution
matrix is generally full rank, which poses a challenge to those exploiting
low-rank label correlation of LDL. Instead of directly exploiting low-rank
label correlation of LDL, this study introduces an auxiliary MLL and exploits
the low-rank label correlation on the MLL process, which can efficiently solve
the above mentioned problem.

## The Proposed Method

Notations: Let
$\mathbf{X}=\left[\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n}\right]\in\mathbb{R}^{n\times
d}$ denote the feature matrix and
$\mathbf{Y}=\left\\{y_{1},y_{2},\ldots,y_{m}\right\\}$ be the label space,
where $n$, $m$, and $d$ denote the numbers of instances, labels, and the
dimension of features, respectively. The training set of the LDL is
represented as
$\mathbf{T}=\left\\{\left(\mathbf{x}_{1},\mathbf{d}_{1}\right),\quad\left(\mathbf{x}_{2},\mathbf{d}_{2}\right),\ldots,\left(\mathbf{x}_{n},\mathbf{d}_{n}\right)\right\\}$,
where
$\mathbf{d}_{i}=\left[d_{\mathbf{x}_{i}}^{y_{1}},d_{\mathbf{x}_{i}}^{y_{2}},\ldots,d_{\mathbf{x}_{i}}^{y_{m}}\right]$
is the label distribution of the $i$th sample $\mathbf{x}_{i}$.
$d_{\mathbf{x}_{i}}^{y}$ is the label description degree of $y$ to
$\mathbf{x}_{i}$, which satisfies $d_{\mathbf{x}_{i}}^{y}\in[0,1]$ and
$\sum_{y}d_{\mathbf{x}}^{y}=1$. The label distribution matrix is denoted as
$\mathbf{D}=\left[\mathbf{d}_{1},\mathbf{d}_{2},\ldots,\mathbf{d}_{n}\right]\in\mathbb{R}^{m\times
n}$. LDL aims to learn a mapping function from $\mathbf{T}$ and predict the
label distribution for unseen instances.

### Transforming Label Distribution into Multi-Label

This subsection will introduce two methods for transforming label distribution
into multi-label. The first one is threshold-based degradation, and the second
one is top-$k$ degradation. Next, we provide details of these two methods.

#### Threshold-Based Multi-Label Generation

To convert label distribution into multi-label, we simulate the labeling
process that users typically follow when assigning labels to images or adding
keywords to texts. Overall, users continue adding the most relevant labels
until they perceive that the labeling is sufficiently comprehensive(Xu, Liu,
and Geng 2019). Based on that, we can degrade multi-label from label
distribution through this iterative labeling procedure. The process is
outlined as follows:

  * •

For each instance $\mathbf{x}$, find the label $y_{j}$ with the highest
description degree $d_{\mathbf{x}}^{y_{j}}$ and add it to relevant labels
(i.e., $l_{\mathbf{x}}^{y_{j}}=1$).

  * •

Calculate the sum of the description degrees of all the currently relevant
labels $H=\sum_{y_{j}\in\mathcal{Y}^{+}}d_{\boldsymbol{x}}^{y_{j}}$, where
$\mathcal{Y}^{+}$ is the set of the currently relevant labels.

  * •

If $H$ is less than a predefined threshold $T$, continue finding the label
with the highest description degree from the labels not included in
$\mathcal{Y}$, and add it to $\mathcal{Y}$. Repeat this process until $H>T$.

Following this process, we can generate multi-label from label distribution
that mimics the way users label data.

#### Top-$k$ Based Multi-Label Generation

Specifically, for any instance $\mathbf{x}_{i}$, we first sort the label
description degrees in descending order. Then, we select the top-$k$ labels
with the highest label description degrees as relevant labels, and assign the
remaining labels as irrelevant labels. That is, the top-$k$ labels with the
highest label description degrees are considered relevant for each instance,
while the rest are deemed irrelevant.

### Auxiliary MLL and Label Correlation

First, we employ the least square method to learn label distribution and
minimize the $L_{2}$-norm loss between the ground-truth label distribution and
prediction, which can be formalized as the following:

$\min_{\mathbf{W}}\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}-\mathbf{D}\right\|_{F}^{2}+\lambda\|\mathbf{W}\|_{F}^{2},$
(1)

where $\mathbf{W}\in\mathbb{R}^{m\times d}$ is the parameter matrix,
$\|\cdot\|_{F}$ represents the Frobenius norm, and $\lambda$ is a
regularization parameter. Next, we establish the mapping relationship between
label distribution and multi-label generated in the previous section. This
linear mapping is formulated as:

$\min_{\mathbf{O}}\|\mathbf{DO-L}\|_{F}^{2}+\lambda\|\mathbf{O}\|_{F}^{2},$
(2)

where $\mathbf{O}\in\mathbb{R}^{n\times n}$ is the transformation parameter
matrix.

Next, we exploit the low-rank label correlation. However, given the full-rank
nature of the label distribution matrix, assuming a low-rank structure does
not suit LDL. To address that, we encourage the low-rank structure on the MLL
process, which has been widely accepted in MLL literature. That is, the
predicted MLL matrix is assumed to be low-rank, which further casts Eq. (2)
as:

$\min_{\mathbf{O}}\|\mathbf{DO-L}\|_{F}^{2}+\alpha\text{Rank}(\mathbf{W}\mathbf{X}^{\top}\mathbf{O})+\eta\|\mathbf{O}\|_{F}^{2},$
(3)

where $\text{Rank}(\mathbf{A})$ represents the rank of $\mathbf{A}$, and
$\alpha$ is a balance parameters. By jointly optimizing Problem (1) and
Problem (3), we obtain the final formulation as follows:

$\begin{gathered}\min_{\mathbf{W,O}}\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}-\mathbf{D}\right\|_{F}^{2}+\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}\mathbf{O}-\mathbf{L}\right\|_{F}^{2}+\\\
\alpha\text{Rank}\left(\mathbf{W}\mathbf{X}^{\top}\mathbf{O}\right)+\lambda\left(\|\mathbf{W}\|_{F}^{2}+\|\mathbf{O}\|_{F}^{2}\right),\end{gathered}$
(4)

$\text{Rank}(\mathbf{W})$ is difficult to solve due to the discrete nature of
the rank function. Fortunately, as suggested by (Candès et al. 2011), the
nuclear-norm (Fazel 2002) is a good surrogate for the rank function. Replacing
the rank function with the nuclear-norm, we obtain the next optimization
problem:

$\begin{gathered}\min_{\mathbf{W,O}}\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}-\mathbf{D}\right\|_{F}^{2}+\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}\mathbf{O}-\mathbf{L}\right\|_{F}^{2}+\\\
\alpha\left\|\mathbf{W}\mathbf{X}^{\top}\mathbf{O}\right\|_{*}+\lambda\left(\|\mathbf{W}\|_{F}^{2}+\|\mathbf{O}\|_{F}^{2}\right).\end{gathered}$
(5)

The method learning multi-label by threshold is called TLRLDL and the other
one learning multi-label from top-$k$ is called TKLRLDL.

### Optimization

We use ADMM to solve problem (5), which is good at handling equality
constraints. First, we introduce an auxiliary variable
$\mathbf{G}\in\mathbb{R}^{m\times n}$ and rewrite Eq. (5) as

$\begin{gathered}\min_{\mathbf{W,O,G}}\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}-\mathbf{D}\right\|_{F}^{2}+\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}\mathbf{O}-\mathbf{L}\right\|_{F}^{2}+\\\
\alpha\left\|\mathbf{G}\right\|_{*}+\lambda\left(\|\mathbf{W}\|_{F}^{2}+\|\mathbf{O}\|_{F}^{2}\right)\\\
\text{s.t. }\mathbf{W}\mathbf{X}^{\top}\mathbf{O}=\mathbf{G}.\end{gathered}$
(6)

We introduce the augmented Lagrangian function for Eq. (6)

$\begin{gathered}\min_{\mathbf{W,O,G}}\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}-\mathbf{D}\right\|_{F}^{2}+\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}\mathbf{O}-\mathbf{L}\right\|_{F}^{2}+\alpha\left\|\mathbf{G}\right\|_{*}\\\
+\lambda\left(\|\mathbf{W}\|_{F}^{2}+\|\mathbf{O}\|_{F}^{2}\right)+\frac{\mu}{2}\left\|\mathbf{G}-\mathbf{W}\mathbf{X}^{\top}\mathbf{O}-\frac{\mathbf{\Gamma}_{1}}{\mu}\right\|_{F}^{2},\end{gathered}$

where $\mu$ is a positive penalty parameter, and $\mathbf{\Gamma}_{1}$ denotes
the Lagrangian multipliers. It can be solved by alternately optimizing three
sub-problems as follows. The whole process is summarized in Algorithm 1.

#### Solving $\mathbf{G}$-Subproblem

The subproblem w.r.t. $\mathbf{G}$ is

$\mathbf{G}^{k+1}=\underset{\mathbf{G}}{\operatorname{argmin}}\alpha\|\mathbf{G}\|_{*}+\frac{\mu}{2}\left\|\mathbf{G-W}\mathbf{X}^{\top}\mathbf{O}-\frac{\mathbf{\Gamma}_{1}}{\mu}\right\|_{F}^{2}.$

It is a nuclear norm minimization problem and has a closed-form solution (Cai,
Candès, and Shen 2010):

$\mathbf{G}^{k+1}=S_{\alpha/\mu}(T),$ (7)

where $T=\mathbf{WX}^{\top}\mathbf{O}+\frac{\mathbf{\Gamma}_{1}}{\mu}$, and
${S(\cdot)}$ is single value thresholding operator. It first performs singular
value decomposition on
$\mathbf{WX}^{\top}\mathbf{O}+\frac{\mathbf{\Gamma}_{1}}{\mu}=\mathbf{U}\hat{\mathbf{\Sigma}}\mathbf{V}^{\top}$,
and then gives the solution as
$\mathbf{U}\hat{\mathbf{\Sigma}}\mathbf{V}^{\top}$, where
$\hat{\Sigma}_{ii}=\max\left(0,\Sigma_{ii}-\alpha/\mu\right)$.

#### Solving $\mathbf{W}$-Subproblem

The subproblem w.r.t. $\mathbf{W}$ is

$\displaystyle\mathbf{W}^{k+1}$
$\displaystyle=\underset{\mathbf{W}}{\operatorname{argmin}}\frac{1}{2}\left\|\mathbf{WX}^{\top}-\mathbf{D}\right\|_{F}^{2}+\frac{1}{2}\left\|\mathbf{WX}^{\top}\mathbf{O-L}\right\|_{T}^{2}$
$\displaystyle+\lambda\|\mathbf{W}\|_{F}^{2}+\frac{\mu}{2}\left\|\mathbf{G-WX}^{\top}\mathbf{O}-\frac{\mathbf{\Gamma}_{1}}{\mu}\right\|_{F}^{2}$

which is a quadratic optimization problem. The optimal solution is obtained by
setting the derivative to zero and equals

$\begin{split}\mathbf{W}=&\left(\mathbf{X}^{\top}\mathbf{X}+2\lambda+\mu\mathbf{X}^{\top}\mathbf{OO}^{\top}\mathbf{X}+\mathbf{X}^{\top}\mathbf{OO}^{\top}\mathbf{X}\right)^{-1}\\\
&(\mu\mathbf{GO}^{\top}\mathbf{X}-\mathbf{\Gamma}_{1}\mathbf{O}^{\top}\mathbf{X}+\mathbf{LO}^{\top}\mathbf{X}+\mathbf{DX})\end{split}$
(8)

#### Solving $\mathbf{O}$-Subproblem

The subproblem w.r.t. $\mathbf{O}$ is

$\displaystyle\mathbf{O}^{k+1}$
$\displaystyle=\frac{1}{2}\left\|\mathbf{WX}^{\top}\mathbf{O}-\mathbf{L}\right\|_{F}^{2}+\lambda\|\mathbf{O}\|_{F}^{2}$
(9)
$\displaystyle+\frac{\mu}{2}\left\|\mathbf{G-WX}^{\top}\mathbf{O}-\frac{\mathbf{\Gamma}_{1}}{\mu}\right\|_{F}^{2}$

which is a quadratic optimization problem. The optimal solution is obtained by
setting the derivative to zero and equals

$\displaystyle\mathbf{O}^{k+1}=$
$\displaystyle\left(\mathbf{XW}^{\top}\mathbf{WX}^{\top}+2\lambda+\mu\mathbf{XW}^{\top}\mathbf{WX}^{\top}\right)^{-1}$
(10) $\displaystyle+$
$\displaystyle\left(\mathbf{XW}^{\top}\mathbf{L}+\mu\mathbf{XW}^{\top}\mathbf{G}-\mathbf{XW}^{\top}\mathbf{\Gamma}_{1}\right)$

#### Updating Multipliers and Penalty Parameter

Finally, the Lagrange multiplier matrix and penalty parameter are updated
based on following rules:

$\left\\{\begin{array}[]{l}\mathbf{\Gamma}_{1}^{k+1}=\mathbf{\Gamma}_{1}^{k}+\mu^{k}\left(\mathbf{G}^{k+1}-\mathbf{W}^{k+1}\mathbf{X}^{\top}\mathbf{O}^{k+1}\right)\\\
\mu^{k+1}=\min\left(1.1\mu,\mu_{\max}\right)\end{array}\right.$ (11)

where $\mu_{max}$ is the maximum value of $\mu$.

Algorithm 1 The proposed methods

Input: training set $\mathbf{T}$, parameters $\alpha$, $\lambda$, and an
instance $\mathbf{x}$
Output: prediction for $\mathbf{x}$

1: Transforming label distribution into multi-label

2: Initialize $\mathbf{W}$, $\mathbf{O}$, $\mathbf{G}$,
$\boldsymbol{\Gamma}_{1}$, and $\mu$

3: repeat

4: Update $\mathbf{G}$ according to Eq. (7)

5: Update $\mathbf{W}$ according to Eq. (8)

6: Update $\mathbf{O}$ according to Eq. (10)

7: Update $\boldsymbol{\Gamma}_{1}$ and $\mu$ according to Eq. (11)

8: until convergence

9: return $\mathbf{d}^{*}=\mathbf{W}\mathbf{x}$.

## Experiments

### Experimental Configuration

#### Experimental Datasets

The experiments are conducted on 16 real-world datasets with label
distribution. The key characteristics of these datasets are summarized in
Table 1. The first 12 datasets are collected by Geng (Geng 2016). Among these,
the first eight ones (from Spoem to Alpha) are from the clustering analysis of
genome-wide expression in Yeast Saccharomyces cerevisiae (Eisen et al. 1998).
The SJAFFE is collected from JAFFE (Lyons et al. 1998), and the SBU_3DFE is
obtained from BU_3DFE (Yin et al. 2006). The Gene is obtained from the
research on the relationship between genes and diseases (Yu et al. 2012). The
Scene consists of multi-label images, where the label distributions are
transformed from rankings (Geng and Xia 2014). Besides, the SCUT-FBP, M2B, and
fbp5500 are about facial beauty perception (Ren and Geng 2017). The last one
RAF-ML is a facial expression dataset (Li and Deng 2019).

ID | Data sets | #Examples | #Features | #Labels
---|---|---|---|---
1 | Spoem | 2465 | 24 | 2
2 | Spo5 | 2465 | 24 | 3
3 | Heat | 2465 | 24 | 6
4 | Elu | 2465 | 24 | 14
5 | Dtt | 2465 | 24 | 4
6 | Cold | 2465 | 24 | 4
7 | Cdc | 2465 | 24 | 15
8 | Alpha | 2465 | 24 | 18
9 | SJAFFE | 213 | 243 | 6
10 | SBU-3DFE | 2500 | 243 | 6
11 | Gene | 17892 | 36 | 68
12 | Scene | 2000 | 294 | 9
13 | SCUT-FBP | 1500 | 300 | 5
14 | M2B | 1240 | 250 | 5
15 | fbp5500 | 5500 | 512 | 5
16 | RAF-ML | 4908 | 200 | 6
Table 1: Details of the dataset.

#### Evaluation Metrics

We adopt six metrics to evaluate the performance of LDL methods, including
Chebyshev ($\downarrow$), Clark ($\downarrow$), Kullback-Leibler (KL)
($\downarrow$), Canberra ($\downarrow$), Intersection ($\uparrow$), and Cosine
($\uparrow$) (Geng 2016). Here, $\downarrow$ indicates that smaller values are
better, and $\uparrow$ indicates that larger values are better.

#### Comparing Methods

We compare the proposed methods with seven LDL methods, including IIS-LDL,
LDLLDM, EDL-LRL, IncomLDL, Adam-LDL-SCL, LCLR, and LDLLC, which are briefly
introduced as follows:

  * •

IIS-LDL (Geng 2016): It utilizes the maximum entropy model and KL divergence
to learn the label distribution and does not consider label correlation.

  * •

LDLLDM (Wang and Geng 2021a): It learns the global and local label
distribution manifolds to exploit label correlations and can handle incomplete
label distribution learning.

  * •

EDL-LRL (Jia et al. 2019): It captures the low-rank structure locally when
learning the label distribution to exploit local label correlations.

  * •

IncomLDL (Xu and Zhou 2017): It utilizes trace-norm regularization and the
alternating direction method of multiplier to exploit low-rank label
correlation.

  * •

Adam-LDL-SCL (Jia et al. 2021): It incorporates local label correlation by
encoding it as additional features and simultaneously learns the label
distribution and label correlation encoding.

  * •

LCLR (Ren et al. 2019): It first models global label correlation using a low-
rank matrix and then updates the matrix on clusters of samples to leverage
local label correlation.

  * •

LDLLC (Zheng, Jia, and Li 2018): LDLLC leverages local label correlation to
ensure that prediction distributions between similar instances are as close as
possible.

The parameters of the algorithms are as follows. The suggested parameters are
used for IIS-LLD, EDL-LRL, LDLLC, and LDL-LCLR. For LDLLDM,
$\lambda_{1},\lambda_{2}$, and $\lambda_{3}$ are tuned from
$\left\\{10^{-3},\ldots,10^{3}\right\\}$, and $g$ is tuned from 1 to 14. For
IncomLDL, $\lambda$ is selected from the range
$\left\\{2^{-10},\ldots,2^{10}\right\\}$, and $\rho$ is set to 1. For Adam-
LDL-SCL, $\lambda_{1},\lambda_{2}$, and $\lambda_{3}$ are tuned from the set
$\left\\{10^{-3},\ldots,10^{3}\right\\}$, and $m$ is tuned from 0 to 14. For
TLRLDL and TKLRLDL, $\alpha$, $\lambda$ is tuned from
$\left\\{0.005,0.01,0.05,0.1,0.5,1,10\right\\}$, $T$ is selected from 0.1 to
0.5, and $k$ is tuned from 0 to $m$. We run each method for ten-fold cross-
validation and tune the parameters on training set.

### Results and Discussion

Table 2 presents the experimental results (mean$\pm$std) of the LDL algorithms
on all datasets in terms of Clark, KL, and Cosine (due to limited space, the
results in terms of other metrics are reported in the supplementary material),
with the best results highlighted in boldface. Moreover, the last row
summarized the top-one times of each method.

First, we conduct the Friedman test (Demšar 2006) to study the comparative
performance among the all methods. Table 3 shows the Friedman statistics for
each metric as well as the critical value. At a confidence level of 0.05, the
null hypothesis that all algorithms achieve equal performance is clearly
rejected. Next, we apply a post-hoc test, i.e., the Bonferroni-Dunn test
(Demšar 2006), to compare the relative performance of TLRLDL against the other
algorithms with it as the control algorithm (due to limited pages, the test
results with TKLRLDL as the control algorithm are presented in the
supplementary material). One algorithm is deemed to achieve significantly
different performance from TLRLDL if its average rank differs from that of
TLRLDL by at least one critical difference (CD) (Demšar 2006). Figure 2
illustrates the CD diagrams for each measure. In each sub-figure, if the
average rank of a comparing algorithm is within one CD to that of TLRLDL, they
are connected with a thick line; otherwise, it is considered to have a
significantly different performance from TLRLDL.

According to Table 2, TLRLDL demonstrates remarkable performance by ranking
first in 70.83% (34 out of 48) of the cases, and it achieves the best mean
performance across all metrics. TLRLDL and TKLRLDL achieve the first place in
85.4% (41 out of 48) of the evaluations, which highlights the effectiveness of
our methods. Besides, we can make the following observations from Figure 2:

  * •

TLRLDL significantly outperforms IIS-LLD in terms of all metrics. This is
because TLRLDL exploits label correlation, but IIS-LLD ignores label
correlation, which proves the importance of label correlation for LDL.

  * •

TLRLDL achieves significantly better performance than IncomLDL and wins ED-LRL
and LCLR by a margin. These three methods all exploit low-rank label
correlation in LDL which may not hold as reported in (Wang and Geng 2021a). In
comparison, TLRLDL exploits low-rank label correlation in the auxiliary MLL
process, which is more appropriate and suitable to LDL.

  * •

Compared with Adam-LDL-SCL, LDLLC, and LDLLDM, TLRLDL also excels, which
justifies that the low-rank label correlation on the auxiliary MLL process is
a competitive way to consider label correlation for LDL.

| Metric | TLRLDL | TKLRLDL | IncomLDL | IIS-LDL | EDL-LRL | Adam-LDL-SCL | LCLR | LDLLC | LDLLDM
---|---|---|---|---|---|---|---|---|---|---
Spoem | Clark | 0.1238$\pm$.0038 | 0.1237$\pm$.0197 | 0.1314$\pm$.0028 | 0.1337$\pm$.0014 | 0.1291$\pm$.0000 | 0.1295$\pm$.0000 | 0.1302$\pm$.0001 | 0.1305$\pm$.0014 | 0.1301$\pm$.0303
KL | 0.0264$\pm$.0311 | 0.0249$\pm$.0032 | 0.0288$\pm$.1709 | 0.0273$\pm$.0011 | 0.0317$\pm$.0000 | 0.0318$\pm$.0000 | 0.0246$\pm$.0001 | 0.0254$\pm$.0007 | 0.0264$\pm$.0061
Cosine | 0.9794$\pm$.0007 | 0.9801$\pm$.0028 | 0.9769$\pm$.0239 | 0.9773$\pm$.0005 | 0.9789$\pm$.0000 | 0.9789$\pm$.0000 | 0.9783$\pm$.0003 | 0.9785$\pm$.0005 | 0.9772$\pm$.0071
Spo5 | Clark | 0.1769$\pm$.0810 | 0.1803$\pm$.0139 | 0.2027$\pm$.0046 | 0.1896$\pm$.0025 | 0.1853$\pm$.0000 | 0.1843$\pm$.0000 | 0.1893$\pm$.0007 | 0.1908$\pm$.0003 | 0.1860$\pm$.0390
KL | 0.0292$\pm$.0343 | 0.0304$\pm$.0408 | 0.0376$\pm$.0148 | 0.0336$\pm$.0008 | 0.0362$\pm$.0000 | 0.0356$\pm$.0000 | 0.0309$\pm$.0000 | 0.0314$\pm$.0000 | 0.0298$\pm$.0336
Cosine | 0.9759$\pm$.0213 | 0.9749$\pm$.0296 | 0.9700$\pm$.0520 | 0.9722$\pm$.0007 | 0.9738$\pm$.0000 | 0.9741$\pm$.0000 | 0.9725$\pm$.0001 | 0.9722$\pm$.0000 | 0.9737$\pm$.0301
Heat | Clark | 0.1790$\pm$.0096 | 0.1809$\pm$.0056 | 0.1940$\pm$.0768 | 0.1998$\pm$.0014 | 0.1831$\pm$.0000 | 0.1826$\pm$.0000 | 0.1874$\pm$.0032 | 0.2717$\pm$.0064 | 0.1848$\pm$.0040
KL | 0.0122$\pm$.0279 | 0.0128$\pm$.0005 | 0.0146$\pm$.0141 | 0.0155$\pm$.0002 | 0.0153$\pm$.0000 | 0.0153$\pm$.0000 | 0.0130$\pm$.0003 | 0.0302$\pm$.0020 | 0.0131$\pm$.0002
Cosine | 0.9884$\pm$.0005 | 0.9882$\pm$.0005 | 0.9865$\pm$.0102 | 0.9855$\pm$.0002 | 0.9879$\pm$.0000 | 0.9880$\pm$.0000 | 0.9878$\pm$.0002 | 0.9695$\pm$.0022 | 0.9875$\pm$.0003
Elu | Clark | 0.2028$\pm$.0024 | 0.2000$\pm$.0103 | 0.2325$\pm$.0612 | 0.2395$\pm$.0022 | 0.1998$\pm$.0000 | 0.1989$\pm$.0000 | 0.2032$\pm$.0028 | 0.4114$\pm$.0089 | 0.2010$\pm$.0011
KL | 0.0062$\pm$.0706 | 0.0063$\pm$.0138 | 0.0066$\pm$.0123 | 0.0091$\pm$.0002 | 0.0073$\pm$.0000 | 0.0072$\pm$.0000 | 0.0064$\pm$.0001 | 0.0296$\pm$.0011 | 0.0063$\pm$.0003
Cosine | 0.9941$\pm$.0010 | 0.9940$\pm$.0077 | 0.9918$\pm$.0049 | 0.9911$\pm$.0002 | 0.9940$\pm$.0000 | 0.9940$\pm$.0000 | 0.9938$\pm$.0001 | 0.9667$\pm$.0013 | 0.9940$\pm$.0005
Cdc | Clark | 0.2142$\pm$.0719 | 0.2094$\pm$.0127 | 0.2243$\pm$.0016 | 0.2537$\pm$.0026 | 0.2168$\pm$.0000 | 0.2161$\pm$.0000 | 0.2172$\pm$.0021 | 0.4259$\pm$.0013 | 0.2147$\pm$.0309
KL | 0.0073$\pm$.0061 | 0.0070$\pm$.0017 | 0.0080$\pm$.0765 | 0.0099$\pm$.0002 | 0.0082$\pm$.0000 | 0.0082$\pm$.0000 | 0.0072$\pm$.0002 | 0.0291$\pm$.0001 | 0.0068$\pm$.0338
Cosine | 0.9935$\pm$.0263 | 0.9934$\pm$.0016 | 0.9926$\pm$.0300 | 0.9905$\pm$.0002 | 0.9933$\pm$.0000 | 0.9933$\pm$.0000 | 0.9932$\pm$.0002 | 0.9680$\pm$.0003 | 0.9934$\pm$.0463
Dtt | Clark | 0.0946$\pm$.0175 | 0.0975$\pm$.0013 | 0.1039$\pm$.0188 | 0.1162$\pm$.0009 | 0.0993$\pm$.0000 | 0.0986$\pm$.0000 | 0.0971$\pm$.0006 | 0.1738$\pm$.0011 | 0.0959$\pm$.0003
KL | 0.0060$\pm$.0076 | 0.0062$\pm$.0008 | 0.0066$\pm$.0765 | 0.0088$\pm$.0002 | 0.0098$\pm$.0000 | 0.0098$\pm$.0000 | 0.0059$\pm$.0000 | 0.0223$\pm$.0005 | 0.0059$\pm$.0012
Cosine | 0.9945$\pm$.0132 | 0.9942$\pm$.0013 | 0.9933$\pm$.0300 | 0.9916$\pm$.0001 | 0.9940$\pm$.0000 | 0.9940$\pm$.0000 | 0.9943$\pm$.0000 | 0.9783$\pm$.0004 | 0.9944$\pm$.0584
Alpha | Clark | 0.2072$\pm$.0042 | 0.2079$\pm$.0314 | 0.2156$\pm$.0775 | 0.2585$\pm$.0015 | 0.2107$\pm$.0000 | 0.2103$\pm$.0000 | 0.2085$\pm$.0012 | 0.4501$\pm$.0019 | 0.2116$\pm$.0236
KL | 0.0052$\pm$.0016 | 0.0054$\pm$.0060 | 0.0058$\pm$.0232 | 0.0084$\pm$.0001 | 0.0063$\pm$.0000 | 0.0063$\pm$.0000 | 0.0054$\pm$.0001 | 0.0267$\pm$.0002 | 0.0055$\pm$.0857
Cosine | 0.9948$\pm$.0015 | 0.9947$\pm$.0069 | 0.9943$\pm$.0052 | 0.9916$\pm$.0001 | 0.9946$\pm$.0000 | 0.9946$\pm$.0000 | 0.9947$\pm$.0001 | 0.9700$\pm$.0001 | 0.9946$\pm$.0362
Cold | Clark | 0.1378$\pm$.0014 | 0.1390$\pm$.0731 | 0.1463$\pm$.0338 | 0.1568$\pm$.0014 | 0.1403$\pm$.0000 | 0.1398$\pm$.0000 | 0.1416$\pm$.0042 | 0.1512$\pm$.0040 | 0.1363$\pm$.0190
KL | 0.0118$\pm$.0042 | 0.0124$\pm$.0565 | 0.0139$\pm$.0056 | 0.0153$\pm$.0002 | 0.0162$\pm$.0000 | 0.0162$\pm$.0000 | 0.0128$\pm$.0009 | 0.0140$\pm$.0006 | 0.0116$\pm$.0154
Cosine | 0.9892$\pm$.0061 | 0.9886$\pm$.0357 | 0.9873$\pm$.0047 | 0.9855$\pm$.0003 | 0.9885$\pm$.0000 | 0.9885$\pm$.0000 | 0.9880$\pm$.0008 | 0.9866$\pm$.0005 | 0.9889$\pm$.0390
SJA | Clark | 0.3602$\pm$.0042 | 0.3657$\pm$.0099 | 0.4567$\pm$.0061 | 0.4516$\pm$.0181 | 0.4232$\pm$.0002 | 1.3730$\pm$.9671 | 0.4049$\pm$.0082 | 0.4369$\pm$.0034 | 0.4153$\pm$.0010
KL | 0.0480$\pm$.0016 | 0.0518$\pm$.0277 | 0.0659$\pm$.0202 | 0.0790$\pm$.0053 | 0.0692$\pm$.0000 | 1.0106$\pm$.9024 | 0.0663$\pm$.0000 | 0.0791$\pm$.0019 | 0.0668$\pm$.0009
Cosine | 0.9558$\pm$.0015 | 0.9509$\pm$.0528 | 0.9321$\pm$.0187 | 0.9208$\pm$.0062 | 0.9319$\pm$.0000 | 0.6503$\pm$.0850 | 0.9372$\pm$.0000 | 0.9245$\pm$.0019 | 0.9363$\pm$.0125
SCU | Clark | 1.0793$\pm$.0061 | 1.4568$\pm$.0000 | 1.5459$\pm$.0016 | 1.5007$\pm$.0064 | 1.5146$\pm$.0000 | 1.4654$\pm$.0000 | 1.3859$\pm$.0062 | 2.6438$\pm$.0000 | 1.3978$\pm$.0009
KL | 0.1779$\pm$.0015 | 0.1503$\pm$.0528 | 2.6539$\pm$.0221 | 0.1824$\pm$.0170 | 9.2314$\pm$.0504 | 7.4655$\pm$.0041 | 0.4248$\pm$.0047 | 16.040$\pm$.1750 | 0.3997$\pm$.0009
Cosine | 0.8208$\pm$.0028 | 0.7436$\pm$.0202 | 0.6108$\pm$.0689 | 0.6627$\pm$.0028 | 0.6477$\pm$.0000 | 0.7436$\pm$.0000 | 0.8126$\pm$.0011 | 0.5144$\pm$.0002 | 0.8375$\pm$.0002
SBU | Clark | 0.3455$\pm$.0043 | 0.3520$\pm$.0016 | 0.3692$\pm$.0011 | 0.4217$\pm$.0029 | 0.4061$\pm$.0000 | 0.3718$\pm$.0000 | 0.3956$\pm$.0039 | 0.4172$\pm$.0003 | 0.4056$\pm$.0071
KL | 0.0502$\pm$.0857 | 0.0552$\pm$.0765 | 0.0619$\pm$.0036 | 0.0776$\pm$.0009 | 0.0726$\pm$.0000 | 0.0604$\pm$.0000 | 0.2008$\pm$.0026 | 0.0845$\pm$.0002 | 0.0791$\pm$.0370
Cosine | 0.9474$\pm$.0012 | 0.9449$\pm$.0300 | 0.9410$\pm$.0013 | 0.9177$\pm$.0011 | 0.9232$\pm$.0000 | 0.9367$\pm$.0000 | 0.9267$\pm$.0011 | 0.9180$\pm$.0002 | 0.9226$\pm$.0010
RAF | Clark | 0.8652$\pm$.0082 | 1.4327$\pm$.0528 | 1.5597$\pm$.0234 | 1.5581$\pm$.0086 | 1.4495$\pm$.0003 | 1.4585$\pm$.0000 | 1.5962$\pm$.0138 | 1.6210$\pm$.0034 | 1.4151$\pm$.0016
KL | 0.0864$\pm$1.8674 | 0.2086$\pm$.0187 | 6.4358$\pm$.0090 | 3.5105$\pm$.0654 | 2.2182$\pm$.0013 | 5.6995$\pm$.0000 | 13.7926$\pm$1.8887 | 0.7347$\pm$.0001 | 0.2699$\pm$.0109
Cosine | 0.9252$\pm$.0034 | 0.9234$\pm$.0115 | 0.5631$\pm$.0101 | 0.7351$\pm$.0020 | 0.9198$\pm$.0000 | 0.8706$\pm$.0000 | 0.7968$\pm$.0047 | 0.6453$\pm$.0007 | 0.8976$\pm$.0002
M2B | Clark | 1.0224$\pm$.0009 | 1.5160$\pm$.0023 | 1.4832$\pm$.0878 | 1.2282$\pm$.0070 | 1.6770$\pm$.0046 | 1.2093$\pm$.0000 | 1.6902$\pm$.1999 | 1.6791$\pm$.0002 | 1.5538$\pm$.0029
KL | 0.6972$\pm$.0108 | 0.7786$\pm$.0109 | 0.8180$\pm$.0301 | 0.8572$\pm$.0354 | 0.8632$\pm$.0044 | 0.8128$\pm$.0000 | 0.9528$\pm$.5826 | 0.9051$\pm$.0000 | 0.7556$\pm$.0083
Cosine | 0.7431$\pm$.0004 | 0.7786$\pm$.0070 | 0.7583$\pm$.0308 | 0.7588$\pm$.0122 | 0.7423$\pm$.0006 | 0.7639$\pm$.0000 | 0.5786$\pm$.0325 | 0.6039$\pm$.0000 | 0.6953$\pm$.0054
Gene | Clark | 2.0077$\pm$.0041 | 2.1086$\pm$.0036 | 2.1110$\pm$.0040 | 2.1734$\pm$.0269 | 2.1102$\pm$.0000 | 2.1144$\pm$.0000 | 2.0677$\pm$.0172 | 2.1162$\pm$.0009 | 2.1374$\pm$.0036
KL | 0.2224$\pm$.0096 | 0.2236$\pm$.0091 | 0.2372$\pm$.0002 | 0.2380$\pm$.0069 | 0.2258$\pm$.0000 | 0.2256$\pm$.0000 | 0.3618$\pm$.0026 | 0.2374$\pm$.0054 | 0.2455$\pm$.0069
Cosine | 0.8387$\pm$.0028 | 0.8376$\pm$.0038 | 0.8342$\pm$.0003 | 0.8274$\pm$.0036 | 0.8347$\pm$.0000 | 0.8345$\pm$.0000 | 0.8374$\pm$.0018 | 0.8338$\pm$.0027 | 0.8290$\pm$.0021
fbp | Clark | 0.5510$\pm$.0036 | 0.6288$\pm$.0155 | 1.2938$\pm$.0011 | 1.5065$\pm$.0023 | 1.6994$\pm$.0001 | 1.2755$\pm$.0000 | 1.4102$\pm$.1809 | 1.8756$\pm$.0857 | 1.2747$\pm$.0122
KL | 0.0904$\pm$.0091 | 0.0800$\pm$.0068 | 0.2017$\pm$.0003 | 4.2207$\pm$.0332 | 1.3803$\pm$.0029 | 0.7725$\pm$.0001 | 0.4208$\pm$.4110 | 1.7692$\pm$.4081 | 0.1149$\pm$.0047
Cosine | 0.9567$\pm$.0038 | 0.9500$\pm$.0047 | 0.9412$\pm$.0005 | 0.6572$\pm$.0017 | 0.7943$\pm$.0000 | 0.9521$\pm$.0000 | 0.8527$\pm$.0051 | 0.8242$\pm$.0123 | 0.9528$\pm$.0056
Sce | Clark | 2.1786$\pm$.0108 | 2.4959$\pm$.0061 | 2.4765$\pm$.0303 | 2.4685$\pm$.0135 | 2.4348$\pm$.0000 | 2.4665$\pm$.0000 | 2.4230$\pm$.0025 | 2.4784$\pm$.0007 | 2.6316$\pm$.0000
KL | 2.1128$\pm$.0061 | 2.2857$\pm$.0015 | 0.2378$\pm$.0061 | 3.0463$\pm$.0490 | 2.3857$\pm$.0002 | 2.7001$\pm$.0000 | 0.8287$\pm$.0227 | 1.0243$\pm$.0027 | 4.4471$\pm$.0000
Cosine | 0.7989$\pm$.0016 | 0.8346$\pm$.0008 | 0.7297$\pm$.0071 | 0.6614$\pm$.0044 | 0.7273$\pm$.0000 | 0.7163$\pm$.0000 | 0.7290$\pm$.0042 | 0.6446$\pm$.0001 | 0.3603$\pm$.0000
top-1 times | 34 | 7 | 1 | 0 | 0 | 0 | 2 | 0 | 4
Table 2: Results (mean$\pm$std) of the comparing methods in terms of three metrics (more results in terms of other three metrics are reported in the supplementary material) on 16 LDL datasets, where the best results are highlighted in boldface.  Critical Value ($\alpha=0.05$) | Evaluation metric | Chebyshev | Clark | Canberra | KL | Cosine | Intersection
---|---|---|---|---|---|---|---
2.8500 | Friedman Statistics $F_{F}$ | 28.0098 | 44.9412 | 46.3235 | 38.0000 | 45.7059 | 44.0158
Table 3: Summary of the Friedman statistics $F_{F}$ in terms of six evaluation
metrics, as well as the critical value at a significance level of 0.05.

(a) KL$\downarrow$ (b) Clark $\downarrow$ (c) Cosine$\uparrow$ (d) Canberra
$\downarrow$

(e) Chebyshev$\downarrow$ (f) Intersection$\uparrow$

Figure 2: CD diagrams of the comparing algorithms in terms of each evaluation
criterion. For the tests, CD equals 2.3296 at 0.05 significance level.

To summarize, the experimental results validate the competitive performance of
the proposed algorithms.

### Ablation Study

Next, we study the advantages of exploiting the low-rank label correlation on
the auxiliary MLL. First, we derive TLRLDL-a by

$\min_{\mathbf{W}}\frac{1}{2}\left\|\mathbf{W}\mathbf{X}^{\top}-\mathbf{D}\right\|_{F}^{2}+\alpha\left\|\mathbf{W}\mathbf{X}^{\top}\right\|_{*}+\lambda\|\mathbf{W}\|_{F}^{2}.$

Second, we derive TLRLDL-b by keeping the first and fourth items of Eq. (5).
That is, TLRLDL-a exploits low-rank label correlation on LDL, and TLRLDL-b
ignores label correlation. We then compare TLRLDL with TLRLDL-a and TLRLDL-b.

Figure 3 presents the comparison results in terms of Clark, KL, Cosine, and
Intersection. Further, we conduct the Wilcoxon signed-rank tests (Demšar 2006)
for TLRLDL against TLRLDL-a and TLRLDL-b and report the results of the tests
in Table 4. According to Figure 3 and Table 4, we can draw the following
conclusions:

  * •

TLRLDL and TLRLDL-a have better performance than TLRLDL-b. TLRLDL-b ignores
label correlation, while TLRLDL and TLRLDL-a consider label correlation, which
improves their performance. This observation further justifies the importance
of label correlation for LDL.

  * •

TLRLDL significantly outperforms TLRLDL-a. Since the different between TLRLDL
and TLRLDL-a lies in that the former (the latter, respectively) exploits low-
rank label correlation on MLL (LDL, respectively), this observation clearly
justifies benefits of exploits low-rank label correlation on the auxiliary
MLL.

(a) Clark

(b) KL

(c) Cosine

(d) Intersection

Figure 3:  Ablation results on seven datasets in terms of Clark $\downarrow$, KL $\downarrow$, Cosine $\uparrow$, and Intersection $\uparrow$. TLRLDL vs. | Chebyshev$\downarrow$ | Clark$\downarrow$ | Canberra$\downarrow$ | KL$\downarrow$ | Cosine$\uparrow$ | Intersection $\uparrow$
---|---|---|---|---|---|---
TLRLDL-a |  $\mathbf{win}$[4.37e-04] |  $\mathbf{win}$[4.38e-04] |  $\mathbf{win}$ [4.37e-04] |  $\mathbf{win}$[4.46e-03] |  $\mathbf{win}$[4.38e-04] |  $\mathbf{win}$[4.38e-04]
TLRLDL-b |  $\mathbf{win}$[4.38e-04] |  $\mathbf{win}$ [3.20e-03] |  $\mathbf{win}$[1.61e-03] |  $\mathbf{win}$[4.38e-04] |  $\mathbf{win}$ [4.38e-04] |  $\mathbf{win}$[4.38e-04]
Table 4: The results (Win/Tie/Lose[$p$-value]) of the Wilcoxon signed-rank
tests for TLRLDL against TLRLDL-a and TLRLDL-b at a confidence level of 0.05.

### Parameter Sensitivity Analysis

TLRLDL has two trade-off parameters, including $\alpha$ and $\lambda$. Next,
we analyze the sensitivity of them.

First, we run TLRLDL with $\alpha$ varying from the candidate set
$\\{0.005,0.01,0.05,0.1,0.5,1,10\\}$ and report its performance on SCUT-FBP,
M2B, SJAFFE, SBU_3DFE, and Alpha in Figure 4. As can be seen from Figure 4,
TLRLDL shows robustness w.r.t. $\alpha$. As a result, we can set $\alpha$ to
0.1 to get satisfying performance. Likewise, we also run TLRLDL with $\lambda$
ranging from the same candidate set and present its performance in Figure 4.
From Figure 4, TLRLDL is robust w.r.t. $\lambda$. We may expect satisfying
performance for $\lambda=0.1$.

(a) Chebyshev with varying $\alpha$

(b) Clark with varying $\alpha$

(c) Chebyshev with varying $\lambda$

(d) Clark with varying $\lambda$

Figure 4: The performance of TLRLDL with $\alpha$ and $\lambda$ varying from
$\left\\{0.005,0.01,0.05,0.1,0.5,1,10\right\\}$ in terms of Chebyshev and
Clark on SCUT-FBP, M2B, SJAFFE, SBU_3DFE, and Alpha.

## Conclusion

LDL has an exponential-sized output space–with a size of
$\mathbb{R}^{m}$–which may decrease the performance of existing algorithms. To
solve that, many LDL studies have proposed to exploit label correlation. Among
these, some have proposed to exploit low-rank label correlation of label
distribution, which may not hold as disclosed by (Wang and Geng 2021a) because
LDL matrices are typically full-rank. To address this problem, we introduce an
auxiliary MLL process in LDL and exploit low-rank label correlation on the MLL
process instead of LDL. By doing this, the low-rank label correlation is
implicitly exploited in our LDL methods. We conduct extensive experiments and
show that our proposed methods achieve remarkably better performance than
several state-of-the-art LDL methods. Moreover, further ablation studies
justify the advantages of exploiting low-rank label correlation on the
auxiliary MLL process.

We believe our work brings a new perspective for modeling the label
correlation for LDL. In future work, we will extend our work to exploit local
low-rank label correlation.

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