Divide and Conquer: Learning Label Distribution with Subtasks
Authors: Haitao Wu
Weiwei Li
Xiuyi Jia
Venue: ICML 2025 poster (2025)
Keywords: label distribution learning
subtask
label polysemy
Primary area: general_machine_learning->supervised_learning

Abstract: Label distribution learning (LDL) is a novel learning paradigm that emulates label polysemy by assigning label distributions over the label space. However, recent LDL work seems to exhibit a notable contradiction: 1) existing LDL methods employ auxiliary tasks to enhance performance, which narrows their focus to specific applications, thereby lacking generalizability; 2) conversely, LDL methods without auxiliary tasks rely on losses tailored solely to the primary task, lacking beneficial data to guide the learning process. In this paper, we propose S-LDL, a novel and minimalist solution that generates subtask label distributions, i.e., a form of extra supervised information,
to reconcile the above contradiction. S-LDL encompasses two key aspects: 1) an algorithm capable of generating subtasks without
any prior/expert knowledge; and 2) a plug-andplay framework seamlessly compatible with existing LDL methods, and even adaptable to derivative tasks of LDL. Our analysis and experiments demonstrate that S-LDL is effective and efficient. To the best of our knowledge, this paper represents the first endeavor to address LDL via subtasks.

Divide and Conquer: Learning Label Distribution with Subtasks
Haitao Wu
Weiwei Li Xiuyi Jia
Abstract
dz) each label of each instance. This form introduces a quantitative manner to address label polysemy and extends LDL's practical applications to wider range, e.g., counting (or grading) (Geng et al., 2013; Wu et al., 2019), sentiment analysis (Chen et al., 2020; Le et al., 2023), segmentation (Gao et al., 2017; Li et al., 2023b), etc.
Label distribution learning (LDLL) is a novel learn- ing paradigm that emulates label polysemy by assigning label distributions over the label space. However, recent LDL work seems exhibit a notable contradiction: 1) existing LDL methods employ auxiliary tasks to enhance performance, which narrows their focus to specific applications. thereby lacking generalizability; 2) conversely, LDL methods without auxiliary tasks rely on losses tailored solely to the primary task, lack- ing beneficial data to guide the learning process. In this paper, we propose S-LDL, novel and minimalist solution that generates subtask label distributions, i.e., form of extra supervised in- formation, to reconcile the above contradiction. S-LDL encompasses two key aspects: 1) an al- gorithm capable of generating subtasks without any prior/expert knowledge; and 2) a plug-and- play framework seamlessly compatible with exist- ing LDL methods, and even adaptable to deriva- tive tasks of LDL. Our analysis and experiments demonstrate that S-LDL is effective and efficient. To the best of our knowledge, this paper repre- sents the first endeavor to address LDL via sub- tasks.
However, LDL encounters spectrum of challenges: 1) dz interfere with each other since el is subject two con- straints, non-negativity (i.e., dy 2 0) and sum-to-one (i.e., Egey d% = 1), and d is often formed from mixture distri- butions, posing significant hurdles for fitting, particularly when employing a maximum entropy model (Shen et al., 2017); 2) ds are usually obtained via crowdsourcing, which is consuminn and labor-intensive, so one often copes with scarce and low-quality datasets (Wang et al., 2023). These two key issues stand as formidable barriers to perfor- mance improvement in LDL.
With the widespread use of multi-task learning, some LDL work tries to compensate for performance from the perspec- tive of auxiliary tasks, which are learned concurrently along- side the primary task, thereby refining its representations and ultimately boosting performance. Unfortunately, though these methods exploit additional supervised information, they 1) neglect the first key issue mentioned above; and 2) require prior/expert knowledge (e.g., facial characteristics (Chen et al., 2020), pathology criteria (Wu et al., 2019), emotion wheel theory in psychology (Yang et al., 2017a), etc.), limiting their generalizability to those corresponding specific applications. Conversely, LDL methods that do not take advantage of auxiliary tasks, despite their efforts in loss function engineering and network structure design, they 1) neglect the second key issue mentioned above; and 2) focus solely on one aspect of label correlations (e.g., correlation of local instances (Jia et al., 2019), ranking relation (Jia et al., 2023), suboptimal label (Wang, Jing and Geng, Xin, 2019), etc.), each with its own set of limitations.
1. Introduction
Multi-label learning (MLL) (Zhang& & Zhou, 2013) handles label polysemy in a binary manner, whereas label distribu- tion learning (LDL) (Geng, 2016) offers more nuanced perspective by answering: "How much does each label y describe the instance x?",, This is accomplished through the concept of a label distribution d, which is a form of probabil- ity simplex that assigns a real value (i.c., description degree
'School of Computer Science and Engineering, Nanjing Uni- versity of Science and Technology, Nanjing, China "College of Computer Science and Technology, Nanjing University of Aero- nauties and Astronautics, Nanjing, China. Correspondence to: Xiuyi Jia <jiaxy @ njust.edu.cn>.
The generalizability appears to conffict with the ability to ex- ploit additional data, so benefiting from both simultaneously seems elusive. However, we can still see the light from some MIL methods, which partition the label space and apply operations on the subspaces (Tsoumakas et al., 2008; 2010). These methods construct subtasks without involving extra knowledge. Intuitively, in the context of LDL, reliable su-
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Divide and Conquer: Learning Label Distribution with Subtasks
pervised information can be generated from these subtasks, which can eventually be aggregated and reconstructed to the information of the primary task via ensemble strategies. In this paper, we introduce S-LDL, novel and minimalist LDL. algorithm that constructs and exploits subtasks, to rec- oncile the contradiction between the generalizability across various domains and the ability to exploit extra data. Serving as auxiliary tasks, subtasks of LDL can 1) provide different views of the primary task distribution, rendering the mixture of distributions more traceable (i.e., the key issue one); 2) furnish constructed supervised data to mitigate the scarcity and ambiguity inherent in LDL datasets (i.e., the key issue two); 3) require no prior/expert knowledge from specific applications; and 4) emphasize various label correlations via partitioning of the label space.
Hayashi criterion, a rule for counting and grading in acne le- sions, which results in their method being only applicable in small branch of the dermatology field. Yang et al. (2017a) employ a multi-task framework for image emotion classifi- cation, designing constraints inspired by Mikel's wheel, a psychological emotion model. It also suffers from similar limitations. The need for prior/expert knowledge signifi- candly narrows the application scenarios of these methods.
MLL with partitioning of the label space For reference, there exist MLL methods based on partitioning of the la- bel space, which can construct multi-label subtasks without involving additional knowledge and can be widely used in various domains. The most classic related work is that of HOMER (Tsoumakas et al., 2008) and RAKEL (Tsoumakas et al., 2010), the former forms a hierarchy of label subspaces while the latter randomly selects label subspaces. Many sub- sequent papers have been inspired by them (Prabhu et al., 2018: Read et al., 2013; Wang et al., 2021). Read et al. (2014) present a general framework of label subspaces and provide theoretical justification for it. Since label distribu- tion contains rich knowledge, we can follow the patterns of these methods to construct label distribution subtasks.
The main contributions of this paper are outlined below: 1) we propose S-IDII, which is considered the first endeavor to address LDL via subtasks; 2) our analysis shows the va- lidity and reconstructability of these subtasks; 3) we present a plug-and-pla -lay framework seamlessly compatible with ex- isting LDL. methods, and adaptable to derivative tasks of LDL; and 4) the code will be available on GitHub soon, facilitating reproducible research endeavors.
2. Related work
LDL with ensemble strategy It is imperative to aggre- gate the output of subtasks. Fortunately, ensemble- based LDL methods have demonstrated promising performance. For instance, LDLFs (Shen et al., 2017) learns different la- bel distributions on the leaf nodes of differentiable decision trees and learns weights that aggregate these label distribu- tions. DF-LDL (González et al., 202 aggregates the label distribution of output of multiple base models by simple averaging, while Zhai et al. (2018) focus on aggregating the results of various neural networks via a combining learner. However, 1) the above methods are not suitable for incom- plete label spaces (i.c., subtask label spaces); and 2) none of them involve label space partitioning, therefore no extra supervised information of label distributions is constructed.
LDL Our work is mainly related to LDL. Initially em- ployed to tackle age estimation (Geng et al., 2013), LDL has evolved into a novel machine learning paradigm (Geng, 2016), which is supported by theoretical underpinnings (Wang& & Geng, 2019). Concurrently, more and more deriva- tive tasks of LDL (González et al., 202 Lu & Jia, 2022; Wang, Jing and Geng, Xin. 2019; Xu & Zhou, 2017; Xu et al., 2019) are emerging to offer assistance in various real-world dilemmas. Most methods focus on improving performance via loss function engineering (Jia et al., 2019; 2023; Ren et al., 2019; Wen et al., 2023) or efficient model structures (González et al., 2021a; Jin et al., 2024; Shen et al., 2017: Yang et al., 2017b), while some work is dedi- cated to practical application scenarios (Gao et al., 2017; Li et al., 2023a; Shirani et al., 2019). However, the scarcity of LDL datasets and the complexity of the label distribution itself make it difficult to further improve performance, at which point one may think of leveraging auxiliary tasks.
3. Subtask construction
3.1. Preliminary
Notation Vectors are denoted by lowercase bold letters, e.g., v, and the corresponding regular letter with subscript i, i.c., Us indicates its i-th element. Matrices are denoted by uppercase bold letters, e.g., A, with a, as the i-th row and a.j as the j-th column. Auj is the element in i-th row and j-th column of A. The superseript (t) indicates that symbol corresponds to the t-th subtask. Table 1 outlines the key notation in this paper.
LDL with auxiliary tasks While there are LDL methods that leverage auxiliary tasks to enhance performance, they often rely on prior/expert knowledge from specific appli- cations, extending beyond the scope of the LDL task. For example, LDL-ALSG (Chen et al., 2020) designs auxiliary tasks dedicated to facial emotion recognition, necessitat- ing the use of external tools to extract facial points and action units from human faces. Wu al. (2019) exploit the
'HA, SA, SU, AN, DI, FE, and NE represent 7 common emo- tions in sentiment analysis datasets, namely happiness, aadness, surprise, anger, disgust, fear, and neutral, respectively.
2

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Divide and Conquer: Learning Label Distribution with Subtasks
Table 1. Key notation and terminology in this paper
Symbol
Description
Example
y=fu;
Label spaee (with L labels)
y {HA, SA, SU, AN, DI, FE}
y
Subtask label spaces
y = {.... )x(4) (HA, SA, SU, FE), ...
dz)
Description degree of æ; about yo
d 0.4, i... HA describes æ; by 0.4
d. = (dzi)) j=1
Label distribution of æ;
d. = (0.4, 0.05, 0.3, 0.1, 0.1, 0.05)
D = (di) = (d=:)5=1
Label distribution matrix (with N samplls)
= (..., d.....)
d() (de) (()
Subtask label distribution
d (0.5. 0.0625, 0.375, 0.0625) D {..., D ...) M (.... m. (1, 1. 0,
D
Subtask label distribution matrices
M- (m=) (Mss)
Mask matrix (with T anticipated tasks)
.6
majority of instances. because 1) theoretically, there is no objective standard for the degree of negative correlation; and 2) empirically, weakly or negatively correlated information is easily overlooked by human annotators in crowdsoureed datasets. To mitigate this, we incorporate the description degree as metric for the reliability of supervised infor- mation in guiding the generation of subtask masks. The third challenge is avoiding analogous extra supervised in- formation, i.e., to generate label distributions containing new knowledge (González et al., 2021b). This necessitates fostering richness and diversity both subtask label spaces and distributions. To achieve this objective, we minimize pairwise similarity among subtask masks; and 2) normalize each subtask label distribution yield brand new insights distinct from the label distribution of the primary task. Fig. portraits an illustrative example of these challenges.
Labels: over 10
partitions
-4
when 10.
Drscetppcion degrns
Real Fig. (a)
-6
FE
A3
20
82
SA
D1
(a) Huge number of partitions (b) Ignorance & homogeneity
Figure 1. (a) is sourced from the emotion6 (Yang et al., 2017b) dataset, which has only labels, but the number of potential parti- tions is huge. (The upper part of) (b) exemplifies subtask label space (FE, AN, DI}, which challenging to describe (a). (The lower part of) (b)'s 's' two subtask label spaces encompass all descrip- tive information, meaning no new knowledge generated about (a). This example vividly illustrates the limitations that may result from local ignorance and lack of diversity in subtask label spaces.
3.2. Learning subtask masks
Let M E {0, 1)Txx. be the subtask mask matrix, where T represents the number of anticipated tasks. To ensure that the subtask label spaces contain as reliable informa- tion as possible, the learning of the subtask mask matrix can be converted into this problem: max Everry dm.". where d = Eiein) m dd//N. Obviously, senseless solution is m, for all t E [T], i.e., all extra supervised information is equivalent to the primary task information. Therefore, solving the above problem alone inappropriate. To ad- dress this, we consider pairwise similarity among subtask masks. We also employ exponential tricks to convert maxi- mization into minimization. Finally, M is calculated as
Problem definition Let x E X' = Rº denote the feature of the instance and E A L-1 denote the label distribution, where - A (o E R 11v = 1, v > 0} is the (K- didimennioonal probability simplex. L.DI's goal to find mapping <: X' - In this paper, we partition the label space y' to obtain the subtask label space set y then accordingly generate extra supervised information, i.e., subtask distribution matrix set D to guide Ç's learning.
Technical challenges Our first challenge arises from the exponential growth in partitions as the number of labels in- creases (Tsoumakas et al., 2010). When generating T tasks from a label space with labels, the number of unique par- titions given by -22=2221// -L-2-1)) This makes impractical calculate metric for each case to select sub- tasks. We tackle this challenge in mask matrix learning manner. The second challenge lies in discerning reasonable partitions. Since the label distribution matrix is usually im- balanced in average description degree (Zhao et al., 2023), some partitions exhibit unreasonable local ignorance. As a result, the corresponding spaces struggle to handle the
M = arg min M
Σ exp (-dm,) + LE[T]
T
(I)
2
m&m
(T(T - 1))
m,
mm
(.jE[T] b>
s.t. Maj E (0, 1}; t [T]; j [4].
where À a trade-off parameter. Eq. (1) is slightly more
3

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Divide and Conquer: Learning Label Distribution with Subtasks
Algorithm 1 Subtask construction
Algorithm 2 S-LDDL (shallow regime)
Input: Input matrix D, trade-off parameter X, anticipated number
Input: Feature matrix X, label distribution matrix D, testing instance x'.
of subtasks
Output: Subtask distribution matrices DD' (with corresponding
Output: Predicted label distribution d' for instance x'.
subtask label spaces )²))
1: Initialize parameter of each estimator;
1: Initialization: y 4- {2], D < (Ø):
2: D t SC());
2: Calculate M using L-BFGS;
(Eq. (1))
3: for f 1 to [D"| do
3: for f. 4-110 to T do
4: Fit an estimator f(() on dataset (X, D(O)):
4: yy(1 <- {
7: Concatenate X and all D to get Z, where E [ dataset {Z, D): 5: d < (t))
5: for j + 1 to L do
6: end for
6: if Mr = then
7:
yot) + ydu) n {us}:
8: Fit an estimator
8: end if
9: Concatenate x' and all d to get z', where t E [[D9]]
9:
end for
10: d' + f(z');
10:
if [T>00 I or | S then
continue: Ignore invalid subtask masks. */
12:
end if
13:
1s The clip(æ, a, b) function restricts to (a, b/a. */
pectations? How do they compare to those generated by random selection?
14:
D + [I:
15:
for + to I do
y; (xt))) then
Qz: Are D provided by Eq. (2) reconstructable? Can one replace it with other normalization functions?
17:
D concatenate(D (t)), clip d e, 1)):
18:
is very small positive number. */
23: What is the overall time complexity of SC? Is it practical for large-scale datasets?
19:
end
20:
end for
21: for e 1 to N do
The validity, reconstructability. and complexity analysis are conducted for 21, Q2, and 23, respectively.
22:
Normalization: de (d
(Eq. (2))
23: end for
24: y + y-t) UY" D + DOOUD
25:
end for
4.1. Validity analysis
Eq. (1) manages the intricate task of selecting y(O's via A and T', and the main idea is to suppress local ignorance and increase diversity of each y(()) simultaneously. To check whether the subtasks are valid and meet expectations, we design two metrics, the information index and the diversity index, both of which are based on the mask valid rate.
complicated than conventional integer programming. For convenience, we solve it using the L-BFGS method, with its constraint enforced via sigmoid (a conversion threshold is set, where outputs greater than it are set to while those below are set to 0). Refer to Section 4.1 for an analysis of the validity of Eq. (1).
Definition 4.1 (Mask valid rate). For all E [T]. the fol- lowing are considered invalid masks: 1) m,
2)
3.3. Generating subtask distributions
|m,|! 1, or, 3) excluding masks in cases 1) and 2), for any remaining mask index i, m, = m, where j E [1]. The mask valid rate is defined as:
We slice D according to y)t)s. To generate diversified s we perform normalization on each slice with
m, 1,
(u)
U/
Us
((M)
1
Σ
0.
|ma/ = 1.
T
0.
e.(mm) > 1,
EE|T
o/w.
The rationale for utilizing NSUM as the normalization func- tion can be found in Section 4.2. The overall subtask con- struction process, denoted by SC, is illustrated Alg. Then, one can naturally come up with an adaptive LDL pipeline based on the shallow regime, as depicted in Alg. 2.
1, o/w,
where (m = Ej I(m = m2) and II is the indicator function.
Definition 4.2 (Information index). We call it informative if My = 1 where [T] and j E [L]. The information
4. Analysis about subtask construction
In this section, we analyze the subtask construction algo- rithm SC by studying the following questions:
"These three cases correspond to 1) masks that are exactly the same as the primary task; 2) masks that fail to form label distributions; and 3) duplicate masks among the remaining masks, respectively.
Q1: Are ylt)s provided by Eq. (1) valid and meet ex-

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Divide and Conquer: Learning Label Distribution with Subtasks
vertices of g are not zero.
00
11
06
65
20
Proef. We solely discuss the extreme case where two sub- task label spaces overlap with just one label. Further spe- cialized cases can be deduced by the reader via induc- tion. With a little bit of symbol abuse, let the general normalization function be defined as N(w)) - plv)//(w). Assume that there is label distribution = (d1, 1.,, dr) and its corresponding label space is y {y1, 1.. VL The decompositions of y are Va { ya and 3%b {yk Vr. respectively. It is obvious that Va U3% = y and Va n = {yr}. Let the subspace label distribution corresponding to these two decompositions be
Sclecion
10
20
33
6
70
90
100
(15)
11
16
16
16T
23
(00))
Figure2 2. Informatioon/ivvesittyy index w.r.t. À on emotion6.
a (0), 0x) and = (b2, ... ....,...,,,) According to our assumptions, ds + O. Then, for any integer j E [1, klz,
index is defined as:
we have
.(d), p(d); g(d))
a a(M) P
Σ dm, T
(4)
(6)
ak N(d)x q(d), p(d))
EE(T)
Typically, for most normalization functions, q(-) is a nor- malizing constant, i.e., q(d); g(d).. Thus Eq. (6) can be rewritten into a,p(d); axp(d),. Plug it into Σ/= j= a, = 1, and do the same for b as well, and get
Definition 4.3 (Diversity index). We call it diverse if two m,s reflect disjoint labels. The diversity index is defined as:
T-1 T
as p(d)
by Ef= =1 (7)
Σ B(M)- T(T 1) 1=1 J=
2p
amm @ m,))_
(5)
p(d)
p(d)
Add these two equations together, we have
We calculate the two metrics of M generated by SC with varying A, results on the emotion6 dataset are shown in Fig. 2 (red curve). For comparison, we also study these two metrics of masks generated by random selection with varying probability y (blue curve). Results other datasets are provided in the appendix. All results indicate that the red curve is always higher than the blue curve, suggesting that the subtasks generated by SC are more informative and diverse than those generated by random selection, as long as the parameters are appropriate. Parameter selection is discussed in Section 6.
p(d)x
p(d)a
p(d) + Σ r(dd) J=1
(8)
ak bb
Eq. (8) implies that Σ/= p(d), must be given a priori, and pl d))x is related to dę, and only dy. To make it possible, the only thing we can exploit is the sum-to-one constraint of i.e., Σ/= dj Note that any nonlinear operations here will lead to inconsistencies with the problem constraints. Therefore p(v), = Vy for all is the simplest nontrivial solution. Since Σ wl M((v); = 1, we have q(v) = -E V/. i.e., the finally deduced normalization function Eq. (2). In this case, for any integer j E [1, L2z, we have
4.2. Reconstructability analysis
a
Reconstructability ensures the effectiveness of the path from d to d. We strive to choose normalization function that d retain more information, even efficacious enough to reconstruct d. Theorem 4.4 illustrates that Eq. (2) is the only possibility.
1, k.
d,
a + by ag
(9)
=
I.
0k + bj
agbb
which illustrates that the original label distribution d can be reconstructed by subtask label distributions and b. This possible thanks to the use of N
Theorem 4.4. Let each subtask label space form a con- nected graph with its each label as node. Then merge these graphs according their respective labels to form G. If and only if NSu is used for normalization, the primary label distribution can be reconstructed from these subtask label distributions, when the following conditions are satis- fied: 1) G is connected; 2) G covers all labels the label space, and 3) corresponding description degrees of all cut
Theorem 4.4 also states that not appropriate to replace Eq. (2) with the min-max softmax function because doing so destroys the reconstruction information.
from
Maintaining non-zero deseription degrees for all cut vertices is essential for preserving the proportional information.
5

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Divide and Conquer: Learning Label Distribution with Subtasks
4.3. Complexity analysis
Subtask Construction
Primary Task Matrix
Subtask Matrices
The overall time cost of SC is primarily influenced by the calculation of M and the normalization process. The time complexity of computing and updating M are OLL + T))) and (LT)), respectively. The time complexity of the normalization process is (LTN). The overall time complexity of cach iteration of SC is OLL(TN + I*)), which is linear with respect to the number of instances and labels. Therefore, it is clear that SC can be applied to large- scale datasets.
D"
Primary Task Loss
Subtask Loss
D
ao )
x
pp))
R
O
R
Identity
Figure 3. The overview of S-I.DI (deep regime). White, red and gray highlight our proposed, existing methods, and loss functions, respectively.
5. S-LDLL of the deep regime
The aforementioned analysis has exposed the problems of the shallow regime: 1) shallow methods as base estima- tors have low potential in themselves; 2) there is training gap between the primary task and subtasks, i.e., no repre- sentation learning is involved. Therefore, it is necessary to introduce our proposed S-LDL of the deep regime, the overview of which is illustrated in Fig. 3. We illustrate our framework by introducing the learnable parts one by one.
Note that ERR changes as the primary task changes. Finally, we can learn the model parameters @ by
arg min(CrRI + plsuB).
(12)
where ja is a trade-off parameter. Compared with the shallow regime, S-LDL of the deep regime has the following advan- tages: 1) There no two-stage training gap, which makes the representation contain insights from both the primary task and the subtasks; 2) the framework not only serves LDL, but can also be directly applied to derivative tasks of LDL, e.g.. LDL for classification (LDL4C) (Wang. Jing and Geng, Xin, 2019), incomplete LDL (IncomLDL) (Xu & Zhou, 2017), label enhancement (LE) (Xu et al., 2019). The modifications involved are shown in Table 2, where Cx indicates the set of losses for adaptable methods in the task of type "X". Special mathematical procedures of LDL4C and IncomL are defined as
pp() is guided by subtasks to learn a powerful representa- tion, i.e., R o (X).
(-) is responsible for predicting subtask label distribu- tions, i.e., (D . ((R). To ensure the precise prediction of subtask label distributions for reconstruc- tion, we employ the mean absolute error function for subtask learning. The loss is weighted by the summation of the description degrees corresponding to the primary tasks, allowing more reliable label spaces to receive more attention. The subtask learning loss has the following form:
(D())
if yj = arg max dg 0. o/w,
1.
4
(13)
fSUR
(D'; (D):
N N
Σ
Σ
(Ro (D))
(D)s, if wz) Ω
(14)
yo
iE[N]
o/w,
(10)
respectively, where (.); represents the element in i-th row of the matrix corresponding to label yj. and Ω represents
Σ BA (
Th
Σ [x
g(t)ys
observed elements sampled uniformly at random from D in IncomLDL. Such modifications are rational since: 1) targets of L.L4CC and IncomL.D., i.e., D and (D()), are essentially different forms of degradation of the label distribution matrix; and 2) the target of LE is a logical label matrix L, the same as the target of MLL, which actually special case of LDL.
w-)) can be any existing method that can be expressed as network structure theoretically. Since the concatenation of the representation and subtask label distributions, we have Z = (R, & R)) and D- = w( Z). In the case of the primary task being vanilla LDL., the primary task loss EpRi can be
6. Experiments
= I 3 N
di lk.. E LLDE.
LKL.
D, D.
d In
In this section, we evaluate S-LDL via a series of experi- ments. Due to page limitations, datasets, comparison meth- ods, parameter settings, and full experimental results are
[E(N)
(11)
6

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Divide and Conquer: Learning Label Distribution with Subtasks
Table Modifications of different task adaptations
Type
Subtask construction
EeRI
ÉSUB
Vanilla LDL.
(D -.) + SC)
E(D: D) E LIDI.
PSuB (D()
LDL4C
( SC(D)
((D; D, D) E LIDACC
&sum (D
D (1)
IncomLDL
(D ......))))). - (C(RR(()
((Rn(D), Ra(D)) E LtecmllL
SSU (D
Ra(D
LE
(L ...) ) LC
[(L, D) LLE
ÊSUB (L
D (
+.004
-00
+.003
Speattin ollast
+.003
Speatman eifort
(On v.l. Di Sol 1d
+.002
+.002
+.001
[1ox ve]
SKKI
+.001
(st (h. du Su. s:l1
S-KLD
8-KLD w//a
10:
regrtn)
8.121 Idualie regltee
SKI snja Ferk
.05
15
5
10
15
20
-00
.65
5.0
(a) Spear. offiet w.r.t
(b) Spear. offset w.r.t.
(c) Spear. offiett w.r.t. f
Figure 4. (a) and (b) visualized results of the parameter sensitivity analysis of SC, which is on the emotion6 dataset. The blue lines in (a) and (b) represent performance without auxiliary tasks. (c) is visualized results of the ablation study and the parameter sensitivity analysis of deep regime S-LDL on the Natural _Seenee dataset.
introduced in the appendix. Details of all implementations are openly accessible at GitHub.*
& Hou, 2015). is important to highlight that S-LDL of the shallow regime (Alg. 2) relies on naive concatenation operations and is not tied to representation learning. Con- sequently, any performance improvement over f is solely attributed to the effects of ( and the original performance of f. The results in Tables 3 and 4 show that S-LDL sig- nificantly improves the performance of the base estimator LDSVR.
Metries For LDL, we use the same metries suggested by Jia et al. (2023). Due to page limitations, we mainly present results Cheby. 1 (Chebyshev distance), Clark (Clark distance), Cosine t (cosine similarity), and Spear. t (Spearman's 's coefficient) in the main paper, where (t) indicates "the lower (higher) the better". Note that these metrics are not as intuitive as accuracy or error rate, i.e., small changes can mean large performance differences. For LDIAC, objective of which is different from LDI., we use 0/1 loss i (zero one loss) and Err. prob. | (error proba- bility) metrics (Wang. Jing and Geng, Xin, 2019).
In Tables 5 to 6, LRR focuses on the label ranking rela- tionship, which also emphasized by each subtask. We believe this is why S-LDL and LRR fit well. Note that our method has the least improvement in SCL, which may be attributed to its reliance on shallow regime methods in the prediction phase. It is also worth noting that the improve- ment in vanilla KLD is considerable, which just illustrates the limitations of loss function engineering that considers label correlation one-sidedly. QFD² and CJS are tailored for ordinary LDL,, and may have better results than LRR this regard. Powered by S-LDL, these methods can all achieve better level. For LDL4C, S-LDL significantly im- proves both HR and LDLM. However, it can be observed that S-LDL4C is unstable on the Flickr dataset, which is not surprising since LDL4C itself fails on it. This is caused by the combined effect of the sparsity of the dataset and the information entropy operation involved in LDLAC
Results and discussion We apply S-LDL to existing methods to demonstrate performance improvements. For each dataset we conduct ten-fold experiments repeated 10 times, and the average performance recorded. Tables 3 show representative results of the shallow/deep regime S-LDL and the remainder are in the appendix, where (o) indicates that more than half of the metrics support that "S-X" is statistically superior (inferior) to the correspond- ing methods "X" (pairwise t-test at 0.05 significance level); there is no significant if neither * nor o is shown.
For S-LDL of the shallow regime, f(t)s and f are imple- mented by a representative LDL method, LDSVR (Geng
Parameter sensitivity analysis First, we employ S-LDL of the shallow regime for parameter sensitivity analysis of Sc (Alg. 1). With varying and fixed at 10, we conduct
hhttps://gi www.hhcc..oce.eeecc Sprr saka/PyL.L.
7

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Divide and Conquer: Learning Label Distribution with Subtasks
Table 3. Experimental results LDL on the JAFFE dataset formatted as (mean + std)
Algorithans
Cbeby. +
Clark 4
Can. 4
KLD J
Cosine
Int.
Spear.
Ken.
LDSVR
0959 4.033
.2800 =.027
.6778 + 058
.476 +.011
.549
4.00
8838 + 4.012
.5175 +.102
.4508 + 4.0
S-LDSVR (shallow)
.0859 1.012 .024 2.003
.6143 1 008
.0449 2.012
9580 2.011
.941 2 012
.5237 2.0m
.4800 1.079
Table 4. Experimental results of I.DI on the Natural Scene dataset formatted as (mean + std)
Algorithmss
Chaby
Clark 4
Can. 4
KLD 4
Cosine 1
Int. 1
Spear. 1
Ken.
I.DVR
4899 016
2.0831
+.025
5.7724 4.002
2.0862 + oss
.5740 +. 4.017
.4430
4.01
4997 ..016
.3695 4.012
S-LDSVR (shallow)
4692 +.07
1.9231 +.084 5,2307 4.118 2.2261 +.121
.5893 +.01s
.4602 +.018
.5236 4.017
.3888 +.018
Table 5. Experimental results of LDL on JAFFE and Yeast diau formatted as (mean std(rank))
JAFFR yons al.,
1998)
Yeast: diau (Geng, 2016)
Algorithms
Algorithms
Clark 4
Comine
Cheby.
Spear.
LDSVR (Ging Hou, 2015)
.3200 2. 0.07 (6)
9949 1:010 (7)
CPNN (Clng et al., 2013)
0085 1 001 (9)
.262 1: 1.044 (10)
AA-NNN (Geng, 2016)
.483 +.082 (8)
9497 +.010
AA-kNN
.085 4.00
.3674 4..09 (9)
LDLFs (Shen et al., 2017)
.367 #.002 (10)
.9944 + 0.09 (10)
LDLFs
0077 +.00 (9)
4088 4.021 (8)
DF BFGS (Gonzälez et al., 2021a)
.3062 L. 025
.9633 007
DF DFBBFS
.038 001
.4161 4.077 (6)
KLD (Geng, 2016) *
.3088 2.031 (0)
.9538
LRR
.030
.4154 (6)
S-KLD
.3007 (2)
.9625 000 (9)
S.L.RR
.336 000 (1)
.4198 (2)
SCL (Jia al., 2019)
.3358 2.024 (7)
.9922 1 006 (6)
QED (Wen al., 2023)
0369 o01 (6)
.4118 4.8.:.020 (7)
S-SCL
.31841.023 (4)
.604 1 008 (6)
S- S-QFD2
.036 om (1)
.4203 1.00 (1)
LRR (Jia et al., 2023)
.3230 +.027 (5)
.616 + +.00 (4)
CJS (Wen al., 2023)
0367 + 4.00 (4)
.4164 (4)
S-LRR
.2934 +. 2.28 (1)
.9635 +. .008 (1)
S S-CIS
0336 001 (1)
4.024 (B)
Table 6. Experimental results of LDL4C on sBU 3DFE and Plickr formatted as (mean 1 std(rank))
aBU 3DFE (Geng, 2016)
Flickr (Yang et al., 201
Algorithms
Algorithms
0/1 loss
prob.
0/1
Err. prob.
LDL4C (Wang. Jing and Geng. 2019)
.5578 2. 028 (9)
.7671 = 007 (5)
LDL4C
8971 2.00 (6)
.884 a 004 (6)
S-LDL4C
.5266 2.025 (5)
.7686 0.06
S-LDLAC
.8705 +. 138 (5)
.8022 1.00
HR (Wang& & Geng. 2021a)
.1677 2.027 (3)
.7596 2. 000
HR *
.4513 .4513 (4)
.523 2.007
S-HR
.5069
(2)
.7598 + 000
S-HR
.2199 .219 +.05
.639 +. 4.007 (1)
LDL. (Wang & Geng, 2021b)
.5258
(1)
.7619 000
LDLM
.384 2884+.0144
.740 4.07 (3)
S-LDLM
.8099 (1)
.7524
S-LDLM
.432
.5667 1. 1.007 (2)
ten-fold experiments repeated 10 times on the emotion6 dataset and record the average Spearman's 's coefficient. Re-
primary task and subtasks, as a good trade-off between them can enhance the performance.
sults are shown in Fig. 4(a). The panels from left to right display examples of (o(t)s when the X is 0.01, 0.05, 0.2, 1, and 10, respectively. When À is suitable, y0(0ss are diverse and do not have excessive local ignorance; as À decreases, (1()) tend to be homogeneous, and invalid masks account for the majority; as increases, the local ignorance of each ()()) becomes significant.
Ablation study Here we are interested in the importance of each part of S-LDL, thus an ablation study is performed with S-KLD: 1) we replace NSuu in SC with the min-max function to examine the importance of the subtask distribu- tion reconstruction, and this model is denoted as S-KLD (min-max); 2) we remove the identity mapping in Fig, 3 to examine the importance of the prediction via subtask repre- sentation, and this model is denoted as S-KIDD w/o id.; 3) we train without the term of fSUB (i.e., setting μ = 0) to ex- amine the importance of subtask learning, and this model denoted as S-KLD w/o {SUB. Results are also shown in Fig. 4(c), which confirms that each part of S-LDL contributes as long as there is good trade-off.
Besides, we also study the parameter sensitivity of with A fixed at 0.2. Results are shown in Fig. 4(b), illustrating that having a plethora of auxiliary tasks are detrimental to performance, which may be due to overfitting.
For S-LDL of the deep regime, we check the sensitiy- ity of the trade-off parameter μ on the LDL. task with the Natural Scene dataset by varying the parameter in {0.01, 0.05, 0.1, 0.5, 1, 5). Results are shown in Fig, 4(c). Spearman's 's coefficient of S-LDL first increases and then decreases as y varies, demonstrating a desirable bell-shaped curve. This justifies our motivation of jointly learning the
7. Limitations and conclusion
Limitations First, S-LDL of the shallow regime is pro- posed out of intuition, and in Section 5, we have discussed
8

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Divide and Conquer: Learning Label Distribution with Subtasks
its limitations, which are addressed via the designing of S-LDL of the deep regime. Second, when the label space is large, especially when labels are continuous and result in unimodal label distributions (e.g,, age estimation), our proposed cannot be rationally applied. Fortunately, one pos- sible workaround is to use binning tricks for preprocessing, and then construct subtasks.
Geng, X., Yin, C., and Zhou, Z.-H. Facial age estimation by learning from label distributions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(10); 2401-2412, 2013.
González, M., González-Almagro, G., Triguero, I., Cano, J.-R., and Gareía, S. Decomposition fusion for label dis- tribution learning. Information Fusion, 66:64 75, 2021a.
Conclusion We propose S-LDL, a subtask learning frame- work nested into LDL.. S-LDL is generic: it generates extra supervised information via subtask construction without any extra knowledge; S-LDL is minimalist: it can be attached to existing methods and handle derivative tasks; S-LDI. is efficient: it captures a wide variety of label correlations. The analysis shows the validity and reconstructability of subtasks, and experiments show the superiority of our frame- work.
González, M., Luengo, J., Cano, J.-R., and Gareía, S. Syn- thetic sample generation for label distribution learning. Information Sciences, 544: 197-213, 2021b.
Jia, X., Li, Z., Zheng, X., Li, W., and Huang, S.-J. Label dis- tribution learning with label correlations on local samples. IEEE Transactions on Knowledge and Duta Engineering, 33(4):1619-1611, 2019.
Jia, X., Shen, X., Li, W., Lu. Y., and Zhu, J. Label distribu- tion learning by maintaining label ranking relation. IEEE Transactions on Knowledge and Data Engineering, 35 (02):1695-170) 2023.
Impact statement
This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Jin, Y., Gao, R., He, Y., and Zhu, X. Gidl: Graph label distri- bution learning. In Proceedings of the AAAI Conference on Artificial Intelligence, pp. 12965-12974, 2024.
Acknowledgements
Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. In Proceedings of the 3rd International Conference on Learning Representations, 2015.
This research is partially supported by Shanghai Cen- tral Guidance Science and Technology Development Fund (YDZX20233 100004000,, the Natural Science Foundation of Jiangsu Province (BK20242045), and the National Natu- ral Science Foundation of China (62176123, 62476130).
Le, N., Nguyen, K., Tran, Q., Tjiputra, E., Le, B., and Nguyen, A. Uncertainty-awan label distribution learn- ing for facial expression recognition. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 6088-6097, 2023.
References
Chen, S., Wang, J., Chen, Y., Shi, Z.,, Geng, X., and Rui, Y. Label distribution learning on auxiliary label space graphs for facial expression recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13984-13993, 2020.
Li, X., Liang, X., Luo, G., Wang, W., Wang, K., and Li, S. Ambiguity aware breast tumor cellularity estimation via self-ensemble label distribution learning. Medical Image Analysis, 90:102944, 2023a.
Li, X., Luo, G., Wang, W., Wang. K., and Li, S. Curriculum label distribution learning for imbalanced medical im- age segmentation. Medical Image Analysis, 89: 2023b.
Gao, B.-B., Xing, C., Xie, C.-W., Wu, J., and Geng, X. Deep label distribution learning with label ambiguity. IEEE Transactions on Image Processing, 26(6):2825- 2838, 2017.
Liang. L., Lin, L., Jin, L., Xie, D., and Li, M. Scut-fbp5500: A diverse benchmark dataset for multi-paradigm facial beauty prediction. In Proceedings of the 24th Inter- national Conference on Pattern Recognition, pp. 1598- 1603, 2018
Geng, X. Label distribution learning. IEEE Transactions on Knowledge and Data Engineering, 2847):1734-1748, 2016.
Geng, X. and Hou, P. Pre-release prediction of crowd opin- ion on movies by label distribution learning. In Pro- ceedings of the 24th International Joint Conference on Artificial Intelligence, pp. 3511 1551-335 1-317,, 2015.
Lu, Y. and Jia, X. Predicting label distribution from multi- label ranking. In Proceedings of the 36th Annual Con- ference on Neural Information Processing Systems, pp. 36931-36943, 2022.

---

Divide and Conquer: Learning Label Distribution with Subtasks
Lyons, M., Akamatsu, S., Kamachi, M., and Gyoba, J. Cod- ing facial expressions with gabor wavelets. In Proceed- ings of the 3rd IEEE International Conference on Auto- matic Face and Gesture Recognition, pp. 200-205, 1998.
Wang, J. and Geng, X. Label distribution learning machine. In Proceedings of the 38th International Conference on Machine Learning, pp. 10749-10759, PMLR, 2021b.
Wang, K., Xu, N., Ling, M., and Geng, X. Fast label enhancement for label distribution learning. IEEE Trans- actions on Knowledge and Data Engineering. 35(02):
Prabhu, Y., Kag, A., Harsola, S., Agrawal, R., and Varma, M. Parabel: Partitioned label trees for extreme classifica- tion with application to dynamic search advertising. In Proceedings of the 2018 World Wide Web Conference, pp- 993-1002, 2018.
1502-15144 1514, 2023.
Wang. R., Kwong, S., Wang. X., and Jia, Y. Active k- labelsets ensemble for multi-label classification. Pattern Recognition, 109:107583, 2021.
Read, J., Bielza, C., and Larrañaga, P. Multi-dimensional classification with super-classes. IEEE Transactions on Knowledge and Data Engineering, 26(7):1720-1733, 2013.
Wang, Jing and Geng, Xin. Classification with label distri- bution learning. In Proceedings of the 28th International Joint Conference on Artificial Intelligence, pp. 3712- 3718, 2019.
Read, J., Puurula, A., and Bifet, A. Multi-label classifica- tion with meta-labels. In Proceedings of the 2014 IEEE International Conference on Data Mining, pp. 941-946, 2014.
Wen, C., Zhang, X., Yao, X., and Yang, J. Ordinal label distribution learning, In Proceedings of the IEEE/CVE International Conference on Computer Vision, pp. 23481- 23491, 2023.
Ren, T., Jia, X., Li. W., Chen, L., and Li, Z. Label distribu- tion learning with lahel-specific features. In Proceedings of the 28th International Joint Conference on Artificial Intelligence, pp. 3318-3324, 2019.
Wu, X., Wen, N., Liang, J., Lai, Y., She, D., Cheng. M., and Yang. J. Joint acne image grading and counting via label distribution learning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 10642- 10651, 2019.
Shen, W., Zhao, K., Guo, Y., and Yuille, A. Label distribu- tion learning forests. In Proceedings of the 31st Annual Conference on Neural Information Processing Systems, pp. 834-843 2017.
Xu, M. and Zhou, Z.-H. Incomplete label distribution learn- ing. In Proceedings of the 26th International Joint Con- ference on Artificial Intelligence, pp- 3175-3181, 2017.
Shirani, A., Demoncourt, F., Asente, P., Lipka, N., Kim, S., Echevarria, J., and Solorio, T. Learning emphasis selection for written text in visual media from crowd- sourced label distributions. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 1167-1172, 2019.
Xu, N., Liu, Y., and Geng, X. Label enhancement for label distribution learning. IEEE Transactions Knowledge and Duta Engineering, 1632)1163211643 2019.
Xu, N., Shu, J., Zheng, R., Geng, X., Meng, D., and Zhang, M.-L. Variational label enhancement. IEEE Transac- tions on Pattern Analysis and Machine Intelligence, 45 (5):6537-6551, 2023.
Tsoumakas, G., Katakis, I., and Vlahavas, I. Effective and efficient multilabel classification in domains with large number of labels. In ECMI/PKDD 2008 Workshop on Mining Multidimensional Data, volume 21, pp. 53-59, 2008
Yang. J., She, D., and Sun, M. Joint image emotion classi- fication and distribution learning via deep convolutional neural network. In Proceedings of the 26th International Joint Conference on Artificial Intelligence, pp. 3266- 3272, 2017a
Tsoumakas, G., Katakis, I., and Vlahavas, I. Random k- labelsets for multilabel classification. IEEE Transactions on Knowledge and Data Engineering, 23(7):1079-1089, 2010.
Yang, J., Sun, M., and Sun, X. Learning visual sentiment distributions via augmented conditional probability neu- ral network. In Proceedings of the AAAI Conference on Artificial Intelligence, pp. 224 224000002220 2017b.
Wang, J. and Geng, X. Theoretical analysis of label distribu- tion learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 5256-5263. 2019.
Wang, J. and Geng, X. Learn the highest label and rest label description degrees. In Proceedings of the 30th International Joint Conference on Artificial Intelligence, pp. 3097-3133, 2021a.
Zhai, Y., Dai, J., and Shi, H. Label distribution learning based on ensemble neural networks. In Proceedings of the 25th International Conference on Neural Information Processing, pp. 593-6022 2018
10

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Zhang, M.-L. and Zhou, Z.-H. A review multi-label learning algorithms. IEEE Transactions Knowledge and Data Engineering,
2013.
Zhao, X., An, Y., Xu, N., Wang. J., and Geng, X. Imbal- anced label distribution learning. In Proceedings of the AAAI Conference Artificial Intelligence, pp. 11336- 11344, 2023.
Zheng. Q.. 7hu, J., and Tang, H. Label information bot- tleneck for label enhancement. In Proceedings of the IEEE/CVF Conference on Computer Vision and Patlern Recognition, pp. 7497-7506, 2023.

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Divide and Conquer: Learning Label Distribution with Subtasks
A. Appendix: details of experiments
Here we try our best to provide as much information as possible for reproducible research.
Table 7. Summary of the metrics
Name
Formula
Name
Formula
Cheby. J
Disi(u, (u, maxy Juy Uy
Cosine
Sim (u, v) V Vij= "I/===
Clark 4
Dis v) V //j= (u (uytoy)
Int. f
Simz(u, v) = E}=z min (z,, nt)))
Disa(u, v) = E/=1 lay-osl
Rnki (u, v) = 6
Can. 4
Spear. f
Mytus
1112
KLD +
(us v) = EE= Uj In 2
Ken. f
Rnkg (u,, v) 2Ijen u sgn(uy )am(
0/1 loss 4
Ci v) ó(arg max(u), arg max(v))
Err.
prob.
C2(u, v)
Marg max(1))
A.1. Metries
For LDL, IncomLDL, and LE, we use the same metrics suggested by Geng (2016), which are Cheby. + (Chebyshev distance), Clark J (Clark distance), Can. | (Canberra distance), KLD J. (Kullback-Leibler divergence). Cosine (cosine similarity), and Int. t (intersection similarity), respectively. Here 4. (1) indicates "the lower (higher) the better". For LDL and IncomLDL, we additionally use two ranking metrics: Spear. 1 (Spearman's 's coefficient) and Ken. 1 (Kendall's coefficient) (Jia et al., 2023). Note that these metries are not as intuitive as accuracy or error rate, i.c., small changes can mean large performance differences. For LDL4C, objective of which is different from LDL, we use 0/1 loss (zero one loss) and Err. prob. & (error probability) metrics (Wang, Jing and Geng, Xin, 2019). Let the real distribution be denoted by {uy}f=1- and the predicted distribution be denoted by v {u, 2 then the above metrics can be summarized in Table 7, where p(-) and 8(,, -) are the ranking function and the Kronecker delta function, respectively.
Table 8. Summary of datasets
Dataset
Instances N
Features P
Labels
JAFFE
213
243
6
sBU 3DFE
2500
243
6
Movie
7755
1869
5
Nature Scene
2000
294
fbp5500
5500
512
Yeast heat
2465
24
6
Yeast_diau
2465
24
7
Yeast.cold
2465
Yeast_dtt
2465
emotion6
1980
168
7
Twitter
10045
168
8
Flickr
11150
168
8
A.2. Datasets
We adopt several widely used label distribution datasets, including: JAFFE (Lyons et al., 1998); fbp5500 (Liang et al., 2018);6 sBU 3DFE, Movie, Natural Scene, Yeast heat, Yeast diau, Yeast cold, and Yeast ått provided by Geng (2016);7 emotion6, Twitter, and Flickr provided by Yang et al. (2017b).* The information these datasets are summarized in Table 8.
Phttps://zencer wttp://woo/e..///ee////.. "https: seu. wwwp.cn/ggegg//a////eee/ dow load htm
Reae
'https://ov.ne htw.///vv..naaiiioeaaaaa wtw.:aaaaaa..ee..cooooooooeeeeee
12

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Divide and Conquer: Learning Label Distribution with Subtasks
A.3. Comparison methods
On the one hand, we apply our proposed S-LDL to existing methods demonstrate performance improvements in the LDL task (denoted by the "S-" prefix). These methods are BFGS-LLD (KLD) (Geng, 2016), SCL (Jia al., 2019), LRR (Jia et al., 2023), QFD (Wen et al., 2023), and CJS (Wen et al., 2023) (the losses of these methods constitute the set LDDL). On the other hand, we compare S-LDL with methods that have specialized structure, which our proposed cannot directly adapt to. These methods are CPNN (Geng et al., 2013), LDSVR (Geng & Hou, 2015), AA-kNN (Geng, 2016), LDLFs (Shen et al., 2017), and DF-LDL (denoted by DF-BFGS since we use BFGS- Ls as base estimators) (González et al., 2021a), Moreover, we apply our proposed to derivative tasks of LDL (i.c., LDL4C, IncomLDL, and LE) and the comparison methods involved are LDL4C (Wang. Jing and Geng, Xin, 2019), HR (Wang & Geng. 2021a), LDLM (Wang & Geng, 2021b), IncomLDDL (Xu & Zhou, 2017), LP (Xu al., 2019), GLLE (Xu al., 2019), LEVI (Xu et al., 2023), and I LIBLE (Zheng al., 2023).
Table 9. Summary of algorithms and parameter settings
Algorithms
Parameter
Value (Range)
AA-kNN
Neighbors
5
# Estimators (trees)
5
LDLFs
Depth
6
Latent units (leaves)
64
Convergence criterion
10-6
BFGS-LLD
Max iteration
600
mc # Clusters
5
SCL
\1 A2 Ag: Trade-off
10-3 1000000-00000 0.1
A: Trade-off (ranking loss)
10
LRR
3: Trade-off (regularization)
10 10,-3,-2,-1
-1000111))
C. Cz: Balance coefficients
10 10-3.10 10-
LDL4C
p: Margin
10-2
X1, λ2 As: Tradc-off
10 10
HR
p: Margin
10-2
A1 λ2, Ag: Trade-off
10 6
10
(3,
10
LDLM
p: Margin
10-2
Convergence criterion
10
IncomLDL
7: Factor of Lipschitz constant
2
A: Trade-off
1
LP
0: Balance coefficient
0.5
d1. A2: Trade-off'
10
10
GLLE
σ: Width parameter for similarity calculation
10
LEVI
A: Trade-off
LIBLE 0, B: Tradc-off
10
-1,00,,,1
S-LDL. P, T
0.1, 0.2, 10
A.4. Parameter settings and experimental environment
The parameter settings of the proposed S-LDL and comparison algorithms summarized in Table 9. Note that DF-LDL is fare,,ter-free, and we use BFGS-LLDs as its base estimators, parameter settings of which are the same BFGS-LLD as the comparison algorithm. We use Adam (Kingma & Ba, 2015) for the optimization of S-LDL. For all methods of the deep regime, the learning rate is chosen among (1, 2, 5) x (0(-4, -3, -2), and the selection of the number of epochs is nested into a ten-fold cross validation. All the results are obtained on Linux workstation with Intel Core i9 (3.70GHz), NVIDIA GeForce RTX 3090 (24GB), and 32GB memory.
A.5. Full analyticaalexxeeiimental results
Here we provide complete results of all conducted amalysisexxperiment.. Figures 5 to 9 are full results of validity analysis in Section 4.1. Tables 10 to 21 are results on the L.I. task with different datasets, For IncomI.LL, we follow the incomplete
13

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Divide and Conquer: Learning Label Distribution with Subtasks
settings (Xu & Zhou, 2017) and vary the observed w% from 20% 40%. Tables 22 to 23 results the IncomL.D. task. Tables 24 25 the LDL4C task. For LE, follow the settings of the recovery experiment (Xu et al., 2019). Tables 26 to 27 show results on the LE task.
10
41
50
70
501
100
90
70
80
50
100
(14)
Figure 5. Informmation/diversityy index w.r.t. À on emotion6.
10
05
06
24
35
20
167
10
24
86
06
15
20
107
0-
Figure 6. Informatioo/diveersty index w.r.t. À on JAFFE.
(15)
Figure 7. Informaation/diversitty index w.r.t. À on SBU_3DFE.
10
194
86
35
55
1/7
11
28
12
36
45
10
2
30
02
70
50
00
100
Figure Information//diveritty index
Twitter,
x(%)
Figure 9. Information/diversityy index w.r.t. À on Flickr.
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Divide and Conquer: Learning Label Distribution with Subtasks
Table 10. Experimental results of LDL on the JAFFE dataset formatted as (mean 1 std)
Algorithms
Cheby 4
Clark +
Can. +
KLD +
Cosine
Int. 1
Spear. 1
Ken. 1
LDSVR
.3280 2.15
.6778
.0476
.9549
.8838 2.12
4.102
.4508 086
S-LDSVR
.8859.002
.3024 +.
.6143 4.008
.0449
.95804.01
.8941
.5237
.4580
AA-ÀNN
.0988
1.02
.7164 1..06
.0527
.9497 1.010
8766
.4111
1.083
.3514
LDLFs
00900
2.022
.7355
1.066
.0550
.94941.000
.8766
.4364 1.
.3749 .093
DF-BFGS
.0827 2.000
.3062
.6239 2.052
.0388
2.00r
96333..0m
2.14
.5244
2.087
.4493 2.77
KLD
.0925
1.55:0012
.3608
1.031
.7363
.0508
.9538 1..00.000
.8777
:.011
.4572
1.097
.3873 .084
S-KLD
.0818 2.11
.3007
2.02
.6132 1.067
.0395
.9625 2.00
.8960 2.12
105 .5461
.4769 096
SCL
0873 2008
.3358
2.024
..074
.0439
2.006
29592
18851 +.00
.4744
4.002
.4020 .800
S-SCL
0814
.3184 1944-1.22
.6526
1..03
.0420
+ .088
.9604 11.4..000.
.8896 1.00
.5110 1.095
.4388
084
LRR
0853 2010
.3230 2.07
.6560 2.055
.0412
008
06161.008
8906
.1177.000
.4420
084
S-LRR
.0804=.
.099
.2934
+.028
+.059
.0383
.000
.9635 2.008
.8981 2.011
.5448
.5448
4.002
.4819 0844
Table 11. Experimental results of LDL on the sBU_3DFE dataset formatted as (mean = std)
Algorithms
Cheby.
Clark
Can. +
KLD 4
Cosine
Int. f
Spear. f
Ken.
LDSVR
12200.00o
..110
.8009 2.021
.0720
.004
.9298 0.00
.8559 + 004
352244.08
.3011 = #.026
S-LDSVR
.1193
.3573
4.010
.7600
4.022
.0672 .000
.9338 4.04
.8625 u004
.3759 4.030
.3236 s. .028
AA-&NN
.12722.004
.4001
2.009
.8281
+.020
.0801
-0004
.9217 2.004
.8488 004
.2053 4:030
.1767 2.026
LDLFs
.0016..003
.3262
+.008
.6841
..017
.0504
.0003
.9499
#.003
.8776
.003 003
.4212 +.023
.3620 0199
DF-BEGS
.1146 ..04
.004
.3616
+.008
.7627
4.019
.0618 +000
.9388
4.003
.8626 + 004
.3026 4.001
.2621
026
KLD
.1147 2.004
.3697
2.008
.7804
#.019
.0624
..003
.9387
2.003
.8604
.00
.3021
4.026
,2643
..022
S-KLD
#.044
.3203
0.00
.6736
4.018
.05
.0003
.9487
+.003
.8789
000
.4334 4.u25
.3729
022
SCL
.1145
#.004
.3688
+.008
.7748
4.018
.0605
+.003
.9404
+0003
.8614
..003
.3091
4.026
.2701
+ #.021
S-SCI.
.1041
1.004
.3301
1.009
.6936
1.019
.0535
0.0
.9468
1 10003
.8754
1.004
.3956
+.030
.3381
1 1.027
LRR
.1067
2.003
.3476
2.08
.73204.011
1.017
.0543
0.0
.9465
1 003
.8695
+ .00
.3626
4.026
.3123
022
S-LRR
.0996
#000
.3157
4.0008
.6610
4.017
.0499
.0003
.9502
4.003
.8812
0.003
.4455
4.026
.3837
#.023
Table 12. Experimental results of LDL on the Yeast heat dataset formatted as (mean = std)
Algorithms
Cheby.
Clark
Can. 4
KLD 4
Cosine
Int. 1
Spear. t
Ken.
CPNN
.0419 1.001 0011
.1818 0.0
.633 1.009
.0125 1.001
.9881 1.001
.9404 + .011
.1507 1.034
.1221 :: 1.028
AA-KNN
.0441 =.001
.1913
0.05
.3840
.0140 =.001
.9867
1 1.001
.9370 + 002
.1678 4.081
.1384
026
LDLFs
..40
.1818
+ 005
.3627 1.009
.0125
=.001
.9881 2.001
.9405 0.00
.1731
4.032
.1409 .22
DF-BFGS
.040 2.001
.1816 + 000
3624 4.09
.0125 .0001
.9881 4.001
.9405 4.00
.1964 4.084
.1624 .028
LRR
.0423 1.001
.1828 + oos .644
10000
.0126 =.001
.9880 2.00
9402+.00
.1655 03.
.1351 = .088
S-LRR
.0417 4.001
.1806 +. .05 3609 1.000
.0124
:..001
.9882 à: 4.001
.9408 #.001
.1882
1.004
.1548 .028
QFD²
,0423 001
.1827
005
.3644
4.009
.0126
=.001
.9880 1
.9402 001 001
.1677 +.032
.1351 027
S-QFD2
.0417
.1808
005
.361
11.009
.0124
.9822
.9088 001
.1880
1.802.003
.1544 027
CIS .
.0423 =
.127
+ 005
.343 1.000
.0126
.9880 001
.9402
000
.1632
.1329 027
S-CIS
.0417 2.001
.1804
4.03
.0124
2.001
.9882 2.01
.9409
#0001
.1940
.1589 2.025
15

---

Divide and Conquer: Learning Label Distribution with Subtasks
Table 13. Experimental results of I.LL the Yeast diau dataset formatted as (mean std)
Algorithms
Cheby.
Clark
Can. /
KLD +
Cosine
Int. f
Spear. 1
Ken.
CPNN
.038 13855.000
.2069 + 006
4439
.0138 =0001
.9872.000
.9383 2.002
4.034
.2427
AA-ENN
0385
22255 006
4487 2.014
.0145
.9867 2.001
.9377 4.002
.2976 4.029
024
LDLFs
0371
2014 0.00
.4324
4.012
.0132
.9879
.9401
.4088
.3254
DF-BEGS
0368
.1999 1.0006
.4294 4.013
.0131 1..001
.9879
.9405
.4161
4.,27
.3404 .022
LRR
0370
22007
0006
.4307
4.012
.0131
.9879 4.001
.9403 0.0
.4154
.3343
.2000
S-LRR
.036
0001
.1983
.4577
4.012
.0129
.9881
.9410
.4198
4.023
.3389
QFD
.0369
40001
.2000
000
.4296
#:012
.0131
0.001
.9879
.000
.9004
0002
.4118 4:22
.3326 .011
S-QFD²
.0366 .001 0011
.1985
1 000
.4261
1.012
.0129
:.001
.9881
1.001
.9409
1: 0.002
.4203 1.021
.3387
#.018
CIS
.0367
0000
.1989 000
#.122
.0130
..0
.9880
#.00
.9408
0.0
.4164 +.02
.3366
2221
S-CIS
.0366 = 0011
.1984 + 000
.4260 4.012
.0130
+.001
.9881
+.001
.9409
+: 4.002
.4198 +.024
.3392 #: .019
Table 14. Experimental results of LDL on the Yeast.cold datasct formatted as (mean = std)
Algorithms
Cheby.
Clark
Can. +
KLD 4
Cosine
Int. f
Spear. f
Ken.
CPNN
.0510 = o02
+1392
2.3964.00s
.0121
:.001
.9866 2.00
.9410 + 002
.2651 4.0
.2263
0032
AA-kNN
.0542 suu2
.1476 o.s
.2549 +.00s
.0135
+.001
+.02
.9371 +.002
.2189 4.0
.1866 s. .031
LDLFs
.0511 2.002
.1396
2.05
:.099
.0122
00001
.9885
2.00
.9408 2.02
.2482
2.038
.2112 2.33
DF-BEGS
.0514 =.14 .002
.1404
0005
.2424
.0123 000
.9885
#.000
.9403
0.02
.2581
+.030
.2190
0000
LRR
.051
.0002
.1395 + 005
.2402
#0009
.0122
000
.9886
2.001
.9408
2002
.2490
2.035
.2111 2.030
S-LRR
.0510-
.002
.1391 + 000
.2395 +.009
.0121 +.001
.9866..
40001
.9410
=.02
.2618
4.037
.2238 003
QFD?
.0513 #.00
.1401
.2413
.0123
.9855
4.001
.9405
+
.2534
.2158
.032
S-QFD"
.0510
= .002
.1391
000
.2396 :000
.0121 ..001
9886
4.000
.9100
.257
+.039
.2197
= .033
CJS
.0513
.000
.1401
+.005
.2412 +.0us
.0123
+.001
9884
4000
.9406
4.002
.2535
4.00s
.2152
.032
S-CIS
.0510
..002
.1392
1.005
2.3961.003
.0121
:.001
.9886
1.001
.9410
.2621
1.037
.2241
1: 2.031
Table 15. Experimental results L.LL on the Yeast dtt dataset formatted (mean std)
Algorithms
Cheby.
Clark
Can. 4
KLD 4
Cosine
Int. 1
Spear. f
Ken.
CPNN
.0361 = 0001
.0984 + 004
.1690 1.006
.0063 1.001
.9941 +: 1.000
.9583 + 0011
.1735 4.035
.1494 = 000
AA-KNN
.0386..om
.047 + .04
.1797
2.006
.0071
+.001
.9933
4.000
.9556
.1591
4.033
.1399
.900
LDLFs
.0360 =.0m
.0981
+ 0004
.1689
4.000
.0063
=.001
.9941 4.000
.9583 4.001
.1986 +.08s
.1727 .034
DF-BEGS
.0365 #.001
.0995
4004
.1712
4.o06
.0064
t0001
.9939
40000
.9578 4.001
.1804
4.00s
.1592 .080
LRR
.0360 =.0m
.0982
000
16900..om
.0063 =.001
.9941
4.000
.9583 +.001
20164.085
.1738
.032
S-LRR
.0359 2.001
.0977
+ 0004
.1680
4.006
.0062 .000
.9911
40000
.9585 + 001
.2068 4.035
.1811 .031
QFD
,0362 .00
.0986 0.00
.1696 +0006
.0063 =.001
.9940 4.000
.9582 + 001
.1917 4.035
.1665 .011
S-QFD2
.0559
.0977
.1681 1.006
.0062
.9911
.000
.9585
.2086 036
.1822 002
CJS
.0361 001
.0984
19884+.004
.1692 4.006
.0063
=.001
.9941 1.000
.9582
.1975 4.040
.1722 .035
S-CJS
.0359
.0978 0004
.1682 4.005
.0062 0.001
.9941 0.00
.9585
000
.2080 4.03..
.1804 .031
16

---

Divide and Conquer: Learning Label Distribution with Subtasks
Table Experimental results of LDL on emotion6 dataset formatted as (mean + =add)
Algorithms
Cheby.
Clark 4
Can. +
KLD
Cosine
Int. f
Spear. f
Ken. 1
LDSVR
2.10
1.8217
220
1.0744 2.04
2.011
..773 _6106_.m
.3915 2.00
.3235
2.025
S-LDSVR
+.10
1,8236
1.016
4.04
1.0672 4.07
69222.u.
4.192
.3952
220
.3280 420
AA-ANN
.3288 4.011
1.7116
#2026
3.8757
4.076
.9512 1.1
.6632 = .013
.5564 4.010
.2920
+.027
.2401 4.022
IDLFs
.3120..010
1.010
1.6625
i. 1.0
3.7330
1.075
.5871 1.024
.7143 0.01
.5802
1.010
.3631
1.029
.3025
1.024
DF-BFGS
.3026 4.010
1.6765
+ u25 3.7675 9.071
.5805 0.02
.7206 .013
.5909 4.010
.3940
+ .227
.3256 0.022
KLD
3037
1.6774 = 025
3.77294..07
.5863
.7191
2.013
.5897
2.011
.3959
= .22
.3259 1.023
S-KLD
.302 0.10
1.6548 + 026
3.6984
2.076
.5631 +.024
.7282
2.012
5926+..10
.4063
= .227
.3361 0.023
SCL
+.200.010
1.6750 +: .25
3.7642
4.073
.5803 4.027
.7219
.003
.5977
4.011
.4003
#.028
.3299 4.023
S-SCI.
.3088 1.010
1.6554
1.022
3.6993
..076
.5631
1.025
.7281
.5936 4.010
.4089
1.027
.3383 1.023
LRR
1.6736
+ 025
3.7601
1.073
.5804
.7212
.5899
.3941
1.027
.3243
1.023
S-LRR
.3028 0.00
1.6524 = 026
3.6923
.5607
.7299
.5923
.4078
2. 2.077
.3373
2.02
Table 17. Experimental results of LDL the Twitter dataset formatted (mean + std)
Algorithms
Cheby.
Clark +
Can. +
KLD 4
Cosine
Int. f
Spear. f
Ken. 1
LDSVR
.4236 1.008 2,6722
1.002
7.3015 009
5.0018 +. 1.1
.7627 = 20088
.5237 1 1.008
.4246 1.007
S-LDSVR
.4288 + 000
2.6723
002
7.3018
5.0051 124
.7623 = .000
.5229 +.097
.5229
007
.4238 006
AA-KNN
3172
2.0142
2012
4.5597
..043
3.1429 1.148
.7926 .06
.60244.o00
.5014
.4432 4.08
LDLFs
4035
4.014
2.5461
4.010
6,8269
1.6884
.6756_u14
.5318
.4164
.3349 010
DF-BFGS
.2982 4.004
2.4025
+
6,2416
.6304 4.012
.8250
.6220 4.004
.5467
.4454 +.007
KLD o
.2966 +.04
2.4059
4.005
6,2558
.2000
.6307
.8243 = .06
.6249 4.00
.5470
.4456
S-KLD
.2995 4.955..00
2.4112
+: 0000
6,2883
.6491
.8203
.6205 4.005
.5384
.4385
SCL +
.2977 4.004
2.4028
6,2435
.6262
.8256
6623 4.005
.5488
.4471 +.007
S-SCI.
.2940 4.005
2.4059
006
6,2589
-. .023
.6203 4.018
.8268
.6281
t.006
.5518
.00
.4497 000
LRR
.2984
4.004
2.4046
uus
6,2525
9.019
.6351 +.012
.8232
.UDG
.6220
4.004
.5443
4.00s
.4636
0.0
S-LRR
.2937
1.004
2.4056
a: t..05
6,2589
4.019
.6189
1.013
.8271
1.008
.6283
1.005
.5519
1: 4.008
.4498
1.007
Table 18. Experimental results
LDL on the Flickr dataset formatted as (mean std)
Algorithms
Cheby.
Clark 4
Can. 4
KLD |
Cosine t
Int. f
Spear. 1
Ken. t
LDSVR *
.5174..000
2.6364
002
7.2094 011
5.0366 +.086
.6636 008
.4683 1.006
.4622 .009
.3811 1.008
S-LDSVR
.4991 4.006
2.6201 +.003
7.1402 21.4022.011
4.9196 2.085
.6721 _007
4814
.4667 2.009
.3850 +.008
AA-ENN
.3266 4.00s
2.0685
.009
4,9363 ..03
2.2172 107
.7200 .006
.5582 0000
.4265
4.009
.3465 0.0
LDLFs
4051 1.011
2,4012 + .12
6,3262
250
1.4274 +.077
.6073 20155
4822.+.011
.3478
2.014
.2847 2.012
DF-BFGS
.307 4.00s
2.1995 007 5,4900
+.000....20 ..06
.6309 +.011
.7801 .0005
.5979
004
.5102 4.000
.4226 0.8
KLD
3015 +. .os
2,2008 +. 007 5,4969
.25
.6348 +.012
.7787 .00
.5973 4.04
.5113
.00
.4234 4.008
S-KLD
.0522 0.0
600
2.2044 +. n07 5.5222 =:: .026
.6485 2.012
.7720 = .005
.5926
+: 004
.5030
4.009
.4166 0.0
SCL
3280
2.2986
.24 5,9247
.099
.8301 +.00s
.7268 =.018
5713+.015
,4566
4.024
.3748 022
S-SCL
.2929
4.005
2.2045
5,5289
.028
.6113 4.012
.7862 .uu
.6070 4.000
.5265
4.00s
.4373 007
LRR
.3057
4.005
2.1969
+. 4.007
5.4763
.025
.6431 4.012
.7752
.008
.5929
4.004
.5047
4.. +.009
.4229 4.008
S-LRR
.2938
4.006
2.2013
n06
5.5138
.023
.6105 +.012
.7864
.005
.6058
+.00
.5261
.009
.4369 4.008
17

---

Divide and Conquer: Learning Label Distribution with Subtasks
Table 19. Experimental results of LDL on the Natural Scene dataset formatted (mean std)
Algorithms
Cheby.
Clark 4
Can. +
KLD +
Cosine
Int. f
Spear. 1
Ken.
LDSVR
.4899 4.016
2.0831
4.025
5.7724 1044..000 .092
2.0862 + 086
.5740=mm .17
4.10
.4997 2.016
.3695 #.012
S-LDSVR
.4692 4.017
1.9231
4.004
5.2307
4.113
2.2261
+.121
.5893 .011
.5236 + 4.017
.3888 4.0
AA-ANN
.3113 1.014
1.9066
1..04
4,5413
1.1
1.0874 1.082
.7113 2.015
.6366
.4921 1. 1.021
.3518 1.016
LDLFs
22808 1.034
2.4329
+: 024
6.6027
1
.6464 i. 118
.7679 046
.5839
18399.004
.5406 1. 058
.4072 1.045
DF-BFGS
_30744.00
2.4126
= 2017
6,5896
.7603 + 033
.7381 2.013
.5568
-51102.01
.3837 2.013
KLD
.3201
1.013
2.4242
1.017
6.6560 1.070
.8285 1.04
.7172 2.015
.5485
.4958 11 1.016
.3715 1.012
S-KLD
2.18
2.3866
+ 020
6,4733
.6608 +.0
.7751 = 004
.6133
.5592
.4221 1.014
SCL
.3379
4.014
2.4800
6,8659
4.076
.8867 4.035
.7014 .014
.4801 4.014
.4109 4.018
.3025 4.013
S-SCL
.2733
1.013
2.3734
018
6,4376 1. .72
.6703 1.043
.7744 0055
.6156..01
.5573
1.031
.4207 1.014
LRR
.3138
2.013
2.4469
018
6.7118
074
.7703
2:032
.7363
=013
.5456..011
.5056
.3782 2.01
S-LRR
.2740 4.018
2.3461
2.0
6.3467
1.34672.087 2.087
.6867 +.070
1.067
.7715 021
.6199 #.017
.5595 2.023
.4228 2.018
Table
Experimental results LDL on the Movie dataset formatted as (mean std)
Algorithms
Cheby. 4
Clark 4
Can. 4
KLD
Cosine f
Int.
Spear.
Ken.
CPNN
.1337 4.003
.5639
1.010
1.0746 1 1.0
.1191 1.005
19194..00
.8164 1.0
.66104.013
.7080 :.002
AA-kNN
..02
.5451
0.0
1.0445 2.018
.I .122 1.0
.9254
1.003
.8250 = 000
.6557 2.011
.5710 2.010
LDLFs
..02
.5233
2.133
1.0134
.1066 + 006
.9305
2.003
.8324 = 000
.6929
.6051 2.012
DF-BFGS
1210 +.00
.5282
+ 009
1.0158 ..19
.1084 + .o0
.9289 0.003
.8301 + 0003
.6848 4.012
.5963 2.01
LRR
1135
.5101
:00
.9770 2.11
.0957
+: 004
.9369
2.002
.8385 = 0003
.7119
.6203 2:011
S-LRR
.1125 2.00
.5086
000
.9717 2.018
.0945
000
.9766
#.002
.8398 = .003
.71264.011
.6227 2.011
QFD
..09
.5200
000
.9920 2.010.
.0975
0004
.9355 2.002
.8357 2.003
.7075
.6158 2.011
S-QFD2
.123
.5073 =
.9700
.0945
0.0
.9376
..002
.8401 + .0003
.7125+.011
.6224
CJS
1.03
.5127 +: 009
.9445
.0984 1.0
.9352
002
.8368
000
.7103
.6178
S-CJS
.1233 +.002
.5072 2009
.9699..01
.0945
0.004
.9376 2.022
.8401
20003
.7125
.6223 2.011
Table 21, Experimental results of LDL on the fbp5500 dataset formatted as (mean std)
Algorithms
Cheby. 4
Clark 4
Can.
KLD
Cosine f
Int.
Spear
Ken. 1
CPNN
4.04
1.3367
000
2.3604 = 2.20
.I .1664 2.0
.9281 #.004
.7958 +.005
.8688 2.005
.7831 200
AA-ANN
1515 1.004
1.0443
2.015
1.7295 ..011
.1 1846 1 016
.9419 + 1.004
.8317 1.005
.8865 : .006
.8123 =. 0.0 008
LDLFs
.3077 1.003
1.2787
1.10
.004 2.1703
.1002 +: 000
.9575
2.009
.8552
+ 0.004
.9600 = oos
.8352 = 007
DF-BFGS
.1341 +.003
1.2889
010
2.1982
..03
.1050 .00
.9551 2.003
.8523 + 004
.9047 .oo0
.8337 007
LRR
.1312 1.00
1.2767
2.1655 ..024
.1004 .04
.9575 4.002
.8547 + 1.003
.9059 =: .044
.8350 = 006
S-LRR
.1002 2.003
1.2796
2.1717 2.024
.0997 + .000
.9566
+.002
.8588 + 003
.9063 .0044
.8425 200
QFD .
.1880 1.00
1.2803 +
2.1858
.24
.1084 + 000
.9535 1: .003
.8466
1.004
.9021 = 004
.8297 008
S-QFD2
.1321 4.004
1,2811 D10
2.1779
.1027 +: 000
.9561 4.003
.8537
0.004
.9044 .005
.8330
CIS
.1343
4.003
1.3057
2.2374
.1084 +. 000
.954
+ t003
.827 4.004
.9044
.8334
S-CJS
.1302 4.003
1.2802
+.010
2.1731
.0997 4.005
.9755
.8559
+.003
.9066 4.004
.8429
Table 22, Experimental results IncomLDL on the JAFFE dataset formatted as (mean std)
Algorithms
w
20%
Cheby. J
Clark L
Can.
KLD
Cosine t
Int.
Spear. 1
Ken. 1
IncomL.D..
.0898 + 010
.3304
1.024
.6742 0.0
.0425 2.007
.9598 007
.8861
+.009
.4742 = .91
40177.083
S-IncomLDL .0863 013
.3799 4.036
.6525 077
.0433 2.012
.9590 2.10
.883 2.1
.5034..11 2.44.114. 114
.4401 1000
w 40%
Algorithms
Cheby. J
Clark L
Can.
KLD
Cosine f
Int.
Spear. 1 .423
Ken. 1
IncomLDL
0946 000
.354 2.026
.7073 053
.0465
4.007
.9558 007
8801 1.010
088
..544..07
S-IncomLDL
08668 ..0
.3111 +.038
.6568 081
.0439
2.014
.9585 2.012
.886 2.015
.5075 115
..434..098
18

---

Divide and Conquer: Learning Label Distribution with Subtasks
Table 23. Experimental results of IncomLDL on the 3BU_3DFE dataset formatted as (mean std)
w
20%
Algorithms
Cheby |
Clark L
Can.
KLD L
Cosine t
Int.
Spear. 1
Ken. '
IncomLDL . .1088 2.003
.3586 1.088
.7586
.0574
.439 = 003
8655 2.003
.3171 027
.2746 2.028
S-IncomLDL .1014 2.00
.3208 2.0
.6727 .09
.0516 4.003
.4855 = 0003
.790 2.004
.2555 2.026
.3679 +.023
Algorithms
w
40%
Cheby. |
Clark L
Can. |
KLD |
Cosine f
Int.
Spear. t
Ken. f
IncomLDL .
.1104 2.033
.3621 2.008
.7673 2017
.0086
94262.008
8638 2.003
.3003 2.024
.2595 1.000
S-LncomLDL 10166.000
.3213 +.009
.6748 .19
.0516
.9885 = .003
.886 0.004
.4232 025
.3659 +.022
Table 24. Experimental results of LDL4C on JAFFE and Twitter formatted as (mean + std)
JAFFE
Twitter
Algorithms
Algorithms
0/1 loss
Err. prob.
0/1 loss |
Err. prob, 4
LDL4C
.4973 108
.7665 1.020
LDL4C
.9081 = 000
.8846 1.005
S-LDL4C
.4453 101
.7600 2.019
S-LDLAC
.8714 2.207
.8729 151
LDL-HR
.4786 2.097
.7676
2.020
LDL-HR
.3656 2.017
.4928
2.011
S-HR
.6533 1.105
.6655
4.019
S-HR
.2753 4.013
.4250
+.00s
LDLM
.487
1.109
.7687
4.021
LDLM
.2814 4.013
.4291
4:008
S-LDLM
.437 2.097
.7689
1.010
S-LDLM
.2753 1.014
.4250
11.008
Table 25. Experimental results LDL4C on 3DFE Flickr formatted as (mean std)
_BF_
Flickr
Algorithms
Algorithms
0/1 loss 1
Err. prob.
0/1 loss |
Err. prob. |
LDL4C
.5578 #.028
.7671
#.007
LDL4C
.8971 #.00s
.8884 4.004
S-LDL4C
.5526 #.025
.7686
1.008
S-LDLAC
.8705 1.138
.8702 1.100
LDL-HR
.5167 2.027
.7596
1.006
LDL-HR
.4513 2.015
.5823 1.007
S-HR
.5069
2.025
.7598
2.006
S-HR
.4219 2.015
.5639 2.007
LDLM
.5258
..0
.7619
2.009
LDLM
.4384 2.014
.5740 4.007
S-LDLM
.4809 +.024
.7524
+.005
S-LDLM
.4321 #.016
.5667 4.007
Table 26. Experimental results of LE on the JAFFE dataset formatted as (mean + std)
Algorithms
Cheby. 4
Clark L
Can. 4
KLD 4
Cosine t
Int. f
LP
0812 1.001
.3466
1.002
.7125
.0424 1.001
.9618 1.001
.8808
GLLE
.0821 1.002
.3196
2.013
.6518
.0386 1.003
.9688
2.002
.8901
LEVI
.0787 1.003
.3316
2.013
.6864
2.028
.0391 1.003
.9649 2.002
.8800 2.004
LIBLE .
.0813 1.006
.20 .3106
.6358
.0370 1.005
005 .6522
.8929
S-LIBLE
.0770 1.003
.2942
.0007
.5997 2.016
.0332 1.002
.9685 1.002
.8987
Table 27. Experimental results of LE the Yeast.heat dataset formatted (mean + std)
Algorithms
Cheby. +
Clark +
Can.
KLD +
Cosine 1
Int. f
LP
.0421 4.000
.2148
4.000
.4711
+.001
.0153
4.000
.8800
4.000
.9235
+0000
GLLE
.0481 4.001
.2114
4.005
.4282
#.011
.0168
4.001
.9822
4.001
.9998
#.002
LEVI
.0494 1.007
.2125
1.027
.4307
1.056
.0169
1.004
.9838
1..004
.9289
1.009
LIBLE
(0453 4.000
.1973
4.001
.3982
+.000
.0148
4.000
.9859
4.000
.9346
+0000
S-LIBLE
.0445 1.000
.1901
1.002
.3790
1.005
.0137
1.000
.9869
1.000
.9376 1.001
19


Reviews:
[{"review_id": "Vw21lqe82c", "reviewer": "Reviewer_sNTH", "summary": "This paper investigates the problem setting of Label Distribution Learning (LDL). In particular, the authors propose the concept of leveraging subtasks to facilitate learning.\n\n## update after rebuttal\nI reviewed the rebuttal and further responses. I am still not fully convinced. I did not have more time to check the motivations of the LDL setting, but it seems that other reviewers approve of this setting. I still have reservations, but the AC may wish to ignore my comments because I did not read the paper in full.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your comments! Responses to your questions are as follows.\n\n**Comment 1:** How does it (LDL model) differ from a typical classification network?\n\n**Response:** Although typical classification networks can formally produce outputs similar to label distribution learning (LDL) through Softmax, they differ significantly in essence, specifically manifested in the following aspects:\n\n+ Learning Objectives: Typical classification networks with Softmax outputs aim to learn a single-peaked probability distribution, emphasizing the certainty that a sample belongs to a specific category (i.e., the \"one true label\"). Their limitation lies in the inability to express the varying degrees of association between a sample and multiple labels. In contrast, LDL seeks to learn a multi-modal (possibly single-peaked or multi-peaked) label distribution, describing the degree of association between a sample and all labels. It is primarily suited for scenarios where semantic overlap or granularity differences exist among labels. The core challenge of LDL *never* lies in ensuring outputs satisfy a probability simplex, but rather in precisely fitting the description degrees.\n\n+ Semantic Interpretation: The Softmax output ensures that probabilities sum to 1, but only the highest probability label is meaningful (winner-takes-all). The label distribution output, however, determines probability distribution based on the relative descriptive strength among labels, allowing all labels to retain interpretability simultaneously.\n\n+ Underlying Philosophy: The classification model is decision-oriented, prioritizing clear classification boundaries. LDL model is description-oriented, pursuing fine-grained quantification of label associations.\n\n**Comment 2:** What form does the training set take? Does each sample come with a soft label?\n\n**Response:** In short, each sample $\\boldsymbol{x}$ come with a label distribution $\\boldsymbol{d}$, which is *not* a soft label. Soft labels are often introduced to improve the generalization of classification tasks, and they are still only related to a specific dominant class. Label distributions model ambiguity, where diversity is a property of the data itself. Therefore, label distributions may inherently follow a continuous or multi-peaked distribution (e.g., subjective human annotations). We have some rigorous statements about LDL in the problem definition in **Section 3.1**. These do not conflict with your observations based on ERM.\n\nTo clarify, our work follows a pure LDL paradigm, which is fundamentally distinct from traditional classification tasks. We believe this clarification adequately addresses the reviewer's concerns. We sincerely hope these issues will not negatively impact the overall rating of our contribution.", "claims_and_evidence": "See weakness and strength.", "methods_and_evaluation_criteria": "See weakness and strength.", "theoretical_claims": "See weakness and strength.", "experimental_designs_or_analyses": "See weakness and strength.", "relation_to_broader_scientific_literature": "See weakness and strength.", "essential_references_not_discussed": "See weakness and strength.", "other_strengths_and_weaknesses": "The concept of Label Distribution Learning (LDL) feels quite unfamiliar to me. How does it differ from a typical classification network? What I mean is, even the most common classification networks can use softmax predictions as the label distribution output in this context.\n\nMoreover, what form does the training set take? The authors only describe the samples as $\\mathcal{X}$ without mentioning the labels. Does each sample come with a soft label? Even if that’s the case, it still strikes me as odd. As we know, the ERM essentially guarantees that, given a sufficient sample size, a classification network trained with cross-entropy loss will eventually produce softmax predictions that converge to the true posterior distribution $P(Y|X)$. In simpler terms, for the same sample point $X$, we would effectively be sampling multiple labels $Y$, which is indeed equivalent to training with soft labels.\n\nPlease first address the points mentioned above, after which I will proceed with my review.", "other_comments_or_suggestions": "See weakness and strength.", "questions_for_authors": "See weakness and strength.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ybsstet3rs", "reviewer": "Reviewer_75Xa", "summary": "This paper studies the Label distribution Learning (LDL) problem. It first claims the disadvantages of existing works, pointing out their contradiction between auxiliary tasks and the generalizability. To mitigate this issue, the authors propose a new method S-LDL, which generates subtask label distributions to help LDL. The proposed S-LDL consists of two simple key components: 1) generating subtasks without prior knowledge; 2) solving subtasks using off-the-shelf LDL algorithms. The experiments demonstrate the effectiveness of S-LDL on multiple datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your precious comments! We have provided point-by-point responses to your questions below.\n\n**Comment 1:** It is better to provide some evidence that existing methods cannot deal with the contradiction.\n\n**Response:** In **Section 1 & 2**, we have provided such evidence. On the one hand, we highlight how existing methods relying on prior/expert knowledge (e.g., [Chen et al., 2020; Wu et al., 2019; Yang et al., 2017a]) suffer from poor generalization, as they cannot adapt to new domains. On the other hand, we demonstrate that conventional LDL techniques use solely the primary task distribution (e.g., [Jia et al., 2019; 2023; Ren et al., 2019; Wen et al., 2023]) as supervised data. These comparisons directly show why existing solutions cannot resolve the contradiction, and the essential reason behind this is highly related to the design logic of these methods.\n\n**Comment 2:** The claim of \"furnish additional supervised data\" seems not rigorous and might cause misunderstanding, since you have not involved more external data. Maybe it is better to use \"supervision\" or \"supervised information\".\n\n**Response:** The claim is indeed not meant to imply external data but rather newly constructed data for supervision. We will revise the wording to clarify this in the manuscript.\n\n**Comment 3:** Some related works in MLL with partitioning of the label space provided theoretical justification. However, there is no theoretical support for the proposed method.\n\n**Response:** We sincerely appreciate the reviewer's insightful comment regarding theoretical justification. While prior MLL works with label space partitioning do offer theoretical analyses, our method provides distinctive theoretical support through the following aspects. In **Section 4.1**, we have discussed the validity of subtask construction, ensuring the subtasks remain meaningful for the primary LDL objective. In **Section 4.2**, we prove that the primary label distribution can be theoretically reconstructed from subtask distributions, demonstrating information preservation. Additionally, in **Section 4.3**, we have provided a detailed analysis of the time complexity of subtask construction, establishing its scalability and efficiency. While our theoretical framework differs from related works in MLL with partitioning of the label space, these three pillars collectively justify the rationality of our design.\n\n**Comment 4:** The performance gains on some datasets are marginal (less than 0.01). Is there any explanation?\n\n**Response:** For LDL, even small changes in metrics can indicate significant performance improvements. This is because the values involved in the calculation are constrained by probability simplex, which limits the range of results and leads to small divergence. For example, on the $\\mathtt{Yeast\\\\_diau}$ dataset, the K-L divergence between the ground truth and a uniform label distribution matrix (values are all $\\frac{1}{L}$, where $L$ is the number of labels) is **0.0158**. This can be considered as the worst performance. DF-LDL, a strong baseline, achieves **0.0131**. To ensure the reliability of the experimental results, we repeat the experiments and provide the standard deviation of the results, and perform significance tests (pairwise *t*-test at 0.05 significance level) to confirm that the observed changes are statistically significant.\n\n**Comment 5:** It seems that the references are a bit out-of-date. Most references are from or earlier than 2023. Is there more recent related work?\n\n**Response:** We appreciate the reviewer's suggestion. While there are indeed some recent works in the field of LDL [1, 2], they currently *lack official open-source implementations*, making their effectiveness and reproducibility unverified. For reliability and fairness in comparison, we chose to focus on well-established methods with publicly available implementations.\n\n[1] Label Distribution Learning Based on Horizontal and Vertical Mining of Label Correlations. *TBD*, 2024.\n\n[2] Exploiting Multi-Label Correlation in Label Distribution Learning. *IJCAI*, 2024.", "claims_and_evidence": "Strengths:\n- The proposed idea is derived from some multi-label learning (MLL) problems, but is also suitable for the LDL problem. This is a minimalist and interesting motivation for LDL.\n\nWeaknesses:\n- It is better to provide some evidence that existing methods cannot deal with the contradiction. Otherwise the motivation will be a bit unconvincing.\n- The claim of \"furnish additional supervised data\" seems not rigorous and might cause misunderstanding, , since you have not involved more external data. Maybe it is better to use \"supervision\" or \"supervised information\".", "methods_and_evaluation_criteria": "Strengths:\n- The proposed method is minimalist and scalable. The authors have properly provided the algorithm procedure.\n- The evaluation metrics are introduced in Section 6. And the detailed formulations are provided in the appendix.", "theoretical_claims": "Weaknesses:\n- Some related works in MLL with partitioning of the label space provided theoretical justification. However, there is no theoretical support for the proposed method.", "experimental_designs_or_analyses": "Strengths:\n- The experiments demonstrate the effectiveness of the proposed method on multiple datasets.\n- The authors have provided proper ablation study and parameter sensitivity analysis.\n\nWeaknesses:\n- The performance gains on some datasets are marginal (less than 0.01). Is there any explanation?", "relation_to_broader_scientific_literature": "Strengths:\n- This paper is partially related to multi-label learning. The authors have provided proper explanations and discussions.", "essential_references_not_discussed": "Weaknesses:\n- It seems that the references are a bit out-of-date. Most references are from or earlier than 2023. Is there more recent related work?", "other_strengths_and_weaknesses": "I have no extra concerns. Please see the above comments.", "other_comments_or_suggestions": "Please see above.", "questions_for_authors": "Please see above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zLhNzzT0kB", "reviewer": "Reviewer_BGkH", "summary": "The authors introduce S-LDL, a novel label distribution learning (LDL) method that generates/utilizes label distribution subtasks. This method can seamlessly integrate with existing LDL methods without any prior/expert knowledge, and is suitable for some derived tasks. The paper also conducts analyses and experiments to demonstrate that S-LDL is effective and efficient.\n## update after rebuttal\nI have carefully read the rebuttal. The rebuttal provided a detailed explanation of the role of downstream tasks and clarified the three assumed conditions in Theorem 4.4, which addressed some of my concerns. Therefore, I maintain my vote for \"accept\".", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your precious comments! Responses to your concerns are as follows.\n\n**Comment 1:** There are limited performance improvements in some cases, particularly with the Yeast_ series datasets.\n\n**Response:** For LDL, even small changes in metrics can indicate significant performance improvements. This is because the values involved in the calculation are constrained by probability simplex, which limits the range of results and leads to small divergence. For example, on the $\\mathtt{Yeast\\\\_diau}$ dataset, the K-L divergence between the ground truth and a uniform label distribution matrix (values are all $\\frac{1}{L}$, where $L$ is the number of labels) is **0.0158**. This can be considered as the worst performance. DF-LDL, a strong baseline, achieves **0.0131**. To ensure the reliability of the experimental results, we repeat the experiments and provide the standard deviation of the results, and perform significance tests (pairwise *t*-test at 0.05 significance level) to confirm that the observed changes are statistically significant.\n\n**Comment 2:** There are some non-intuitive parts of the analysis. The notation $[\\cdot]$ is used to denote the set of natural numbers, but $[\\cdot]_i$ is used to represent the element at the $i$-th position of a vector, which could be confusing for readers sometimes.\n\n**Response:** We appreciate this observation. To avoid ambiguity, we will revise the notation by using parentheses for indexing (i.e., $(\\cdot)_i$ for the $i$-th element of a vector) and reserve square brackets exclusively for set definitions.\n\n**Comment 3:** Regarding Section 4.2, what benefits does the reconstructability of subtask label distribution bring?\n\n**Response:** The purpose of **Section 4.2** is to demonstrate that, under certain conditions, the subtask label distributions can indeed reconstruct the primary task label distribution. In **Section 5**, we concatenate the subtask label distribution with the representation to serve as input to a label distribution estimator, simulating this reconstruction process. In essence, reconstructability guarantees that *subtasks are not just arbitrary auxiliary objectives but are structurally aligned with the primary LDL problem*. Therefore, it is crucial for the overall validity of the $\\mathcal{S}$-LDL framework to prove that the subtask label distributions can yield the original distribution.\n\n**Comment 4:** Regarding Theorem 4.4, why there are three conditions? More explanations would be helpful.\n\n**Response:** Below, we provide explanation of the conditions:\n\n+ $\\mathcal{G}$ is connected: This ensures that the proportions between all description degrees of the primary label distribution are known. If $\\mathcal{G}$ is not connected, the proportions of description degrees corresponding to labels on different connected components would remain unknown.\n+ $\\mathcal{G}$ covers all labels in the label space: The union of the subtask label spaces must encompass the entire primary label space. If any labels are not included in any subtask label space, the description degrees for those labels would be unknown. Hence, this is a necessary condition.\n+ The corresponding description degrees of all cut vertices of $\\mathcal{G}$ are not zero: When there are cut vertices in $\\mathcal{G}$, the non-zero description degrees corresponding to all the cut vertices are necessary to keep the proportions known.", "claims_and_evidence": "Yes. The analyses and experiments in this paper show that the additional beneficial data claimed by the authors can indeed be mined from subtasks and improve the performance of the LDL method.", "methods_and_evaluation_criteria": "Yes. The proposed methods and/or evaluation criteria make sense for the problem at hand.", "theoretical_claims": "Yes. The correctness of the proofs for the theoretical claims is checked, and they appear to be correct without any significant issues.", "experimental_designs_or_analyses": "Yes. The experimental designs and analyses are sound and valid. I have doubts about the analysis part. See the question below for details.", "relation_to_broader_scientific_literature": "The idea of using subtasks in the field of LDL is novel, and the proposed S-LDL is compatible with existing LDL methods. This paper provides new insights into how subtasks can be leveraged to refine label distribution predictions, contributing to the ongoing evolution of LDL methodologies.", "essential_references_not_discussed": "No. This paper adequately discusses related work.", "other_strengths_and_weaknesses": "Strengths:\n1. This paper proposes an LDL method that generates/utilizes label distribution subtasks, and this idea seems novel.\n2. There are some good properties of the proposed method: independence from expert/prior knowledge, compatibility with existing methods, and applicability to derived tasks, etc.\n3. There are creative analyses of the proposed method, illustrating the effectiveness of the subtask label spaces/distributions.\n4.  There are sufficient experiments to demonstrate the performance of the proposed method, and the results seem promising.\n\nWeaknesses:\n1. There are limited performance improvements in some cases, particularly with the Yeast_ series datasets.\n 2. There are some non-intuitive parts of the analysis. See the questions below for details.", "other_comments_or_suggestions": "The notation $[\\cdot]$ is used to denote the set of natural numbers, but $[\\cdot]_i$ is used to represent the element at the $i$-th position of a vector, which could be confusing for readers sometimes.", "questions_for_authors": "1. Regarding Section 4.2, what benefits does the reconstructability of subtask label distribution bring?\n2. Regarding Theorem 4.4, why there are three conditions? More explanations would be helpful.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GTXPUI6Hcv", "reviewer": "Reviewer_yDqw", "summary": "The paper proposes a new plug-in method for label distribution learning, based on auxiliary tasks derived from the original dataset label distribution. The subtasks are defined by a label mask optimized before the optimization procedure, encouraging diversity between the subtasks and alignment with the original label distribution.\n\nUPDATE AFTER REBUTTAL:\nI keep my recommendance of acceptance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your precious comments! We have provided point-by-point responses to your questions below.\n\n**Comment 1:** Why do the authors define the diversity as in Definition 4.3, instead of the cosine similarity used in Equation 1?\n\n**Response:** In **Definition 4.3**, we use $\\bar{\\boldsymbol{d}}$ to assign weights to each subtask pair, which can make the two subtask label spaces with different high-frequency labels considered more differentiated. This helps *evaluate* diverse subtask label distributions. In **Equation (1)**, we use cosine similarity instead for *optimization* purposes, since element-wise XOR operator is a discrete, non-continuous operation. We will clarify this distinction in the manuscript to improve readability.\n\n**Comment 2:** Some steps in Theorem 4.4 could be made more clear. For example, the proof relies on the assumption that $q$ is a normalization constant, but does not discuss what happens if this is not the case. Moreover, the steps following Equation 8 are not clear: what does it mean that $\\sum [p(d)]_{j}$ is \"given\"? Furthermore, there is implicitly a linearity assumption made to deduce that $p(v) = v$, hence more details on the derivation would be helpful.\n\n**Response:** Below, we provide a detailed explanation of the points raised:\n\n+ If $[q(\\cdot)]_j$ is not a constant for all $j$, each element has its own scaling factor, which is atypical and it is difficult to output a probability simplex.\n+ What we mean is that the term $\\sum_{j=1}^{L} [p(\\boldsymbol{d})]_{j}$ must be known *a priori* when solving for $d_k$ in Equation (8).\n+ The choice $[p(\\boldsymbol{v})]_j = v_j$ for all $j$ is the simplest nontrivial solution consistent with the problem constraints. We will explicitly call out the linearity assumption and justifies why it’s necessary (to avoid cross-dependencies and preserve normalization).\n\n**Comment 3:** The writing of the paper could be improved, for example the notation used is not always properly introduced (cf. Algorithm 1 where $j$ appears without proper definition, making it hard to understand the algorithm).\n\n**Response:** In this context, $j$ denotes the index of the label space. While we initially used a more compact pseudocode style for brevity, we acknowledge that this could lead to ambiguity. We will revise the algorithm to provide a more detailed and unambiguous formulation in the updated version of the paper.\n\nSpecifically, **Line 4** in **Algorithm 1** can be expanded into:\n\n> $\\mathcal{Y}^{(t)} \\leftarrow \\lbrace \\varnothing \\rbrace$;\n>\n> **for** $j=1$ **to** $L$ **do**\n>\n> &nbsp;&nbsp;&nbsp;&nbsp;**if** $M_{tj}=1$ **then**\n>\n> &nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;$\\mathcal{Y}^{(t)} \\leftarrow \\mathcal{Y}^{(t)} \\cap \\lbrace y_j \\rbrace$;\n>\n> &nbsp;&nbsp;&nbsp;&nbsp;**end if**\n>\n> **end for**\n\n**Line 9** in **Algorithm 1** can be expanded into:\n\n> $\\boldsymbol{D}^{(t)} \\leftarrow []$;\n>\n> **for** $j=1$ **to** $L$ **do**\n>\n> &nbsp;&nbsp;&nbsp;&nbsp;**if** $y_j \\in \\mathcal{Y}^{(t)}$ **then**\n>\n> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$\\boldsymbol{D}^{(t)} \\leftarrow [ \\boldsymbol{D}^{(t)} | \\text{clip}(\\boldsymbol{d}_{\\bullet j}, \\varepsilon, 1) ]$;\n>\n> &nbsp;&nbsp;&nbsp;&nbsp;**end if**\n>\n> **end for**\n\n**Comment 4:** Furthermore, there is no clear intuition or theoretical explanation for why subtasks should help with the LDL primary task, and why diversity and information should be maximized when designing good subtasks.\n\n**Response:** First, the reason *why subtasks help with the primary task* can be explained from the perspective of shared representation learning, which forces the model to learn more general and discriminative feature representations. These features may capture patterns that are beneficial to the primary task. Moreover, empirically speaking, simple techniques that are similar to our method (e.g., label powerset, dummy variable, label smoothing, etc.) have been demonstrated to achieve performance improvements. Second, the reason *why diversity and information should be maximized* is to ensure that each subtask contributes unique knowledge, avoiding trivial or overlapping learning signals. Also, diverse and informative subtasks can act as implicit regularization, preventing the model from relying too heavily on spurious label correlations in the primary task.", "claims_and_evidence": "Claim 1: The optimization procedure (Eq. 1) can find subtasks which are informative and diverse.\n\nThis claim is verified using the metrics defined in Definition 4.2 and Definition 4.3, and the empirical evidence is supported on the different datasets, comparing against random baseline. One question: why do the authors define the diversity as in Definition 4.3, instead of the cosine similarity used in Equation 1? \n\n\nClaim 2: The NSUM is the only normalization function which permits to reconstruct the original label distribution.\n\nThis claim is proved in Theorem 4.4, with ablations comparing it to a a min-max normalization.\n\nClaim 3: S-LDL leads to better performance\n\nThis claim is supported by empirical evidence in Section 6, with a sufficient number of baselines and metrics.", "methods_and_evaluation_criteria": "The datasets and evaluation metrics are borrowed from the label distribution litterature.", "theoretical_claims": "Some steps in Theorem 4.4 could be made more clear.\nFor example, the proof relies on the assumption that q is a normalization constant, but does not discuss what happens if this is not the case. Moreover, the steps following Equation 8 are not clear: what does it mean that \\sum [p(d)]_{j} is \"given\"? Furthermore, there is implicitly a linearity assumption made to deduce that p(v) = v, hence more details on the derivation would be helpful.", "experimental_designs_or_analyses": "Yes, the experimental section is quite comprehensive and covers a wide range of baselines", "relation_to_broader_scientific_literature": "This paper proposes a plug in which can be used on top of certain LDL methods.", "essential_references_not_discussed": "I am not familiar enough with the litterature to discuss this.", "other_strengths_and_weaknesses": "The writing of the paper could be improved, for example the notation used is not always properly introduced (cf. Algorithm 1 where $j$ appears without proper definition, making it hard to understand the algorithm).\n\nFurthermore, there is no clear intuition or theoretical explanation for why subtasks should help with the LDL primary task, and why diversity and information should be maximized when designing good subtasks.", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper proposes ​​$\mathcal{S}$​​-LDL, a novel framework for Label Distribution Learning (LDL) that reconciles the trade-off between generalizability and auxiliary task reliance by generating subtask label distributions without prior knowledge. S-LDL introduces two key components: (1) an algorithm to autonomously construct subtasks, and (2) a plug-and-play framework compatible with existing LDL methods and derivative tasks. Theoretical analysis validates subtask validity and reconstructability, while experiments demonstrate consistent improvements across multiple datasets and tasks.

The paper makes a ​​timely and practical contribution​​ to LDL by introducing a ​​generalizable subtask-learning framework​​ that does not rely on expert knowledge. There are some strengths mentioned by the reviewers: **(1) Novelty​​**: Reviewer BGkH noted S-LDL is the first to address LDL via subtasks, with a "novel" and "plug-and-play" design. **(2) ​​Empirical Validation​​:** All reviewers acknowledged the comprehensive experiments (10+ datasets, 4 tasks). **(3)​​Theoretical Rigor​​:** Reviewer yDqw initially found Theorem 4.4’s proof unclear, but the authors clarified the linearity assumption and normalization uniqueness, satisfying BGkH’s request.

While minor issues remain, the ​​rebuttal and revisions have adequately addressed concerns. Reviewer sNTH​​ initially questioned the distinction between LDL and standard classification and the authors provided detailed responses from different aspects. Reviewer sNTH expressed that he still has reservations about the LDL setting, but doesn't have time to thoroughly check its the details and doesn't object if the paper is accepted. Thus, I recommend ​​acceptance​​.
Decision: Accept (poster)