Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
Authors: Teng Huang
Bin-Bin Jia
Min-Ling Zhang
Venue: ICML 2025 poster (2025)
Keywords: Multi-Dimensional Classification
Class Dependency Modeling
Primary area: general_machine_learning->supervised_learning
TL;DR: We propose DCOM, a novel MDC approach that decouples dimension interactions. It identifies a latent factor to achieve partial conditional independence among class variables.  Experiments show DCOM outperforms state-of-the-art approaches..

Abstract: In multi-dimensional classification (MDC), the semantics of objects are characterized by multiple class variables from different dimensions. Existing MDC approaches focus on designing effective class dependency modeling strategies to enhance classification performance. However, the intercoupling of multiple class variables poses a significant challenge to the precise modeling of class dependencies.  In this paper, we make the first attempt towards escaping from class dependency modeling for addressing MDC problems. Accordingly, a novel MDC approach named DCOM is proposed by decoupling the interactions of different dimensions in MDC. Specifically, DCOM endeavors to identify a latent factor that encapsulates the most salient and critical feature information. This factor will facilitate partial conditional independence among class variables conditioned on both the original feature vector and the learned latent embedding. Once the conditional independence is established, classification models can be readily induced by employing simple neural networks on each dimension. Extensive experiments conducted on benchmark data sets demonstrate that DCOM outperforms other state-of-the-art MDC approaches.

Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
Teng Huang 12 Bin-Bin Jia
Min-Ling Zhang 12
Abstract
the analysis of multiple class spaces carrying distinct seman- tics aspects. For example, Figure 1 (Liu et al., 2016) illus- trates cloth image characterized by three dimensions in- cluding Texture, style and El Elast ilaaticityy each with corresponding possible labels. In reality, the necessity for learning from MDC objects is prevalent across a variety of real- real-world applications. including computer vision (Lian et al., 2020; Shi et al., 2025), text mining (Lertnattee & Theeramunkong, 2004; Serafino et al., 2015), ecological informatics (Dzeroski et al., 2000; Verma et al., 2021), etc.
In multi-dimensional classification (MDC), the semanties of objects are characterized by multiple class variables from different dimensions. Exist- ing MDC approaches focus designing effective class dependency modeling strategies to enhance classification performance. However, the inter- coupling of multiple class variables poses a sig- nificant challenge to the precise modeling of class dependencies. In this paper, we make the first attempt towards escaping from class dependency modeling for addressing MDC problems. Accord- ingly, a novel MDC approach named DCOM is proposed by decoupling the interactions of dif- ferent dimensions in MDC. Specifically, DCOM endeavors to identify a latent factor that encap- sulates the most salient and critical feature infor- mation. This factor will facilitate partial condi- tional independence among class variables condi- tioned on both the original feature vector and the learned latent embedding. Once the conditional independence is established, classification mod- els can be readily induced by employing simple neural networks on each dimension. Extensive experiments conducted on benchmark data sets demonstrate that DCOM outperforms other state- of-the-art MDC approaches.
Formally speaking, let X' = R2d be the input (feature) space and y = C1 x xC x Cg be the output space which corre- sponds to the Cartesian product of q class spaces. Each class consists of K possible class labels (1 S S q). In addition, denote the d-dimensional feature variable by X defined on X' and the associated g- dimensional class variable by Y = (Y,, Y2. Yg)) defined on V, where each component Y, is scalar class variable defined on class space Cj (1 I j S q). Given an MDC data set = { yr)l1 S i S m] consisting of m i.i.d. train- ing examples sampled from the Cartesian product of input space and output space X' x Y, the aim of discriminative MDC approaches is to estimate the conditional joint proba- bility Py (yo|or)) (n.) Py, Y2I..,,,|I (Vo1, yo2 ... Vog (a:) by training on the data set D.
space C, { (c], de........c
Specifically, for an unseen sample C., the predicted class vector ÿ. exactly arg max Pyxx (y, lr. x..). One straightforward strategy is to transform the multi-variant class vector yo into a scalar variable y((P) with a predefined bijective mapping function ò ) + 21. K₁ K2 K2}. In other words, each distinct class combination is regarded as a new class and multi-class classification algo- rithms can be utilized subsequently, which is known as class powerset (CP) (Read et al., 2014b). Then the targeted proba- bility transformed into PyJx (d(Wo1..... Moog xx.)) How- ever, the considered sample space is still the whole output space y, which contains tremendous number of elements (i.e., the cardinality of ), denoted by |}*] 11 K,) and
1. Introduction
In practical applications, objects are always characterized by diverse semantics across multiple dimensions. To achieve a more nuanced and comprehensive depiction of these ob- jects, multi-dimensional classification (MDC) focuses on
'School of Computer Science and Engineering, Southcast Uni- versity. Nanjing 210096, China Key Lab. of Computer Network and Information Integration (Southeast University), MOE, China College of Electrical and Information Engineering, Lanzhou Uni- versity of Technology, Lanzhou 730050, China. Correspondence to: Min-Ling Zhang <zhangml @ seu.edu.en>.
NNote that v. is a randomly sampled value of the feature vari- able X and æ; represents the i-th feature vector in the data set D. Similarly, yo, Vol...., yog are randomly sampled values of class variables Y, Vi...., Yg and y; represents the i-th class vector the data set D. Additionally, Vis denotes the j-th elements of y:-
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
Dimensions
Texture
Style
Elasticity
sions involved. To address this intricate class dependency modeling problem from dual perspective, we propose novel MDC approach named DCOM (i.e., DeCOupling muulti-dimessionll classification). Instead of modeling class dependencies directly, we attempt to identify a latent factor that encapsulates the most essential and critical feature in- formation, which enables partial conditional independence among class variables conditioned on both the original fea- ture variable and the learned latent variable. Furthermore, once the conditional independence is established, classifi- cation models can be induced easily by employing simple neural networks on each dimension. The main contributions of DCOM are summarized as follows:
florel
denim
graphic
chiffon cotton
striped
Light
Labels
embroidered
loose
leather
pleated
conventional
(solid)
faux
knit
lattice
Figure 1. An image example of multi-dimensional classification. Labels in each column on a blue background belong to the corre- sponding dimension the first row. The ground truth labels solid,
denim and right are in red circles.
We make a first attempt towards escaping from class de- pendency modeling in MDC. The proposed decoupling strategy can also be generalized to other multi-output learning paradigms, such as multi-task learning (Feng & Chen, 2023) and multi-label classification (Shi et al., 2024; Zhang & Zhang, 2024; Sun et al., 2024).
is prone to -lass-imaalance and overfitting problem. An- other natural strategy is to focus on each class space inddpen- dently so that the conditional joint probability Py (zo( transformed to q independent conditional marginal prob- ability PY, x x(Vo (z:)) (1 S j S q). known as binary rele- vance (BR) (Zhang et al., 2018). Although BR significantly reduces the cardinality of probability space to be measured, the class dependencies among dimensions are thoroughly ignored, which may result in suboptimal prediction per- formance. This highlights a critical need for exploring an accurate approach with theoretical interpretation to model class dependencies.
From probabilistic standpoint, we present an efficient approximation method. This method fosters the condi- tional independence among class variables conditioned on the informative latent factor and feature vector, ac- companied by a theoretical analysis for dealing with the vast sample space.
Accordingly, existing MDC approaches generally dedi- cate to appropriately modeling class dependencies as well as avoiding confronting the vast original output space y directly. On the one hand, some approaches opt to ac- count for local class dependencies among a subset of class variables, rather than the entire set, such as probabilistic graph-based approaches (Bielza et al., 2011; Gil-Begue et al., 2021; Nguyen et al., 2023), super-class modeling approaches (Read et al., 2014a) and pairwise dependency modeling approaches (Jia & Zhang, 2020; Huang et al., 2024). On the other hand, alternative approaches are de- signed to transform the original input or output space into novel representations that facilitate more implicit depen- dency modeling processes (Ma& & Chen, 2018; Wang al., 2020; Jia & Zhang, 2023).
Comprehensive experiments over seventeen bench- mark data sets demonstrate that DCOM outperforms other of-te-of-the-ar MDC approaches.
The rest of this paper is organized as follows. Firstly, re- lated works are briefly reviewed in Section 2. Secondly, the details of the proposed DCOM approach Section 3. Thirdly, experimental results of comparative stud- ies on benchmark dilei-dimensional data sets are reported in Section 4. Finally, Section 5 concludes this paper.
presented in
2. Related Work
Most existing MDC approaches dedicate to modeling the class dependencies well as avoiding confronting the huge original output space y directly. Amongst them, cbain- models (Zaragoza et al., 2011; Read et al., 2014b) disassembles the conditional joint distribu- tion into the product of q distributions by the chain rule of probability, i.c., Pr,, (Mo1, Yo2..... Yog (o))
However, modeling dependencies in MDC remains a signifi- cant challenge. Based on the above probabilistic analysis on BR and Cp, the hardness of modeling class dependencies stems from the typical intercoupling within multiple dimen- sions. This multi-dimensional intercoupling arises from the conditional dependency among class variables given the feature variable, namely the discrepancy between the condi- tional joint probability py (yo) (m)) ditional marginal probability II Sisis PyşIx (Hoy (x.). As a matter of fact, given an MDC instance, it is highly improba- ble that these two forms of probability are identical, unless there is a complete absence of correlation among dimen-
I[;-1 Py, .X.Y.....,,.11 (Voj ( Vol...... Mo(j-1)) and esti- mates each individual probability term with multi-class classifier. Such step-by-step strategy skillfully makes learn- ing on each classifier achievable by considering single class space on each step but leads to the propagation and ac- cumulation of errors (Read et al., 2014b). Probabilistic graph models (Bielza et al., 201 1; Gil-Begue et al., 2021;
(yo) and the product of con-
2

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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
Nguyen et al., 2023) seek for potential relationship among class variables by directed acyclie graphs (DAG). Thus the global joint distribution can be partitioned into some local joint distributions involving class variables and their respec- tive parent variables. Nevertheless, the computational com- plexity associated with determining DAG structure remains substantial. Dimension-specifie MDC approach (Huang et al.. 2024) transforms the original feature vector æ into dimension-specific feature vectors d6) corresponding to the j-th dimension. Then the modeling problem is transformed into PY, D(oojld" ( with each class space being treated as an individual sample space.
where O is the model parameters. Recall that for the j- th class variable y, there exist K2 possible class labels c/. C2...., .., Consequently, the cardinality of the proba- bility space is given by [ = I Kj, which is exceed- ingly large and poses a significant challenge for precise estimation. Considering the equivalence between maxi- mizing likelihood estimation (MLE) and minimizing the cross entropy loss, the loss item regarding classification in existing deep MDC approaches (Huang et al., 2024; Shi et al., 2025) is equivalent to the summation of conditional marginal log-likelihood function defined as follows:
172
Another category of MDC approaches aim to transform the original input or output space into new representations, thereby facilitating implicit dependency modeling processes. gMML (Ma& & Chen, 2018) decomposes the class spaces into binary-valued label space via one-vs-rest strategy and solves the resulting problem via a metric approach. LEFA (Wang et al., 2020) learns latent label embeddings based on attentional factorization machines to augment the original feature space. ADVAE. (Zhang et al., 2022) encodes both feature and class variables into probabilistic latent spaces by normalizing flows. DLEM (Jia & Zhang, 2023) enables modeling alignment in an encoded label space derived from one-vs-one decomposition and transforms the output space into a ternary encoded label space.
(1
PY, PY, Ix (Ui B0)),
1=
where O = 10 ((2) 0(0) is the set of model pa- rameters and @(i) represents the model parameter of the j-th dimension. Here, the predictive model of the j-th di- mension only considers the j-th class space Cj and the total cardinality of probability space is Σ}= Kj, which is much less than [[A= Kj. However, the equality of E1.(I) and (2) requires an implicit assumption on the partial? condi- tional independence among class variables conditioned on the feature variable, which is formulated as:
PrIx(Us(044) II Py.ix lx;),
However, no existing MDC approaches attempt to escape from the complicated dependency modeling problem. In the next section, we will elaborate the technical details of the proposed DCOM approach, which seeks for partial condi- tional independence based on the instance-based conditional mutual information.
where E f1,2....,m)) m). Unfortunately in the context of MDC, this strong assumption is rarely tenable. To facil- itate the validity of the partial conditional independence and thereby reduce the cardinality of the probability space, DCOM seeks to identify a high-level informative latent factor denoted by variable Z that enhances the original and basic feature information. Specifically, we assume that the prior over the latent variable distribution is the centered isotropic multivariate Gaussian (z.))) (O, I). We then employ simple encoding network G to derive the latent vector Zo corresponding to each feature vector v.. Nevertheless, such direct mapping from X' is incapable of altering the conditional dependence since the conditional probability p( (x:)) (z,) is inherently describing a deterministic event. To address this issue, we introduce a minor perturbation on v. prior to applying the encoding network G. Consequently, the latent vector can be computed follows:
3. The DCOM Approach
Notation. In this paper, we use capital letters to denote random variables (e.g., Y, X, Z), lowercase bold letters to denote vectors (e.g.. D2) and lowercase non-bold letters to denote scalars (e.g., Moj.
Moj.Wty). Way). A detailed summary of notations can be found in Table 8 of Appendix C.2.
3.1. Latent Factor Introduction
Given the MDC training data set D = (a,, y:)l1 < i m} consisting of m i..dd. training examples sampled from X' x Y. The conditional log-likelihood function in terms of class variables conditioned on the feature variable can be given as follows:
Zo
(
(4)
where ã. = x. + e and e is a random noise vector. Further- more, in order to comprchensively assess the influence of the
m
Lo (0o) = D log 0o)
We use "partial" here because the assumption does not require (3) to hold universally for all possible values of Y and X. but rather only under the observed training distribution (which is finite and empirically sampled).
O Viglo;;;01 U. Qu), (1)
3

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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
latent variable Z both feature variable X and class vari- able Y, we extend the original conditional log- oigllikeihoodd function conditioned on X (i.e., Eq.(I)) to conditional log-like ligliikeihood function conditioned on Z as follows:
mutual information is defined as:
I( Y2,,Y,,.. ................................ yg|X, 23
KL
p(it) log plit)
Σ
plpa) dzo
((
=
Yo Ey o 2o no
Σ
Ilyo: Doo 2o 20)dzo
(8)
1ogpy1x.2 0a)
ye
+ (0
(5)
For convenience, the above integrand has been denoted as:
púit) (yo, To. Ro)
Lo, Ro = (066) (yoo (yy: Co: 2o) log p(pd) yoo Bos 20)
where θ = (0, is the set of model parameters. Og and 0, correspond the parameters of the discriminant component and reconstruction component, respectively.
Given the non-negativeness of KL divergence, it is evident that the condition wherein (q.(8) equals 0 constitutes a sufficient but not necessary condition for the satisfaction of (7) since a multitude of possible values of the feature and latent variable are intractable and not considered in (7) Therefore, for a given MDC data set, we employ an Instance-based conditional mutual information defined as follows:
3.2. Conditional Independence Achievement
Although the challenge posed by the vast probability space associated with the first term in (5) persists, the introduc- tion of the latent variable Z offers the potential for achieving the partial conditional independence among class variables conditioned on the feature variable X as well as the latent variable Z. The desired partial conditional independence is formulated as
Definition 3.2. For a given MDC data set D {( y,)l S S m), the instance-based conditional mu- tual information is defined as:
172
(1) xi) =
Py, x 2 2;),, II 1=1
ID(Y,, ... X,I,,,,
Σ
Ilyo, 2.,
(9)
B.EY
In this paper, we show a feasible and efficient way to es- timate the instance-based conditional mutual information with Theorem 3.3.
where i E (1,2....,m)} m]]. For notational concise- ness, we further denote the joint probability of [yo:
all variables py,x.z(yo. xo. by p Ro, 2o) and the conditional marginal probability multi- plicd by the joint probability of feature and latent variables I Px, (Vo, 2o) (Fo. 2o) as p(Dd) To, 2.).² Then (6) is equivalent to:
Theorem 3.3. If the joint probability of class vari- ables py (Uo1, Vo2,.... .oo)) is small enough, i.e., 0 S
py (No1, Vo2...., & 1, for 48 E R s.t. 0 < 8 < (g)) To. 20), then we have I[yo, Do, 20) - 0 under Assumption 3.4.
(it) (2r, zi) ( (yd, 21).
Assumption 3.4. Given 4 E R. s.t. 0 05S5
(y), Do. zo). then the joint probability (yo, Co 20) and the product of conditional marginal probability
p(nd) (yo Co, 2o) satisfy that plit) Lo, 2o) 8 S
To provide a theoretical support, we define the distance between the two sides of (q. i.e., p(it) and (Pd), in- spired by Kullback-Leibler (KL) divergence and conditional mutual information as follows:
p(nd) (Yo, Lo. zo).
As a matter of fact, the joint probability p p(tt) (No, To. 2o, 2o)
fa, J=, p(it) (yo, To. Zo )dz, I (y Ylgg) when the condition of Theorem 3.3 is satisfied. Conse- quently, fiit) 5 is also small probability that tends to 0. This observation suggests that the Assumption 3.4 is highly likely to be valid. Theorem 3.3 provides an efficient strategy to reduce the practical size of probability space considering a specific MDC data set, which shows that when the prior
Definition 3.1. Given g class variables Y1, Y2...., Yg de- fined on y = C xC2 x... C.p, feature variable X defined on X and a latent variable Z I), the conditional
'henn no ambiguity arises, we abbreviate p (it)) (yo; Ro Zo) as p (t) and p (rd) (yo, Ro. 20) as (id))

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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
joint probability of class variables py (1o1).... )oog) is small enough, the corresponding summand in (q. can be disre- garded. We defer the proof of Theorem 3.3 to Appendix A. Moreover, this prior which only regards to class variables can be computed easily based on the m class vectors of the i..d. MDC data set. Specifically, given an MDC data set D- = {(a,; y; S i S m}. we can approximate the instance-based conditional mutual information as follows:
where Ç denotes the soft-max function and the subscript and b denote the a-th and b-th element of the corresponding output vectors, respectively.
For the second term of Eq1(1 regarding the reconstruction component, we assume Px|z (a., (2,) is a multivariate Gaus- sian with a diagonal covariance structure N Mi,, oi Here, μi and or are the d' d-dimensional mean and the standard de- viation vectors, which are the output of a reconstruction network R. Then for the i-th training example, the logarith- mic posterior probability can be computed as:
(Y,Y,, Y2,......... 2) = Σ
Σ 1=1 VoED
Ilyo: Li Zi) (10)
Here, D' is defined as
d log 2n
1 Σ
D' #y
c, yo D}.
logpx12(0:(22) =
-2
2
log o
m
d
where #ys indicates the count of class vectors in data set D that are equal to Yo and cs a small constant that approaches O. The transformation on the sample space of class variables considered from Y/ into D' makes the instance-base -aased condi- tional independence computable. In real-world MDC data sets, the cardinality of D' is typically deemed acceptable since the inherent class dependencies across dimensions necessitate the presence of highly correlated class vectors. Now, the intractable conditional joint log-likelihoond (i.e., E5)( can be transformed into an accessible form with the minimization of (10) as follows:
Fia uiaa)
(13)
2
a=1 Va
where d and Zia denotes the dimensionality and the a-th element of I; respectively. To simplify the model, it is common to assume that the variances {o²a|1 I a S d} are identical and constant. Under this assumption, the recon- struction network R only needs to output the mean param- eters μ, Then the reconstruction loss can be computed as follows:
[()) =
Σ
m
log logPr,1x.2(00/zz 2;;03" )
Lre
(Tia
Hia) )2
(14)
2
i=1
1=1
a=1
+ log px (x,
(11)
For the approximated instance-based conditional mutual in- formation, i.e., Eq.(, need to estimate p(it) and p(pd), According to the chain rule of probability, pfit) can be de- composed as pz (2(zz)))p)))) (z.))xzz (z; (z; ( z;), where the first term can be computed as p2(z;) = N(2 0,I) following the normal distribution and the second term (x)z(xx|22)) can be computed as (q3) without the log- arithmic function. As for the last term pY|x,2 (Vo/ /2; z;), we employ a neural network T with [D| output nodes to estimate this probability with soft-max function as (12).
where ê = (0(1) 0 8 0,] is the set of model
parameters. o and 0, correspond the parameters of the discriminant component for the j-th dimension (1 I j S q) and the reconstruction component, respectively. The detailed construction of loss functions based on (11) will be discussed in the next section.
3.3. Loss Function Construction
For the first term of (11( regarding the discriminant component consisting of the conditional probabilities (Pv) zo) < j S q), we utilize g independent neural networks (Hr S j S 4) to model the correspond- ing conditional probability. Specifically, considering the i-th training example, for the j-th dimension, the input of H2 will be set as the concatenation of v; and 2; and the output of H, is transformed to normalized probabilities as follows:
Then the classification loss regarding classification networks
(Hl 1 < j < 4} and T can be computed as the cross-
entropy loss, formulated as:
m Σ
Lee =
log (ogpy1x,2(111e)) 21)
q +
log logpy, ( 22 1xx x1)] (15)
P9,1X.2(13) = eel zi)
ISHH, (1E,, 211))la
Similarly, each term in p(rd)) (y., 2;, 2;) can be estimated by existing neural networks. Therefore, the minimization of (Q.(10) can be regarded as the minimization of the following
exp( M4 [2,, 21))aa)
(12)
Ed, exp(Ht, (1:;, =i])6)
5

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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
loss function Lei, formulated
et al., 2013; 2014),² DeepFashion (Liu al., 2016) and SEWA (Kossaifi et al., 2021).? Table summarizes basic characteristics, including the number of examples (#Exam.), the number of dimensions (#Dim.), the number of labels in each dimension (#1 abels//mmmm and the number of features (#Feat.).
m
||00;
11,112+
||
Les
exp
2
1=1
Σ Yo ED
Prix (3)
4.1.2. EVALUATION METRICS
Pr 33 Dz. xi) -1 PY, x.2
log
(16)
(Noz (oo Bis zi)
In this paper, we adopt three commonly used metries for performance evaluation, i.c. hamming score (HS), exact match (EM) and sub-exact match (SEM) (Read et al., 2014a; Zhu et al., 2016). Given the test set S {( y:) S s p} and the MDC model f to be evaluated, the definitions of these three evaluation metrics are given as follows:
The overall loss function L is the combination of loss func- tions defined above:
C 1 (Coe afre + BCci),
(17)
m
HSs(f)
1()
where a and 9 are trade-off parameters.
Table Basic information for data sets, Here, n, x and image in last column represent numeric, nominal type and unstructured features.
EMs(f)
Σ
(r
p i=1
1 P
SEMs(f)
Σ
I O 2 1].
data set
#Exam.
#Dim.
#Labels/Dim. #Feat.
p
Flarel
323
3
3,4,2
10x
Ocs97
334
16
3
263n
Here, (()) Ii ú.y denotes the number of
Jura
359
2
4,5
9n
dimensions which are predicted correctly, where Vij and ūi denote the ground-truth and predicted label w.r.t. the j-th dimension for the i-th test sample æ, respectively. [*] returns if x holds and otherwise.
Ocs10
403
16
3
298n
Enb
768
2
2,4
6n
Song
785
3
98n
BeLaE
1930
5
1n, 44x
Voice
3136
2
4.2
19n
4.1.3. COMPARED APPROACHES
Thyroid
9172
7
5,5,3,2,444,33
7n, 22x
In this paper, the proposed DCOM approach is compared with seven sfate-of-the-ar MDC approaches including BR, Cp, DLEM, EDCC, LEFA, PIST and SLEM. Specifically, BR independently learns a multi-class classifier for each dimen- sion, while Cp learns multi-class classifier via treating each distinct label combination as new label. DLEM (Jia & Zhang, 2023) solves the MDC problem by enabling mod- eling alignment for MDC in encoded label space derived from one-vs-one (OvO) decomposition. EDCC (Jia & Zhang, 2022) builds chain of binary classifiers and augments the feature space by predictions generated by preceding clas- sifiers. LEFA (Wang al., 2020) introduces a cross corre- lation aware network learn latent label embeddings and augments the original feature space by the learned label
CoIL2000
9822
5
6,10,10,4,2
81x
TIC2000
9822
3
6,4,2
83r
Flickr
12198
5
3,4,3,4.4
1536n
Adult
18419
4
7,7,5,2
5n, 5x
Default
28779
2,7,4,2
14n, 6a
BP4D
16037
6,6,6,6,6,2,88
image image
DeepFashion
20000
6
7,3,3.4,6,3
SEWA
19275
3
21,20,19
image
4. Experiments
4.1. Experimental Setting
4.1.1. DATA SETS
'nttps: www. www.biinnnn....... adu/-1ijum Besearch/3DFE/ http///FEeaaaaaaaaaaaeeaaaaaa Analysis html. See more detailed descriptions in Appendix B.
In this paper, we use seventeen real-world MDC data sets for experimental studies, including fourteen structured data sets and three unstructured data sets: BP4D (Zhang
'nttps: //wpla///b////////// ic cuhk edu. hk/pproeets//
watrraasioon/ae DeepFashion/Attee
ctionn html. See more
detailed descriptions in Appendix B.
httpp:: httpp/aa.mmmm.mmmm... seu, edu. en/ edw.cc/zhanggm// //nnggga/ Resourcee rces. htm#MDC_datt
7https: ///bb. httpa///...swwwoeeeet....... sewaproject. .wrrojecctee/....
See more detailed
descriptions Appendix
6

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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
Table Experimental results (scan=sstd.) of each MDC approach in Hamming Score. In addition, fo indicates whether DCOM
is significandl superior/inferior to other compared approaches on each data set with pairwise t-test 0.05 significance level.
Data Set
DCOM
BR
CP
DLEM
EDCC
LEFA
PIST
SLEM
Flare1
920+.0033
923+--031
881=
-897=.035
.923+-035
.113-.004
.923+.031
.645+.262
Ocs97
7799.022
.6071.031 .6861.026e
.738=.026
.747:.021
.730:.022
.737=.023
.659=.016*
Jura
.7404.050 .586::.065
.675+.011 .720=.057
.6161.055
.542:t.05
.6021.004e
.695-.066.
Ocs 10
8131.013 .61.1001...644
.7662.018+
805=.015.
.7972.018
.8101.008
.8062.012
.758=.022
Enb
9699..01 711::.028+
.98==.022 .935=.023+
.792+.033.
.701#.047
.773#.041
.769=.082
Song
.7881.021 .71:1.024e
7881.021,.71111022222
.922...3.
789
.7931.026
.753:t.03*
.7841.022
.744=.05|e
BeLaE
4511.019 4233..021
.357=.019
4491..066
4101.012e .4522.015
.341=.012+
Voice
9711.005 .9401.009
938
.509..099
.3221.115
.9441.08
.33
Thyroid
971 961=.002 .968 09w.:00.........0000 9632.0002 .60 .960=.003+ 9000.0000..900..003 .939=.061
Co1L
.00.0000... 007.--9-0---- 0744 -90--900+----- .938--000......0000033000000 .938+-0033
.949+0099
.957=-004+
-000 .006
TIC2000
0115:.004..9221.0000
89224.007 .875=.006+ .885-.805. 00020000000001110000008001011000 999-1.005 0000
.9321.006
.945..004
.23
Flickr
.7981.005 715:1.005 .6751.006+ .355..005. - 000.55....00500011000000 80011.004
.7481.007 .7951.003
.755
Adult
.728=.004 7 .03.=.00. w7w.:.699.0002000.. .733+ .0w.:/....56...000 .07
.735=.003+
.697
Default
.677+.003
663+.003
0007 6611 0alfffe62225-000 .255
(0..663++0000
6676+.003 .632+.013
Table 3. Experimental results (mcantsstd.) of cach MDC approach in terms of Exact Match. In addition, e indicates whether DCOM significantly superior/inferior to other compared approaches on each data set with pairwise t-test at 0.05 significance level.
Data Set
DCOM
BR
Cp
DLEM
EDCC
LEFA
PIST
SLEM
Flarel
8114.076
821+.069
.718+.071
7622+..74
8184-076
.805+.074
.821..069
.410-t.18
Oes97
.069:.052
030:1.027 .030=.030 .060=.046
.057:.045
.033:.036.
.0602.048
.018:1.031
Jura
5544:.054
.329:.104e .512+.052 .540=.076
.382:t.99.
.315..041
.359:.096
.549=.062
Oes10
.097:.046 ,064:.033.
099.:.04..044..033........
.077:.042
.094:.043
.099:.003
.084+.037
.102#.038
.062:t.34
Enb
939+.002 .311+0055
9394.033)431+.0555.e..
.95+..024
.870=.045+
5584+.067
.402+.094e
.546+.083+
.611+.091
Song
.880:.054 4491..077
.43::.04e
.478=.059
4933.049
.29:.071
.467:.047
.437=.065
BeLaE
1336:.013).0281...000 .013=.009
0077=0033 +
.033:.012
.017:.008
.035#.019
.006=.004
Voice
00211.010.8842..1166
.9022.017
.8211.021
.902..06+
.28
Thyroid
.815..012 815+:.012.443::.0114
.77::.0144
.737=.019
.738=.016+
+.88
CoIL2000
.8111.011 515:..01e 0141:00lee.662200013.66000.1..10 044.
747:1015
.7861.036
.8221.014
1.09
TIC2000
.8422.011 11921..01,,992..1188...666
6985:.018..6655.000006 .6632.610+ 000.0000...88.00000 .888=.008.........
.7991.014e
.819.001+
.843.0013
.723=.034e
Flickr
.328+.013 .187+.010 wtl.e1158000005 11F+0000...181100002226661110111 20..2222
0332 .332+:005
.246+.010
.300.001
2247
Adult
.303..009 228::.00e 200.2.000200000000 20.-2299..0
00000...00...229000000 2881+.008
2022
2888.006e
.288=.013
Default
.199+005
124 1774400000
177
181
007 .185+ 018
134+
1955
148 -012
embeddings. Multi-class algorithms are used for subsequent classification. PIST (Huang et al., 2024) learns pairwise dimension-specifie features to consider both the specific characteristics in each dimension and class dependencies among different dimensions. SLEM (Jia & Zhang, 2021) learns a multi-output regression model within an encoded label space by considering the sparse property. For all com- pared approaches, the recommended parameter setting in the corresponding literatures are used.
sionality of latent variable set as 512. the overall loss function, i.e., (17() the trade-off parameter and 33 are both set as 1 (please refer detailed discussions param- eter sensitivities in Section 4.3.2). All activation functions are fixed as ReLU followed by dropout layer (Srivastava et al., 2014) with dropping probability of 0.5. For network optimization, we utilize SGD with a batch size of 512, mo- mentum of 0.9 and weight decay 10-1, We only adopt experimental results of the last epoch and the number of epoch is uniformly set 500 for all data sets. Moreover, for the three unstructured MDC data sets, ResNet-18 (He et al., 2016) 6) pretrained on ImageNet dataset (Deng et al., 2009) is implemented as the feature extractor for the proposed approach as well as all compared approaches.
4.1.4. 1. IMPLEMENTATION DETAILS
For all neural networks G, R, T and (H, < q). we employ the multi-layer perceptron (MLP) with one hidden layer, configured with hidden dimension 512. The dimen-

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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
Table Experimental results tst.) of each MDC approach in terms of Sub- Eub-Exact Match. In addition, /o indicates whether DCOM
is significandl superior/inferior to other compared approaches on each data set with pairwise t-test at 0.05 significance level.
Data Set
DCOM
BR
CP
DLEM
EDCC
LEFA
PIST
SLEM
Flare1
954
951++033
.935+.032e .938=.042
.954+-037
.941+.038
.951=.035
.658-t.332
Ocs97
.1232.088
007::1004 1123:.008..72::.0448.
.075=.045
.111=.064
.123:.072
.0901.062e
.102:2.059
.039=..08
Jura
.925..07 84441.056e
0255..077.444..556
..0
.900=.081
.850:t.041
.769:t.104
.8441..04e
.841=.075
Ocs 10
2218:.00011191.06666
.139=.048
.191=.070
.201
.216:.046
.208=.058
.152.t
Enb
1.00#.000 1.00:1.000
1.00=.000
1.00=.000
1.00+.000
1.00++000
1.00=.000
.927=.093+
Song
.8881.034 .88::..030 .633...44...
.878=.038
.8871.045
.841:.0555
.888..036
.818:.066
BeLaE
1111.007 1551:.02711222.0022 11....2233322..
.070=.021 .136=.030
.157:.021
.1172.017
.160=.024
.068=.014
Voice
1.002.000 -99.99999.... .03
.9991.002
.9771.003+
992.2.01
.97=.003+
.930=.033
Thyroid
.9..00004993::.444 000 000.77000004...33110000
.982#0004
.982=0004
.896=.213
Coll
9699.007
.948
.93++0007
.966+0006
-903+.009
TIC2000
09931.003,999-1.0044
000.0010.0000011110.000000000000 0.6. .003
9894.004
.993=0002
.959=.030
Flickr
.729:.015 543:L.015e.483:1.0100 .83:2.100 (0..60000......... .377H.011 737:10011 040.000445000.000000000000000011
.2771.021
.723+.009
.600
Adult
.689=.007 ..77:..09. .532=.010 ww..6.100000.....
0tt.::000...............0....... 606
.555+.011
.693.0070
.09
Default
.612+.005 590+.008 446+.440. 586-
604 5188
.610=.007 .531
Figure 5, Figure 6 and Figure 7 in Appendix C.4 presents the t-SNE visualizations (van der Maaten Hinton, 2008) for the original features and latent factors in the first fold of dataset Voice, TIC2000 and Flickr w.r.t. the first dimension respectively, which shows the learned latent features capture more compact manifold representations than the original features. According to the reported experimental results, we can make some observations as follows:
Table 5. Experimental results of each MDC approach on the three unstructured MDC data sets (i.e., BP4D, DeepFashion, and SEWA) with best results shown in boldface.
(a) Hamming Score
Data DCO
BR
Cp DLEM EDCC LEFA
PIST SL.MM
BP4D
.785
.755 6044
.744
.754
.649
.728
.547
DeepFa.
.785
.771
.749
.764
.774
.782
.780
.713
SEWA
.580
.537 .416
.474 .559
.448
.435
.617
(b) Exact Match
For structure data sets, DCOM significantly outper- forms the seven compared approaches in 79.6%, 73.5% and 60.2% cases in terms of the three evaluation met- ric, respectively. For unstructured data sets, DCOM surpasses all compared approaches on data set BP4D and DeepFashion and ranks only second to SLEM on data set SEWA.
Data DeM
BR
Cp DLEM EDCC
LEFA PIST SLEM
BP4D
.274
219.195
.246
.231
.117
.186
.135
DeepFa.
.285
249
.27
.242
.266
.26
.262
.166
SEWA
.290
.211 .22
.199 .665
.178
.121
.369
(c) Sub-Exact Match
Data DCOM
BR
Cp DLEM EDCC LEFA PIST
SLEM
There are only cases where DCOM achieves inferior performance against all the other compared approaches, Two of these cases occur data Enb when com- pared with Cp. This suggests that the two dimensions of Enb are strongly correlated, as evidenced by the superior performance of CP in accurately learning the possible class combinations directly. Consequently, it is more appropriate to study the entire probability space for data set Enb.
BP4D
.537
.458 .355
.458
.479
.299
.422
.244
DeepFa.
.616
.911
.548
.568
.592
.611
.616
.478
SEWA
.590
540.393
.464
.563
.428
.422
.630
4.2. Experimental Results
We report the detailed experimental results in Table 2, Ta- ble 3. Table and Table 5. For structured datasets, ten- fold cross validation are conducted where the mean metric value as well the standard derivation recorded for comparison. Furthermore, pairwise t-test (Demšar, 2006) at 0.05 significance level is conducted to show whether DCOM achieves significantly superior/inferior performance against compared approaches. Accordingly, the resulting win/tie/loss counts are summarized in Table Moreover,
For data set SEWA, SLEM performs the best, likely due to its consideration on label sparsity. This is par- ticularly relevant given the number of classes in each class space of SEWA is exceptionally great (21, 20, 19 corresponding to each dimension, respectively).
8

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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
Table 6. Win/tic/loss counts of pairwise t-test (at 0.05 significance level) between DCOM and each compared approach.
Evalu.
DCOM against
Metric
BR
Cp
DLEM
EDCC
LEFA
PIST
SLEM
In Total
HS
13/1/0
13/0/1
11/3/0
9/5/0
12/2/0
7/7/0
13/1/0
78/19/1
EM
12/2/0
12/1/1
10/4/0
8/6/0
12/2/0
7/7/0
11/3/0
72/25/1
SEM
10/4/0
12/2/0
7/7/0
5/9/0
10/4/0
3/10/1
12/2/0
59/38/1
In Total
35/7/0 37/3/2
28/14/0
22/20/0
34/8/0
17/24/1
36/6/0
209/82/3
and Exact Match, a strategy that accounts for both condi- tional marginal and joint probability is preferable.
Table Summary of the Wilcoxon signed-ranks test DCOM against its variants in terms of each evaluation metric at 0.05 significance level. The p-values are shown the brackets.
4.3.2. PARAMETER SENSITIVITY
DCOM against
HS
EM
SEM
We show how the the performance of DCOM fluctuates with different dimensionality of latent variable Z, different values of trade-off parameter a and / in Figure 2, Figure 3 and Figure 4 of Appendix C.I, respectively. It is demonstrated that DCOM achieves relatively stable performance as the three hyperparameters vary within a broad range. However, it is recommended that the dimensionality of latent variable Z should not be set too low, considering its necessity to carry sufficiently rich feature information.
DEVI
www.::0002winn17009eee.0 tie[5.52e-02]
DEV2
winl.077-00//inn3336--03 win[1.07e-04]
4.3. Further Analysis
4.3.1. 1. ABLATION STUDIES
In this section, we further compare the performance of DCOM with its two degenerated versions on all the seven- teen MDC benchmark data sets. The two variants, denoted as DEVI and D2 represent two possible model config- urations when discarding the proposed class dependency escaping strategy, i.e. the minimization of (16).
5. Conclusion
In this paper, we propose a novel MDC approach DCOM as a first attempt towards escaping from class dependency modeling. This approach provides effective method to estimate the discrepancy between the conditional joint prob- ability and the product of conditional marginal probabili- ties. The effectiveness of the minimization of the proposed instance- conditional mutual information is rigorously validated via comprehensive experiments conducted on sev- enteen real-world data sets.
DEV1. Predictions are derived solely from the condi- tional marginal probability. In other words, the first term of Lee (15) and Las (Eq,(16)) are discarded.
DEV2. Predictions are derived solely from the con- ditional joint probability. In other words, the second term of Lee (15) and Lai (16) are discarded. However, the considered sample space remains D' as defined in (10) given that the cardinality of original output space y is excessively large.
A Acknowledgements
The authors wish to thank the anonymous reviewers for their helpful comments and suggestions. This work was supported by the National Science Foundation of China (62225602, 62306131), the SEU Innovation Capability En- hancement Plan for Doctoral Students (CXIH SEU 25134) and the Big Data Computing Center of Southeast University.
We conduct Wilcoxon signed-ranks test (Demšar, 2006; Jia et al., 2025) at significance level 0.05 to analyze whether DCOM performs statistically better than variant models. Ta- ble 7 summarizes the p-value statistics on each evaluation metric. Compared with these two variant models, we ob- serve that DCOM achieves statistically superior performance against them in terms of each metric. The only comparable case DEV1 in terms of Sub-Exace Match is notable. It indicates that classification approach relying exclusively on conditional marginal probability can achieve satisfactory performance when evaluated by metrics with relatively low demands. Nevertheless, when considering evaluation met- rics that require higher accuracy, such as Hamming Score
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.
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Towards Escaping from Class Dependency Modeling for Multi-Dimensional Classification
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Towards Escaping from Class Dependency Modeling for Multi-Dimensinnal Classification
A. Proof of Theorem 3.3
Theorem 3.3 shows that when the joint probability of class variables py (Not, Vo....... Moo) small enough, the absolute value of integrand [[yo To. 2o) in (q.(8) tends 0 under Assumption 3.4. Now we are at a position prove Theorem 3.3.
Proof. Assumption 3.4 indicates that ( Lo, 20) 8 S p(ld) wo 2o).
Consider cases:
When p p0t) yo: Lo. 20) s p(od)((1 (yo, 2or 20), we have
I[yo, Co. To,
Co. 20)1 - plit) log pladj
plit)
= p plit) log p(pd)
pbtj S ppit) log plit)
1
-p log logp00g (it).
(18)
And
lim -p log log (1)) = 0, p09-+0
(19)
With the condition that py (Vo1. Vo2..... Vog) + O. we have put)) (yo, Io. 20) = x,, (Yo,, Lo, 2o) Jep Jzo P 2 (Vol,.... Vog, Dio 2o) ddz = py (yol, Vo2, Bog). Therefore, [T(yo: Do 2o)l - 0.
(yo
When ((it((y Do, =o) 8 s p(od) (yo: ao. 2o) < plit) Lo, zo), we have
I(yo Vo. 2o. 20)1 =(tt) log plod) S plit) log plit)
p(it)
(20)
plit) 5
And
lim lim (())) log plit)
lim lim 8plit)
p0t) +5 5-+0 plit) 0.
(21)
p00 5- púit 8
Therefore, [I(yo: Ro: 20)1 + O.
B. Detailed Information about Data Sets
In this paper, all structured datasets with detailed descriptions are available in z /eeeml/leessssssseeeeee /tt.e/oorrreeeeettttteteeeeeddee htttMMC_ddaa..
seu edu. en/
The unstructured dataset BP4D collected from https://www cs
3DFE/3DFE 3at.aaaaanaaaiisseeaaaaaaassaaee htmll..l................m.m....m html. The original SpoDtSpootaneous dataset (Zhang et al., 2013; 2014) is a 3D video database of spontaneous facial expressions diverse group of young adults. Well-validated emotion inductions were used to elicit expressions of emotion and paralinguistic communication. Frame-level ground-truth for facial actions obtained using the Facial Action Coding System. Eight tasks were covered with an interview process and series of activities to elicit eight emotions. As well, the Metadata include manually annotated action units (FACS AU). automatically tracked head pose, and 2D/3D facial landmarks. Given the size the original BP4D-Spontaneous dataset is too large to he used for training efficiently, we sampled part of frames (16037 images) and kept label combinations as much as possible. The first five dimensions correspond to AU06, AU10, AU12, AU14, and AU17 coded with intensity respectively. The intensity codes themselves are either 0 for absent, for present at the A level, 2 for present the level, for present at the C level, for present at the D level, 5 for present at the level; The sixth dimension corresponds to gender where 0 denotes male and I denotes female; The seventh dimension corresponds to tasks where 0-7 represents task 1-8.
The unstructured dataset DeepFashion is collected from https: /tt:://mmmall ie cuhk edu, www.hk//....eeeeeeeeeeeee hk/projects/ DeepFaske aashiiiio Atttriiiiii ributer redictiiii ctliiction html. The original DeepFashion (Liu et al., 2016) deals with the Cat- egory and Attribute Prediction task in Large-scale Fashion (DeepFashion) Database. Category and Attribute Prediction Benchmark evaluates the performance of clothing category and attribute prediction. This a large subset of DeepFashion,
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Towards Escaping from Class Dependency Modeling for Multi-Dimensinnal Classification
0.95
Ops10
Des10
1.0
Jura
0.8-
0.90
Oes97
Exact Match March
0es97
Sub-Exact Match
0.8
Enb
0.6
Oe<10
Hamming
0.85
0.6
Jura
Oes97
0.80
0.4
0.4
Enb
0.75
0.2
0.2
2a
2
2
2
2
2
2
2
2
2
2
2
2
The dimensionality of latent variable
The dimensionality latent variable
The dimensionalitt of latent variable
(a) Hamming Score
(b) Exact Match
(c) Sub-Exact Match
Figure 2. Performance of DCOM changes as the dimensionality of the latent variable r varies in the range of 22,,20 2 2", )²)
20,
Oes10
Des10
1.0
0.95
Jura
Ons97
0.8
Oes97
Sub-Exact Match
0.8
0.90
Emb
Exact Match
0.6
Enb
Des10
Hamming
0.6
Jura
0.85
De:97
0.4
Enb
0.80
0.2
10
10
10
10
10
10
10
10
The trade-off parameter
The trade-off parameter
The trade-off parameter a
(a) Hamming Score
(b) Exact Match
(c) Sub-Exact Match
Figure 3. Performance of DCOM changes as the trade-off parameter varies in the range of (10-3, 1002, 2,00,
1,,00,100000 1,00, ..1)".00).
containing massive descriptive clothing categories and attributes in the wild. We only adopted the fine version of DeepFash- ion dataset given its multi-dimensional characteristic. The "elasticity" dimension is not mentioned in the original paper. However, in the "iis_atttr_lloh..xx" in the "Anno_fine" folder of Category and Attribute Prediction Benchmark, we found 3 labels "tight", "loose" and "conventional" which are denoted by "6". We named the sixth dimension as "elasticity".
The unstructured dataset SEWA is collected from https: //db. db. sewapro eu/. The original SEWA (Kossaifi et al., 2021) is database of more than 2000 minutes of audio-visual data of 398 people coming from six cultures, 50% female, and uniformly spanning the age range of 18 to 65 years old. The annotation was performed in real-time using joystick. Specifically, the annotators were asked to push / pull the joystick based on their perception of the subject's level of valence, arousal, or liking/ / disliking (toward the advert) while being presented the recording. The joystick position (a value between -1000 and 1000) was sampled at 66 Hz and saved into the result file. Given the size of the original SEWA dataset is too large to be used for training efficiently, we sampled part of frames. Finally we kept 19275 images and their corresponding kinds of labels "arousal"" "liking" and "valence"". We transformed the continuous levels in the range of (-1,1) into discrete labels. For Arousal dimension, we transform (-1.-
:;...: 0.95. 1) - 20; for Liking dimension, 0; (0.95, -0.85)
we transform (-1, -0.55) - 0085, -0.75) + +1 1-750. 0.65) - 2; .. :0.95, 1) + 19; for Valence dimension,
transform (-1,-0.85 -0.55) + 0; 008.5, -0.65) + 1:1-0.65, -0.55) 1-055, -0.45)
3; ... [0.95, 1) - 18.
Note that in dimension I.i k ng and Valer encee we merged some intervals of length 0.1 as there are too few examples in the corresponding range.
C. Additional Tables and Figures
Due to the page limit, we present additional but still important tables and figures in this section.
C.1. Parameter Sensitivity
Figure shows how the performance of DCOM fluctuates with different dimensionality of latent variable Z.
13

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Towards Escaping from Class Dependency Modeling for Multi-Dimensinnal Classification
0.95
Oes10
Des10
1.0
Jura
Ocs97
0.8
Oes97
Sub-Exact Match
0.8
0.90
Enb
Exact Match
0.6
Des10
Hamming
0.85
0.6
Jura
0es97
0.80
0.4
0.4
Enb
0.75
0.2
10
10-2
10-3
109
161
10-3
10-2
10
10
107
10
10-2
10
10
107
The trade-off parameter 8
The trade-off parameter
The trade-off parameter $
(a) Hamming Score
(b) Exact Match
(c) Sub-Exact Match
Figure 4, Performance of DCOM changes as the trade-off parameter 33 varies in the range of (10-3, 10-2, 1001 ,,,0,,00,,,,,,,,,,,,,,,,,,,,,,,, 10*)
Figure and Figure 4 shows how the performance of DCOM fluctuates with different values of trade-off parameter a and B.
C.2. Notation Table
To facilitate understanding, Table 8 summarizes the notations used in Section I, Section 2 and Section 3.
C.3. Pseudo Code
Algorithm presents the pseudo code of the proposed DCOM approach.
Algorithm 1 The DCOM approach
Input: MDC training set D, an unseen instance æ., the trade-off parameter and | Output: Predicted label vector ý- for æ.
I: Count the number each class vectors repeat
training set and obtain the of frequently occurring class vectors D
3: Randomly sample batch of examples from D
4: for example in the batch do
5: Corrupt æ; by random noise and obtain a;
Input Ø; to encoding network to obtain the corresponding latent vector z,
7: Input the classification networks (H,lgg S j S and T to get the conditional marginal probability p(pd) (yo, v. 2,) and yoo mi, 2;), respectively
the conditional joint probability (
8: Input the reconstruction network R get the estimated mean vector μa; of the posterior probability px (m, Compute the classification loss L.se by E15() the reconstruction loss L... hy (q. and the loss for minimizing instance-hased conditional mutual information by (16 Compute the overall loss C by (17)
10:
11:
Update the trainable parameters with SGD optimizer end for
12:
13:
until Converge
14:
for j = 1 to q do
15:
Feed æ, to the trained encoding network G and output the latent vector z,
16: Feed to the classification network Hj output the conditional marginal probability PYşIX,2 (Vey lr,, (x., 2.)
17:
Get the prediction result on the j-th dimension by ش., = arg maX EC, PY,IX, (3., (x.,
18:
end for
19: Return ÿ. 10-1, De2. "ng]"
C.4. Visualization of Latent Factors
14

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Towards Escaping from Class Dependency Modeling for Multi-Dimensonnal Classification
40
40
30
20
20
10
0
0-
label
-20
20
1
-40
- 30
2
(a) Original Features X
(b) Latent Factors Z Learned by DCOM
Figure 5. I-SNE Visualization of original feature (left) and latent factors (right) in the first fold of dataset Voice w.r.t. the first dimension.
100
label
label
60
75
50
40
2
25
20
0
-2
-20-
-50
-40
-75
60
-100
100
75
50
25
25
50
75
100
60
40
20
20
40
60
(a) Original Features X
(b) Latent Factors Z Learned by DCOM
Figure 6. t-SNE Visualization of original feature (left) and latent factors (right) in the first fold of dataset 71C2000 w.r.t. the first dimension.
80
label
60
label
60
40
20
20
-20
-20-
-40
60
-40
-80
-75
-50
-25
25
50
75
100
-60
-40
-20
60
(a) Original Features
(b) Latent Factors Learned by DCOM
Figure 7. I-SNE Visualization original feature (left) and latent factors (right) in the first fold of dataset Flickr w.r.t. the first dimension.
15

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Towards Escaping from Class Dependency Modeling for Multi multi-Dimensional Classification
Table 8. Summary of the notations used in our paper.
Notation
Descriptions
d
number features imput space number class spaces (dimensions) in output space number of class lahels the j-th class space (1 5 9)
q
K
number of MDC training examples
X
the d-dimensional input (feature) space, i.c., X R2"
C
the j-th class space where C₁- (2. 2x (1 s 9)
cl
the a-th class label C₁
K2)
y
the output space where y C1 x C2 x ... x Cg
D
the set of MDC samples where = [ y S Sis m m)
f
the MDC predictive model: X' + 3 the class variable defined on the j-th class variable defind on Cj
Y
yo
randomly sampled value of Y in a probability function the class vector of the i-th example in D randomly sampled value of Y, in probability function the j- element y:
Mos
My
the feature variable defined on X
Zo
randomly sampled value of X in a probability function the feature vector of the i-th example in D the i-th corrupted feature vector where a; æ; + and is random noise vector
D'
the set of frequently occurring class vectors where D' {vol m c, Vo D)
p0t)
(yo, To: x
the conditional joint probability where ( (y., Ro, zo) = py, x (Yo, Do; zo)
(pd) (Yo, To, %o)
the product of conditional marginal probabilitics and joint probability feature and latent variables
where (Ddd) (yo, Ro, 2.) [] Px, (Noy ),x))xxx_ (Ro. 2o)
p00 (Mo. the integrant conditional mutual information where I(yo, 2o. 2o) = = (Yo, Tos 2o) log plad) (vo moo
I(yo: Do. 20)
g
The encoding network
H4
The classification network to model conditional marginal probability for the j-th dimension
R
The reconstruction network
The classification network model conditional joint probability
latent variable which the prior is the centered isotropic multivariate Gaussian pz = ((0,) randomly sampled value of Z in a probability function the i-th latent vector where the mean parameters of the assumptive posterior probability which is multivariate Gaussian with diagonal covariance structure A ((u,, of1)
Zo
G(E.)
u.
the standard deviation parameters of the assumptive posterior probability
S
the soft-max function
Lee
the classification loss regarding classification networks (H, S j S q) and T the reconstruction loss
L.
the loss function for minimization of the proposed instance-based conditional mutual information the overall loss function where L 1 (Lee + aorre + BLa))
C
16


Reviews:
[{"review_id": "cQOCKVkV9i", "reviewer": "Reviewer_Kvke", "summary": "This paper mainly focuses on multi-dimensional classification tasks and points out that existing works mainly focus on designing effective class dependency modeeling strategies but fail to solve the intercoupling of multiple classes. To solve this problem, this paper proposes a method, Dcom, to identify a latent factor that encapsulates the most salient and critical feature information.", "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": "We want to express our sincere gratitude for your invaluable comments and suggestions. The point-to-point responses are given as follows:\n\n- **Question 1: Could you please provide more explanations for the assumptions mentioned in Eq. (3)?**\n\nAnswer for Q1: Eq.(3) serves as a sufficient yet non-necessary condition for ensuring the equivalence between Eq.(1) and Eq.(2). Recall that in MDC, each feature variable $X$ is associated with $q$ class variables $Y_j (1\\le j\\le q)$, forming a composite class variable $Y=(Y_1,\\dots,Y_q)$. Therefore, the class variable considered in Eq. (1) is $Y=(Y_1,\\dots,Y_q)$, which models the **joint class structure**. However, existing MDC approaches mainly adopt Eq.(2) as the loss function which considers $q$ scalar class variables $Y_j (1\\le j\\le q)$ independently and aggregates errors across dimensions via summation (i.e., $\\sum_{j=1}^{q}$ over the dimension index). \n\nOne straightforward way to impose equivalence between Eq.(1) and Eq.(2) is to assume that each summand is equivalent, i.e., Eq.(3), which actually assumes the partial conditional independence among class variables. We use “partial” here because the assumption does not require Eq.(3) to hold universally for all possible values of $Y$ and $X$, but rather only under the observed training distribution (which is finite and empirically sampled). We will make this clearer in the revised version.\n\n\n- **Question 2: Does Theorem 3.3 mean that the candidate set will be reduced with the mild assumptions (Assumption 3.4)?**\n\nAnswer for Q2: Yes, precisely. Eq.(9) involves modeling the class space $\\mathcal{Y}$, whose cardinality $|\\mathcal{Y}|=\\prod^{q}_{j=1}K_j$ grows exponentially with the number of dimensions $q$, resulting in a computationally intractable hypothesis space. By focusing on high-frequency class combinations, the full joint space $\\mathcal{Y}$ will be reduced to a sparse subset $\\mathcal{D}’$ which is computable in practical algorithms. As shown in the following table, the candidate set is reduced dramatically (with the frequency threshold $c$ set as $0.1$% when constructing $\\mathcal{D}’$).\n| Dataset | $\\|\\mathcal{Y}\\|$ | $\\|\\mathcal{D}’\\|$ |\n|---|---|---|\n|Adult|490|37|\n|BeLaE|3125|17|\n|CoIL2000|4800|140|\n\n- **Question 3: I am curious about whether VLMs/MLLMs can perform such tasks better than the conventional deep learning methods. An intuition here is that the powerful foundation models can directly handle the multimodal data, like the image and text labels in this paper. It would be interesting to discuss this in the paper.**\n\nAnswer for Q3: We sincerely appreciate the insightful suggestion regarding the potential of VLMs and MLLMs for MDC. We fully agree that foundation models like CLIP or GPT-4V could revolutionize MDC by unifying visual and textual reasoning—a direction we are actively exploring in ongoing work. \nHowever, our current work faces a practical limitation: most existing MDC data sets (except for the exemplar data set *DeepFashion*) represent class labels as numerical indices (0, 1, $\\dots$) rather than rich semantic descriptors. This format significantly constrains our ability to leverage the text-image alignment capabilities that make VLMs/MLLMs so powerful. \n\nBesides, we do believe VLMs/MLLMs will likely demonstrate superior performance with more semantically-rich MDC data sets in the future and will incorporate this important discussion in our revised manuscript to better highlight both the current challenges and future opportunities in applying VLMs/MLLMs models to MDC.", "claims_and_evidence": "The claims in the paper are well supported by evidence.", "methods_and_evaluation_criteria": "Yes, the proposed method and evaluation criteria make sense.", "theoretical_claims": "I did not carefully check all theoretical claims in the paper, but the theory parts overall look sound.", "experimental_designs_or_analyses": "The experimental designs and analyses are reasonable.", "relation_to_broader_scientific_literature": "The contributions are good.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Pros:\n- The motivation of the paper is good and clear.\n- The paper is well written.\n- The paper is well supported by both theoretical and empirical results.\n\nCons:\n- I did not notice the obvious weakness of the paper.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- Could you please provide more explanations for the assumptions mentioned in Eq. (3)?\n- Does Theorem 3.3 mean that the candidate set will be reduced with the mild assumptions (Assumption 3.4)? \n- I am curious about whether VLMs/MLLMs can perform such tasks better than the conventional deep learning methods. An intuition here is that the powerful foundation models can directly handle the multimodal data, like the image and text labels in this paper. It would be interesting to discuss this in the paper.", "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": "oQ6D9TtNFJ", "reviewer": "Reviewer_tGjW", "summary": "In this paper, the authors propose a new method called DCOM to avoid class dependence modeling in multi-dimensional classification tasks. DCOM introduce an additional estimation of the gap between the joint probability and the product of marginal probabilities. Empirically, the authors verify the effectiveness of DCOM across various multi-dimensional classification tasks.", "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": "We want to express our sincere gratitude for your invaluable comments and suggestions. The point-to-point responses are given as follows:\n\n- **Q1: Is it possible to design a strategy and dynamically adjust the coefficient of the three loss terms?**\n\nA1: Thanks to the comments. It is indeed possible. In our current experiments, it is shown in Figure 3 and Figure 4 of Appendix C.2 that our approach achieves relatively stable performance as the coefficients vary within a broad range. Therefore, the coefficients of the three loss terms are fixed as 1 during training. This static strategy is chosen to simplify the optimization landscape.\n\nAs for possible dynamical strategies, we may suggest adopting gradient balancing techniques [1][2] to automatically align gradient magnitudes across loss terms. This is indeed a promising direction that could further enhance our model's adaptability.\n\n- **Q2: Is it possible to evaluate the influence of dependence modeling in multi-dimensional classification tasks?**\n\nA2: Thank you for raising this critical question about evaluating the influence of dependence modeling in MDC. Your observation highlights a fundamental limitation in current methodological practices. Existing MDC approaches mainly evaluate the influence of dependence modeling implicitly through the downstream **classification metrics** in terms of HS, EM and SEM compared to the baseline approach (BR [3], a classic approach which ignores the dependence modeling completely).\n\nGiven the indefinability of dependence, explicit and direct evaluation methods for the influence of dependence modeling may be inaccessible. We suggest seeking possible theoretical definitions of dependence from information theory [4] and further exploring appropriate evaluation methods.\n\n***\n[1] Yu T, Kumar S, Gupta A, et al. Gradient Surgery for Multi-Task Learning[J]. 2020.DOI:10.48550/arXiv.2001.06782.\n\n[2] Chen Z , Badrinarayanan V , Lee C Y ,et al. GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks[J].  2017.DOI:10.48550/arXiv.1711.02257.\n\n[3] Zhang, M.-L., Li, Y.-K., Liu, X.-Y., and Geng, X. Binary relevance for multi-label learning: An overview. Frontiers of Computer Science, 12(2):191–202, 2018\n\n[4] Tishby N. The information bottleneck method[J]. 1999. DOI: 10.1145/345508.345578.", "claims_and_evidence": "Most of the claims in this paper are convincing with theoretical and empirical evidence. However, I think it would be better to empirically verify that the performance of previous methods is limited by dependence modeling and DCOM successfully escapes from this issue.", "methods_and_evaluation_criteria": "The introduction of an additional estimation is an interesting idea to solve the dependence modeling, and the proposed methods make sense. Besides, the authors verify their method both theoretically and empirically.", "theoretical_claims": "The proof of Theorem 3.3 is clear and I do not find errors.", "experimental_designs_or_analyses": "The experiments focus on the performance in multi-dimensional classification tasks. I think it would be better to introduce additional experiments to empirically verify that the performance of previous methods is limited by dependence modeling and DCOM successfully escapes from this issue.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "I think most of essential references are discussed.", "other_strengths_and_weaknesses": "1. The paper is easy to follow and the core idea is straightforward.\n2. The theoretical motivation and empirical results cooperate well.\n3. The emprical improvements are not marginal, which verifies the effectiveness of DCOM.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. Is it possible to design a strategy and dynamically adjust the coefficient of the three loss terms? \n2. Is it possible to evaluate the influence of dependence modeling in multi-dimensional classification tasks?", "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": "aiNIZF2u8B", "reviewer": "Reviewer_pkvi", "summary": "This submission proposes a feature augmentation approach for multi-dimensional classification (MDC). I think the key motivation is to seek a set of augmented features $\\mathbf{Z}$ to fulfill the partial conditional independence (6). In theory, it might work thanks to the notion of conditional independence of random variables. In practice, it might be hard to analyze whether that partial conditional independence is fulfilled or not.  \n\nUnder the partial conditional independence (6), the joint conditional log-likelihood (5) becomes decomposable as given in (11). Under the assumption of parameter independence, i.e., the parameters of local models used to estimate $\\mathcal{H}_j$, $j = 1, \\ldots q$, are independent, we can train the local models separately. Section 3.3 details three components of the training loss and introduces the trade-off parameters $\\alpha$ and $\\beta$. \n\nExperiments are conducted on seventeen data sets to assess the potential advantages of the proposed approach, namely DCOM, compared with other MDC approaches recalled in Section 4.1.3. The empirical evidence suggests that DCOM can provide more promising results, compared to other competitors, and the performance of DCOM seems to be robust to the change of $\\alpha$ and $\\beta$ and the cardinality of the augmented feature set $\\mathbf{Z}$. However, it is not entirely clear to me which type of encoding network has been employed in the experiments.", "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": "We want to express our sincere gratitude for your invaluable comments and suggestions. The point-to-point responses are given as follows:\n\n-**Q1: I couldn't find details of the encoding network. If I missed some part of the paper, could you give me a pointer? Otherwise, could you add it in the next version?**\n\nAnswer for Q1: Thanks to the comments. Please kindly refer to **​the right column of page 7 (lines 375–384) in Section 4.1.4 (*IMPLEMENTATION DETAILS*)**, the encoding network $\\mathcal{G}$ is a **multi-layer perceptron (MLP) with one hidden layer, configured with hidden dimension of 512**. The output dimension, i.e., the dimensionality of latent variable $Z$ is also set as 512. All activation functions are fixed as ReLU followed by a dropout layer with dropping probability of 0.5. We truly hope this clarification can help elucidate details of the encoding network and we will ensure that the implementation details are more prominently highlighted in future revisions.\n\n\n-**Q2: Could you make the relations between $\\log p_{X|Z}(\\boldsymbol{x}_{i}|\\boldsymbol{z}_i)$ in equation (13) and $\\boldsymbol{z}_i$ clearer?**\n\nAnswer for Q2: We sincerely apologize for the lack of clarity and thank you for identifying this ambiguity. Below we provide a detailed clarification of the relationship between $\\log p_{X|Z}(\\boldsymbol{x}_i|\\boldsymbol{z}_i)$ in Eq.(13) and the latent variable $\\boldsymbol{z}_i$:\n\nThe term $\\log p_{X|Z}(\\boldsymbol{x}_i|\\boldsymbol{z}_i)$ represents the logarithmic posterior probability of reconstructing input $\\boldsymbol{x}_i$ given the latent variable $\\boldsymbol{z}_i$, where $\\boldsymbol{z}_i$ serves as the **input to reconstruction network $\\mathcal{R}$**. \nThe reconstruction network $\\mathcal{R}$ will then output the mean vector $\\boldsymbol{\\mu}_i$ \nand standard deviation vector $\\boldsymbol{\\sigma}_i$, i.e., key parameters of the assumed multivariate Gaussian $\\mathcal{N}(\\boldsymbol{\\mu}_i,\\boldsymbol{\\sigma}_i^2 \\mathbf{I})$.\n\n-**Q3: How do $\\mu_{ia}$ and $\\delta_{ia}$ related to $\\boldsymbol{z}_i$?**\n\nAnswer for Q3: Please kindly refer to the answer for Q2. \n\nConsidering that there is no $\\delta_{ia}$ in Eq.(13), we suppose $\\delta_{ia}$ in your question is $\\sigma_{ia}$.\n$\\boldsymbol{z}_i$ serves as the **input to reconstruction network $\\mathcal{R}$** and $\\mathcal{R}$ will **output** the mean vector $\\boldsymbol{\\mu}_i$ and standard deviation vector $\\boldsymbol{\\sigma}_i$. \n\n$\\mu_{ia}$ and $\\sigma_{ia}$ are the $a$-th element of $\\boldsymbol{\\mu}_i$ and $\\boldsymbol{\\sigma}_i$ respectively. \nSo they are related directly through the reconstruction network $\\mathcal{R}$ as the input and output vectors.\n\nBesides, Eq.(13) aims at inducing the reconstruction loss, i.e., Eq.(14). Thus in the practical algorithm, the reconstruction network $\\mathcal{R}$ only needs to output mean parameters $\\boldsymbol{\\mu}_i$.\n\n-**Q4: Which kind of reconstruction network has been used in the experiments?**\n\nAnswer for Q4: Please kindly refer to the answer for Q1. All implementation details are presented in **​the right column of page 7 (lines 375–384) in Section 4.1.4**.  All neural networks ($\\mathcal{G}, \\mathcal{R}$, $\\mathcal{T}$ and {$\\mathcal{H}_j|1\\le j\\le q$}) are employed as the **multi-layer perceptron (MLP) with one hidden layer**, configured with hidden dimension of 512 (except for the feature extractor for image data sets). \n\n-**Q5: Have you assessed the robustness of DCOM under the presence of either noisy features or small data sets? Please all refer to \"Claims And Evidence\" for my detailed comments on this point.**\n\nAnswer for Q5: Thank you for raising this critical question about the robustness of DCOM under noisy features or limited size of data sets. Actually, considering the conditional probability $p(\\mathcal{G}(\\boldsymbol{x})|\\boldsymbol{x})$ is inherently describing a deterministic event, we introduce a minor perturbation on $\\boldsymbol{x}$ before inputting it into the encoding network $\\mathcal{G}$ (please kindly refer to Eq.(4)). In other words, we have added random noise to the original MDC data sets as Denoising Auto-Encoders [1]. We will make this clearer in the revised version.\n\nAs for the small data sets, We have rigorously evaluated DCOM across datasets with varying sizes to assess its generalization capability (please kindly refer to Table 5 in Appendix B). Small data sets such as *Flare1*, *Oes97* and *Jura* only involve less than 400 samples. Though these results demonstrate DCOM's ability to handle limited data, we acknowledge (as correctly noted by you) that the performance advantage is more pronounced on larger datasets. This observation aligns with fundamental learning theory about the data requirements of deep models.\n\n***\n[1] Vincent, Pascal, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. “Extracting and Composing Robust Features with Denoising Autoencoders.” In Proceedings of the 25th International Conference on Machine Learning (ICML ’08)", "claims_and_evidence": "I think the claims made in the submission are supported by rather clear evidence. However, I would recommend the author(s) to add experiments to assess the robustness of DCOM under the presence of either noisy features or small data sets.\n\nThis is because these factors, which can appear in relevant applications, may affect the quality of the augmented feature set $\\mathbf{Z}$. I guess under the presence of either noisy features, the augmented feature set $\\mathbf{Z}$ may amplify the noisy level of the data. In the case of small data sets, the augmented feature set $\\mathbf{Z}$ may amplify the overfitting of the local models $\\mathcal{H}_j$, $j = 1, \\ldots q$.", "methods_and_evaluation_criteria": "I think the key assumptions of the proposed DCOM have been either stated in the submission or can be derived from the submission. Commonly used evaluation criteria in the MDC setting have been used in the submission.", "theoretical_claims": "I did my best to check the correctness of the theoretical claims. I haven't found any major issue.", "experimental_designs_or_analyses": "On one hand, I think the soundness/validity of any experimental designs and analyses is good and seems to be in favor of the proposed DCOM. \n\nOn the other hand, I would recommend the author(s) to add experiments to assess the robustness of DCOM under the presence of either noisy features or small data sets.", "relation_to_broader_scientific_literature": "It seems to me that considerable efforts in seeking scalable MDC approaches are based on the notion of conditional independence of random variables, decomposability of the loss/utility and parameter independence. Therefore, this submission may reasonably complement the existing literature on MDC.", "essential_references_not_discussed": "I think essential references are discussed.", "other_strengths_and_weaknesses": "To my knowledge, DCOM can be seen as a probabilistic graphical model learning approach with multiple latent variables. However, employing parametric models to estimate the local probability distribution may greatly facilitate the optimization of the loss/utility and scalability of the learning phase under suitable assumptions on the structure constraints and (decomposability of) the loss/utility. Therefore, the authors might consider making a connection between DCOM and the literature on that research topic.", "other_comments_or_suggestions": "I would use bold capital letters to denote sets of random variables, such as $\\mathbf{X}$, $\\mathbf{Y}$, and $\\mathbf{Z}$ (instead of $X$, $Y$, and $Z$)", "questions_for_authors": "Q1: I couldn't find details of the encoding network. If I missed some part of the paper, could you give me a pointer? Otherwise, could you add it in the next version? \n\nQ2: Could you make the relations between $log p_{\\mathbf{X} | \\mathbf{Z}}(\\boldsymbol{x}_i | \\boldsymbol{z}_i)$ in equation (13) and $\\boldsymbol{z}_i$ clearer? \n\nQ3: How do $\\mu_{ia}$ and $\\delta_{ia}$ related to $\\boldsymbol{z}_i$? \n\nQ4: Which kind of reconstruction network has been used in the experiments?\n\nQ5: Have you assessed the robustness of DCOM under the presence of either noisy features or small data sets? Please all refer to \"Claims And Evidence\" for my detailed comments on this point.", "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": "UvplV1soJG", "reviewer": "Reviewer_y3pL", "summary": "The paper proposes an approach to multi-dimensional classification (MDC), named DeCOupling Multi-dimensional classification (DCOM). Different from most MDC methods which explicitly model class dependencies through classifier chains or probabilistic graphical models (PGMs), DCOM captures partial class dependencies by conditioning on original features and latent variables computed from the original features. DCOM is evaluated on a set of benchmark datasets and shows its effectiveness in MDC.", "questions": "", "limitations": "", "rating": 1, "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": "We want to express our sincere gratitude for your invaluable comments and suggestions. According to your comments in *Claims And Evidence*, *Experimental Designs Or Analyses* and *Other Strengths And Weaknesses*, we summarize the following questions.\nThe point-to-point responses are given as follows:\n\n-**Q1: The assumption of Theorem 3.3 may not be easily satisfied in practice. Can the authors provide examples to demonstrate how it can be held?**\n\nAnswer for Q1: We sincerely appreciate your insightful question regarding the practical validity of the assumption of Theorem 3.3.\n\nThe basic condition of Theorem 3.3 is that the joint probability $p_{Y}(y_{\\diamond 1},y_{\\diamond 2},\\dots,y_{\\diamond q}) \\to 0$, which can guide selecting high-frequency class combinations when constructing $\\mathcal{D}'$ in Eq.(10) (please refer to the answer for Q2 for Reviewer Kvke). Take dataset *Flare1* as an example, the 323 examples present 14 distinct class combinations, where only 3 class combinations correspond to over 10 examples. In other words, the probability of most class combinations is less than 0.03. \n\nAs for the low-frequency class combinations involved in Theorem 3.3, we have \n$$p^{\\mathrm{(jt)}}(y_{\\diamond},x_{\\diamond},z_{\\diamond})\\le \\int_{x_{\\diamond}}\\int_{z_{\\diamond}}p^{\\mathrm{(jt)}}(y_{\\diamond},x_{\\diamond},z_{\\diamond})dz_{\\diamond}\\le p_{Y}(y_{\\diamond 1},\\dots,y_{\\diamond q})\\to 0,$$\nAccordingly, $p^{\\mathrm{(jt)}}-\\delta$ also tends to 0, which makes the Assumption 3.4 easily satisfied. \n \n-**Q2: The authors claim that the learned latent factor can capture \"critical feature information\". However, only empirical results are provided and there are no theoretical results or visualizations for justifying its effectiveness.**\n\nAnswer for Q2: Thanks to the comment. Considering that most MDC data sets are tabular data sets, we have not provided visualizations of latent vectors before. For image data sets, we implement pretrained ResNet-18 as the feature extractor and then conduct the proposed approach. So the obtained latent vectors are still tabular. \n\nFigure 1-3 at **anonymous** link <https://anonymous.4open.science/r/DCOM-C8FE/ICML25_authors_response.pdf> present the t-SNE visualizations for the first fold of dataset *Voice*, *TIC2000* and *Flickr*, which show latent factors capture more compact manifold representations than the original features. We will incorporate these additions in the next version.\n\nThis dilemma urges us to develop CNN-based MDC approaches to obtain feasible feature maps directly in the future. Thanks again for this insightful comment.\n\n-**Q3: The authors claim that DCOM is computationally superior to MDC methods based on learning a graphical model. However, no experiments are performed to support this.**\n\nAnswer for Q3: Thanks to the comment. In Section 2 (*Related Work*), we empirically mention the limitations of existing graphical MDC approaches which are widely recognized in previous works [1][2]. However, computational efficiency is neither the primary focus nor the core contribution of this paper. For comparision with the latest MDC approach *PIST* [3], which shares similar deep learning architecture, we list running times (500 epochs for training) on some data sets as follows which can show our efficiency:\n| Dataset | Time for DCOM (s) | Time for PIST (s) |\n|---|---|---|\n|Flare1|85.97|421.94|\n| Enb|112.94|471.63|\n| Jura|89.88|306.05|\n|Song|167.77|765.10|\n| Oes10|699.84|10219.91|\n\n-**Q4: The proposed method lacks significant technical novelty.**\n\nAnswer for Q4: Thanks to the comment. We would like to provide a more comprehensive response addressing this important concern.\nExisting MDC approaches mainly consider modeling class dependencies, where the hardness stems from the typical intercoupling within multiple dimensions. To address this issue, this paper proposes a dependency-free framework and establishes a novel technical route to solve MDC problems. As shown in our experimental results, the proposed approach which only employs conventional MLPs has yielded statistically significant performance improvements over state-of-the-art dependency-modeling baselines.\n\n-**Q5: The authors did not compare DCOM to MDC methods built upon PGMs. The claim that DCOM is more computationally efficient is not supported by experimental evidence.**\n\nAnswer for Q5: Please refer to the answer for Q3. \n\n-**Q6: The learned latent variable lacks interpretability.**\n\nAnswer for Q6: Please refer to the answer for Q2.\n***\n[1] B.-B. Jia, M.-L. Zhang. Decomposition-based classifier chains for multi-dimensional classification. IEEE TAI, 2022, 3(2): 176-191\n\n[2] M. Zhu, S. Liu, and J. Jiang, “A hybrid method for learning multidimensional Bayesian network classifiers based on an optimization\nmodel,” Applied Intelligence, vol. 44, no. 1, pp. 123–148, 2016.\n\n[3] Huang, T., Jia, B.-B., and Zhang,M.-L. Deep multi-dimensional classification with pairwise dimension-specific features. IJCAI'24, pp.  4183–4191.", "claims_and_evidence": "- The assumption of Theorem 3.3 may not be easily satisfied in practice. Can the authors provide examples to demonstrate how it can be held?\n- The authors claim that the learned latent factor can capture \"critical feature information\". However, only empirical results are provided and there are no theoretical results or visualizations for justifying its effectiveness.", "methods_and_evaluation_criteria": "The evaluation criteria (HS, EM and SEM) are appropriate for the MDC problem.", "theoretical_claims": "I did not find issues in the derivations in the main text.", "experimental_designs_or_analyses": "- The experimental design is mostly sound for validating DCOM's core contributions. \n\nWeaknesses:\n- No further analysis or visualization is performed for the learned latent factor.\n- The authors claim that DCOM is computationally superior to MDC methods based on learning a graphical model. However, no experiments are performed to support this.", "relation_to_broader_scientific_literature": "DCOM links some aspects of latent variable models to MDC. The effectiveness of latent variable models (e.g. Gaussian mixture models and variational auto-encoders) has been demonstrated in generative modeling tasks in previous studies.", "essential_references_not_discussed": "To my knowledge, no.", "other_strengths_and_weaknesses": "Strengths:\n- The paper is well-written.\n\nWeaknesses:\n- The proposed method lacks significant technical novelty.\n- The authors did not compare DCOM to MDC methods built upon PGMs. The claim that DCOM is more computationally efficient is not supported by experimental evidence.\n- The learned latent variable lacks interpretability.", "other_comments_or_suggestions": "N/A.", "questions_for_authors": "Please see above sections.", "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: This work proposes a new approach for dealing with multi-dimensional classification where one avoids modelling the relations among class variables and instead rely on some assumptions about their structure (or lack of it). The work is built on some interesting ideas, even though the theoretical part is not strong. Some committee members question whether the assumptions are often satisfied in practice (including wrt the joint class probability), and also whether the experimentation is sufficient to convey all the desired points (including choices of parameters and other decision when putting the experimentation together). Those points are indeed unclear to some extent, but one may say that the overall message remains.
Decision: Accept (poster)