Revisiting multi-dimensional classification from a dimension-wise perspective

doi:10.1007/s11704-023-3272-9

Front. Comput. Sci.

2025, Vol. 19

Issue (1) : 191304 https://doi.org/10.1007/s11704-023-3272-9

Artificial Intelligence

Revisiting multi-dimensional classification from a dimension-wise perspective

Yi SHI¹, Hanjia YE¹(

), Dongliang MAN^2,³, Xiaoxu HAN^2,³, Dechuan ZHAN¹, Yuan JIANG¹

¹. National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210046, China
². Department of Laboratory Medicine, The First Hospital of China Medical University, Shenyang 110001, China
³. National Clinical Research Center for Laboratory Medicine, The First Hospital of China Medical University, Shenyang 110001, China

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Abstract

Real-world objects exhibit intricate semantic properties that can be characterized from a multitude of perspectives, which necessitates the development of a model capable of discerning multiple patterns within data, while concurrently predicting several Labeling Dimensions (LDs) — a task known as Multi-dimensional Classification (MDC). While the class imbalance issue has been extensively investigated within the multi-class paradigm, its study in the MDC context has been limited due to the imbalance shift phenomenon. A sample’s classification as a minor or major class instance becomes ambiguous when it belongs to a minor class in one LD and a major class in another. Previous MDC methodologies predominantly emphasized instance-wise criteria, neglecting prediction capabilities from a dimension aspect, i.e., the average classification performance across LDs. We assert the significance of dimension-wise metrics in real-world MDC applications and introduce two such metrics. Furthermore, we observe imbalanced class distributions within each LD and propose a novel Imbalance-Aware fusion Model (IMAM) for addressing the MDC problem. Specifically, we first decompose the task into multiple multi-class classification problems, creating imbalance-aware deep models for each LD separately. This straightforward method performs well across LDs without sacrificing performance in instance-wise criteria. Subsequently, we employ LD-wise models as multiple teachers and transfer their knowledge across all LDs to a unified student model. Experimental results on several real-world datasets demonstrate that our IMAM approach excels in both instance-wise evaluations and the proposed dimension-wise metrics.

Keywords multi-dimensional classification dimension perspective class imbalance learning

Corresponding Author(s): Hanjia YE

Just Accepted Date: 14 November 2023 Issue Date: 14 March 2024

Cite this article:

Yi SHI,Hanjia YE,Dongliang MAN, et al. Revisiting multi-dimensional classification from a dimension-wise perspective[J]. Front. Comput. Sci., 2025, 19(1): 191304.

URL:

https://academic.hep.com.cn/fcs/EN/10.1007/s11704-023-3272-9
https://academic.hep.com.cn/fcs/EN/Y2025/V19/I1/191304

Fig.1 In a fashion recommendation system, images encapsulate rich semantic information. For example, the left image can be labeled across multiple dimensions such as “type”, “color”, and “style”. Each of these dimensions encompasses several potential classes, of which only one is accurate. For the left image, the appropriate labels across the three dimensions are respectively “boot”, “black”, and “trendy”

Measure	Formulation	Note
Instance Accuracy (I-Acc)	$1 N ∑ i = 1 N [[n (i) = L]]$	The probability that labels from all L LDs are correctly predicted
Hamming Accuracy (H-Acc)	$1 N ∑ i = 1 N 1 L ? n (i)$	The probability that labels from any L LDs are correctly predicted
meanMacroF1 (M²F1)	$MacroPrecision (j) = ∑ k = 1 K j ∑ i = 1 N [[y^i j = y i j = k]] ∑ i = 1 N [[y^i j = k]]$ $MacroRecall (j) = ∑ k = 1 K j ∑ i = 1 N [[y^i j = y i j = k]] ∑ i = 1 N [[y i j = k]]$ $∑ j = 1 L 2 1 / MacroPrecision (j) + 1 / MacroRecall (j)$	Averaged MacroF1, the harmonic mean of Macro-Precision and Macro-Recall, across LDs
meanMacroAUC (M²AUC)	$∑ j = 1 L (∑ k = 1 K j ∑ i = 1 N [[f^i j k = max (f^i j 1, f^i j 2, …, f^i j K j)]] ∑ i = 1 N [[y i j = k]] ? ∑ i = 1 N [[y i j ≠ k]])$	Averaged MacroAUC across LDs

Tab.1 Definitions of four MDC performance measures. The later two are our newly proposed dimension-wise criteria

Fig.2 Imbalanced class distribution on different dimensions of the Zappos dataset (with 11 and 13 classes, respectively)

Fig.3 We show the imbalance shift from one LD to another. The color map counts the number of instances over two LDs on Zappos. The numerical values annotated on the colored blocks in the figure represent values post logarithmic transformation. Many major class instances become minor ones when the LD changes. In other words, the major/minor class property of an instance is difficult to be kept across LDs

Tab.2 MDC performance comparison on Med and Zappos. H-Acc/I-Acc and M

2

F1/M

2

AUC are instance-wise and dimension-wise criteria, respectively

Med	H-Acc	I-Acc	M²F1	M²AUC
M-Head	90.66	77.29	76.78	95.20
M-Emb	91.83	78.08	78.66	96.98
M-Head $I m b$	92.05	78.73	78.59	97.00
M-Emb $I m b$	91.77	77.95	78.32	96.73

Zappos	H-Acc	I-Acc	M²F1	M²AUC
M-Head	66.41	47.89	43.24	87.35
M-Emb	67.16	48.87	45.58	88.36
M-Head $I m b$	68.26	49.76	45.75	88.75
M-Emb $I m b$	66.95	48.12	45.45	88.00

Tab.3 Deep MDC performance comparison on Med and Zappos. H-Acc/I-Acc and M

2

F1/M

2

AUC are instance-wise and dimension-wise criteria, respectively

Fig.4 An illustration of our proposed IMAM approach based on an MDC problem with two LDs. In the decomposition step (a), we construct imbalance-aware deep models for each LD. In the fusion step (b), we use the models in the former step as fixed teachers and fuse their knowledge into a compact student. Both embedding (green) and classifier (red) distillations help in matching knowledge between models. Subscript ‘T’ denote the component of teacher. ‘CE’ means the cross-entropy

Tab.4 Statistics of MDC datasets

Method	Med				Zappos				Calligraphy				Letter
Method	H-Acc	I-Acc	M²F1	M²AUC	H-Acc	I-Acc	M²F1	M²AUC	H-Acc	I-Acc	M²F1	M²AUC	H-Acc	I-Acc	M²F1	M²AUC
BR	90.63	74.62	70.09	92.69	66.54	47.89	43.35	85.87	80.65	72.32	78.71	96.62	71.00	44.25	67.37	92.07
ECC	87.91	68.35	51.25	84.69	60.74	37.86	35.44	69.53	74.27	71.94	71.19	87.38	65.45	40.06	59.34	86.53
KRAM	91.11	74.62	72.38	94.33	67.23	48.65	44.46	78.69	81.40	73.60	78.88	94.53	72.03	44.05	68.12	92.07
LEFA	90.75	75.27	69.88	92.84	67.66	48.36	44.85	80.87	80.42	72.85	77.06	91.86	72.33	42.80	66.88	91.75
M-Head	90.66	77.29	76.78	95.20	66.41	47.89	43.24	87.35	81.12	73.01	78.97	97.30	72.66	47.31	68.75	94.30
M-Emb	91.83	78.08	78.66	96.98	67.16	48.87	45.58	88.36	83.21	75.68	81.39	97.54	74.49	49.57	71.17	94.87
M-Head $I m b$	92.05	78.73	78.59	95.60	68.26	49.76	45.75	88.75	82.86	75.24	81.49	97.54	75.33	50.73	71.46	94.60
M-Emb $I m b$	91.77	77.95	78.32	96.73	66.95	48.12	45.45	88.00	82.22	74.56	80.40	97.47	73.52	49.33	70.58	94.37
IMAM	93.10	80.82	81.79	97.71	68.33	50.01	45.90	89.32	83.44	76.06	81.98	97.61	81.48	61.43	79.21	96.70

Tab.5 Performance of four traditional MDC methods, four deep MDC methods, and our IMAM on four real-world datasets. Four evaluation metrics from instance and label aspects are calculated

	H-Acc	I-Acc	M²F1	M²AUC
M-Head $I m b$	92.05	78.73	78.56	95.60
M-Emb	91.83	78.08	78.66	96.98
IMAM $1 s t$	93.05	80.11	78.88	97.42
IMAM $C L S ? D i s t i l l$	92.77	80.53	78.95	97.63
IMAM $E M B ? D i s t i l l$	92.14	78.51	78.74	97.52
IMAM	93.10	80.82	81.79	97.71

Tab.6 Performance comparison between IMAM and its variants on Med dataset

Fig.5 TSNE of the learned feature embeddings of IMAM on Med dataset. Different colors denote different classes

Fig.A1 Imbalanced class distribution on different LDs of three datasets. Different colors represent different LDs. (a) Med; (b) Calligraphy; (c) Letter

Fig.A2 Examples of instances

x i

and multiple dimensions of labels

y i

in different datasets. (a) Zappos; (b) Letter; (c) Calligraphy; (d) Med

Med	H-Acc	I-Acc	M²F1	M²AUC
M-Head	90.66	77.29	76.78	95.20
M-Emb	91.83	78.08	78.66	96.98
M-Head $C D T$	92.05	78.73	78.59	97.00
M-Emb $C D T$	91.77	77.95	78.32	96.73
M-Head $D R W$	91.46	78.02	76.93	95.39
M-Emb $D R W$	91.63	77.87	78.34	96.21
M-Head $B S$	91.01	77.93	77.03	95.37
M-Emb $B S$	91.93	78.45	78.37	96.93

Zappos	H-Acc	I-Acc	M²F1	M²AUC
M-Head	66.41	47.89	43.24	87.35
M-Emb	67.16	48.87	45.58	88.36
M-Head $C D T$	68.26	49.76	45.75	88.75
M-Emb $C D T$	66.95	48.12	45.45	88.00
M-Head $D R W$	67.03	48.02	44.12	87.94
M-Emb $D R W$	66.35	47.32	44.45	87.63
M-Head $B S$	67.03	48.63	43.45	87.82
M-Emb $B S$	67.35	49.26	45.34	87.49

Table A1 Deep MDC performance comparison on Med and Zappos. We incorporate various class imbalance learning methods such as CDT, DRW, and BS, with M-Head and M-Emb

$λ 1$	H-Acc	I-Acc	M²F1	M²AUC
0.1	92.33	78.26	81.02	96.49
1	92.68	78.45	81.33	96.36
5	93.10	80.82	81.79	97.71
10	90.25	76.32	79.93	95.28

$λ 2$	H-Acc	I-Acc	M²F1	M²AUC
0.1	90.03	77.93	78.36	95.26
1	93.10	80.82	81.79	97.71
5	91.25	78.61	78.02	95.39
10	88.04	74.37	73.68	92.47

Table A2 Influence of

λ 1

and

λ 2

of IMAM on the Med dataset

Zappos	H-Acc	I-Acc	M²F1	M²AUC
IMAM $1 s t$	66.86	47.41	45.72	88.31
IMAM $C L S ? D i s t i l l$	68.13	49.32	45.24	88.66
IMAM $E M B ? D i s t i l l$	67.95	49.20	44.27	88.34
IMAM	68.33	50.01	45.90	89.32

Letter	H-Acc	I-Acc	M²F1	M²AUC
IMAM $1 s t$	86.48	67.16	85.63	96.72
IMAM $C L S ? D i s t i l l$	80.29	61.28	78.34	96.20
IMAM $E M B ? D i s t i l l$	80.74	61.35	78.39	96.12
IMAM	81.48	61.43	79.21	96.70

Table A3 Performance comparison between IMAM and its variants on Zappos and Letter datasets

Table A4 Performance comparison when we start the classifier distillation after different numbers of training epochs followed by the embedding distillation

Zappos	H-Acc	I-Acc	M²F1	M²AUC
KRAM + M-Head	67.23	48.65	44.46	78.69
KRAM + M-Head $I m b$	67.68	49.04	45.12	80.33
LEFA + M-Head	67.66	48.36	44.85	80.87
LEFA + M-Head $I m b$	67.78	49.07	44.95	80.94
IMAM	68.33	50.01	45.90	89.32

Letter	H-Acc	I-Acc	M²F1	M²AUC
KRAM + M-Head	72.03	44.05	68.12	92.07
KRAM + M-Head $I m b$	72.59	45.35	68.40	92.64
LEFA + M-Head	72.33	42.80	66.88	91.75
LEFA + M-Head $I m b$	73.30	43.47	67.63	92.82
IMAM	81.48	61.43	79.21	96.70

Table A5 Performance comparison between KRAM/LEFA trained with features extracted by M-Head and M-Head

I m b

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