一种非线性表征的概率潜在因子张量模型

doi:10.16180/j.cnki.issn1007-7820.2025.03.002

Abstract

Abstract:

In view of the filling problem of non-negative incomplete data with extremely sparse and unbalanced data, a probabilistic potential factor tensor model is proposed. The data sparsity is mitigated by reasonably assuming the probability distribution of the data as a priori information. Nonlinear mapping is used to realize nonlinear characterization of each non-negative element in the data and improve the characterization ability of the model. Considering the unbalance of data, the weights based on instance frequency are added to the traditional regularization terms to increase the effectiveness and pertinence of regularization terms. The experimental results show that the proposed model has obvious improvement over the existing model in terms of completion accuracy and time cost.

Key words: nonlinear representation, probabilistic factorization tensor model, frequency of known entries, nonlinear mapping, data sparsity, CP decomposition, unbalanced distribution, regular term

CLC Number:

TP393

DONG Jiaying, SONG Yan, LI Ming. A Nonlinear Representation-Based Probabilistic Latent Factorization Tensor Model[J].Electronic Science and Technology, 2025, 38(3): 7-15.

Figures/Tables 11

输入:Λ,R,η,γ₁,γ₂,γ₃

初始化:

U -

V -

T -

初始化:l=1,max_l=L
while not converge and l≤L do
1. for each Y_ijk∈Λ

Y^i j k l

∑ r = 1 R

u - r i l

)f(

v - r j l

)f(

t - r k l

)
er

r i j k l

=Y_ijk-

Y^i j k l

end for
2.根据式(10)更新

u - r i l + 1

3.根据式(11)更新

v - r j l + 1

4.根据式(12)更新

t - r k l + 1

5.l=l+1
end while
6.根据式(13)更新u_ri、v_rj、t_rk

Θ((|I|+|J|+|K|)×R)
Θ(1)
×N
×|Λ|
Θ(R)
Θ(1)

Θ(Λ_i×|I|×R)
Θ(Λ_j×|J|×R)
Θ(Λ_k×|K|×R)
Θ(1)

Θ((|I|+|J|+|K|)×R)

输出:U,V,T

Table 1.

Table 2.

Figure 1.

Figure 2.

Figure 3.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

References 25

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数据集	D₁	D₂
数据类型	电影评分	电影评分
用户数	6 040	27 221
项目数	3 706	19 367
时隙数	35	230
评分数	1 000 209	4 000 053
数据密度/%	0.130 0	0.003 3
评分范围	{1.0,2.0,…,5.0}	{0.5,1.0,…,5.0}

数据集	序号	训练-测试比/%	训练数据量	测试数据量	数据密度/%
D₁	D₁_.₁ D₁_.₂ D₁_.₃	10∶90 40∶60 70∶30	100 021 400 084 700 146	900 188 600 125 300 063	0.013 0 0.051 0 0.089 0
D₂	D₂_.₁ D₂_.₂ D₂_.₃	10∶90 40∶60 70∶30	400 006 1 600 022 2 800 037	3 600 047 2 400 031 1 200 016	0.000 3 0.001 3 0.002 3

数据集	η	β	γ
D₁_.₁	1.6×10^-4	2.3	2.0×10^-6
D₁_.₂	4.0×10^-5	2.3	2.0×10^-7
D₁_.₃	1.1×10^-5	2.3	3.0×10^-7
D₂_.₁	6.0×10^-4	2.3	2.0×10^-5
D₂_.₂	1.5×10^-4	2.3	4.0×10^-6
D₂_.₃	1.0×10^-4	2.3	1.5×10^-6

序号	名称	是否考虑概率分布	是否仅定义在已知数据上	是否考虑非负	是否为张量模型
M₁	BNLFT^[13]	否	是	是	是
M₂	NNCP^[4]	否	否	是	是
M₃	WSPred^[3]	否	否	否	是
M₄	NLF^[24]	否	是	是	否
M₅	PLFT	是	是	是	是

数据集	评价指标	M₁	M₂	M₃	M₄	M₅
D₁_.₁	MAE RMSE	1.061 3 1.425 4	3.579 2 3.749 5	8.248 5 12.056 8	4.089 3 16.085 1	1.060 9 1.342 5
D_1.2	MAE RMSE	1.097 3 1.489 4	3.498 1 3.668 8	6.113 8 7.757 1	2.205 9 8.173 6	1.072 8 1.369 0
D_1.3	MAE RMSE	1.102 8 1.496 7	3.406 6 3.578 8	6.509 1 8.210 4	1.653 2 5.954 7	1.093 0 1.388 2
D_2.1	MAE RMSE	2.670 1 2.852 1	16.888 2 32.291 1	6.816 4 9.137 4	5.352 4 26.023 9	1.096 7 1.401 8
D_2.2	MAE RMSE	2.666 8 2.847 7	3.291 0 3.448 8	6.283 1 7.901 3	2.037 7 11.578 8	1.102 2 1.409 8
D_2.3	MAE RMSE	2.667 2 2.847 9	3.517 0 3.671 4	6.041 0 7.603 9	1.483 2 7.823 7	1.084 5 1.388 9

A Nonlinear Representation-Based Probabilistic Latent Factorization Tensor Model

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References 25

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数据集	评价指标	M₁	M₂	M₃	M₄	M₅
D₁_.₁	总耗时/s 单次迭代耗时/s 迭代次数	156 2.771 56	926 1.852 500	5 376 10.752 500	942 1.884 500	124 3.284 38
D_1.2	总耗时/s 单次迭代耗时/s 迭代次数	623 17.800 35	3,405 6.809 500	22,775 45.550 500	3,903 7.805 500	515 13.271 39
D_1.3	总耗时/s 单次迭代耗时s 迭代次数	893 29.767 30	5 908 11.817 500	33 788 67.575 500	4,187 8.373 500	330 22.630 15
D_2.1	总耗时/s 单次迭代耗时/s 迭代次数	2 545 55.324 46	12 088 24.176 500	19 278 38.557 500	22 855 45.709 500	1 010 31.959 32
D_2.2	总耗时/s 单次迭代耗时/s 迭代次数	11 888 258.433 46	62 759 125.519 500	81 501 163.002 500	87 127 174.254 500	4 273 164.350 26
D_2.3	总耗时/s 单次迭代耗时/s 迭代次数	20 236 439.921 46	109 709 219.418 500	138 219 276.439 500	117 500 235.000 500	14 894 273.788 54

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