Tensor Product Neural Networks for Functional ANOVA Model
Authors: Seokhun Park
Insung Kong
yongchan Choi
Chanmoo Park
Yongdai Kim
Venue: ICML 2025 poster (2025)
Keywords: Interpretability
Trustworthy AI
Functional ANOVA model
Generalized additive models
Tensor product neural network
Primary area: social_aspects->accountability_transparency_and_interpretability
TL;DR: In this paper, we propose a novel neural network which guarantees a unique functional ANOVA decomposition and thus is able to estimate each component stably and accurately.

Abstract: Interpretability for machine learning models is becoming more and more important as machine learning models become more complex. 
The functional ANOVA model, which decomposes a high-dimensional function into a sum of lower dimensional functions (commonly referred to as components), is one of the most popular tools for interpretable AI, and recently, various neural networks have been developed for estimating each component in the functional ANOVA model. 
However, such neural networks are highly unstable when estimating each component since the components themselves are not uniquely defined. 
That is, there are multiple functional ANOVA decompositions for a given function. 
In this paper, we propose a novel neural network which guarantees a unique functional ANOVA decomposition and thus is able to estimate each component stably. 
We call our proposed neural network ANOVA Tensor Product Neural Network (ANOVA-TPNN) since
it is motivated by the tensor product basis expansion.
Theoretically, we prove that ANOVA-TPNN can approximate any smooth function well.
Empirically, we show that ANOVA-TPNN provide much more stable estimation of each component and thus much more stable interpretation when training data and initial values of the model parameters vary than existing neural networks do.
Our source code is released at https://github.com/ParkSeokhun/ANOVA-TPNN

Tensor Product Neural Networks for Functional ANOVA Model
Scokhun Park Insung Kong²
Yongehan Choi
Chanmoo Park
Yongdai Kim
Abstract
(Ribeiro et al., 2016) and SHAP (Lundberg, 2017) have been suggested to interpret complex black-box Al models. However, these methods explain given black-box model through a locally approximated interpretable models and thus often fail to provide faithful global view of the model (Slack et al., 2020).
Interpretability for machine learning models is becoming more and more important machine learning models become more complex. The func- tional ANOVA model, which decomposes a high- dimensional function into a sum of lower dimen- sional functions (commonly referred to as compo- nents), one of the most popular tools for inter- pretable AI, and recently, various neural networks have been developed for estimating each compo- nent in the functional ANOVA model. However, such neural networks are highly unstable when estimating each component since the components themselves are not uniquely defined. That is, there are multiple functional ANOVA decompositions for given function. In this paper, we propose a novel neural network which guarantees unique functional ANOVA decomposition and thus is able to estimate each component stably. We call our proposed neural network ANOVA Tensor Prod- uct Neural Network (ANOVA-TPNN) since is motivated by the tensor product basis expansion. Theoretically, we prove that ANOVA-TPNN can approximate any smooth function well. Empiri- cally, we show that ANOVA-TPAA -PNN provide much more stable estimation of each component and thus much more stable interpretation when train- ing data and initial values of the model parameters vary than existing neural networks do. Our source code released at https: //gttbbb..//b///////// com/ ParkSeokhun/ANOO TPNN
The functional ANOVA model, which approximates a given complex nigh-dimensional function by the sum of low- dimensional (e.g., one or two dimensional) functions, is well known transparent-box AI model. One of the most representative examples of the functional ANOVA model is the generalized additive model (GAM, Hastie & Tibshi- rani (1987)), which consists of the sum of one-dimensional functions, each corresponding to each input feature. Low- dimensional functions are easier to understand, and thus the functional ANOVA model is popularly used for interpretable AI (Lengerich et al., 2020; Märtens & Yau, 2020).
Recently, several specially designed neural networks for the functional ANOVA model have been proposed, including NAM (Agarwal et al., 2021) and NBM (Radenovic et al., 2022). These neural networks can be learned by standard stochastic gradient descent algorithms and thus can be ap- plied to large sized data compared to traditional learning algorithms based on basis expansions (Wood, 2006) and regularizations (Gu & Wahba, 1993). However, existing neural networks are not good at estimating each component (each diweennnniooal function in the functional ANOVA model) mainly due to unidentifiability. Here, 'unidentifia- bility" means that there exist multiple different functional ANOVA decompositions of given function and we do not know which decomposition gradient descent algorithm converges on. Note that poor estimation of the components would result in poor interpretation.
1. Introduction
Interpretability has become more important as artificial in- telligence (AI) models have become more sophisticated and complicated in recent years. Various methods such as LIME
In this paper, we develop new neural networks for the func- tional ANOVA model such that each component is identifi- able but they are learnable by standard stochastic gradient descent algorithms. Identifiability makes our proposed neu- ral networks be good at estimating the components and thus provide reliable interpretation. In addition, it is robust to outliers and easy to reflect monotone constraints.
'Department of Statistics, Seoul National University, Seoul, Re- public of Korea DDepartment of Applied Mathematics, University of Twente, Enschede, Netherlands "Toss Bank, Seoul, Republic of Korea "Samsung Electronics Co., Ltd, Republic Korea. Corre- spondence to: Yongdai Kim <ydkim0903 @ 0otim/09000gmal..ommmmmm gmail.com>.
To develop the proposed neural networks, we begin with the tensor product basis expansion (Wood, 2006) and replace each basis function by specially designed neural networks so
Proceedings of the nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Tensor Product Neural Networks for Functional ANOVA Model
that each component becomes identifiable and robust out- liers. We call our proposed neural networks Tensor Product Neural Networks (TPNN). Finally, we propose ANOVA- TPNN, which estimates each component in the functional ANOVA model using TPNNs. Theoretically, we prove that ANOVA-TPNN has the universal approximation property in the sense that approximates any Lipschitz function well.
model with only main effects included, that is
f(x) - So + Σ fz,Ø)
Similarly, G is defined as the functional ANOVA model including all of the main effects and second order interac- tions,
Overall, our contributions are summarized as follows.
f(x) 80 +
Σ
fs(xs),
We propose novel neural networks (TPNN) for the functional ANOVA model with which we can estimate each component stably and accurately by use of a stan- dard stochastic gradient descent algorithm.
or more generally can consider GA MM defined as
f(x) = 80
E
fs(xs).
We prove the universal approximation property in the sense that TPNN can approximate any Lipschitz func- tion up to an arbitrary precision.
SEPLISISa
Several learning algorithms for the functional ANOVA model have been proposed. Gu & Wahba (1993) applied the smoothing spline to learn the functional ANOVA model, Lin & Zhang (2006) developed a component sparse penalty, Kim al. (2009) proposed a boosting algorithm for the func- tional ANOVA model, and Wood (2006); Rügamer (2024) proposed methods for estimating the functional ANOVA model using basis expansion.
By analyzing multiple benchmark datasets, we demon- strate that TPNN provides more accurate and stable estimation and interpretation of each component com- pared to the baseline models, including NAM (Agarwal et al., 2021), NBM (Radenovic et al., 2022), NODE- GAM (Chang et al., 2021) and XGB (Chen & Guestrin, 2016) without losing prediction accuracy.
In addition, the functional ANOVA model has been applied to various problems such as sensitivity analysis (Chastaing et al., 2012), survival analysis (Huang et al., 2000), diag- nosties of high-dimensional functions (Hooker, 2007) and machine learning models (Lengerich et al., 2020; Märtens & Yau, 2020).
2. Background
2.1. Notation
Let x = -(I1,..,,P)) .pp) E X'= = 21 x x Xp be vector of input features, where we assume X C [--a, a]" for some a > O. We denote [p] = {1.....P) and denote its power set as P(p))). For x X' and S [p). let xs = ( E T) We denote fs function of xg. For real-valued function |l/ supxex [xx).
Recently, learning the functional ANOVA model using neu- ral networks has received much attention since gradient descent based learning algorithms can be easily scaled up. Examples are Neural Additive Model (NAM, Agarwal et al. (2021)), )), Neural Basis Model (NBM, Radenovic et al. (2022)) and NODE-GAM (Chang et al., 2021) name just few. NAM models each component of GAM by DNNs, and NBM achieves a significant reduction in training time com- pared to NAM by using basis DNNs to learn all components simultancously. NODE-GAM extends Neural Oblivious De- cision Ensembles (NODE, Popov et al. (2019)) for GAM.
f: X R. we denote
2.2. Functional ANOVA model
The functional ANOVA model (Hoeffding, 1992: Sobol, 2001) decomposes dig-diimeniooal function f into the sum of low-dimensional functions
((x) = 9o
Σ Isls) + E (s(((),) .... j=1
JCCk
2.3. Tensor Product Basis
which is considered one of the most important XAI tools Agarwal et al. (2021); Radenovic et al. (2022); Lengerich et al. (2020). Typically, f is defined as f fx) = ((E(Y|XX = x)) where is a link function and Y is target variable. In the functional ANOVA model, fy.j E [p] are called the main effects, and fe k) E [2] are called the second interaction terms and so on. In practice, only interactions of lower orders (e-g., the main and second order only) are included in the decomposition for easy interpretation.
Let B, = = L,..,, Mj} be given set of basis functions on X, (e.g. truncated polynomials or B-splines). Then, the main effect is approximated as follows:
=
M
f,,,))
Σ
aakk (z,
=B,(I,) a.
The Generalized Additive Model (GAM, Hastie & Tibshi- rani (1987)) is a special case of the functional ANOVA
where B,(,;) = (B),(1,), (x)) and a, = (as,1...,
2

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Tensor Product Neural Networks for Functional ANOVA Model
For the second order interactions, J5alarg is approxi- mated by
where fi(21) z1, fa(x2) = 22, f12((11)2)) = 21 can be expressed as
(B,(2))
@ (x(x)))a
f(11, = fi (x1) ) + (x2) + (211
M,
Me
Σ
where (11 = -21, (xz2) ) x2, fi2 22) xn (22 2 + 2) (Lengerich et al., 2020). Without the identifiability condi- tion, each component cannot estimated uniquely and thus interpretation through the estimated components be- comes unstable. See empirical evidences of instability of estimated components by NAM Agarwal al. (2021) and NBM Radenovic et al. (2022) are given in Appendix E.
Here, (zy) & (T( =
are called the tensor product basis functions and agj (a.,A.L.. E [M], lo E [M&]) E R Mr.
Similarly, for S S [p]. a general order interaction fs(xs) is approximated by
A simple remedy to ensure the identifiability of each com- ponent to put constraint. One of the most popular con- straints for identifiability of each component in the func- tional ANOVA model is so called the sum-lo-zero condition (Hooker, 2007; Kim et al., 2009). The sum-to-zero condition requires that each component is should satisfy
fs(xs) (8yesB,
where (z,) and as are ILs My-dimensional ten- sors.
problem of using tensor product basis functions is that the number of learnable parameters (i.e. s1 23 E [M2)) increases exponentially in |S] and thus estimating higher order interactions would be computationally demanding. Thus, they are typically used for www-dimesiiona datasets.
Vj S, AZ E Ksvor
(I)
fs(xs(b) = 2, 2,z3), = =0
for some probability measure Jáj on X3. For PJy the empiri- cal distribution of the input feature 2; or the uniform distri- bution on X, can be used in practice. With the num-to-zero in (1), the functional ANOVA model becomes identifiable, as can be seen in proposition 3.1. Let p= -I, Pb-
3. Tensor Product Neural Networks
The aim of this section is to propose a specially designed neural models called Tensor Product Neural Networks (TPNN). The most important advantage of TPNN is that they satisfy an identifiability condition of each component in the functional ANOVA model and so we can estimate each component accurately and stably by use of a gradient descent algorithm. Additionally, unlike the conventional ten- sor product basis approach (Wood, 2006; Rügamer, 2024), our model does not lead an exponential increase in the number of learnable parameters with respect to (S), making it applicable to nigh-dimensional datasets. Finally, TPNN is robust to input outliers and easy to accommodate the monotone constraint on the main effects.
I1,
Proposition 3.1. (Hooker, 2007) Consider two component sets U$S. S S [S| S d} and (J3, C Is] < d} which satisfy (I). Then, Es = f(() almost everywhere (with respect to μ)) if and only if fs(() f2(() almost everywhere (with respect (1) for every S C
[p] with |S| S d.
Herren Hahn (2022) demonstrated that there is an interest- ing relation between the sum-to-zero condition and SHAP (Lundberg, 2017) which is a well known interpretable AI method. That is, the SHAP value of a given input can be cal- culated easily from the prediction values of each component of the functional ANOVA model, For a given function f and input vector x, the SHAP value of the jth input variable is defined as
In this section, we first explain the sum-to-zero condition for the identifiability of each component in the functional ANOVA model and then propose TPNNs that always sat- isfy the sum-to-zero condition, and develop correspond- ing learning algorithm for the functional ANOVA model. Moreover, we show theoretically that TPNNs are flexible enough so that they can approximate any Lipschitz continu- ous function well. A modification of Neural Basis Models with TPNNs is also discussed.
SS - [S|-1); (XSU()) vy
(f.x))
Σ SCIeM
p!
where My (xs) = =E[[(X) = xs), where X ~ μ.
Proposition 3.2. (Herren & Hahm, 2022) For a given GA f satisfying the sum-to-zero condition, we have
$f(,))
Σ
fs(xs)/\S)
(2)
3.1. Sum-to-zero condition for identifiable components
The functional ANOVA model itself is not identifiable. That is. there are multiple functional ANOVA decompositions for a given function. For example,
Equation (2) provides an interesting implication - the func- tional ANOVA model satisfying the sum-to-zero condition also decomposes the SHAP value. That is, the contribu- tion of the interaction between Hj and x5 to the SHAP
x2) = =(())) + f2(22) + #2)
3

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Tensor Product Neural Networks for Functional ANOVA Model
value (X, is f(xx) 1/S" where S' U This result could provide a new way of calculating the SHAP value approximately. Specifically, we first approximate given function by a GA"M satisfying sum-to-zen condition and calculate the SHAP value of the approximated GA MM by use of (2). We refer this approximated SHAP value as ANOVA-SHAAP. The results of numerical studies in Appendix H confirm that ANOVA- SHAAP is similar to Kernel-SHAP calculated from the shap python package (Lundberg. 2017) One obvious advantage of ANOVA-SHAP is that it is signif- icantly faster to compute compared to existing methods of calculating SHAP values such as Deep-SHAP and Kernel- SHAP proposed by Lundberg (2017), as well as Tree-SHAP proposed by Lundberg et al. (2018).
Higher order interactions. For Ss, we consider the fol- lowing tensor product neural network (TPNN)
K
fs(xs) R3
B (x5
(4)
where (x{xx)( = ss 9 (1, (x,10,,e). Since (2, (0,)) satisfies the sum-to-7ero condition, ( (0,k)) does and so does fs. As is done for the main effect, we learn 85,48 and fs while we let Kş hyper-parametes
Remark. As the traditional tensor product basis expansion approaches (Wood, 2006; Rügamer, 2024) do, we may con- sider
fs(xs) ~ (0
3.2. ANOVA-TPNN
where b,(2,) = (1)(1))) E [Kj]), As is a Hysss Ký Ky-dimensional tensor, and 0, are those used for the main effect. This expansion shares the parameters be- tween the components and thus the number of parameters 0yuS is smaller than the expansion in (4). However, the num- ber of parameters in By is exponentially proportional [S] and thus the number of the total learnable parameters would be much larger than that in the expansion (4). This is an important advantage of the proposed TPNNs compared to the traditional tensor product basis expansions.
We first propose basis neural networks for the main effects and extend these basis neural networks for higher order in- teractions through tensor product which we call tensor prod- uct neural networks (TPNN). Finally, we propose ANOVA- TPNN which estimates each component by TPNN.
Main effects. We first consider the main effects. Let (x(()) be a basis neural networks for the main effect fy parametrized by 0. That is, we approximate fI,, by
f10,)) ~ B (3,,)
(3)
ANOVA-T'PNN. Finally, propose PNOVA-TTPNN that estimates each component of the functional ANOVA model using TPNN as follows.
for 3jk E R. To ensure the sum-to-zero condition, we choose o such that Saex, (dz)) for all 0.
Our choice of (z is as follows:
Ke Σ
f(x) 30 +
Σ Sak-
(s (s,x).
o (x|10)) = { (1000 7 (m=²) }
x
x (#-D) +c((0,)00 }
}
?
In PNN the learnable parameters are 3g and (3s.u, 0g & E [Ks]. S S. [p]. I d. Unless there is any confusion, we use ANOVA-TPNN and ANOVA-T'PNN for general d interchangeably.
for b E R and- > 0, where 0 = (b,7), o.)) is an ac- tivation function and (b) = -(1 (11 (b.
with (b) = - o (3)) (dr). We introduce the term 7)) to ensure oz (=(0) satisfies the sum-to-zero condition. For (-) function, any differentiable function can be used but in this paper we use the sigmoid function for axx (i.e. axx) 1/(1 + exp(-x))), since it is robust to input outliers and provides nice theoretical property: an universal approximation theorem given in Section 3.3.
Training. For given training data (x1, 81), (Xn, yrr and a given loss function e, we learn the parameters by min- imizing the empirical risk Σn_ {(y.. fxx)) by a gradient descent algorithm. Overfitting can be avoided by selecting the number of epochs and learning rate carefully.
The standard basis function approach can be understood as method of fixing the parameters 0 a priori and only learning Sau in (3). In contrast, we learn 0,x as well as 3,k- That is, the terms o5 (=)(0,.)) can be considered as data- adaptive basis functions. Since there is no constraint (except the nonnegative constraint on ?) on the parameters, gradi- ent descent algorithm can be used for learning. The number of basis functions K₁ is fixed in advance as is done in the basis expansion approaches.
Data preprocessing. The term ca 7) could be too large when r7(6,) 7) is close to which can happen when μy is the empirical distribution and there exist outliers. To avoid this problem, we transform each input feature based on the marginal ranks to make μy of the transformed data be similar to the uniform distribution. Since ANOVA- ANOVA-TPNN is nonparametric, this data transformation does not affect much to the final prediction model.
4

---

Tensor Product Neural Networks for Functional ANOVA Model
Table I: Stability scores on real datasets. Lower stability score means more stable interpretation. The bold faces highlight the best results among GAM and GA
GA
GA-M
Dataset
ANOVA
T'PNN
NA M
NB M
TPPNN
NA
NB²M
CALIOUSING (Podrggoa et la)
0.012
0.045
0,039
0,035
0:771
0.075
Corie 2009)
0.011
0.058
0.065
ONU (Femandes al., 2015)
0.031
0.076
0.054
0.072
0.072
ABALONE (Warwick et al. (995)
0.008
0.013
0.036
0,025
0.047
0,038
FICO (iic, 2018)
0.035
0.016
0.016
0,048
0.089
0.075
CHURNichu, 2017)
0.017
0.027
0.017
0,047
0.089
0,080
CREDIT
0.021
0.069
0.053
LETTER (Slate, 1991)
0.017
0.022
0.014
0.036
0.075
0,081
DRYBEANIdry 2020)
0,028
0.074
0,070
0.053
0,088
0.081
3.3. Universal approximation thorem
3.4. Extension to Neural Basis Models
An interesting theoretical property of ANOVA-TPNN is the universal approximation property the standard neural network has (Hornik et al., 1989). That is, ANOVA-TPNO can approximate any arbitrary GA M function to desired level of accuracy, as stated in the following theorems.
Similarly to NBM (Radenovic et al., 2022), we extend ANOVA-TPNN to NBM-TPNN, which estimates each com- ponent linear combination of common basis neural networks.
Let Kj.J E [p] be all equal to X (e.g. X, = [-1, 1) and let Ky.J E [p] be also all equal to Ho (e.g. the uniform dis- tribution on [-1, 1]). Let (xxx(0) be a basis neural network on X satisfying the sum-to-zero condition with respect to Pg- Then, NBM-TPNN approximates fs by
Theorem 3.3. Suppose that J E (p) have lower and upper bounded densities with respect to the Lebesgue measure. Then, for any L-Lipschitz continuous function' go.s I X3 + R. satisfying the sun-to-zera condition, there exists a TPNN with Kg many basis neural networks such that
K
Ks
fs(xs) 2
Σ 3s.k II ((y)(0ss).. JES
(0,s() Σ (s
<ss
|
K
+
That is, NBM-TPNN shares basis neural networks across the components with the same cardinality, which substantially reduces the number of learnable parameters. The experimen- tal results for NBM-TPNX, are provided in Section 4.6.
for some constant Cs > 0 and S S [p]-
Theorem 3.3 shows that TPNN can approximate any Lips- chitz continuous function satisfying the sum-to-zero condi- tion. with an arbitrary precision by choosing Kss sufficiently large, and the required Kg for a given precision should increase |S] increases. An obvious corollary of Theo- rem 3.3 is that ANOVA-TPNN can approximate any GA model where each component is Lipschitz continuous and satisfies the
4. Experiments
This section presents the results of numerical experiments. More results along with details about data, algorithms and selection of hyper-parameters are provided in Appendices B to M.
condition.
Corollary 3.4. Let gx( = Eschhiisa (xs) be given GA function satisfying the sum-to-zero condition. If U.. ip) be probability measures having bounded den- sities with respect the Lebesgue measure. and each go is L-Lipschitz continuous, then, there exists JANOVA.' P'W such that
4.1. Stability in component estimation
Similarly to NAM (Agarwal et al., 2021) and NBM (Raden- ovic et al., 2022), ANOVA- TPNN provides interpretation through the estimated components. Thus, if the components are not estimated stably, the interpretations based on the esti- mated components would not reliable. In this subsection, we investigate the stability of the component estimation of ANOVA-TPNN compared with the other baseline models including NAM and NBM. For this purpose, generate randomly sampled training data and estimate the compo- nents of the functional ANOVA model. We repeat this pro- cedure 10 times to obtain 10 estimates of each component, and measure how similar these 10 estimates are. For the similarity measure, we use
[|ao()) - -aavoca Olloo <C Σ
IS
SSIPLISIE KJ +1
for some constant C > 0, where Kg is the number of basis neural networks for component S.
A given function defined on 2 is L--Lipschitz continuous if (o(2a) x(22]] < 21 z2| for all 21, 22 Z, where || is certain norm defined on
5

---

Also, the plots of the functional relations of the main effects are provided in Appendix 1, which amply demonstrate the instability of NAM and NBM in component estimation. Tensor Product Neural Networks for Functional ANOVA Model
Table 2: Performance of component selection. We report the averages (standard deviations) of AUROC' of the estimated importance scores of each component on y(), ,(9) ((3) synthetic datasets. The bold faces highlight the best results.
True model
p(ay
Models
ANOVA
T'PNN
NAPM
NB
ANOVA T'PNN
NAPM
ND
ANDVA PPPNN
NA
NB
AUROCT
1,000
0.330
0,522
(0.01)
11.44
0.311
0.481
0.956
0.381
(0,07)
0.477
Table Prediction performance. We report the averages (standard deviations) of the prediction performance measure. addition, we report the averages of ranks of prediction performance of cach model on nine datasets. The optimal (or suboptimal) results are highlighted in bold (or underlined).
GA'M
GA-M
Black
Datasct
Measure
ANOVA
T PNN
GA M
NODE
NA M
NB M
ANOVA
TOPNN
GA²M
NODE
NA
NB²M
XGB
DNN
0.614
0.381
0.639
0,394
0,12
0.315
0.325
0.502
0.452
0,318
CALHOUSING
RMSE
(0.01)
(0.01)
(0.01)
(0.08)
(0:01)
(0.01)
(0.02)
(0.03)
(0.01)
(0.01)
0.725
0.723
0.733
0,724
0.704
0.730
0,720
0.702
0.635
0.696
WINE
RMSE
(0.02)
(0.02)
(0.02)
(0.02)
(0.02)
(0.02)
(0.02)
(0.03)
(0.03)
(0.01)
1.111
L1211
1.350
1.187
1.111
1.137
1.313
1.179
1.122
1.123
ONLINE
RMSE
(0:25)
(0.27)
(0.57)
(0.25)
(0:25)
(0.26)
(0.46)
(0.21)
(0.26)
(0.26)
ABALONE
RMSE
(0.09)
2.135
2.141
(0.09)
2.171
(0.08)
2.167
(0.09)
2.087
(0.08)
2.100
(0.10)
2,088
(0.08)
(0.08)
2,083
(0.09)
26164
2.071
(0,10)
FICO
AUROC 1
0.799
0.795
0.788
0.797
0,800
0.793
0.799
0.799
0.796
0.793
(0.007)
(0.009)
(0.006)
(0.006)
(0.007)
(0.007)
(0.007)
(0.008)
(0.008)
(0.008)
0.824
0.846
0.815
0.842
0.844
0.842
CHURN
AUROC T
(0.012)
(0.012)
(0.011) (0.012)
(0.012)
(0.011)
(b.011)
(0.011)
(0.012)
(0.013)
0,983
0.983
0.976
0,972
0.984
0,980
0.985
0,980
CREDIT
AUROC 1
(0.005)
(0.005)
(0.012) (0.011)
(0.006)
(0.006)
(0.007)
(0.004)
(0.004)
(0.006)
0,900
0.910
0.904
0.910
0.984
0.988
0.986
0.990
0.997
0.9%6
LETTER
AUROC 1
(0.003)
(0.002) (0.001) (0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0,001)
0.995
0.996
0.996
0.994
0.998
0.996
0.995
0.995
0.997
0.997
DRYBEAN
AUROC 7
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
(0.001)
Rank avg
6.22
5.44
8.11
7.44
3.11
5.56
5.33
3,56
3.11
4.33
the one in Tsang et al. (2017). We consider the I norm of each estimated component (i.e. II (ls))li) as the important score, and select the components whose important scores are large. We generate synthetic datasets from Y = f (xx + 6, where f is the true prediction model defined in Appendix B.1 for k 1,2,3. Then, we apply ANN N and NB²M to calculate the importance scores of the main effects and second order interactions and examine how well they predict whether a given component is signal. Table 2 compares the AUROCs of ANOVA-T PNNN, 2 and NB3M, which clearly indicates that ANOVA- outperforms the baseline models in component selection. More details regarding component selection with ANOVA- TNN are given in Appendix B.4.
1 11 プド= (f((x)) fs(x)))
SC(fs) = 12 Σ E1-1(1((x))
for given pre-selected n many input vectors x,, I....,7, n, where f&g j = I,..., 10 are the 10 estimates of fs and fs is their average. A smaller value of SC(fs) means a more stable estimation (and thus more reliable interpretation).
We compare the overall stability score SC()) Sc(ss)\\S For each of nine benchmark datasets, we calculate the overall stability scores of ANOVA-TPNN, NAM, and NBM, whose results are given in Table 1. The re- sults again confirm that ANOVA-TPNN is superior in terms of the stability of component estimation.
4.3. Prediction performance
We compare prediction performance of ANOVA- with baseline models. We randomly split the train, validation and test data into the ratio 70/10/20, where the validation data is used to select the optimal epoch and the test data is used to measure the prediction performance of the estimated models. We repeat this random split 10 times to obtain 10 performance measures for prediction. For the performance measure, we use the Root Mean Square Error (RMSE) for regression datasets and the Area Under the ROC curve (AU- ROC) for classification datasets.
The stability of ANOVA-TPNN with respect to the choice of initial values are illustrated in Appendix C.2.
4.2. Performance in component selection
To investigate how well ANOVA-TPNN selects the true signal components, we conduct an experiment similar to
6

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Tensor Product Neural Networks for Functional ANOVA Model
Table 4: Prediction performance on high-dimensional datasets. We report the averages (standard deviations) of the prediction performance for 10 randomly sampled training data from the ligh-dimensional datasets. The bold faces highlight the best results.
GA'M
GA-M
Dataset
Micasure
ANOVA
T'PNN
NA'M
NB'M
NID
ANOVA
TºPNN
NID N
NID NB²M
MICROSOFT (Qin & Liu, 2013)
RMSE 4
0.756 (0.061)
0.774 (0.001)
0.770 (0,001)
0.754 (0,001)
0.761 (6.001)
0.755 (0.001)
YAHOO (Yahoo, 2010)
RMSE 1 AUROCT
0.787 (0.002)
0.797 (0.002)
0.783 (0.002)
0.779 (0.001)
0.793 (0.002)
0.779 (0.002)
MADELON (Cayon, 2004)
0.587 (0.02)
0.587 (0.02)
0.582 (0.03)
0.605 (0.01)
0,568 (0.03)
0.594 (0.02)
Table 5:
Stability scores the high-dimensional datasets. For each dataset, stability scores of of GA M (GA²M) models are presented. Lower stability scores imply more stable interpretation. The bold faces highlight the best results.
GA MM
GA GA'M
Dalaset
ANOVA
T'PNN
NA M
NB' M
NID +
ANOVA TPPNN
NID NA
NID + NB²M
MICROSOFT
0.030
0.089
0.118
0.040
0.083
0.089
YAHOO
0.049
0.088
0.126
0.049
0.090
0.074
MADELON
0.070
0.137
0.141
0.076
0.090
0.086
Table 3 presents the results of prediction performance of ANOVA-TPNN, NODE-GAM, NAM, and NBM as well as two black box models including deep neural networks (DNN, Rosenblatt (1958)) and XGB (Chen & Guestrin, 2016)). At the final line, the average ranks of each model over the nine datasets are presented, which shows that -NOVA-TTPNN exhibits comparable or superior predic- tion performance compared to the baseline models. Details about the experiments are given in Appendix B.2.
Table 6: Results of the prediction performance and stability scores of ANOVA-TPNN and Spline-GAN on the CALHOUS- ING dataset. The optimal results are highlighted in bold.
Measure
ANOVA-T PNN
Spline-GA 'M
RMSE (sid) Stability sonre J
2,614(0,01)
0.636 (0.03)
0.012
0.033
Mcasure
ANOVA-T PPNN
Spline-GA 'M
RMSE i (std)
0.512 (0.01)
1.326 (2.058)
Stability score J
0.035
0.052
4.4. Application to high-dimensional data
To see whether ANOVA-TPNN is applicable to high- dimensional data, we analyze three additional datasets with input dimensions ranging from 136 to 699. See Table 13 of Appendix for details of these three datasets. For ANOVA- PNN,,, we include all main effects into the model. For ANOVA-T*PNN. however, the number of second order in- teractions too large so that considering all the main effects and second order interactions would be difficult unless very large computing resources are available. A simple alterna- tive is to screen out unnecessary second order interactions a priori and include only selected second order interactions (and all the main effects) into the model. In the experiment, we use Neural Interaction Detection (NID) proposed by Tsang et al. (2017) for the interaction screening. The num- bers of selected interactions are given in Appendix B.2.
4.5. Comparison between ANOVA-TPNN and the basis expansion approach
In this section, we conduct experiments to compare ANOVA- TPNN and Spline- Apline-GAM GAM which estimates each component by using cubic B-spline basis functions. We evaluate the pre- diction performance and stability of the component estima- tion in ANOVA- -ROVA and Spline-GAM on CALHOUSING dataset. We implement Spline- GAMM using pygam python package (Servén & Brummitt, 2018).
Table 6 presents the results of the prediction performance and stability scores of ANOVA-TPNY and Spline- GAM on CAL.HOSSING dataset. A surprising result is that ANOVA- TPNN is superior to Spline-GAM in both of prediction per- formance and stability of component estimation. By investi- gating details of the empirical results, we find that Spline- GAM is vulnerable to input outliers. That is, when there is an outlier input (i.e. an input vector in the test data locating outside the range of input vectors in the training data), the prediction the outlier could be inaccurate since the B- splie basis uses the linear extrapolation outside the domain of the training data. In contrast, the basis neural networks in TPNN use the sigmoid activation function which is bounded outside the range of input vectors and so robust to input out- liers. The details of experimental results for Apline-GAM are presented in Appendix N.
From Table 4 and Table 5, we observe that ANOVA-TPNN shows favorable prediction performance compared to NAM and NBM and estimates the components more stably on high-dimensional datasets. Note that the reported RMSE of 2 M with all second order interactions on MICROSOFT by Radenovic et al. (2022) is 0.750, which indicates that screening interactions using NID does not hamper prediction performance much.
7

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Tensor Product Neural Networks for Functional ANOVA Model
Contribation
0
05
0.5
05
0.
05
0,5
05
0.5
MedIne
HouseAge
AveRooms
AveBedrms
Population
AveOccup
Latibude
Longitude
Figure I: Plots of the functional relations of the main effects in ANOVA-T PNNN PNN CALHOUSING dataset.
Table 7: Importance scores of ANOVA-T' PNN CALHOUSING dataset.
Feature indes
4
5
Importance score
0.906
0.284
0.107
0.093
0.057
0,049
Table 8: Importance scores of ANOVA-T" T PNN on CALHOUSING dataset.
index
(7.8)
(1.7)
(3.8)
(1.8)
(4.8)
(2.7)
(1.5)
Importance
1,000
0.347
0.324
0.268
0.258
0.247
0.212
0.194
0.193
0.178
4.6. Experiment results for NBM-TPNN
is done in Section 4.2, and the functional relation identified by the functional form of each component tools for global interpretation.
We conduct experiment to evaluate NBM-TPNN. Table 9 shows the prediction performance and stability score of NNM-TT PNN on CALHOUSING dataset and ABALONE dataset. We observe that NBM-T' PNN also exhibits similar prediction performance and stability to ANOVA-T'PNN The plots of estimated main effects are given in Appendix F and the results computation time are given in Appendix We can observe that relative computation times of NBM- TPNN compared ANOVA-TPNI decreases as the number of input features increases.
4.7.1. ILLUSTRATION OF INTERPRETABILITY ON CALHOUSING DATASET.
PNN
Table 10: Feature descriptions of CALHOUSING dataset.
Feature name
Index
Description
Feature type
MedIn:
Median ingoiee block
Numerical
HouseAge
2
Median house age block
Numerical
AveRoams
Average number of rooms
Numerical
AveBedrms
Average
bedrooms
Numerical
Population
5
Population in block
Numerical
Table 9: Results of prediction performance and stability scores of ANOVA-T PNN and NBM-T -NN.
AveOscup
6
Average house occupancy
Numorical
atitude
Latitude block
Numerical
Longitude
8
Longitude of house block
Numerical
Datasct
Mcasure
ANOVA-T' PNN
NBM-T PNN
CALHOUSING
RMSE (std)
0.614 (0.001)
0.604 (0.001)
Stability score.
0,012
0,00
Local Interpretation on CALHOUSING dataset. We con- duct an experiment on CALHOUSING (Pedregosa et al., 201 la) dataset to illustrate local interpretation of ANOVA- PNN.. Note that ANOVA-T PNN is given as
RMSE 1 (sid)
0.725 (0.02)
(0.02)
WINE
Stability score J.
0.017
4.7. Interpretability of ANOVA-TPNN
We consider the two concepts interpretation: Local and Global which are roughly defined as:
i JANOVA PNN
Σ M,,r) j=1
Local Interpretation: Information about how each fea- ture of given datum affects the prediction. SHAP (Lund- berg, 2017) is notable example of local interpretation. For the functional ANOVA model, the predictive values of each component at a given datum would be considered as local interpretation.
Thus, it is reasonable treat (x;;) as the contribution of Tj to fx)). In fact, we have seen in Section 3.1 that this contribution is equal to SHAP (Lundberg, 2017). As an illustration, for given datum
x 0.2378, 0044 0.0036, -01036660.111 10.153,
0.3814, 00.067, 0.5541, -0.1111)
Global Interpretation: Information about how each fea- ture is related to the final prediction model. The importance scores of each feature (e.g. global SHAP (Molnar, 2020)) and the functional relations between each feature and the prediction model (c.g. the dependency plot of SHAP (Mol- nar, 2020) are examples of global interpretation. For the functional ANOVA model, the importance score, which can be defined as the 1 norm of the corresponding component as
the contributions of each feature to fx) are
(f..... fa) = 4.9900, 0.3278, 111.8.8-00000 -0.0456, 0.4432, 0.1730, 001173,, - 11.6190, 6.5184).
That is, the 7th variable contributes most to the prediction value of fxx), which can be interpreted as 'the housing price is low because the latitude not good'.
8

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Tensor Product Neural Networks for Functional ANOVA Model
Global Interpretation on CALHOUSING dataset. Figure 1 and Table 7 present the functional relations of each input feature to the prediction model learned by ANOVA-T' PNN and their importance scores. From these results, we can see that the location is the most important features and the south-west area is the most expensive.
Among the attributes, we set 'gender' as the target label and the remaining attributes are set as concepts for images. Since 'male' is labeled as 1 and 'female' as O, higher value of each component results in a higher chance of being classified as 'male".
Figure 2 presents the functional relations of the two main effects corresponding to the concepts 'No Beard' and "Wear- ing Lipstick', estimated on a randomly sampled training dataset with and without the monotone constraint, Note that the functional relations are quite different even though their prediction performances, which are given in Table 18 of Appendix D, are similar. It is a common sense that an image having the concept of 'No Beard' and "Wearing Lipstick' has higher chance of being female and thus the functional relations are expected to be decrease. Figure 2 illustrates that a completely opposite result to common sense could be obtained in practice. Implications of the opposite func- tional relations to interpretation of each image are discussed in Appendix D.
Table 8 describes the 10 most important components with de- scending order of the importance scores of ANONA-T'PNN normalized by the maximum importance score. The results are bit different from those of ANOVA-T In particular, the interaction between 'latitude' and 'longitude' emerges as a new important feature while the main effects of 'latitude' and 'longitude' become less important.
4.8. ANOVA-TPNN with monotone constraints
Monotone constraint. In practice, prior knowledge that some main effects are monotone functions is available and it is needed to reflect this prior knowledge in the training phase. notable example is the credit scoring model where certain input features should have monotone main effects (Chen& & Li, 2014; Chen & Ye, 2022).
4.9. Additional Experiments
An additional advantage of ANOVA-TPNN is to accommo- date the monotone constraints in the model easily. Suppose that fj is monotonically increasing. Then, ANOVA-TPNN can estimate/ f, monotonically increasingly by letting the 35k in equation (3) be less than or equal to 0.
In Appendix K. we explore ANOVA-TPNN with ReLU activation function. In Appendix L, we confirm that the component estimation of PNNN becomes highly unstable when the sum-to-zero condition is not imposed. Finally, in Appendix M, we discuss a method for enforcing the sum-to-zero condition after training NAM or NBM.
Without
5. Conclusion
In this paper, we propose a novel XAI model called ANOVA- TPNN for estimating the functional ANOVA model sta- bly by use of TPNNs. We theoretically demonstrate that ANOVA-TPNN can approximate a smooth function well. We also empirically show that prediction performance of ANOVA-TPNN is comparable to its competitors. Addition- ally, we proposed NBM- that improves the scalability of ANOVA-TPNN using the idea from NBM (Radenovic et al., 2022). One advantage of the basis function in NBM- TPNN is that the number of basis functions does not depend on the dimension of the input feature.
Ch
No Beard
Weoting Liprrc
No Beard
Weating Upskk
Figure 2: Plots of the functional relations of 'No Beard' and "Wearing Lipstick' on CELEBA dataset estimated by ANOVA- T'PNN with and without the monotone constraint.
Application to Image data. Monotone constraint helps avoiding unreasonable interpretation. To illustrate this ad- vantage, we conduct an experiment with image dataset. We use CELEBA (Liu et al., 2015) dataset which has 40 binary attributes for each image. To apply ANOVA-TPNN to CELEBA dataset, we consider Concept Bottleneck Model (CBM, Koh et al. (2020)) similar to the one used in Rade- novic et al. (2022). In CBM, rather than directly inputting the embedding vector derived from an image data through a CNN into a classifier, the CNN initially predicts each con- cept accompanied with each image. Then, these predicted values of each concept are subsequently used as the input of a DNN classifier. For our experiment, we use a pretrained ResNet18, where the last layer consists of a linear transfor- mation with a softmax activation function, and we replace the final DNN classifier with ANOVA-TPNN
Even though it is computationally more efficient that stan- dard basis expansion approaches, ANOVA-TPNN with high order interactions is still computationally demanding to be applied to high-dimensional data. A remedy would be to develop a tool to select and estimate signal components simultaneously in the functional ANOVA model, which makes it possible to detect higher order interactions in high- dimensional data.
9

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Tensor Product Neural Networks for Functional ANOVA Model
Acknowledgements
Chen, C.-C. and Li, S.-T. Credit rating with monotonicity- constrained support vector machine model. Expert Sys- tems with Applications, 41(16):7235-7247 2014.
This research was partly supported by Institute of Informa- tion & communications Technology Planning & Evalua- tion (IITP) grant funded by the Korea (MV)) RAS-2022-I122001 84, Development and Study of AI Techh nologies to Inexpensively Conform to Evolving Policy Ethics), the National Research Foundation of Korea(NRFF grant funded by the Korea govermmen((MSTT) (RS-2025- 00556079), the Institute Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea govemment(MSIT) ficial Intelligence Graduate School Program (Seoul National University)], and Samsung Electronics Co., Ltd.
Chen, D. and Ye, W. Monotonic neural additive models: Pursuing regulated machine learning models for credit scoring. In Proceedings of the third ACM international conference on Al in finance, pp. 70-78, 2022.
Chen, T. and Guestrin, C. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd acm sigkdd inter- national conference on knowledge discovery and data mining, pp. 785-794, 2016.
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Cortez, P., Cerdeira, A., Almeida, F., Matos, T. 2009. DOI:
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 feel must be specifically highlighted here.
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Fernandes, K., Vinagre, P., Cortez, P., and Sernadela, P. On- line News Popularity. UCI Machine Learning Repository, 2015. DOL: https://bo.org//1.244333/33
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Tensor Product Neural Networks for Functional ANOVA Model
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Tensor Product Neural Networks for Functional ANOVA Model
Supplementary material
Appendix
A. Proof of Theorem 3.3
A.1. Case of |S| =
Without loss of generality, we assume that S {j}. The proof is done by the following steps.
Firstly, we find function fe.j that approximates the true component 9o.; well.
Next, we decompose the approximation function fe. into a sum of TPNNs.
Step 1. Finding approximation function SE.j
Let 0 < pr. < pR- < do be the lower and upper bounds of the density of pýj respectively, Let [ [xx (xx)= be an interval partition of X3 such that Py (52) [K]. For 73 1/K3, we define (-)
K for some positive integer K. Then, we have Ix X2-11 S P for
1 Xi
(1((1)= = =1
Yj
1 X
Xt
(x(())
e 12,..., - 1}
7j
7
11 XK-1
(xx(x))
7
Note that for every x E Xz. Ek= (z) 1 and for every le E [K], 0 - fyx(+) < holds. Also, {l satisfy Lemma A.1 below, whose proof is provdied in Section A.2.
Lemma A.1. For any [K], we have
2pu K2
Ex, (X, $ 0n)] -
and
Ex, (X,)(XXXX 2 Co
K
for some positive constant Co-
Now, we consider the function defined
(x) Σ 8 (gl),((z)
k=1
where ōgs are defined
Ex,
6g
Ex, (..M((
13

---

Tensor Product Neural Networks for Functional ANOVA Model
for k E [K]. Then, for any x E Rz. we have
K
(0)(2) (e)
= (0) Σ (ol))(o)
k=1
K
Σ (z))(z) - 5c) )x)l(((()))
[00)(()) 8ul()a(z))
K
(05(1)) Ex, (X,) 1000,5 (x²]] (2)((2)
Ex; 11 (X,)]
Exx, (X,
190,3 (x) ] Ex, (X,,
Lia
Ex, 1
Ex, L (X3) 10 Xill
Ex, (x, (2)(())
(5)
where L is the Lipschitz constant for (+))
For a given x E Kz, let r E [K] be the index such that x E [x--1, xx] For E fr - 1,7,r + 17. we have
Ex, 12 (X,) X,ll
Ex, 1x X,)T(X E SR)
Ex, Ex -X,,),,,x,,,,,,,,,, & 2D]
Ex, ((.M((x))
Ex, 19. [X))[(X, E e
Ex, 1x, 23)
Ex, (X))( 2 p [Xx E e2/]
2Ex, 16x, (XX) $ 201
Ex, (IXX, 241
[x IXX, 2
2
4pu
+
PLK
CoK
C'
K
for C² 1pu Co
Also, for l: + 2, we have S X and X X and hence
PL
x x
(()((z)) So
7/
X -x+1 Xr+1
2o
7/3
+ exp(K)
To sum up, we have
K
Ex, (X,, ) x X,1| (x)(())
(x0,)(x) - ses( L
Ex, 1 (X,))
1=1
Ex, Ie (Xx) - Xx51
Ex, (X,x X,s| Ex, (x,))
Σ
(2x(T)+ +
Σ
(2a(())
L
Ex, (xx()
kE(r-Lnr+11
ke(r-Lnrt1)
3C"
1
< L
1 1 ()x((()))
K
14

---

Tensor Product Neural Networks for Functional ANOVA Model
Step Decomposing the approximation function fe.j into of TPNNs. Now, for fæ.5(z) = Ex= )z)),(()), our goal is to show that there exists By.k and (5(1888,), A = 1,.... K 1 such that
K-1 (z)) E (
for every x E Xy. To derive this result, we use Lemma A.2 below whose proof is provided Appendix A.2. Lemma A.2. For every T (2....,R. Kj, we define
xi
(1,T(2) 1
73 1
Xi-1
(T((1))
Xi
t E (2. .2....,T --11
1}
%
1 XT-1
(z))
7i
Then, for any given T (3...., K} and for any pI,Ts-... PP. satisfying
Ex,
Σ Exrllr((X) 0.
(6)
there exist PI,T-1:....
PT-1,T-11 -.T-----.-PP-1---1 1) and T such that
Σ P(T()T(1)= t=1
T-1 Σ
P
+ 97. (1-0 (x-x+-1)
)) + +
(x-xT-1
x
x²-
(7)
Yj
7/
(8)
and
(x-o(xx-xtx)) )) +
xx--
(x - (X,-Xe X7-1
Ex,
Σ PeT P =0, Ex, 17. ["
99)
7
73
Since
Ex, (X,)) Ex, Σ (rlga(()))
1x,; (X,)90X)
Ex (X,))))
Ex, (X,,
K
Ex, [eg [ei (X,,)
Ex,
Σ
(g, (X),
k=1
Ex,, [90.; (X₂)]
we can decompose fe.(.) into the sum of functions using Lemma A.2 and mathematical induction, i.c.,
X, [x,- (X,=Xa) )) TzO
( (X (X)-X( Xa
(x,5(x) (01,2l12(x) + (2)(22(z))
Σ
11k
75
7
k=2
15

---

Tensor Product Neural Networks for Functional ANOVA Model
Here, since
Ex, (21,2(1,2(x)) + p2.2222(Xi)1 o,
we can express (1,2(12(0) + 22,222,2 (2) using Bn, and 83,1 such that
(12211,(2) + p2.622((x) (=100,)
Similarly, since
Xj xx xx )) + Tho
Xj {x,- (x)=x
}
Exy ?k
1 -
78
7
we can express T]K
1 1-o
x-x2
+
Tha
x-xi
using 33.k and fgR such that
x Xx
))
-=xx
I
1 1-
TAOT
Yi
7
for k 2, ...y K- 1.
Completion of the proof.
By summarizing the results obtained in Step 1 and Step and letting K, K 1, we have
K,
3C
1
(0.5()
Σ
3
SE())
ST. T.
exp(K))
K
1
Cj
Ký+1 +1
where Cj is positive constant.
16

---

Tensor Product Neural Networks for Functional ANOVA Model
A.2. Proof of Lemma A.I For S x RT. have
I ---1
[((((2)
So
K2ng
1 + exp(K)
1. Also, for x 2 x² + K2. we have
x X
13x((1) 51
%
1
51 -
K2ng
1 + exp(KK)
Hence, we obtain
K2 x U
1
U
k²2 1
1
1
(X, 06)1 SP
X3 E
I exp(())
2p
S K
Also, for [xb_1 + Re,, Xk
11
have
k²
Xk-1
X
(xx(2)
20
7
7
20(K)
o o-(K)
1
2
for sufficiently large K. Hence, we obtain
1
1
Ex, (X,)(,,,,, E 011 >P X, E
No- +
K2
X
K2
2
Co
K
for some positive constant Cy.
17

---

Tensor Product Neural Networks for Functional ANOVA Model
A.3. Proof of Lemma A.2
We define
1) == Ex - pr-1),
T-1
Σ
Ex, (xx]exr(x;))orxxrrr..
==
Kr = Pt 11
fort E [T' 1]. Then, we obtain (7) by
E - X(-1 (110)(=-x))))
))
x - XT-1
Σ Keli (1x) + 17.
= 0
74
T-2
XT-2
-X--1
Σ (pi (2) + (ps-1- -11-1111 11) .0
(T 11))
+ r7
7/3
75
T-2
-1) 11). (0 2
(0(2- -0--2
XI-2
--xt-1)
Σ (pi - (x))) + (pr-1-11)
0 9i
7
+ +pr-1 -n)) . I ---x--1 (
2 XT-1
+ (pr - p1-
+ +1]
7i
T-1
(x-xク-1
Σ (pr (1)))(()) + (PT - -1) 10
+ 17
%
Σ (pr (x) +1
Σ (x).
18

---

Tensor Product Neural Networks for Functional ANOVA Model
A.4. Case of |S| d
Without loss of generality, we consider S {1, ..., d3. Similarly the case of S {j}, we define an interval partition of 23 as {12 1==1 fixk-11 xyl such that Hij ( and 1xk Xe S PLK for 1,..,, d. Additionally, let ke Sh x ... x S for ky E [K], j ,,,, d. For Sik..... we define ka (XS) as
d II j=1
Ex ke (xs)
Bas (z)))
where Ly,,(( (.) is defined in the same way as in Step 1 for |S| 1.
Consider the approximation function fe.s(Xs) defined as
fx,s(xs)
Σ
ka
(10)
k=...
where
SxS lei...k (xs) (ss)ssss Jxs lxi...k (xs) PP (dxs)
8kg kk.....k
Here, Ps I[ Py. where Py is the population distribution of Xj. Using a similar approach as is done in Step 1 for |S| 1, we have
Csd K
(0.s(xs) (e,s(xs) S
for some positive constant Cs. turn, we decompose fe,s by Lemma A.3, i.e., there exist B5,s and 05.k. for I,,,, (K - 1)d such that
K
fe,s(xs) =
Σ
ka l ta (XSs)
kam..kam
(K-1)a
Σ
Us.oos(xs0s.s)
Finally, we have
Ks (0..(())
5 Csd
Σ
3s.
K$ +1
1=1
where Ks (K 1)d and Cs is positive constant.
19

---

Tensor Product Neural Networks for Functional ANOVA Model
Lemma A.3. For given (x,)(xs) Σ
kiu kis Aps
Ma k (xs) defined in (10), there exist Bs, and 0g.k for
k ==1,,, (K- 1)181 such that
(K (K-1)
fe,s(xs) =
Σ
(s
k=
Proof.)
Without loss of generality, we assume that S = [1,..., Let part(f); be an interval partition for xy used when the function f is defined. For example, for fx.s(xs), we have
part(fe,s); fix x
part(fe.s))) K
for j 1..,, d. Since, Ex, (Xe, = 0, as is done in the proof of Step 1 for |S| 1, by Lemma A.2 and mathematical induction, we decompose (-)s(:) into sum of ((4) (K) (
S
such that
K-1
(x,s(Xs)
Σ (xs):
11
where (.)s satisfy the sum-to-zero condition, part(J (1 2 and part ( part
( )a K for any
l [K- 1]. Similarly, for any k E [K 1), we decompose /() (-) into sum of (1.2)) (),..., (1,2)
k-1) such that
K
(
12 (+)
where fky (.)s satisfy the sum-to-zero condition, part A b
part(J (12.2) = 2 and part p K for
(art(Jk,,N2/ 12,
(1.2)
j ..., and b2 1, I.... K 1. We repeat this decomposition to have (fkmnii k Je such that
K-1
K-1
[,,s()) = Σ
A.m..h.))) (-),
where f....: ke (:)s satisfy the sum-to-zero condition, part (l.....dd)
(arttt Ki..... ks Li)s = 2 for j E [d] and KG E [K - 1] for E [d].
For given i [d] and K; E [K- 1], we can express (l.....d) (.) using (Ther....ka The...kkk ld Ys, and fxk, I such that
(l..dd (xs) Dee..ikk
II
The Ter...n
(
(1-
Tij Xin
) + The bi
Themnika
Iij Xky
(11)
78
SS
5=1
2, x
2i xx-x
Since I1;
T .i.nke 1 1-
75
Th....kk
78
satisfies the sum-to-zero condition, similarly to Step
2 for |S| 1, it expressed
d II j=1
The....ka (1 (1-0
xie
)) + T
Xh. )} (m))
13 -xxx
B Bs 1ssss
75
for some Bs.A,..... by and fs.k .e have
Bay which implies that fe,s(-) is decomposed into the sum of (K 11) TPNNs, i.e.,
(K (K-1)*
fe,s(xs)
Σ
(8.05(xx10s..)
k=
20

---

Tensor Product Neural Networks for Functional ANOVA Model
B. Details of the experiments
All experiments are run with RTX 3090, RTX 4090, and 24GB memory.
B.1. Details for Synthetic datasets
Table 11: Test suite of synthetic functions.
(()
Y= 10X; + 10X, 20 0.3) - 0.6) + 20Xx, 5X5 + 10sin(mx;
((0)
y= >= sin (X44 log(Xx
Xs)
X10
Xe
V
X2X7
t
f(3)
Y exp Xi - X2 X 2 log(
X3
X3
x3)
+
X
10
Table 12: Distributions of input features corresponding each synthetic function.
f0)
X1, X2, X3, Aid U(0,1)
((2)
X2, Xo. X7, Xo and U10, 1) and X4, Xs. Xs, X10 rid U(0.6, 1).
f(
X2, Ns, X4. X5. Na. X7, Xs. Xo. X10 ri U(-1,1)
The synthetic function (()) is slightly modified version of Friedman's 's synthetic function used in (Chipman et al., 2010). f(2) and f(3) are taken from the interaction detection experiments in (Tsang et al., 2017). We generate 15K data samples from the distribution in the Table 12 and functions in the Table 11. Also, we divide them into train, validation and test datasets with ratio 0.7, 0.1 and 0.2, respectively. For all of the synthetic functions, the number of basis neural networks for each component C [p]. Ks, is set be 30.
B.2. Details of the experiments with real datasets.
Table 13: Descriptions of real datasets.
Dataset
Size
Number of features
Problem
Number of Class
CALHOUSING
21k
8
Regression
WINE
4k
11
Regression
ONLINE
40k
58
Regression
ABALONE
4k
10
Regression
FICO
10k
23
Classification
CHURN
7k
39
Classification
CREDIT
284k
30
Classification
LETTER
20k
16
Classification
DRYBEAN
13k
16
Classification
7
MICROSOFT
960k
136
Regression
YAHOO
700k
699
Regression
MADELON
2.6k
500
Classification
CELEBA
200K
Classification
2
Implementation of baseline model. We conduct experiments for all baseline models (NAM, NBM, NODE-GAM) using the official source codes. For DNN, we utilize the pytorch python package (Paszke et al., 2019) and for XGB, we utilize the xgboost package (Chen Guestrin, 2016).
Data descriptions. Table 13 summarizes the descriptions of 9 real datasets analyzed in the numerical studies.
21

---

Tensor Product Neural Networks for Functional ANOVA Model
Data preprocessing. Minimax scaling is applied to NAM and NBM, while the standardization used for NODE-GAM, DNN, and XGB. For ANOVA-TPNN, we transform each input feature based the empirical quantiles make the marginal empirical distribution of the transformed input features be close to the uniform distribution. Additionally, all categorical input variables are encoded using the one-hot encoding.
Learning rate. For all models except XGB, we set the learning rate of Adam optimizer 5e-3 and batch size 4096. We find the optimal learning rate of XGB via grid search.
Model hyperparameters. We set the architectures of NAM and NBM as defined in (Agarwal et al., 2021) and (Radenovic al., 2022). In other words, NA M, the dimensions of the hidden layers each component consists of (64,32,16) for MICROSOFT, YAHOO and MADELON, and [64,64,32] for other datasets. NA the hidden layers consist of [64.32, 16] for the ONLINE, CREDIT and DRYBEAN datasets, [64,16,8 16,8 for MICROSOFT, YAHOO and MADELON, and [64,64,32] for the other datasets.
For XGB, DNN, and NODE-GAM, we randomly split the train, validation and test data into the ratio 70/10/20 and evaluated its performance on the validation dataset using the model trained on the train dataset. We repeated this process 10 times with randomly split data, resulting in 10 prediction performance values for the validation datasets. Then, we selected the optimal hyper-parameters by the grid search based on the average the prediction performance values for the validation datasets.
Finally, with the optimal hyper-parameters selected earlier, fixed the model's 's hyper-parameters and used the 10 train-test dataset pairs obtained from the previous data splitting to train the model the train datasets and evaluate its performance on the test datasets.
For XGB, the range of hyper-parameters for the grid search is as below.
The number of :ree: 155.10022..000000055000000000000
max depth: {3, 5.7)
learning rate : {0.0001, 0.005, 0.01, ,
The hyper-parameters for NODE-GA M and NODE-GA²M is determined through grid search, using similar settings to those employed in (Chang et al., 2021). That is, the range of hyper-parameters for the grid search is as below.
The number of layer: {2, 4, 8}
tree depth: (6, 8}
The number of trees in each layer: {256. 512}
For DNN, we report the best prediction performance among the three architectures, as in (Radenovic et al., 2022).
2 hidden layers with 2048 and 1024 units
3 hidden layers with 1024, 512, and 512 units
5 hidden layers with 128, 128, 64, 64, and 64 units
For Spline-GAM, we implement it using pygam python package (Servén & Brummitt, 2018). We set the number of knot as 20 for each components and set Äs to be the same for all components S, i... Äs À for all S C [p]. Also, similar to the approach taken in (Chang et al., 2021), we find the best À penalty by using grid search on the range as below.
Range of À = [0.001, 0.01, 0.1, 0.3, 0.6, 1, 10, 100, 1000]
Due to the limitation of our computational environment, in ANOVA-TPNN. for all real datasets except ONLINE, CREDIT, MICROSOFT, YAHOO, and MADELON, we select the optimal parametrssssters by the grid search on the range Kş (10, 30, 50, 1003. For other datasets, we use the range Kş: (10, 30, 50) for grid search.
22

---

Tensor Product Neural Networks for Functional ANOVA Model
Selected components by NID for high-dimensional datasets. Table 14 presents the number of components used in training ANOVA-T4PNN and baseline models. All main effects are used, and the second order interactions are selected using NID (Tsang al., 2017). To find the optimal number of second order interactions, we conduct grid search over [100, 300, 500]. For MICROSOFT, 300 second order interactions are used; 500 for YAHOO, 500; and 300 for MADELON.
Table 14: The number components used training ANOVA-T PNN M,, and NB
Dataset
MICROSOFT
YAHOO
MADELON 500(Main) + 300(2nd)
# of selected components
136(Main) 300(2nd)
699(Main) 500(2nd)
B.3. Experiment details for image dataset.
For CELEBA image dataset, we use the Concept Bottleneck Model (CBM) in (Koh et al., 2020). The main idea of CBM(Koh et al., 2020) not to directly input the embedding vector derived from image data through a CNN into a classifier for classification. Instead, CNN predicts given concepts (attributes) for the image, and the predicted values for these concepts then used as an input of the final classifier. (Koh et al., 2020) used DNN for the final classifier which is a black box model. In this paper, we replace DNN with ANOVA-TPNN, NAM, NBM and NODE-GAM. For CNN that predicts concepts, we use linear heads for each concept on the top of the pretrained ResNet18.
All models are trained via the Adam optimizer with the learning rate 1e-3 and the batch size for training 256. For ANOVA- T PNN, Kg is determined through grid search on [10,30,50]. For NA M and NB' M. the neural network consists of hidden layer with sizes (64,64,32) and (256,122 128, 128), respectively. Due to the limitations of the computational environment, for ANOVA-T'PNN, we Ks s 10 for the main effects and Ks 3 for the second order interactions. For NA we use the neural network consisting of 3 hidden layers with sizes (66,168), and for NB the neural network consists of 3 hidden layers with sizes 22,, For NODE-GA' M and NODE- M, the number of trees and the number of layers are determined through grid search on the range as:
The number of layers: {2, 4}
Tree depth: : {4, 6}
The number of trees in each layer: {50, 125, 256}
B.4. Experiment details for component selection
An important implication of stable estimation of the components the ability of selecting signal components. That is, ANOVA-TPNN can effectively identify signal components in the true function by measuring the variations of the estimated components. For example, we consider the l norm of each estimated component (i.e, (xff((s))l) as the important score, and select the components whose important scores are large. This simple component selection procedure would not perform well if component estimation is unstable.
To investigate how well ANOVA-TPNN selects the true signal components, we conduct an experiment similar to the one in (Tsang et al., 2017). We generate synthetic datasets from Y = fx) + e, where f is the true prediction model and is a noise generated from Gaussian distribution with mean 0 and variance o- Then, we apply to calculate the importance scores of the main effects and second order interactions and examine how well they predict whether a given component signal. For the performance measure of component selection, we use AUROC obtained from the pairs of fsls and rs for all S C [p] with |S|] < 2, where fs are the estimates of fs in f and rs = I[]ls || > 0) are the indicators whether fs are signal or not.
PPNN, ? and NB² M
(1)
For the true prediction model, we consider the three functions f()), k = 2, 3 whose details given in Appendix B.I. We set the data size to 15,000 and set of to make the signal-to-noise ratio become 5.
23

---

Tensor Product Neural Networks for Functional ANOVA Model
C. Ablation studies.
C.1. The choice of Ks in ANOVA-TPNN.
Table 15 presents the averages (the standard deviations) of prediction performance of ANOVA-TPNN on 10 randomly sampled datasets from ABALONE for various values of Kg. For simplicity, we set all Ks. 5 S [p] to be equal. We observe that Kg around 50 yields the best results for ANOVA-T-PNN and Kg around 10 for ANOVA-T*PNN, The results suggest that the choice of optimal Kg is important for prediction performance and smaller Kş is required for model with higher order interactions.
Table 15: Results of prediction performance for various Ks on ABALONE dataset.
Ks
10
50
100
ANOVA-T T PSN
2.176 (0.09)
2.163 (0.08)
2.160 (0.09)
2.135 (0.09)
2.159 (0.08)
PNN
2.103 (0.08)
2.102 (0.08)
2,087 (0.08)
2.105 (0.08)
2.122 (0.08)
C.2. Impact of the initial parameter values to stability
We investigate how the choice of initial parameter values affects the stability of the estimated components of ANOVA-TPNN by analyzing synthetic datasets generated from (()), We fit ANOVA -NN on 5 independently generated datasets, and the 5 estimated main effects are presented in Figure 3. We observe ANOVA-T*PNN is insensitive to the choice of initial parameter values.
Concibotion
Trial
30
-39
-1.0
-28
-10
0.
03
Consibationn
Trial
90
68
Concrioatoon
1.
18
30
14
1.8
05
Trial 3
-05
-45
-45
05
0.3
05
05
Contribattinn
4
Contchaaion
10
10
18
14
Trial 5
-05
-25
-65
6.5
X2
X2
X3
X4
X;
Figure 3: Plots of the functional relations of the main effects in AN on synthetic datasets generated from (()
D. Prediction performance and Interpretability on CELEBA dataset.
Comparison with baseline models terms of prediction performance. We consider two types of CBMs: one is the joint concept bottleneck model (JCBM), where the CNN and the classifier are trained simultaneously. and the other the independent concept bottleneck model (ICBM), where the CNN is kept fixed, and only the classifier is trained. In Table 16, we compare prediction performance of various models for the final classifier in the JCBM, which shows the prediction performances of ANOVA-T* and ANOVA-T*PNI are comparable or superior to their competitors. In Table 17, we compare the prediction performance of ANOVA-T PNN, M, NB 'M, DNN, and Linear model. The hidden layers of DNN are consists of five layers with size of (128, 128.64.64.64). In ICBM, the prediction performance is inferior to other nonlinear models when the classifier is linear. However, ANOVA-T' PNN is found to have comparable prediction performance compared to other baseline models including NA M, N M, and DNN
24

---

Tensor Product Neural Networks for Functional ANOVA Model
Prediction performance with and without the monotone constraint. Table 18 presents the prediction performances of ANOVA-TPNN with and without the monotone constraint. For attributes 'Bald', 'Big Nose', 'Goatee' and 'Mustache', we apply the increasing monotone constraint, while for attributes 'Arched Eyebrows', 'Attractive', 'Heavy Makeup', 'No Beard', 'Wearing Earrings', 'Wearing Lipstick', 'Wearing Necklace', 'Wearing Necktie', we use the decreasing monotone constraint. Prediction performances are similar regardless of the monotone constraint but interpretation of the estimated model can be quite different which is discussed in the followings.
Global interpretation on CELEBA dataset. In ANOVA-T'PNN without the monotone condition, we select the two features which have incorrect global interpretations among the 10 most important ones. Table 19 gives the importance scores (normalized by of the maximum important score) and its ranks of 2 components obtained by ANOVA-T'PNY PNN on a randomly sampled data from CELEBA dataset.
Local interpretation on CELEBA dataset. In Table 20, we observe that Image 2-1 of Figure 4 is correctly classified when the monotone constraint is applied, whereas it misclassified without the monotone constraint. Despite Image 2-1 of Figure having 'No Beard', 'Heavy Makeup', and "Wearing Lipstick', the scores of these features without the monotone constraint make the probability of being male increase. However, ANOVA-T' PNN with the monotone constraint avoid these unreasonable interpretations and classifies the image correctly.
In Image 2-2 of Figure 4, we observe that PNN with the monotone constraint assigns a negative score to 'No Beard' that increases the probability of being classified as female. However, ANOVA-T PNN without the monotone condition assigns a positive score to 'No Beard' that increases the probability of being classified as male, even though 'No Beard' is present.
Note that understand why ANOVA-T' PNN with the monotone constraint classifies Image 2-2 of Figure 4 incorrectly PNN
because there is no bear in the image. In contrast, it is not easy to understand why ANOVA-T PNN without the monotone constraint classifies Image 2-1 of Figure 4 incorrectly. That is, imposing the monotone constraint is helpful to learn more reasonably interpretable models.
Attributes to which monotone constraints are applied. For attributes 'Bald', 'Big Nose", 'Goatee' and 'Mustache', we apply the increasing monotone constraint, while for attributes 'Arched Eyebrows', 'Attractive', 'Heavy Makeup', 'No Beard', 'Wearing Earrings', 'Wearing Lipstick', 'Wearing Necklace', 'Wearing Necktie', we used the decreasing monotone constraint.
Table 16: Accuracies (standard deviations) on CELEBA dataset in JCBM.
ANOVA-T'PNN
NODE-GA'M
NA'M
NB'M
NODE-GA"M PNN
NA
NR
0.985 (0.001)
0,981 (0.006)
0.982 (0.002)
0.980 (0.002)
0.986 (0.001)
0.981 (0.006)
0.986 (0.001)
0.980 (0.002)
Table 17: Accuracies (standard deviations) on CELEBA dataset in ICBM.
ANOVA-T'PNN
NA'M
NB'M
DNN
Lincar
0.929 (0.001)
0.977 (0.001)
(0.00))
0.366 (0.001)
0.876 (0.006)
Table 18: Results of prediction performance of ANOVA-TPNN with and without the monotone constraint.
Mensure
ANOVA-T' PNN
ANOVA-T'PM
Without Momolone constraint
Accuracy
0.985 (0.001)
0.986 (0:001)
With Monotone coustraint
Accuracy 7
0.984 (0.001)
0.985 (0.001)
Table 19: Importance scores and ranks for the important components.
Components
Monatone
No Beard
Wenring Lipatick
Score (Rank)
X
0.794 (3)
0.465 (4)
Score (Kank)
O
0.757 (6)
0.738 (7)
25

---

Tensor Product Neural Networks for Functional ANOVA Model
Table 20; Results of local interpretation with and without the monotone constraint.
Image index
Monolone
Hewvy Makeup
No heard
Wearing Lipstick
classified label
True label
0.030
0.035
female
-0.080
-0.161
-0.106
female
female
2.2
X
0.036
0.104
0.095
male
male
2-2
-0.081
-0.183
-0.106
lemale
male
Image 2-1
Image 2-2
Figure 4: Two misclassified images.
26

---

Tensor Product Neural Networks for Functional ANOVA Model
E. Additional experiments for the stability of ANOVA-TPNN.
E.1. Stability the estimated components with respect to variations of training data
We investigate the stability of components estimated by ANOVA-TPNN when training datasets vary. We analyze CALHOUS- ING (Pedregosa et al., la) and WINE (Cortez al., 2009) datasets and compare ANOVA-TPNN with NAM and NBM in view of stability of component estimation. For 5 randomly sampled training and test datasets, train ANOVA-TPNN, NAM, and NBM the training datasets and plot the functional relations of the main effects the test datasets.
Experiment for CALHOUSING dataset. Figures 5, 6 and present the plots of the functional relations of the main effects estimated by ANOVA-T PNN, NA MM, and NB' M for randomly sampled training datasets, and Figures 8, 9 and 10 present the plots of the functional relations of the main effects estimated by ANOVA-T"PNN NA M, and N for randomly sampled datasets. We observe that the 5 estimates of each component estimated by ANOVA -PNN are much more stable compared to those by NAM and NBM. Note that Figure 9, we observe that some components are estimated as constant function in NA M, which would be partly because the main effects are absorbed into the second order interactions.
Experiment for WINE dataset. Figures 11, 12 and 13 present the plots of the functional relations of the main effects estimated by ANOVA-T'PNN, NA'M, and NB'M M for 5 randomly sampled datasets, and Figures 14, 15 and 16 present the plots of the functional relations of the main effects estimated by ANOVA-T'PNN N and 2 for randomly sampled datasets. The results are similar to those of CALHOUSING dataset.
Qustributco
Trial
28
28
Contribution
Trial
6.6
0.6
8.5
0.5
85
6.6
Contribation
Trial
Contebutio
Trial
65
66
6/6
05
05
66
05
Ocstribation
Trial
255
05
255
0.5
25
0.5
3
05 Longitude
02
MedIne
HouseAge
AveRooms
AveBedrms
Population
AveOccup
Latitude
Figure Plots
functional relations of the main effects in ANOVA-T PNN on CALHOUSING dataset.
27

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Tensor Product Neural Networks for Functional ANOVA Model
Centriutio
Trial 1
008
933
63
830
0.03
0100
19
Contribation
Trial 2
85
12
Contriutto
Trial 3
Contribtton
Trial
Controutton
Trial 5
Medine
HouseAge
AveRooms
AveRedrms
Population
AveOccup
Langltude
Figure 6: Plots of the functional relations main effects in NA on CALHOUSING dataset.
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Tensor Product Neural Networks for Functional ANOVA Model
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Figure 8: Plots of the functional relations main effects in ANOVA-T PNN on CALHOUSING dataset.
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Figure 9: Plots of the functional relations the main effects NA 2M on CALHOUSING dataset.
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Tensor Product Neural Networks for Functional ANOVA Model
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Figure 10: Plots of the functional relations of the main effects in NB M on CALHOUSING dataset.
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Figure 11: Plots of the functional relations the main effects ANOVA-T' PNN on WINE dataset.
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Tensor Product Neural Networks for Functional ANOVA Model
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Figure 12: Plots of the functional relations of the main effects in NA M on WINE dataset.
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Figure 13: Plots of the functional relations main effects NB on WINE dataset.
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Tensor Product Neural Networks for Functional ANOVA Model
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Figure 14: Plots of the functional relations of the main effects in ANOVA-T PNN on WINE dataset.
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Figure 15: Plots of the functional relations of the main effects NA?M 2 on WINE dataset.
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Tensor Product Neural Networks for Functional ANOVA Model
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Figure 16: Plots of the functional relations of the main effects NB MM on WINE dataset.
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Tensor Product Neural Networks for Functional ANOVA Model
F. Stability of NBM-TPNN.
Figures and 18 show the plots of the functional relations of the main effects estimated by NBM-T-PNN on randomly sampled datasets from WINE and CALHOUSING dataset, respectively, which amply show that NBM-TPNN is also highly stable in estimation of components.
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Figure 17: Plots of the functional relations of the main effect estimated by NBM-T PNN on randomly sampled training data from CALHOUSING dataset.
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Figure 18: Plots of the functional relations of the main effects estimated by NBM-T PNN 5 randomly sampled training data from WINE dataset.
34

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Tensor Product Neural Networks for Functional ANOVA Model
G. Additional experiments for component selection
Table 21 presents the averages (the standard deviations) of the prediction performance of the models used the component selection experiment in Section 4.2, which indicates that ANOVA- -PNNN, NAM and NBM perform similarly.
Table 21:
The results of prediction performance. We report the averages and standard deviations of RMSEs of ANOVA- PNNN, M and NB²M on 10 synthetic datasets generated from ((), p(a) and f(3)
GA M
Synthttic function
Measure
ANOVA
NA*M
NB*M
TPPNN
f(l)
3,483
3,474
3.511
(0.03)
(0.03)
(0.03)
((2)
RMSE
(0.001)
(0.005)
(0.001)
((3)
RMSE
0.161
0.183
0.137
(0,003)
(0.016)
(0,003)
35

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Tensor Product Neural Networks for Functional ANOVA Model
H. Comparison of ANOVA-SHAP and Kernel-SHAP
In this section, we conduct an experiment to investigate the similarity between ANOVA-SHAP and Kernel-SHAP (Lundberg, 2017). We calculate ANOVA-SHAP from ANOVA-T PNN approximating given black-box model. For the black-box model, we use XGB with 100 trees, a maximum depth of 7, and learning rate of 0.1 trained on WINE dataset. Below, compare the SHPA values for three imputs.
For input (6.9, 0.22, 0.32, 9.3, 0.04, 22, 110, 0.99, 3.34, 0.54, 10.7)", the SHAP values are given as:
Kemel-SHAP (0.057, 0.090, 0.065, 0.088, -..009, -0.136, 0.034, -..119 049 0.040, -0.082) ANOVA-SHAP (-000A--SAPP.0.01,,,..0000 0.104, 0.132, 0.006, -0.118, 0.055, -0.163, 0.093, 0.016, -0.081).
For input X2 (7.00, 0.17, 0.33, 4.00, 0.03, 127, 0.99, 3.19, 0.39, 3.99, 10.6). the SHAP values are given as:
Kemee-SHAA (0.015,0.38) 0.327,0.031, -0.0800 0..4...........,...... 0.050 0.043, -..01, -0.115) ANOVA-SHAP (-0.015, 0.243, 0.093, -0.148, 0.024, -0.11, 0.075, 0.141, -0.061, 00.022, 0.1001.
For input X3 = (6.90, 0.25, 0.35, 9.20, 0.03, 42, 150, 0.99, 3.21, 0.36, 11.5) the SHAP values are given as:
Kemel-SHAP (-0.004, 0.018, 0.060, 0.092, 0.032, 0.047, 0.014, 0.024, -0.123, 0.027, 0.084) ANOVA-SHAP (-0.017,0.00 0.071, 0.128, 0.024, 0.088,0.062, -0.004, -0.062, -0.046, 0.118).
The results suggest that ANOVA-SHAP and Kernel SHAP are similar. At least, the signs are exactly the same. An obvious advantage of ANOVA-SHAL is computation. Only one model fitting for finding ANOVA-TPNE approximating a given black-box model is required for ANOVA-SHAY In contrast, Kernel SHAP requires training linear models for each data point. For illustration, if want calculate the SHAP values for 1000 data points, computation time of ANOVA-SHAP is approximately 6,500 times faster than that of Kernel SHAP.
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Tensor Product Neural Networks for Functional ANOVA Model
I. Additional Experiments for prediction performance comparison with Decision Tree
Table 22: Results of the prediction performance Decision Tree and ANOVA-TPNN.
Dataset
Measure
Decision Tree
ANOVA-T'PNN
ANOVA-T'PNN
CALHOUSING
RMSE
0.671 (0.02)
0.614
0..02
WINE
RMSE 4
0.811 (0.03) )
0.725 (0.02)
(0.02)
ONLINE
RMSE
1.119 (.266 )
(0.25)
1.111
ABALONE
RMSE
2.396 (0,08)
2.135 (0.09)
2.087 (0.08
FICO
AUROC
0.704 )
0.799 (0.007)
0.0007
CHURN
AUROC
0.676 (0.03)
0.839 )0.012)
0.842
CREDIT
AUROC
0.890 (.002
0.983 (0.005
(0.00
LETTER
AUROC
0.745 ( (0.001)
0.900 (0.003)
09.001
DRYBEAN
AUROC
0.975 (0.0002)
0.995 (0000
0.997 (0.001)
Table 22 presents the averages and standard deviations the prediction performance of Decision Tree (Breiman, 2017) for 10 trials. We observe that the performance of ANOVA- is significantly better than that of Decision Tree. We implement Decision Tree by using the seikit-learn python package (Pedregosa al., 201 1b) and turned by using the optuna python package based on below range of
Range of max depth [2 .12]
Range of min samples leaf [2,10]
Range of min min-samples cesppeessspltt-ee... [2,10]
Range of max leaf leae_ [2.10]
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Tensor Product Neural Networks for Functional ANOVA Model
J. Additional Experiments for runtime on various datasets.
Table 23: Results of runtimes of NA' M, NB' M, ANOVA-T PNN, and NBM-T' PNN.
PNN,,
Dataset
Size of dataset
of features
NA'M
NB M
ANOVA-T'PNY
NBM T'PNN
ABALONE
4K
10
6,0.000
3.0 sec
1.6 sec
CALHOUSING
21K
14.1 sec
4.1 sec
3.8 sec
3.5 sec
ONLINE
40K
58
15.6
65
9.8
We conduct experiments to assess the scalability of NBM-TPNN. We consider NA M, which has 3 hidden layers with 16, 16, and nodes; 10 basis DNNs for NB' M, which have 3 hidden layers with 32, 16, and 16 nodes; Kg = 10 for each component ANOVA PNN; and 10 basis functions NBM-T*PNN, Table 23 presents the results runtimes of NA NB* M, ANOVA-T PNN, and NBM-T' PNN on ABALONE, CALHOUSING, and ONLINE datasets. When the dimension of input features is small, there is little difference in runtime between ANOVA-T'PNI and NBM-T' However, the input dimension increases, the runtime gap becomes pronounced.
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Tensor Product Neural Networks for Functional ANOVA Model
K. Additional Experiments for ReLU activation function
Table 24: Results of prediction performance of ANOVA-TPNN with ReLU. We report the averages of RMSE (standard deviation) and stability score for 10 trials.
Dataset
ANOVA T' PNN with ReI.U
ANOVA- T PNN
RMSE 4
ARALONE
2.148 (0.08)
2.135 (0.09)
WINE
0.735 (0.02)
0.725 (0.02)
Stability score
ABALONG
0.016
0.008
WINE
0,018
0.011
We conduct additional experiments to evaluate the performance of ANOVA-TPMN with the ReLU activation function i.e., (xx) = max(O, z). The yss for ANOVA-T' PNNs with ReLU and sigmoid are determined through grid search on the range [10,30,50]] Table 24 presents the results of stability scores and prediction performance of ANOVA-TPNNs with the ReLU and sigmoid activations ABALONE and WINE dataset. We observe that ANOVA-TPNN with Relu is slightly inferior ANOVA-TPNN with sigmoid in view of both prediction performance and stability of component estimation. This would be because ANOVA-TPNN with sigmoid is more robust to input outliers than ANOVA-TPNN with ReLU.
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Tensor Product Neural Networks for Functional ANOVA Model
L. ANOVA-TPNN without sum-to-zero condition
Table 25: Comparison of ANOVA-TPNN and GAM-TPNN. We report the stability score (normalized by the that of ANOVA-TI PNN or ANOVA-T ANOVA-T'PNN) PNN) for 10 trials.
ANOVA-T PIOVA-T'PPN PNN
GAM-T'
ANOVA-T-PNI
GAM-T-PNN
HOLOSIINNG
L,000
1,500
1.000
1.690
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Figure 19: Plots the functional relations the main effects on randomly sampled training data from CALHOUSING datasets.
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Figure 20: Plots the functional relations of the main effects on randomly sampled training data from WINE datasets.
We investigate the stability in component estimation ANOVA-TPNN without the sum-to-zero condition, which we denote
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Tensor Product Neural Networks for Functional ANOVA Model
GAM-TPNN, by analyzing CALHOUSING and WINE datasets. In GAM-T'PNN, approximate Jy (x, by
K
23 (2) byk )) + 13
byt
111z))
8ik 1 0
Pikk
Yijk
k=1
where }) 3R bysn Yyk are learnable parameters. GAM-T'PNN can be easily extended GAM-TTPNN in a similar way to PNN.
Table 25 presents the stability score of ANOVA-TPNN and GAM-TPNN based on 10 randomly selected datasets. Without the sum-to-7ero condition, we observe increasing in the stability score. particular, when the second order interactions are in the model, the main effects are estimated very unstably.
Figure 19 and 20 present the plots of the functional relations of the main effects CALHOUSING and WINE dataset in GNN. We observe that PNN estimates the components more unstable compared to ANOVA-T'PNN.

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Tensor Product Neural Networks for Functional ANOVA Model
M. On the post- post-processing for the sum-to-zero condition
Table 26: Stability scores for 'Latitude' and 'Longitude' of CALHOUSING dataset after post-processing
Model
ANOVA-T'PNN
NA
NBAM
Latitude
0.006
0.067
0.104
Longitude
0.015
0.094
0.103
ANN
NA 2
NB
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0.75
Latitude
Longitude
Lafilude
Longitude
Latitude
Longitude
Figure 21: Plots of the functional relations of 'Latitude' and 'Longitude' of CALHOUSING dataset after post-processing.
We have seen that NA 2M and NB3M are competitive to ANOVA-TPNN in prediction performance even though they are poor estimating the components. There is way to transform any estimates of the components those that satisfy the sum-to-zero condition (Lengerich et al., 2020).
We consider arbitrary estimated GA f(x) 30 + fs fs(ss. We fix and write fs instead of fs(xs) for notational simplicity.
For given S with |||| = d, we first transform Ís into
= Σ Σ (-1)a-k ( Xx, fsdlleevyy
(12)
k= VCS.
Then, Is satisfies the sum-to-zero condition (Hooker, 2007; Lengerich al., 2020). In turn for any S' S redefine ig: into
= --11111 / Xays
fadllesssssp
(13)
We apply the transformation (12) to fs for S = d - 1 to have Ss, and redefine fsy for S' C S by (13). We repeat this process sequentially until |S| = to have Fs for all S < [p] with |.|| < d which satisy the sum-to-zero condition.
Computational complexity of this post- processocns for a fixed input is (dd and thus computational complexity of calculating fs for all training data becomes (dn))) which is demanding when n or d is large. Furthermore, performing the post-processing requires storing thw whole dataset, which causes memory efficiency issues.
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Tensor Product Neural Networks for Functional ANOVA Model
Table 26 compares the stability scores of the main effects of 'Latitude' and 'Longitude' of CALHOUSING dataset estimated by ANOVA-T*PNN and post-processed M and M and Figure 21 draws the 5 functional relations of the estimated main effects of 'Latitude' and 'Longitude' on 5 randomly sampled training data. is observed that NA and NB M are still unstable even after the post-procest procceoccss,ng which suggests that instability in NAM and NBM is not only from unidentifabbility but also instability of DNN.
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Tensor Product Neural Networks for Functional ANOVA Model
N. Details of SAM
20
Spline-Ga Maine-GA'M M
Train
10
Test
Spline-GA'h
0
-10
1000
AveOccup
Figure 22: Seatter plot (y) test f Ssptino-CA va (2) )), and the plot main effect 'AvcOccup' estimated by Spline-GA" M.
Table 27: Contribution of the main effeets in Spline-GA G..
Main effect
'Medlnc'
'oussAgge
"AveRooms"
AveeBedmm
Popplation
AveeCcuup
"Laitdde
"Longtude"
Contribution
.64
0.36
4.18
4.82
0.25
-13.78
-0.77
1.55
In this section, we describe Spline-GAN GAM (Wood, 2017) which estimates each component by tensor product spline basis functions. For j E [p]. let 1..1, "..i....jjj. M..M,+ +5 be sorted knots into non-decreasing order on Xz. For given sequence of knots, the cubic spline basis functions for f.) are defined as
B3,c,(z) = T[(t, & x < (4+1)
B 11,AA1-1(25) + t++++ -x x
(z(x(T) Litk 4i
-1,4418-1
44R41 l++1
for k = 1,2,3 and l,..., My-
For general S S [p], Spline-GAMe GAM estimates fx(s) by
fs(xs) ~ (Bees (I,))as = (xs)
where B (xy) (B,1,(z;),.... M, (I, and the second derivatives of the basis functions the boundaries are set be zero. These basis functions called the Natural Cubic Splines (Hastie, 2009).
Furthermore, to estimate each component in the functional ANOVA model by the smooth function, Spline-GAM employs the penalty term. For S C [p]. the penalty term for (xs) is defined as
( = As Σ /
I (Fs((s) Par
2
dxs
JES
The smoothness of the spline model can be adjusted by tuning As > 0.
Inaccurate prediction beyond the boundary. Spline-GAM is model that sets knots based on the training data and interpolates between the knots using cubic spline basis functions. If the test data contains input outliers. the prediction performance of Spline-GAM may deteriorate for those outliers.
We conduct additional analysis on CALIOUSING dataset to examine the effect of input outliers. Figure 22 shows the scatter plot of (yfest JSptinc-Ga'' (xfest)) i I..... Tust and the plot of the functional relation of the main effect 'AveOccup' estimated by Spline-GAl GA M. In scatter plot, we observe that for given data point x² (10.2, 45, 3.17, 0.83, 7460, 1243, 38.32, -12)) the value of y is approximately 1.3, while the corresponding predic- tion from Spline-G.l M approximately -10.
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Tensor Product Neural Networks for Functional ANOVA Model
To investigate the reason behind such inaccurate prediction for the data point x², we explore the contribution of each main effect in Spline-GA M as shown in Table 27. We observe that the contribution of the main effect 'AveOccup' is abnormally high. Upon examining the plot of the main effect 'AveOccup' estimated by the Spline-GA'Me we conclude that the inaccurate prediction arises because the 'AveOccup' feature value of x" is an outlier and linear extrapolation of the cubic spline basis functions outside the range of inputs is used. In contrast, the TPNN with the sigmoid activation is bounded outside the range of inputs and so robust to input outliers.
45


Reviews:
[{"review_id": "D7OB8224Yi", "reviewer": "Reviewer_6BRT", "summary": "This paper introduces ANOVA Tensor Product Neural Network (ANOVA-TPNN), a novel neural network framework designed to estimate the functional ANOVA model with greater stability and accuracy. Theoretical analysis confirms that ANOVA-TPNN has universal approximation capabilities for smooth functions. Empirical studies across multiple benchmark datasets demonstrate that ANOVA-TPNN provides more stable component estimation and interpretation than existing models like NAM, NBM, NODE-GAM, and XGB. Additionally, the paper introduces NBM-TPNN, a variant that enhances scalability by ensuring the number of basis functions is independent of input feature dimensionality. Despite these advantages, the authors acknowledge computational challenges when handling high-order interactions, suggesting future work on component selection techniques.", "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": "Thank you for your valuable feedback and questions.\nWe have made every effort to address your insightful questions.\n> **W1.** While ANOVA-TPNN improves efficiency ...\n\n**Response to W1.**\n\nWe have conducted runtime experiments for the functional ANOVA model only with the main effects in Appendix K, whose results are summarized in Table 23.\nThese results suggest that ANOVA-TPNN is competitive with other baselines in terms of runtime complexity.\n\nAs the reviewer pointed out, we conducted additional experiments for the runtime complexity of higher order functional ANOVA models.\nWe analyzed Abalone data with the functional ANOVA models with up to the 4th order interactions and compared the runtimes of ANOVA-TPNN, NAM and NBM.\nThe hyperparameters of all models are set identically to those in Appendix K of the paper.\n\n**Table A.1**\n|Maximum order of interaction|1|2|3|4|\n|-|-|-|-|-|\n|NAM|6.6sec|11.1sec|28.3sec|79.3sec| \n|NBM | 3.0 sec|6.8sec|12.2sec|21.1sec| \n|ANOVA-TPNN|1.6sec|5.2 sec|22.7sec|82.7sec| \n|NBM-TPNN|1.5sec|4.1sec|7.8sec|16.4sec| \n\nTable A.1 presents the results, which amply show that ANOVA-TPNN is competitve with NAM and NBM in terms of runtime. \nIn addition, it is interesting to see that the runtimes of NAM and ANOVA-TPNN is super-linear in the order of interactions while those of NBM and NBM-TPNN are linear. \n\nWe emphasize that estimating high-order interactions is a common challenge in all functional ANOVA models, including NAM and NBM. To address this, we used Neural Interaction Detection (NID) to remove unnecessary components before training. Section 4.4 demonstrates its effectiveness through numerical experiments.\n\n> **Q1.** Can NBM-TPNN be further extended...\n\n**Response to Q1.**\n\nOne may consider the group lasso penalty, which is well-suited for selecting meaningful components while promoting sparsity.\nNBM-TPNN models component $f_{S}$ as\n\n$f_S(\\textbf{x}\\_{S} ) =  \\sum_{k=1}^{K} \\beta_{S,k} \\prod_{j \\in S} \\phi(x_{j} | \\theta_{k}),$\n\nwhere $\\beta_{S,k},\\theta_{k}$ are learnable parameters and $\\phi(\\cdot)$ is the basis neural network which is defined in Section 3.4 of the paper. Note that the parameters $\\theta_k$ are shared by the components while\n $\\beta_{S,k}$ are not, which makes it possible to apply a sparse penalty.\n\nGiven observed data ${ (y_{i},\\textbf{x}\\_{i}) }\\_{i=1}^{n}$ with $\\mathbf{x}\\_{i}=(x_{1,i},...,x_{p,i})^\\top \\in \\mathbb{R}^p$ and $y_i \\in \\mathbb{R}$, consider the objective: \n\n $ {1\\over n}\\sum_{i=1}^n\\bigg (y_{i} - \\sum_{S \\subseteq [p],|S|\\leq d}f_S(\\mathbf{x}\\_{S,i}) \\bigg )^2 +  \\sum_{S \\subseteq [p],|S|\\leq d}\\lambda_{S}\\Vert \\mathcal{B}\\_{S}\\Vert_2,$\n\nwhere $\\lambda_{S} > 0$ is a hyperparameter, $\\mathcal{B}\\_{S} = (\\beta\\_{S,1},...,\\beta\\_{S,K})^\\top$ , $\\mathbf{x}\\_{S,i} = (x_{j,i}, j \\in S)$ and $d$ is the highest order of interactions.\nThen, the group lasso penalty makes $\\mathcal{B}\\_{S}$ sparse on the component level, enabling component selection during training. This sparse estimation improves interpretability of higher-order interactions.\n\nHowever, group lasso penalty does not help reduce runtime complexity. The number of parameters remains proportional to $Kp^{d}$, regardless of sparsity. Developing new algorithms using forward selection or random search could be a promising future direction for future work.\n\n> **Q2.** How does the choice...\n\n**Response to Q2.**\n\nThe experimental results on the prediction performance and stability of ANOVA-TPNN when using the ReLU activation function are already given in Appendix L of the paper.\nWe found that ReLU slightly underperforms compared to the sigmoid version in prediction accuracy and component stability, likely because the sigmoid-based TPNN is more robust to input outliers.\n\nAs the reviewer suggested, it is interesting to investigate how the choice of a different network architecture, rather than our proposed TPNN model, affects the stability of component estimation.\nTherefore, we conducted additional experiments to evaluate the choice of the network architecture on the performance of stability.\nWe consider a deep neural network based tensor product model (TPDNN) which assumes \n\n$ f_{S}(\\textbf{x}\\_{S}) = \\prod_{j \\in S}g(x_{j} | \\theta_{j,S})$\n\nfor each component $f_S,$ where $g(\\cdot|\\theta_{j,S}) : \\mathbb{R} \\xrightarrow{} \\mathbb{R}$ is a 3-layer neural network with hidden sizes [32,32,16] and $\\theta_{j,S}$s are the learnable parameters.\nWe refer to the model that estimates components up to order $d$ in the functional ANOVA model using a TPDNN as ANOVA-T$^{d}$PDNN.\n\n**Table A.2**\n|Model|ANOVA-T$^{2}$PNN|ANOVA-T$^{2}$PDNN| \n|-|-|-|\n|RMSE|2.087(0.08)|2.148(0.08)| \n|Stability score|0.028|0.041|\n\nAs in Section 4.3, we ran 10 trials.\nTable A.2 shows the averaged prediction and stability scores for ANOVA-T$^{2}$PNN and ANOVA-T$^{2}$PDNN on the Abalone dataset.\nANOVA-T$^{2}$PNN outperforms ANOVA-T$^{2}$PDNN, likely due to TPNN's robustness to input outliers.\nWe will add these results to the Appendix.", "claims_and_evidence": "All the claims made in the abstract are supported by clear and convincing evident.", "methods_and_evaluation_criteria": "The proposed method makes sense for the problem.", "theoretical_claims": "I have not checked all of the proofs in detail. However, it seems to me that the proof for the universality is correct.", "experimental_designs_or_analyses": "The experimental designs are soundness.", "relation_to_broader_scientific_literature": "This paper introduce ANOVA-TPNN, a novel neural network framework designed to estimate the functional ANOVA model in XAI.", "essential_references_not_discussed": "Relevant works are cited through the papers.", "other_strengths_and_weaknesses": "**Strengths:**\n\n- The paper is clearly presented, and both theoretical and experimental justifications are provided.\n\n- The paper addresses a crucial issue in explainable AI (XAI) by improving the stability of functional ANOVA decomposition.\n\n- The authors provide a universal approximation proof, ensuring the validity of the proposed method.\n\n- The model demonstrates competitive prediction accuracy while offering superior component stability compared to baseline models.\n\n**Weaknesses:**\n\nWhile ANOVA-TPNN improves efficiency over traditional basis expansion approaches, the paper acknowledges that high-order interactions remain computationally demanding. Additional analysis of runtime complexity would strengthen the work. Beside of this concern, I do not see any other major weaknesses of the paper.", "other_comments_or_suggestions": "No.", "questions_for_authors": "- Can NBM-TPNN be further extended to handle higher-order interactions efficiently, perhaps through sparsity-inducing techniques?\n\n- How does the choice of activation functions and network architecture impact the stability of the estimated components?", "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": "7loXDv2fAo", "reviewer": "Reviewer_CFrs", "summary": "The paper proposes an approach for constructing interpretable machine learning models based on the functional ANOVA decomposition. The authors consider a decomposition of small order (1-2), and the decomposition terms are constructed with the basis functions represented by neural networks. To satisfy the condition of uniqueness of the expansion terms, the authors impose a natural restriction that the integral of each of the decomposition terms is equal to zero. The last condition is achieved by a special choice of the coefficient in the basis function.", "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": "Thank you for your valuable feedback and questions.\nWe have made every effort to address your insightful questions.\n\n> **Weakness 1 in Claims And Evidence.** In the introduction to your paper, you explicitly formulate the problem of interpretability of AI models. In this context, the task seems to consist of analyzing already existing and trained large neural network models. It is not immediately clear from the introduction that instead you are developing a directly interpretable model. \nIn this context, the relevant questions raised in \"Questions For Authors\".\n\n**Response to Weakness 1.**\n\nIn Line 12 on the right column of Page 1 of the manuscript, we mentioned that the functional ANOVA model is a transparent box-design model frequently used in explainable AI (XAI). \nThen, in Line 53 on the right column of Page 1, we stated that we propose a learning algorithm that estimates the components of the Functional ANOVA model using Tensor Product Neural Network (TPNN).\n\nIn response to the reviewer’s comments, we will revise the introduction to explicitly state that \"we propose a new transparent box-design model based on the functional ANOVA model and a specially designed neural network called Tensor Product Neural Network (TPNN)''  to improve clarity.\n\n$\\newline$\n\n> **Weakness 2 in Essential References Not Discussed.**\nReferences to relevant works are provided, but perhaps a more detailed discussion of the innovations proposed in comparison with older works is missing. Also in the context of ANOVA decomposition it is probably logical to cite the well-known work of Sobol (2001).\n\n**Response to Weakness 2.**\n\nIn Section 4.5, we compared ANOVA-TPNN with Spline GAM (older work), and found that our model is more robust to input outliers. For more details, please refer to Section 4.5 and Appendix O of the paper.\nAs suggested by the reviewer, we will include references to key works on functional ANOVA decomposition, such as Sobol (2001).\n\n\n$\\newline$\n\n>**Q1.** I would ask you to formulate more clearly what exactly you consider to be the main innovation proposed in your approach (in comparison with previous works).\n\n**Response to Q1.**\n\nThe main innovation is a specially designed neural network called **Tensor Product Neural Network** (TPNN) defined in Equation (4) in Page 4. \nThis neural network is not only flexible enough to satisfy the universal approximation property, as proven in Section 3.3, but also automatically satisfies the sum-to-zero condition without imposing any constraints on the learnable parameters, which allows the use of standard gradient descent algorithms.\nThe sum-to-zero condition theoretically guarantees the uniqueness of the functional ANOVA decomposition (Proposition 3.1 of the paper), and thus it is essential for the stable estimation of each component, as demonstrated in Section 4.1. \nExisting deep neural network-based functional ANOVA models, such as NAM and NBM, do not satisfy the sum-to-zero condition, and thus they are unstable in estimating components.\n**ANOVA-TPNN is the first neural network which is flexible (e.g. having the universal approximation property) but satisfies the sum-to-zero condition without any additional constraints.**\n\nNext, unlike traditional tensor product basis expansion approaches, TPNN does not lead to an exponential increase in the number of learnable parameters when estimating component functions $f_{S}$ as $|S|$ increases.\nFor more details, please refer to the **Remark** on Page 4 of the paper.\n\nFurthermore, since ANOVA-TPNN satisfies the sum-to-zero condition, it allows for fast and accurate computation of SHAP values by leveraging Proposition 3.2 of the paper.\n\n$\\newline$\n\n>**Q2.** Can this approach be used to interpret already trained neural network models (for example, as neural network attribution methods)?\n\n**Response to Q2.**\n\nYes, it is possible. \nA pre-trained neural network can be approximated using ANOVA-TPNN by treating the prediction values of a pre-trained neural network as outputs of the training data, and interpretation can then be provided by analyzing the estimated components as is done in Appendix D of the paper.\nWe will add post-hoc interpretation results of a pre-trained neural network using ANOVA-TPNN to Appendix D.\n\n$\\newline$\n\n>**Q3.** If the model you build is interpretable, then the question arises about demonstrating that interpretability and its usefulness. I did not see any examples of this in the experiments section.\n\n**Response to Q3.**\n\nIn Section 4.7 of the paper, we applied ANOVA-TPNN to image classification and illustrated how the prediction model can be interpreted based on the components estimated by ANOVA-TPNN. \nDue to the page limitations, details related to interpretations using ANOVA-TPNN are provided in Appendix D-E of the manuscript. \nFor example, in Appendix E, we described the results of both local and global interpretations of the image classification model using the components estimated by ANOVA-TPNN.", "claims_and_evidence": "In the introduction to your paper, you explicitly formulate the problem of interpretability of AI models. In this context, the task seems to consist of analyzing already existing and trained large neural network models. It is not immediately clear from the introduction that instead you are developing a directly interpretable model. In this context, the relevant questions raised in \"Questions For Authors\".", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Yes", "experimental_designs_or_analyses": "The authors conduct a comparison with modern alternative approaches on standard test datasets.", "relation_to_broader_scientific_literature": "The authors develop the approach proposed in works \"Neural Basis Models for Interpretability\" (2022) and Scalable Higher-Order Tensor Product Spline Models (2024).", "essential_references_not_discussed": "References to relevant works are provided, but perhaps a more detailed discussion of the innovations proposed in comparison with older works is missing. Also in the context of ANOVA decomposition it is probably logical to cite the well-known work of Sobol (2001).", "other_strengths_and_weaknesses": "--", "other_comments_or_suggestions": "--", "questions_for_authors": "1. I would ask you to formulate more clearly what exactly you consider to be the main innovation proposed in your approach (in comparison with previous works).\n2. Can this approach be used to interpret already trained neural network models (for example, as neural network attribution methods)?\n3. If the model you build is interpretable, then the question arises about demonstrating that interpretability and its usefulness. I did not see any examples of this in the experiments section.", "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": "bt9YG8sILj", "reviewer": "Reviewer_9V8Z", "summary": "The authors propose an approach for learning functional ANOVA decompositions form data. The neural network-based architecture they propose is designed such that it admits a unique functional ANOVA decomposition. The authors prove their architecture is a universal approximator for smooth functions which satisfy the sum-to-zero condition. By approximating a unique functional ANOVA decomposition (rather than a non-unique functional ANOVA decomposition) the authors seek to make the learning process more stable. They show their approach provides comparable performance to standard approaches for XAI on some standard benchmarks while being more stable.", "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": "Thank you for your valuable and insightful feedback.\nWe have made every effort to address your comments.\nDue to character limits, \"Comment\" is abbreviated as \"C\".\n\n>**C1 in Claims and Evidence** : One of the claims which...\n\n>**C2 in Supplementary Material :** I struggled to understand...\n\n**Response to C1 and C2.**\n\nWe do not claim that component estimation is accurate under the sum-to-zero condition, but rather that it is stable.\nThe sum-to-zero condition is not a unique condition to ensure the identifiability of each component in the functional ANOVA model. \nIn turn, different conditions result in different component estimations, and thus considering the accuracy of estimating component would not make sense.\n\nHowever, stability is crucial since we want that the interpretation to be robust to data perturbation.\nAppendix F aims to visually demonstrate the superior stability of ANOVA-TPNN over NAM and NBM, as measured by the stability score in Section 4.1.\nFigures 5–16 show that the main effects estimated by ANOVA-TPNN are consistent across all trials, while those from NAM and NBM vary significantly.\n\nThe main effects in the functional ANOVA model are useful for visual interpretation.\nPost-hoc methods (e.g., PDPs, SHAP) generate interpretation after model fitting.\nIn contrast, ANOVA-TPNN is an in-processing method that jointly performs model estimation and interpretation, ensuring consistency between the model and interpretation plots.\n\n>**C3 in Theoretical Claims.** The sketch of the proof for...\n\n**Response.**\n\nThe basis neural network in Equation (3) is inspired by a smooth version of a decision tree, where the indicator function is replaced with sigmoid functions.\nThus, the sum of TPNNs resembles the sum of smooth decision trees. We used techniques in Lemma 3.2 of [1] to derive the approximation property of TPNN.\n\n>**C4 in Experimental Designs Or Analyses.** Study 4.2 shows...\n\n**Response.**\n\nThe sum-to-zero condition is not a requirement for the true function.\nRather, any functional ANOVA decomposition can be redecomposed into one that satisfies the sum-to-zero condition (See Section 22 in [2]).\n\n>**C5 in Experimental Designs Or Analyses.** Study 4.3 shows...\n\n**Response.**\n\nThe smaller performance difference in Section 4.3 compared to Section 4.2 is not due to the sum-to-zero condition in the synthetic dataset, but rather due to the different evaluation criteria: Section 4.2 focuses on component selection, while Section 4.3 evaluates prediction performance.\n\nNA$^{2}$M and NB$^{2}$M perform poorly in component selection because they fail to properly separate main effects and second-order interactions.\nAs shown in Figures 9 and 10, when second-order interactions are included, main effects in NA$^{2}$M and NB$^{2}$M are absorbed into the interactions, resulting in near-constant main effects.\nIn contrast, ANOVA-T$^{2}$PNN uses the sum-to-zero condition, ensuring mutual orthogonality of components in the $L_2$ space, leading to more accurate identification of component effects, as shown in Figure 8.\n\n>**C6 in Experimental Designs Or Analyses.** \nStudy 4.6 compares ...\n\n**Response.**\n\nSee response to Reviewer 6BRT's W1.\n\n>**C6 in Supplementary Material.** \nAppendix D was very...\n\n>**C7 in Other Strengths And Weaknesses**\nIt would have been...\n\n**Response to C6 and C7.**\n\nIn response to the reviewer’s comments, we will move some contents from Appendix D to the main text and provide a more detailed explanation of the interpretability of the functional ANOVA decomposition in the final version of the manuscript.\n\n>**C8 in Questions For Authors.**\nOne claim you...\n\n**Response.**\n\nAs the reviewer mentioned, other identifiability conditions exist.\nAmong them, the sum-to-zero condition is adopted for two main reasons.\n\nFirst, the sum-to-zero condition is easy to implement during training.\nFor example, consider a functional ANOVA model for main effects: \n$ f(\\mathbf{x}) = \\sum_{j=1}^p f_j(x_j),$\nwhere $\\mathbf{x} = (x_1,..,x_p)^\\top$. \nWe may consider the identifiability condition as $\\forall i \\neq j$, $\\mathbb{E}[f_i(X_i)f_j(X_j)]=0$, but enforcing this for neural networks is difficult and the optimization is impractical.\n\nFinally, since ANOVA-TPNN satisfies the sum-to-zero condition, it enables fast and efficient computation of SHAP values using Proposition 3.2.\n\n>**C9 in Questions For Authors.**\nAs a reader less...\n\n**Response.**\n\nAs shown in Appendix E, by replacing the classifier in CBM with ANOVA-TPNN, the image model can be interpreted through the components estimated by ANOVA-TPNN.\nFor a given image, the contributions of concepts can be determined as in Table 20, and importance scores can be calculated as in Tables 14 and 15 to identify which concepts the model considers important for classification.\n\n**References**\n\n[1]. Ročková et al. Posterior concentration for Bayesian regression trees and forests.\n\n[2]. Christoph, Molnar. Interpretable machine learning: A guide for making black box models explainable.", "claims_and_evidence": "- A central motivating claim of their approach is that it is more stable than other methods for learning functional ANOVA decompositions from data. This claim is well supported by Tables 1, 2, and Appendix C.2.\n- The authors claim that their approach can approximate a class of smooth functions well. This is supported by their theoretical result showing universal approximation as well as the numerical studies where their approach performs similarly to other SOTA methods for XAI. \n- One of the claims which motivates the need for the sum-to-zero condition is that without identifiability, components become unstable and inaccurate. The study provided in Appendix F.1 was not convincing in my opinion. In particular, it wasn't clear to me what was meant by unstable and inaccurate in this case.", "methods_and_evaluation_criteria": "The authors evaluate their proposed approach on a number of synthetic benchmarks (consisting of three test functions) as well as a 13 real-world datasets. These datasets encompass a breadth of classification and regression problems making them well-suited for evaluating the proposed approach in my perspective.", "theoretical_claims": "The sketch of the proof for Theorem 3.3 looks correct but I found the details in Appendix A to be a bit challenging to follow. It would be helpful if you provided references to some standard results that you rely on (even if they are textbook results).", "experimental_designs_or_analyses": "- Study 4.1 shows that the author's approach is preferred in terms of component estimation stability\n- Study 4.2 shows that the proposed approach tends to learn a representation that is close to the true functional ANOVA on synthetic data. While this study is convincing, it would have been helpful to have included a study on a synthetic benchmark which does not satisfy the sum-to-zero condition by construction to understand how your approach might perform in less favorable situations. \n- Study 4.3 shows that the proposed approach achieves comparable or better predictive performance to methods from the literature across a number of standard benchmark problems. The fact the proposed approach is no longer doing far better than standard approaches (like in Study 4.2) calls into question how realistic the assumption of the sum-to-zero condition is in practice. Some discussion on this would have been helpful. \n- Study 4.4 applies the proposed approach to some high-dimensional problems convincingly demonstrating the approach can be useful on moderately sized problems. \n- Study 4.5 compares the proposed approach to Spline-GAM (Serven 2018). This is an interesting study which demonstrates that the proposed approach seems to be more robust to outliers than this prior approach. \n- Study 4.6 compares ANOVA-TPNN to NBM-TPNN. I think this study was comparatively weak. It would be useful to understand the computation time advantages of NBM-TPNN vs ANOVA-TPNN vs some standard approach for learning functional ANOVA decompositions.", "relation_to_broader_scientific_literature": "This work relates broadly to work on learning interpretable representations from data.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "- The proposed approach is a novel contribution as far as I'm aware. As discussed previously, the numerical studies do a good job of supporting the author's main claims.\n- It would have been helpful to include a more complete discussion of why learning such decompositions is useful for interpretability in the main text.", "other_comments_or_suggestions": "NA", "questions_for_authors": "- One claim you make is that a model that provides more stable estimates of component functions is more desired in XAI. It was unclear to me why you might prefer one particular ANOVA decomposition over another if their predictive accuracies are similar. In other words, can you provide some additional insight into why the particular sum-to-zero condition is desirable for XAI over the many other methods for enforcing some form of identifiability (i.e. through regularization, etc.)? \n- As a reader less familiar with XAI, can you provide a practical example of how one might use a learned ANOVA-TPNN model to drive decision making?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper proposes an approach for constructing interpretable machine learning models based on the functional ANOVA decomposition. The paper is clearly presented, and both theoretical and experimental justifications are provided. Some concerns on the runtime analysis and clarity of presentation have been successfully addressed by the authors' rebuttal, which should be incorporated into the camera-ready version.
Decision: Accept (poster)