Minimal Sufficient Representations for Self-interpretable Deep Neural Networks

Zhiyao Tan; Liu Li; Huazhen Lin

stat.MEMar 25, 2026

Minimal Sufficient Representations for Self-interpretable Deep Neural Networks

Authors: Zhiyao Tan, Liu Li, Huazhen Lin

Organizations: Center of Statistical Research, School of Statistics, Southwestern University of Finance and Economics, Chengdu, China · School of Mathematics and Statistics, Wuhan University, Wuhan, China · New Cornerstone Science Laboratory, Southwestern University of Finance and Economics, Chengdu, China

Abstract

Deep neural networks (DNNs) achieve remarkable predictive performance but remain difficult to interpret, largely due to overparameterization that obscures the minimal structure required for interpretation. Here we introduce DeepIn, a self-interpretable neural network framework that adaptively identifies and learns the minimal representation necessary for preserving the full expressive capacity of standard DNNs. We show that DeepIn can correctly identify the minimal representation dimension, select relevant variables, and recover the minimal sufficient network architecture for prediction. The resulting estimator achieves optimal non-asymptotic error rates that adapt to the learned minimal dimension, demonstrating that recovering minimal sufficient structure fundamentally improves generalization error. Building on these guarantees, we further develop hypothesis testing procedures for both selected variables and learned representations, bridging deep representation learning with formal statistical inference. Across biomedical and vision benchmarks, DeepIn improves both predictive accuracy and interpretability, reducing error by up to 30% on real-world datasets while automatically uncovering human-interpretable discriminative patterns. Our results suggest that interpretability and statistical rigor can be embedded directly into deep architectures without sacrificing performance.

Figures & tables

Table 1

$\rho$	Methods	PE	MSE	Dims.	Prop.0	TPR	FPR
0	DNN	0.3559(0.0123)	2.8494(0.0984)	200	0.0392(0.0004)	1	1
	Index model(1)	0.3242(0.0037)	2.6126(0.0315)	1	0.0396(0.0004)	1	1
	Index model(5 ∗ )	0.1665(0.0111)	1.2747(0.0911)	5	0.0390(0.0005)	1	1
	Index model(10)	0.1921(0.0124)	1.4854(0.1043)	10	0.0388(0.0004)	1	1
	DFS	0.3144(0.0517)	2.4939(0.4316)	102.8000(61.4505)	0.6972(0.0295)	0.6770(0.2874)	0.3510(0.3463)
	LassoNet	0.3578(0.0967)	2.8710(0.8036)	107.2600(28.5950)	0.1766(0.0016)	0.8642(0.0957)	0.2084(0.2004)

Table 1: Simulation results for prediction, model complexity and variable selection for Setting 1. Predictive error (PE) and mean square error (MSE) are the model performance metrics. Dimension (Dims.) and proportion of zero elements in model parameters

\hat{\bm{\theta}}

(Prop.0) are the metrics for model complexity. The true positive rate (TPR) and false positive rate (FPR) are the metrics for variable selection. All metrics are reported by the means with standard deviations in parentheses.

Panel of Figure 1 — Figure 1: Test power for informative variables and empirical size for noisy variables in Settings 1–4 under correlation levels $\rho=0.2$ .

Panel of Figure 2 — Figure 2: The distribution under $H_{0}$ and the power of different representations with their singular value in Setting 3 ( $\rho=0.2$ ): (a) The distribution under $H_{0}$ ; (b) The power of representations; (c) The singular value of representations.

Dataset	Methods	PE	MSE	Dims.	#Variables	Prop.0
ADNI	DNN	0.4314(0.1102)	5.5006(1.4050)	192	192	0.0390(0.0009)
	Index model(10)	0.7793(0.1749)	9.9376(2.2303)	10	192	0.0407(0.0023)
	Index model(100)	0.5390(0.1006)	6.8727(1.2824)	100	192	0.0341(0.0004)
	Index model(150)	0.5580(0.0832)	7.1163(1.0608)	150	192	0.0297(0.0008)
	DFS	0.8108(0.0586)	10.3398(0.7477)	170.6667(12.2701)	170.6667(12.2701)	0.4461(0.0350)
	LassoNet	0.6099(0.1651)	7.7778(2.1055)	163.4444(32.7960)	163.4444(32.7960)	0.1719(0.0733)

Table 2: Variable selection and prediction performance for the ADNI and BlogFeedback datasets. “#Variables" indicates the number of selected variables. All metrics are reported by the means with standard deviations in parentheses.

Gene	p-value	Gene	p-value	Gene	Gene	p-value	Gene	p-value	Gene	p-value
RAB3D	0.0000	NOP14	0.0000	TBX5	KMT2C	0.087	XPO1	1.0000	RAB1B	0.5291
STIM1	0.0000	LANCL1	0.0043	CLPTM1	FMO1	0.0029	ARSA	0.0000	DCAF8	0.0000
MICU1	0.3492	MTSS1	0.9971	MAPKAP1	MAOB	0.0245	TSC22D4	0.0000	LOC $\\|$ RAB a	0.0000
TBPL1	0.0000	DIO2	0.0384	EIF2AK3	GBA	0.0048	CHAC1	0.0000	AKAP11	0.4741
PRR13	0.0000	BTAF1	1.0000	SCN1B	NAPEPLD	0.0000	ATP6AP1	0.0000	CLCN1	0.0000
TMEM127	0.0021	IFT74	0.0058	HAPLN2	ACADM	0.0003	PLCH1	0.0000	EPS15L1	0.5907

Table 3: The selected genes and their p-values for ADNI analysis.

Dataset	Methods	Acc.	Dims.	#Variables	Prop.0
MNIST	DNN	0.9583(0.0045)	784	784	0.0360(0.0003)
	DFS	0.9526(0.0101)	430.8000(169.0330)	430.8000(169.0330)	0.5550(0.0226)
	LassoNet	0.9508(0.0092)	413.2000(171.1723)	413.2000(171.1723)	0.2818(0.1219)
	DeepIn (Ours)	0.9730(0.0011)	331.6000(62.6086)	457.4000(3.7736)	0.5916(0.0092)
FashionMNIST	DNN	0.8874(0.0032)	784	784	0.0329(0.0007)
	DFS	0.8786(0.0071)	470.0000(240.4483)	470.0000(240.4483)	0.5022(0.0411)

Table 4: The variable selection and prediction performance of MNIST and Fashion MNIST dataset. “#Variables" means the number of selected variables. All metrics are reported by the means with standard deviations in parentheses.

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Neural network interpretability studies how models represent information and make predictions, and how to explain, test, or control their internal computations.

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