It is desirable to restrict the number of components (i.e., encourage sparsity) for

DNN to represent the collaboration between a set of input variables.

Learning feature interactions can be the key for multivariate predictive modeling. ReLU-activated neural networks create piecewise linear prediction models.

Current sparse autoencoder (SAE) approaches to neural network interpretability assume that activations can be decomposed through linear superposition into sparse, interpretable features. Despite high reconstruction fidelity, SAEs consistently fail to elimi nate polysemanticity and exhibit pathological behavioral errors. We propose that neural networks encode information in two complementary spaces compressed into the same substrate: feature identity and feature integration. To test this dual encoding hypothe sis, we develop sequential and joint - training architectures to capture identity and integration patterns simultaneously. Joint training achieves 41.3% reconstruction improvement and 51.6% reduction in KL divergence errors. This architecture spontaneously d evelops bimodal feature organization: low squared norm features contributing to integration pathways and the rest contributing directly to the residual. Small nonlinear components (3% of parameters) achieve 16.5% standalone improvements, demonstrating para meter - efficient capture of computational relationships crucial for behavior. Additionally, intervention experiments using 2 2 factorial stimulus designs demonstrated that integration features exhibit selective sensitivity to experimental manipulations and produce systematic behavioral effects on model outputs, including significant nonlinear interaction effects across semantic dimensions. This work provides systematic evidence for (1) dual encoding in neural representations, (2) meaningful non-linearly encod ed feature integrations, and (3) introduces an architectural paradigm shift from post - hoc feature analysis to integrated computational design, establishing foundations for next - generation SAEs.

Autonomous Driving (AD) vehicles still struggle to exhibit human - like behavior in highly dynamic and interactive traffic scenarios. The key challenge lies in AD's limited ability to interact with surrounding vehicles, largely due to a lack of understandi ng the underlying mechanisms of social interaction. To address this issue, we introduce MPCFormer, an explainable socially - aware autonomous driving approach with physics - informed and data - driven coupled social interaction dynamics. In this model, the dynam ics are formulated into a discrete space - state representation, which embeds physics priors to enhance modeling explainability. The dynamics coefficients are learned from naturalistic driving data via a Transformer - based encoder - decoder architecture. To the best of our knowledge, MPCFormer is the first approach to explicitly model the dynamics of multi - vehicle social interactions. The learned social interaction dynamics enable the planner to generate manifold, human - like behaviors when interacting with surro unding traffic. By leveraging the MPC framework, the approach mitigates the potential safety risks typically associated with purely learning - based methods. Open - looped evaluation on NGSIM dataset demonstrates that MPCFormer achieves superior social interac tion awareness, yielding the lowest trajectory p red iction errors compared with other state - of - the - art approach. The prediction achieves an ADE as low as 0.86 m over a long prediction horizon of 5 seconds. Close - looped experiments in highly intense interact ion scenarios, where consecutive lane changes are required to exit an off - ramp, further validate the effectiveness of MPCFormer. Results show that MPCFormer achieves the highest planning success rate of 94.67%, improves driving efficiency by 15.75%, and re duces the collision rate from 21.25% to 0.5%, outperforming a frontier Reinforcement Learning (RL) based planner. A. Research motivation During recent years, Autonomous Driving (AD) has demonstrated significant progress within transportation systems [1] [2] . However, AD vehicles still face significant challenges in exhibiting human - like behavior in highly dynamic and interactive traffic scenarios such as off - ramp and unprotected left turns [3] [4] . One critical reason is that AD vehic les lack the understanding of the underlying mechanisms of social interaction between surrounding vehicles.

We pay special attention to the setting where the sample size n is small. This type of tests carries concrete significance in scientific studies.

The simultaneous application of multiple treatments is increasingly common in many fields, such as healthcare and marketing. In such scenarios, it is important to estimate the single treatment effects and the interaction treatment effects that arise from treatment combinations. Previous studies have proposed using independent outcome networks with subnetworks for interactions, or combining task embedding networks that capture treatment similarity with variational autoencoders. However, these methods suffer from the lack of parameter sharing among related treatments, or the estimation of unnecessary latent variables reduces the accuracy of causal effect estimation. To address these issues, we propose a novel deep learning framework that incorporates a task embedding network and a representation learning network with the balancing penalty. The task embedding network enables parameter sharing across related treatment patterns because it encodes elements common to single effects and contributions specific to interaction effects. The representation learning network with the balancing penalty learns representations nonparametrically from observed covariates while reducing distances in representation distributions across different treatment patterns. This process mitigates selection bias and avoids model misspecification. Simulation studies demonstrate that the proposed method outperforms existing baselines, and application to real-world marketing datasets confirms the practical implications and utility of our framework.

We thank reviewers for their valuable comments. We will incorporate all feedback in our final manuscript. EBMs generate more diverse samples. Extending the space of concepts requires retraining the network. This can miss interaction effects between parts (such as shadow casting).