Vision Transformers demonstrate remarkable global modeling capacity but often underperform in data-scarce regimes. Distilling convolutional inductive biases from a CNN teacher provides an effective remedy while leaving the deployed model unchanged. However, general-purpose feature distillation transfers little in this setting. In CNN-to-CNN distillation, pooling, flattening, and logit-space projections remove the spatial grid that encodes locality and translation equivariance. Unlike a convolutional student, a ViT cannot readily reconstruct this structure on its own. In this paper, we propose iBKD, a distillation framework that preserves the spatial grid throughout the entire transfer process. Its core module, the Inductive Bias Attention Module, aggregates features from all student layers onto the teacher's grid using learned weights. It then enhances structural cues through channel and deformable spatial attention and injects them via convolutional cross-attention operating directly between spatial grids rather than token sets. The module is used only during training, leaving the deployed model as an unmodified ViT with no inference overhead. Across seven Transformer backbones and six data-scarce benchmarks, iBKD consistently outperforms both locality-guidance methods and general knowledge distillation baselines, with its advantage increasing as the amount of training data decreases.

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