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arXiv · 2605.04946

Training-Time Batch Normalization Reshapes Local Partition Geometry in Piecewise-Affine Networks

Xuan Qi·Yi Wei·Fanqi Yu·Furao Shen·Vittorio Murino·Cigdem Beyan
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Abstract

Batch normalization (BN) is central to modern deep networks, but its effect on the realized function during training remains less understood than its optimization benefits. We study training-time BN in continuous piecewise-affine (CPA) networks through the geometry of switching hyperplanes and the induced affine-region partition. Conditioned on a mini-batch, we show that BN defines for each neuron a reference hyperplane through the batch centroid, and that breakpoint-switching hyperplanes are parallel translates whose offsets are expressed in batch-standardized coordinates and are independent of the raw bias. This yields an exact criterion for when a switching hyperplane intersects a local $\ell_\infty$ window and motivates a local region-density functional based on exact affine-region counts. Under explicit sufficient conditions, we show that BN increases expected local partition refinement in ReLU and more general piecewise-affine networks, and that this mechanism transfers locally through depth inside parent affine regions where the upstream representation map is an affine embedding. These results provide a function-level geometric account of training-time BN as a batch-conditional recentering mechanism near the data.

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Xuan Qi, Yi Wei, Fanqi Yu, Furao Shen, Vittorio Murino, Cigdem Beyan. 2026-05-06. Training-Time Batch Normalization Reshapes Local Partition Geometry in Piecewise-Affine Networks. https://arxiv.org/abs/2605.04946

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