Learning goal: convolution combines Local and weight sharing, but these are separate choices. Compare dense, local-unshared, and convolutional models across data conditions to see which structure reuses patterns across positions and which suffers when spatial neighborhoods are scrambled. The upper plot separately examines an untrained structural property. Finite-image boundaries can affect the resulting equivariance and invariance.

Left: draggable 12×12 probe image, without noise. Right: features from the same randomly initialized, untrained layer.

Browser-trained test accuracy: 384 images and 480 steps. This shorter experiment differs from the note; compare conditions within the same model rather than expecting identical scores.

Parameters, MLP = — Local, unshared = — Convolution = — Equivariance discrepancy max|f(Tx) − Tf(x)| = — Feature scale max|f| = — Mean trained-output change after a one-pixel shift = —

Observe the displayed equivariance discrepancy. For the convolution under this probe convention it is numerically zero, while the local-unshared discrepancy is around 10⁻¹, comparable to the feature scale. Its weights differ across locations. Dense hidden units have no native spatial translation operator in this construction. Translation equivariance comes from compatible sharing, not locality alone. The shifted-position task probes this reuse: convolution reaches 1.000 in the displayed run, while other models struggle with unseen positions. These five-pixel segments and the short browser training differ from the note's experiment. In the shuffled condition, convolution can lose its spatial advantage. In the position-label task, its score remains above 0.5 here. Boundary effects and position-sensitive features can carry location information; this result alone does not identify a unique mechanism.