CALLIOPE · BCOM

Standard deep networks apply every layer uniformly to every input. We argue that depth should be dynamic and input-dependent, controlled by the compositional complexity of the data-generating process. Building on the Lie-pseudogroup framework of compositional symmetry , we show that the hierarchical flag $G=H_0 H_1 H_L$ of a generative model naturally prescribes which layers an agent needs to engage for a given input: layers corresponding to inactive symmetry scales should be gated off. We formulate a symmetry-gated residual network in which multiplicative gates have precise geometric meaning---they measure the presence of unresolved structure at each scale of the pseudogroup hierarchy. We then lay out a concrete experimental programme using Blender-generated datasets with known compositional structure to validate the theory.

A theory that tells you which layers a neural network actually needs for a given input, derived from the geometry of how the world generates data.

Standard deep networks are wasteful in a specific, principled way: every input — whether it's a static scene or a wildly articulated figure mid-leap — passes through every layer with equal weight. Prior work (Adaptive Computation Time, Mixture-of-Depths, LayerSkip) has noticed this and built engineering fixes that skip layers to save compute. But none of them explain why certain layers should be skipped for certain inputs. This paper argues that the answer lives in the structure of the data-generating process itself.

The key idea draws on a mathematical framework called Lie pseudogroups, developed in Ruffini's prior work. The intuition: any complex scene can be described as a composition of simpler transformations — camera pose, body position, limb articulation, facial expression, lighting, and so on. These transformations nest hierarchically, forming what the paper calls a "flag" of symmetry groups. Each level of the flag corresponds to a different scale of structure in the world. The central claim is that a neural network layer aligned with a symmetry level that isn't active in the current input should produce near-zero updates — and therefore should be gated off. Dynamic depth isn't a trick; it's a geometric consequence of compositional structure.

The paper translates this into a concrete architecture: a standard residual network where each layer gets a scalar gate α ∈ [0,1], learned with a small sparsity penalty. The gate has a precise geometric interpretation — it measures how much unresolved structure exists at that scale of the hierarchy. This is different from a free learned parameter; the theory predicts what the gate pattern should look like for inputs of known compositional complexity.

To test this, the paper proposes using Blender (the 3D rendering software) to generate datasets with fully controlled compositional structure — a rigged cat character where you can independently activate or freeze camera motion, body pose, spine, limbs, face, fur, lighting, and environment. This gives you ground truth over which symmetry levels are active, so you can directly check whether the network's learned gate pattern matches the theory's predictions. The four key predictions are: gates should be lower for layers aligned with inactive levels; effective depth should grow monotonically with the number of active generative levels; pre-gate residuals (not just post-gate outputs) should be small at inactive layers; and a model trained on simpler compositions should generalize its depth profile when a new level is introduced at test time.

The paper is a theoretical proposal plus experimental design — the Blender experiments are laid out in detail but not yet executed. The source is explicit about this. What's novel isn't the gating mechanism itself, but the claim that a principled theory of when and why to gate exists, and that it's testable with controlled synthetic data.