Learning as a Geometric Phase Transition: Renormalization Group Flow and Anisotropic Symmetry Breaking in Deep Networks

Abstract

We formulate feature learning as a geometric critical phenomenon of the lifted tensor-product learning metric. The central object is not a scalar overlap, but the target-active geometry of \[ \mathcal N_{0,L}=\frac1N\sum_{r=1}^{L}Σ_{r\to L}\otimes T_{0\to r-1}, \] which entangles forward pullback survival with backward push-forward visibility. The neutral phase is target-isotropic: after restriction to endpoint target-active states and trace normalization, the lifted metric is proportional to the identity. Learning corresponds to an instability of this target-isotropic fixed point and to the emergence of traceless target-aligned eigentensors. We derive discrete Dyson expansions for local anisotropic insertions and their continuous Callan--Symanzik flow. Crucially, before constructing the full temporal mean-field theory, we identify the local spatial source of the $β$-functions directly from microscopic kinematics: asynchronous gradient updates generate synchronous metric strains, whose target-active symmetric traceless components act as curvature-like defects. The Wilsonian depth RG flow is then governed by the transport, balance, and coarse-grained irrelevance of these defects. Heavy-tailed spectra arise, under a scale-free counting hypothesis, as the spectrum of the target-active lifted geometry, with exponent addition in the matched pullback--push-forward sector. Finally, we relate this depth RG picture to temporal stochastic training dynamics and to the kinematic imprint of the learned channel on empirical weight Gram matrices.