[Verse 1] Seven rules to make the gradients align When cortical columns compute and define First condition, settling dynamics must converge Spectral radius less than one, let the fixed point emerge Contractive flow brings the system to rest Mathematical stability puts us to the test [Chorus] Seven conditions, gradient equivalence Settling, energy, phases for reference Feedback pathways, learning rules contrastive Small perturbations, architecture massive Seven conditions, make the training work When biology and backprop don't shirk [Verse 2] Energy function guides the neural dance Scalar potential gives gradient descent a chance Symmetric connections make the math clean Or random feedback with alignment unseen Derivable dynamics from a single well Energy landscape has stories to tell [Chorus] Seven conditions, gradient equivalence Settling, energy, phases for reference Feedback pathways, learning rules contrastive Small perturbations, architecture massive Seven conditions, make the training work When biology and backprop don't shirk [Verse 3] Two phases dancing, free and target states Or compartments where the error awaits Temporal switching or spatial divide Both representations living side by side Difference between them encodes what we need Error signal plants the learning seed [Bridge] Consistent feedback must reach every weight Symmetric exact or random approximate Sign-symmetric falls between the extremes Feedback alignment fulfills the dreams [Verse 4] Local learning rules with contrastive form Target minus free becomes the norm Delta W proportional to the difference Hebbian plasticity with added significance Small perturbation limit makes it precise Beta approaches zero, mathematics suffice [Verse 5] When cortical circuits learn from experience Credit assignment through neural inference Backprop's precision meets biology's way Gradient flow without backward relay Inference networks solve the learning game Forward and feedback working just the same [Chorus] Seven conditions, gradient equivalence Settling, energy, phases for reference Feedback pathways, learning rules contrastive Small perturbations, architecture massive Seven conditions, make the training work When biology and backprop don't shirk [Outro] Architecture matching, depth correspondence Recurrent unrolling with feedforward concordance Tied weights connect the forward flow Seven conditions make gradients glow
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