[Verse 1] Implementation starts with forward pass design Simple recurrent network, weights align Almeida-Pineda takes the stage tonight Gradient descent through recursive sight Code your function, set the learning rate Numerical differentiation we calculate Compare the outputs, verify they match Mathematical precision in every batch [Chorus] Backward dynamics, J transpose y plus e Contractive mirror of forward harmony Check your gradients, make sure they're right BPTT versus cost per iteration fight Almeida-Pineda, recursive computation Cortical columns, distributed simulation [Verse 2] Forward dynamics stable, eigenvalues small Jacobian matrix controls it all When contraction holds in forward time Backward follows with the same design Spectral radius less than unity Guarantees convergence, can't you see Mirror property reflects the truth Mathematical elegance, living proof [Chorus] Backward dynamics, J transpose y plus e Contractive mirror of forward harmony Check your gradients, make sure they're right BPTT versus cost per iteration fight Almeida-Pineda, recursive computation Cortical columns, distributed simulation [Bridge] Memory allocation, time complexity Fixed point iteration versus history BPTT unrolls the temporal chain Almeida-Pineda breaks computational strain Per step analysis reveals the cost Efficiency gained, no cycles lost [Verse 3] Regression task becomes our testing ground Compare algorithms, see what can be found Iterations count, the clock keeps time Memory usage, performance paradigm Backward error propagation flows Through recurrent paths that network knows Equilibrium reached through patient wait Gradient accuracy worth the computational fate [Verse 4] Biological plausibility in neural design Cortical circuits where connections align Local computation, distributed state Synaptic weights that learn and update Homeostatic balance maintains the flow Neural plasticity helps networks grow From silicon chips to organic brain Learning algorithms break the training chain [Chorus] Backward dynamics, J transpose y plus e Contractive mirror of forward harmony Check your gradients, make sure they're right BPTT versus cost per iteration fight Almeida-Pineda, recursive computation Cortical columns, distributed simulation [Outro] Fixed point convergence, mathematical grace Neural architecture finds its place Distributed units process information Cortical wisdom through computation
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