[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