[Verse 1]
In the cortex where neurons fire and dance
Credit assignment needs a fighting chance
The Jacobian matrix holds the key
Parameters need structure we can see
Rank and sparsity must align
For learning signals to clearly shine
Without this order chaos reigns
No pathway back through neural chains
[Chorus]
Identifiability, the theorem we need
Mixture experts competing, each with different creed
Distributed codes spread wisdom around
But credit gets lost when signals aren't found
Structure the gradient, make the path clear
Mathematical truth is what we hold dear
[Verse 2]
Mixture of Experts divides the load
Each specialist walks their chosen road
The gating network decides who speaks
When input data has what it seeks
Modular minds beat dense arrays
When assumptions hold in provable ways
Efficiency blooms through separation
Smart division beats brute computation
[Chorus]
Identifiability, the theorem we need
Mixture experts competing, each with different creed
Distributed codes spread wisdom around
But credit gets lost when signals aren't found
Structure the gradient, make the path clear
Mathematical truth is what we hold dear
[Bridge]
Hinton saw the parallel dream
Processing spread across the team
Robust to damage, hard to break
But attribution's hard to make
Which unit caused the magic spark
When representations blur and arc
The tradeoff haunts our neural quest
Robustness versus credit's test
[Verse 3]
Cortical columns stand in rows
Each one computing what it knows
The brain's own mixture, gates unknown
Distributed wisdom, locally grown
From theorem to biology's dance
Mathematics guides our neural glance
Credit flows where structure permits
The gradient speaks in precise bits
[Verse 4]
Back-propagation needs its map
To close the learning feedback gap
Each layer's contribution clear
When gradients flow without the fear
Of vanishing into the void
When proper structure is deployed
The chain rule walks the neural tree
Assigning blame mathematically
[Chorus]
Identifiability, the theorem we need
Mixture experts competing, each with different creed
Distributed codes spread wisdom around
But credit gets lost when signals aren't found
Structure the gradient, make the path clear
Mathematical truth is what we hold dear
[Outro]
In cortical mathematics we trust
Jacobians clean away the dust
From chaos comes computational grace
When structure finds its rightful place