[Verse 1] When data flows through neural nets, we need to aggregate Average pooling takes the mean, identity's the gate Zero starts us off just right, addition builds the sum Associative and commutative, that's how monoids run Max pooling finds the largest peak, negative infinity waits As our identity element, while maximum operates [Chorus] Monoids everywhere, structure crystal clear Rho of sum phi x-i, DeepSets theorem here Pooling operations, attention mechanisms too Graph networks passing messages, monoids pull us through Associative combining, with identity so true Mathematical foundations in everything we do [Verse 2] Attention weights the values deep, weighted average flows Addition forms our operation, zero's where it goes Graph neural networks spread the word, messages cascade Sum aggregation builds the monoid, patterns never fade But permutation invariance needs a special form Decompose your function right, weathering the storm [Chorus] Monoids everywhere, structure crystal clear Rho of sum phi x-i, DeepSets theorem here Pooling operations, attention mechanisms too Graph networks passing messages, monoids pull us through Associative combining, with identity so true Mathematical foundations in everything we do [Verse 3] DeepSets reveals the secret that permutation-invariant maps Can always be decomposed as rho applied to sums perhaps First phi transforms each element, then we add them all Finally rho processes the result, answering the call This theorem guarantees the form, no matter what you choose If order doesn't matter, this structure you can use [Verse 4] String concatenation builds our text with empty string as start Logical operations join with truth and false apart Product monoids multiply dimensions, tuples combine with ease Each component runs its own monoid, harmony in the breeze From simple sets to complex graphs, the pattern stays the same Associativity and identity, playing nature's game [Bridge] But what if we need something more, associative but skewed Matrix multiplication shows us how it's viewed Left times right not right times left, order matters here Transformations cascading through, making patterns clear Neural pathways might require this asymmetric flow When sequence matters more than sets, that's the way to go [Chorus] Monoids everywhere, structure crystal clear Rho of sum phi x-i, DeepSets theorem here Pooling operations, attention mechanisms too Graph networks passing messages, monoids pull us through Associative combining, with identity so true Mathematical foundations in everything we do [Outro] Cortical columns computing distributed and wide Monoid structures guide us with mathematical pride
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