Symbol f
f is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
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Why this formula appears here
This decomposition is worth writing out because it names what a reported simulation number actually is: not the true answer f(x) , but a sum of the true answer with a discretization error that shrinks with step size h , a rounding error governed by machine precision u , and — the term simulations share with no other approach on this list — an irreducible modeling error from every simplifying assumption baked into the equations themselves before any computer touches them. A formal proof and a peer-reviewed argument both aim at f(x) exactly; a simulation reports and, if it is honest, reports an estimate of how large the three error terms might be.
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Symbol x
x is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
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Published contexts (4)
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Equation 2 · Mathematics
Comparing the Main Approaches to Mathematics, Proof, and Scientific Computation
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
This decomposition is worth writing out because it names what a reported simulation number actually is: not the true answer f(x) , but a sum of the true answer with a discretization error that shrinks with step size h , a rounding error governed by machine precision u , and — the term simulations share with no other approach on this list — an irreducible modeling error from every simplifying assumption baked into the equations themselves before any computer touches them. A formal proof and a peer-reviewed argument both aim at f(x) exactly; a simulation reports and, if it is honest, reports an estimate of how large the three error terms might be.
Equation guide → · Article →Equation 5 · Mathematics
Comparing the Main Approaches to Mathematics, Proof, and Scientific Computation
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
This decomposition is worth writing out because it names what a reported simulation number actually is: not the true answer f(x) , but a sum of the true answer with a discretization error that shrinks with step size h , a rounding error governed by machine precision u , and — the term simulations share with no other approach on this list — an irreducible modeling error from every simplifying assumption baked into the equations themselves before any computer touches them. A formal proof and a peer-reviewed argument both aim at f(x) exactly; a simulation reports and, if it is honest, reports an estimate of how large the three error terms might be.
Equation guide → · Article →Equation 8 · AI Research
Probing, Sparse Autoencoders, Patching, and Steering: The Main Interpretability Methods, Compared
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
The formal objective has evolved since the earliest versions, and the direction of that evolution is itself informative. A dictionary decoder reconstructs the activation x from a sparse code f(x) :
Equation guide → · Article →Equation 2 · Interpretability
What a Circuit Explains: The State and Limits of Mechanistic Interpretability
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
Now the careful part. Consider what the training objective actually asks for. Writing x for an activation vector, f(x) for the sparse code and for the reconstruction, the objective has the form