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Replacing |x−1| with √((x−1)²+T) removes its cusp when T>0. The resulting score has a zero derivative at x=1, but its maximum is 1/(1+√T), which is below one; with T=0.01 the maximum is about 0.90909. The smooth and original score therefore have different peak calibration.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| T | Positive smoothing parameter |
| x | Activation input |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
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Original context
Read the surrounding source passage
z->inputs = {x};
z->backward_fn = [x, z]() {
x->grad += z->grad * (x->value > 0 ? 1.0 : 0.0);
};
return tape.record(z);
}
// Phi (smooth): z = 1/(1 + sqrt((x-1)^2 + T)), differentiable consciousness
inline GradNodePtr phi_smooth(GradNodePtr x, TapeRecorder& tape, double T = 0.01) {
double diff = x->value - 1.0;
double smooth_abs = std::sqrt(diff * diff + T);
double z_val = 1.0 / (1.0 + smooth_abs);
auto z = std::make_shared<GradNode>(z_val);
z->inputs = {x};
z->backward_fn = [x, z, diff, smooth_abs, T]() {
double denom = 1.0 + smooth_abs;
double dphi = -diff / (smooth_abs * denom * denom);Exact original notation
Phi (smooth): z = 1/(1 + sqrt((x-1)^2 + T)), differentiable consciousness