Manuscript Psy01-A@1 · Psy01-A.tex · section · 94477fa3

A valid differentiable relaxation

Stability, hard-mask discontinuity, and schedule dependence › A valid differentiable relaxation · lines 1326–1342

Psy01-A@1#eq:softcount

For fixed weights and temperature $T > 0$ , define

s_{ℓ, i} (q) = \frac{1}{1 + exp (( q - \abs W _{ℓ, i} ) / T )} .

Choose $q_{ℓ} (p)$ by

\frac{1}{N _{ℓ}} i \sum s_{ℓ, i} (q_{ℓ} (p)) = p, p = d_{e} (1 - f) \in (0, 1) .

The left side is continuous, strictly decreasing from one to zero, so $q_{ℓ} (p)$ exists uniquely. Define $B_{ℓ, i} (p) = W_{ℓ, i} s_{ℓ, i} (q_{ℓ} (p))$ .

Displayed formulas in this passage

Psy01-A@1#eq:softcount · line 1334
$\frac{1}{N _{ℓ}} i \sum s_{ℓ, i} (q_{ℓ} (p)) = p, p = d_{e} (1 - f) \in (0, 1) .$
unlabeled · line 1329
$s_{ℓ, i} (q) = \frac{1}{1 + exp (( q - \abs W _{ℓ, i} ) / T )} .$

Nearby in this section

proposition: Proposition: Smooth soft-density control
section: A valid differentiable relaxation