A good start eases the journey

A good start eases the journey

For every layer,

Input : $x \sim \mathcal{N}(0, 1)$

Output: $y \sim \mathcal{N}(0, V)$

As numbers are encoded for computation, our layer outputs need to be bound to a finite value \(\forall x \sim \mathcal{N}(0, 1), \forall n \in \mathbb{N}, f_1 \circ ... \circ f_n(x) = f^{(n)}(x) \\ where\ f_i(x) = w_i \cdot x + b_i \\\) Xavier init \(\forall x \sim \mathcal{N}(0, 1), \forall n \in \mathbb{N}, \\ E[f^{(n)}(x)] = 0 \iff \forall i \in [\![1, n]\!], E[f_i(x)] = 0 \\ \iff \forall i \in [\![1, n]\!], E[w_i \cdot x + b_i] = 0 \\ \iff \forall i \in [\![1, n]\!], b_i = 0\)

\[\forall X \sim \mathcal{N}(0, 1), \forall n \in \mathbb{N}, var(f^{(n)}(X)) = var(f^{(n+1)}(X)) \\ \iff \forall X \sim \mathcal{N}(0, V), \forall n \in \mathbb{N}, var(W_n \cdot X) = var(X) \\ \iff E[X]^2 \cdot var(W_n) + E[W_n]^2 \cdot var(X) + var(X) var(W_n) = var(X) \\ \iff var(W_ n)= n \cdot var(w_n^{(i)}) = 1 \\ \iff var(w) = \frac{1}{n}\]




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