Architecture, data, and stochastic forward map

Let

Write

and $b_{\ell }\in {\mathbb{R}}^{n_{\ell }}$.

There are

weight positions, totaling

There are $N_b=1284$ bias parameters, so the total trainable parameter count is $396548$.

Fix $a\in [0,1]$, activation $\phi \in \{\mathrm{ReLU},\tanh \}$, input-noise intensity $\xi \ge 0$, and hidden-noise intensity $\sigma \ge 0$. For independent standard Gaussians $G_x,G_1,G_2,G_3$, define

The smallest maximizing index resolves ties. The population accuracy is

All accuracies in this paper are fractions, not percentages.

The input law is

with $Z\sim N(0,I_2)$ and

Additional input noise of intensity $\xi$ gives conditional variance $(0.8^2+{\xi }^2)I_2$. The clean-center test is a different input law with $X={\mu }_Y$.

Equation (eq:forward) is the source's ordering: activation, additive noise, then multiplicative damping. The output is a logit vector. An explicit softmax is unnecessary for its argmax or the cross-entropy training interface.