The residual Jacobian I + ∂f/∂x

Background

A residual block computes y = x + f(x), so its Jacobian gains an identity term:

$$\frac{\partial y}{\partial x} = I + \frac{\partial f}{\partial x}$$

That I is the gradient highway: even if ∂f/∂x collapses to zero, the gradient through the skip is still 1, so it can't vanish.

Problem statement

Implement residual_jacobian(df_dx) returning I + df_dx.

Input

• df_dx — the sublayer Jacobian ∂f/∂x, a square array (n, n).

Output

Returns an np.ndarray (n, n): the identity matrix plus df_dx.

Examples

Example 1 — vanished sublayer gradient still gives identity

Input:  df_dx = [[0, 0], [0, 0]]
Output: [[1, 0], [0, 1]]

Example 2

Input:  df_dx = [[1, 0], [0, 1]]
Output: [[2, 0], [0, 2]]

Constraints

• Return eye(n) + df_dx.
• Shape (n, n).

Notes

• The whole point: a zero sublayer Jacobian leaves the full Jacobian at I, never 0 — the gradient survives any depth.