Newton's method optimization for Deep Learning
If you take a look at section 2, it says


  The central idea motivating Newton’s method is that $f$ can be locally
  approximated around each $\theta$, up to 2nd-order, by the quadratic: $$ f(\theta + p) \approx q_\theta(p) \equiv f(\theta) + \nabla f(\theta)^Tp + \frac{1}{2} p^TBp \, \, (1) $$  where $B = H(\theta)$ is the
  Hessian matrix of $f$ at $\theta$. Finding a good search direction then reduces
  to minimizing this quadratic with respect to $p$.


To minimize, you need to take the derivative of (1) with respect to $p$ and set it to zero:

$$\Rightarrow \nabla f(\theta) + Bp = 0$$

which is equivalent to $Bp = -\nabla f(\theta)$.