Side-by-side 3-D phase portraits of position \(X\) and memory \(V\), in rescaled coordinates.
Loss. RMSProp is run on the quadratic loss
\[ f(X)=\tfrac12\,X^\top A X,\qquad \nabla f(X)=A X, \]
where the loss matrix is a diagonal spectrum rotated by three Euler angles,
\[ A=Q\,\Lambda\,Q^\top,\quad \Lambda=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3),\quad Q=R_z(\theta_z)R_y(\theta_y)R_x(\theta_x), \]
with \(R_x,R_y,R_z\) the standard rotation matrices (angles in degrees below).
Map (unscaled coordinates). With learning rate \(h\), decay \(\beta\), and stabilizer \(\varepsilon\) (all operations component-wise):
\[ V_{n+1}=\beta\,V_n+(1-\beta)\,\big(\nabla f(X_n)\big)^2,\qquad
X_{n+1}=X_n-\dfrac{h\,\nabla f(X_n)}{\sqrt{V_{n+1}}+\varepsilon}. \]
Map (rescaled coordinates). Writing \(\tilde X=X/h\) and \(\tilde V=\sqrt{V}/h\), the same dynamics become
\[ \tilde V_{n+1}=\sqrt{\,\beta\,\tilde V_n^{2}+(1-\beta)\,\big(\nabla f(\tilde X_n)\big)^2\,},\qquad
\tilde X_{n+1}=\tilde X_n-\dfrac{\nabla f(\tilde X_n)}{\tilde V_{n+1}+\varepsilon/h}. \]
The plots and the initial-condition inputs use these rescaled variables \((\tilde X,\tilde V)=(X/h,\ \sqrt{V}/h)\). The learning-rate field is given in units of \(10^{-6}\).