moves uniformly within this space and away from the “near intersection" of $A$ and $B$. This behavior is also desirable as it locates points $x_A=P_A(x^+)$ and $x_B=P_B(2 P_A(x^+)-x^+)$, associated to the asymptotic orbit $x^+$, that are proximal on the two constraint sets (minimize $\|F\|$). In the neural network setting finding a best approximate solution arises in the training of autoencoders.

![RRR orbits (blue) for feasible (left) and infeasible (right) instances.[]{data-label="fig:fig1"}](fig1.eps){width="6.in"}

An interesting contrast between RRR and gradient descent algorithms is how they manage to avoid getting stuck. In strict gradient descent with loss function $\mathcal{L}$ and flow $F=-\nabla \mathcal{L}$, the loss is monotonically decreasing. The leading strategies for avoiding local minima where $\mathcal{L}>0$ are (i) the stochastic estimation of $F$ (SGD algorithm), thereby relaxing monotonicity, and (ii) exploiting application-specific structure when initializing the flow [@candes2015phase]. A third strategy, having special relevance to machine learning, is to use SGD only for highly over-parametrized models, where the loss landscape may be free of traps.

The speed of the flow, $v=\|F\|$, is weakly analogous to loss for the RRR algorithm. Like a restless shark, RRR keeps moving while $v>0$, ceasing only when the solution ($v=\|F\|=0$) is in its maw. However, unlike $\mathcal{L}$ in 