Full-rank invariance conjecture for gradient-flow synaptic dynamics

Let $X \in \mathbb{R}^{n \times T}$ satisfy $C_X \succ 0$, and let $W(t)$ solve the continuous-time gradient-flow feedforward synaptic dynamics

$$\dot W = - \nabla_W S(W, M^{\star}, Y^{\star}) = -4W + 4\bigl(WC_XW^{\top}\bigr)^{-1/3}WC_X, \qquad W(0)=W_0.$$

Write $\mathcal{R}_{m,n}$ for the set of full-row-rank matrices in $\mathbb{R}^{m\times n}$. Full-rank invariance conjecture. If $W_0\in\mathcal{R}_{m,n}$, then $W(t)\in\mathcal{R}_{m,n}$ for all $t>0$. The conjecture is motivated heuristically by the divergence of the cost and its gradient near rank-deficient matrices, and is empirically validated in the paper; a general proof is not provided.