Conjecture on two-dimensional loss landscapes of wide neural networks

Let $F$ be the loss of a neural network and let $\theta^*$ be a global minimum. Choose two vectors $v_1,v_2$ according to a specified rule, for example a Gaussian distribution, and define

$$G_{v_1,v_2}(\alpha,\beta)=F(\theta^*+\alpha v_1+\beta v_2),\qquad \alpha,\beta\in\mathbb{R}.$$

A sub-optimal basin is a basin of attraction whose loss is greater than the global minimum value. Two-dimensional landscape conjecture. For standard neural nets with width above a threshold $c_1$, or ResNet with width above a threshold $c_2<c_1$, $G_{v_1,v_2}$ has no sub-optimal basins. In addition, for standard neural nets with width below a threshold $c_1'<c_1$, $G_{v_1,v_2}$ has many basins. This conjecture is motivated by empirical visualizations relating width and trainability; the thresholds and the claimed basin behavior are not established in the source.