Approximation of statistical dimension by distance to the scaled subdifferential

Let $f$ be one of the objectives $f_2$, $f_3$, $f_5$, or $f_6$ described in the preceding remark, let $\overline{x}$ be the signal under consideration, and let $X\sim\mathcal{N}(0,I)$. For $\tau\geq 0$, define

$$J(\tau):=\mathbb{E}\left[\operatorname{dist}^2(X,\tau\partial f(\overline{x}))\right].$$

Approximation conjecture. For these choices of $f$, the statistical dimension is well approximated by the squared Euclidean distance to the scaled subdifferential:

$$\min_{\tau\geq 0}J(\tau)\approx\delta(\partial f(\overline{x})).$$

The claim is motivated by empirical results and is intended to provide an approximation when explicit formulas for $J$ and its minimum are unavailable for the constrained objectives. Its validity is not established in the supplied text.