Phase-transition conjecture for support vector proliferation in ℓ₁-SVMs

Let $(\bX,y) \in \mathbb{R}^{n \times d} \times \mathbb{R}^n$ be an isotropic Gaussian sample, and let SVP denote support vector proliferation, namely coincidence between the solutions of the corresponding SVM and minimum-$\ell_1$-norm interpolation problems. Phase-transition conjecture for $\ell_1$-SVMs. The probability of SVP occurring for an $\ell_1$-SVM with $\bX$ and $y$ undergoes a phase transition around $d=f(n)$, for some $f(n)=\omega(n\log n)$. Formally, there exist positive constants $c$ and $c'$ with $c\leq c'$ such that

$$\lim_{n\to\infty}\mathbb{P}(\text{SVP occurs for an $\ell_1$-SVM})=\begin{cases}0&\text{if }d<cf(n),\\1&\text{if }d>c'f(n). \end{cases}$$

The conjecture predicts that support vector proliferation for $\ell_1$-SVMs requires a substantially higher-dimensional regime than the corresponding phenomenon for $\ell_2$-SVMs; the precise growth rate of $f(n)$ is not established by the conjecture.