19996         33.4 min         25.2 min
  -------------------- ------------------ ---------------

  : Simplex-constrained screening, clock time[]{data-label="tab:tab1"}

  -------------------- ------------------------ ------------------------
      **Dataset/**         **No Screening**          **Screening**
   **No. of Samples**   **($\ell_1$-constr.)**   **($\ell_1$-constr.)**
     *Synth1* 5000               13.1                   12.2 sec
     *Synth2* 10000            28.3 sec                 24.7 sec
      *RCV1* 20242             18.6 min                 14.9 min
    *news20B* 19996            33.4 min                 27.1 min
  -------------------- ------------------------ ------------------------

  : $L_1$-constrained screening, clock time[]{data-label="tab:tab2"}

Now we apply our Theorems \[thm:l1\_smooth\_constrained\] and \[thm:simplex\_smooth\] on variable of ${\bm{x}}$ and ${\bm{x}}_{{\triangle}}$ respectively to screen, in order to compare the two alternative screening approaches on the same problem. Note that the Wolfe gap is identical in both parameterizations, for any ${\bm{x}}$. One important point to note here is that dimension of ${\bm{x}}_{{\triangle}}$ is the double of the dimension of ${\bm{x}}$, and any $L_1$-coordinate value $x_i$ is zero if and only if both “duplicate” variables $x_{{\triangle},i}$ and $x_{{\triangle},n+i}$ are zero, where $n$ is the dimensionality of ${\bm{x}}$.

Therefore, the simplex variant (with more variables) performs a more fine-grained variant of screening, where we can screen each of the sign patterns separately for each variable. In Fig \[fig:fig1\], the blue curve illustrates the screening efficiency for the $L_1$-constrained screening case, while the red curve illustrate simplex constrained screening. Our theorems \[thm:l1\_smooth\_constrained\] and \[thm:simplex\_smooth\] are 