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### arxiv suppmat

parent 7dd94710
 ... ... @@ -246,7 +246,7 @@ "metadata": {}, "source": [ "#### Figure 4: \n", "Two-way punctured matrices $K$ for **(left)** $(\\varepsilon_S,\\varepsilon_B)=(.2,1)$ or **(right)** $(\\varepsilon_S,\\varepsilon_B)=(1,.04)$, with $c_0=\\frac12$, $n=4\\,000$, $p=2\\,000$, $b=0$. Clustering setting with $x_i\\sim .4\\mathcal N(\\mu_1,I_p)+.6\\mathcal N(\\mu_2,I_p)$ for $[\\mu_1^T,\\mu_2^T]^T\\sim \\mathcal N(0,\\frac1p[\\begin{smallmatrix} 20 & 12 \\\\ 12 & 30\\end{smallmatrix}]\\otimes I_p)$. **(Top)** first $100\\times 100$ absolute entries of $K$ (white for zero); **(Middle)** spectrum of $K$, theoretical limit, and isolated eigenvalues; **(Bottom)** second dominant eigenvector $\\hat v_2$ of $K$ against theoretical average in red. **As confirmed by theory, although (top) $K$ is dense for $\\varepsilon_B=1$ and sparse for $\\varepsilon_B=.04$ ($96\\%$ empty) and (middle) the spectra strikingly differ, (bottom) since $\\varepsilon_S^2\\varepsilon_Bc_0^{-1}$ is constant, the eigenvector alignment $|\\hat v_2^T v_2|^2$ is the same in both cases.**" "Two-way punctured matrices $K$ for **(left)** $(\\varepsilon_S,\\varepsilon_B)=(.2,1)$ or **(right)** $(\\varepsilon_S,\\varepsilon_B)=(1,.04)$, with $c_0=\\frac12$, $n=4\\,000$, $p=2\\,000$, $b=0$. Clustering setting with $x_i\\sim .4\\mathcal N(\\mu_1,I_p)+.6\\mathcal N(\\mu_2,I_p)$ for $[\\mu_1^T,\\mu_2^T]^T\\sim \\mathcal N(0, \\frac1p[ {\\tiny \\begin{matrix} 20 & 12 \\\\ 12 & 30\\end{matrix}}]\\otimes I_p)$. **(Top)** first $100\\times 100$ absolute entries of $K$ (white for zero); **(Middle)** spectrum of $K$, theoretical limit, and isolated eigenvalues; **(Bottom)** second dominant eigenvector $\\hat v_2$ of $K$ against theoretical average in red. **As confirmed by theory, although (top) $K$ is dense for $\\varepsilon_B=1$ and sparse for $\\varepsilon_B=.04$ ($96\\%$ empty) and (middle) the spectra strikingly differ, (bottom) since $\\varepsilon_S^2\\varepsilon_Bc_0^{-1}$ is constant, the eigenvector alignment $|\\hat v_2^T v_2|^2$ is the same in both cases.**" ] }, { ... ...
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