Commit 1b5b1885 by Florent Chatelain

### up nb

parent 184521a1
 ... @@ -42,21 +42,23 @@ ... @@ -42,21 +42,23 @@ ## Two-way puncturing of the kernel matrix ## Two-way puncturing of the kernel matrix The data matrix is \$X\in \mathbb{C}^{p \times n}\$ where \$p\$ and \$n\$ are the feature and sample size resp. The data matrix is \$X\in \mathbb{C}^{p \times n}\$ where \$p\$ and \$n\$ are the feature and sample size resp. Then Then \$\$ K = \left[ \frac 1p (X \odot S)' (X \odot S) \right] \odot B \$\$ \$\$K = \left[ \frac 1p (X \odot S)' (X \odot S) \right] \odot B\$\$ is the \$n\times n\$ *two-way punctured* kernel matrix where: is the \$n\times n\$ *two-way punctured* kernel matrix where: - \$S\$ is the Bernoulli iid \$(p \times n)\$ random matrix to select the **data** entries with rate `eS` - matrix \$S\$ is the Bernoulli iid \$(p \times n)\$ random matrix to select the **data** entries with rate `eS` - \$B\$ is the Bernoulli iid \$(n \times n)\$ random matrix to select the **kernel** entries with rate `eB` - matrix \$B\$ is the Bernoulli iid \$(n \times n)\$ random matrix to select the **kernel** entries with rate `eB` %% Cell type:markdown id: tags: %% Cell type:markdown id: tags: ## Simulations ## Simulations #### Figure 1. #### Figure 1. Eigenvalue distribution \$\nu_n\$ of \$K\$ versus limit measure \$\nu\$, for \$p=200\$, \$n=4\,000\$, \$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 \left[\begin{smallmatrix} 10 & 5.5 \\ 5.5 & 15\end{smallmatrix}\right]\otimes I_p)\$; \$\varepsilon_S=.2\$, \$\varepsilon_B=.4\$. Sample vs theoretical spikes in blue vs red circles. The two ``humps'' remind the semi-circular and Marcenko-Pastur laws. Eigenvalue distribution \$\nu_n\$ of \$K\$ versus limit measure \$\nu\$, for \$p=200\$, \$n=4\,000\$, \$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 \left[\begin{smallmatrix} 10 & 5.5 \\ 5.5 & 15\end{smallmatrix}\right]\otimes I_p)\$; \$\varepsilon_S=.2\$, \$\varepsilon_B=.4\$. Sample vs theoretical spikes in blue vs red circles. The two "humps" remind the semi-circular and Marcenko-Pastur laws. %% Cell type:code id: tags: %% Cell type:code id: tags: ``` python ``` python # Generation of the mean vectors for each sample # Generation of the mean vectors for each sample ... ...
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