Commit 1b5b1885 authored by Florent Chatelain's avatar Florent Chatelain
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up nb

parent 184521a1
......@@ -42,21 +42,23 @@
## 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.
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:
- $S$ 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 $S$ is the Bernoulli iid $(p \times n)$ random matrix to select the **data** entries with rate `eS`
- matrix $B$ is the Bernoulli iid $(n \times n)$ random matrix to select the **kernel** entries with rate `eB`
%% Cell type:markdown id: tags:
## Simulations
#### 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. <b>The two ``humps'' remind the semi-circular and Marcenko-Pastur laws.</b>
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. <b>The two "humps" remind the semi-circular and Marcenko-Pastur laws.</b>
%% Cell type:code id: tags:
``` python
# Generation of the mean vectors for each sample
......
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