# Properties¶

## Generic DPPs as mixtures of projection DPPs¶

Projection DPPs are the building blocks of the model in the sense that generic DPPs are mixtures of projection DPPs.

Consider $$\mathcal{X} \sim \operatorname{DPP}(K)$$ and write the spectral decomposition of the corresponding kernel as

$K = \sum_{n=1}^{\infty} \lambda_n \phi(x) \overline{\phi(y)}.$

Then, denote $$\mathcal{X}^B \sim \operatorname{DPP}(K^B)$$ with

$K = \sum_{n=1}^{\infty} B_n \phi(x) \overline{\phi(y)}, \quad \text{where} \quad B_n \sim \mathcal{B}er(\lambda_n) \text{ are independent},$

$$\mathcal{X}^B$$ is obtained by first sampling $$B_1, \dots, B_N$$ independently and then sampling conditionally from $$\operatorname{DPP}(K^B)$$, the DPP with orthogonal projection kernel $$K^B$$.

Finally, we have $$\mathcal{X} \overset{d}{=} \mathcal{X}^B$$.

## Linear statistics¶

### Expectation¶

$\mathbb{E}\left[ \sum_{X \in \mathcal{X}} f(X) \right] = \int f(x) K(x,x) \mu(dx) = \operatorname{trace}(Kf) = \operatorname{trace}(fK).$

### Variance¶

$\begin{split}\operatorname{\mathbb{V}ar}\left[ \sum_{X \in \mathcal{X}} f(X) \right] &= \mathbb{E}\left[ \sum_{X \neq Y \in \mathcal{X}} f(X) f(Y) + \sum_{X \in \mathcal{X}} f(X)^2 \right] - \mathbb{E}\left[ \sum_{X \in \mathcal{X}} f(X) \right]^2\\ &= \iint f(x)f(y) [K(x,x)K(y,y)-K(x,y)K(y,x)] \mu(dx) \mu(dy)\\ &\quad + \int f(x)^2 K(x,x) \mu(dx) - \left[\int f(x) K(x,x) \mu(dx)\right]^2 \\ &= \int f(x)^2 K(x,x) \mu(dx) - \iint f(x)f(y) K(x,y)K(y,x) \mu(dx) \mu(dy)\\ &= \operatorname{trace}(f^2K) - \operatorname{trace}(fKfK).\end{split}$
1. Hermitian kernel i.e. $$K(x,y)=\overline{K(y,x)}$$

$\operatorname{\mathbb{V}ar}\left[ \sum_{X \in \mathcal{X}} f(X) \right] = \int f(x)^2 K(x,x) \mu(dx) - \iint f(x)f(y) |K(x,y)|^2 \mu(dx) \mu(dy).$
2. Orthogonal projection case i.e. $$K^2 = K = K^*$$

Using $$K(x,x) = \int K(x,y) K(y,x) \mu(dy) = \int |K(x,y)|^2 \mu(dy)$$,

$\operatorname{\mathbb{V}ar}\left[ \sum_{X \in \mathcal{X}} f(X) \right] = \frac12 \iint [f(x) - f(y)]^2 |K(x,y)|^2 \mu(dy) \mu(dx).$

## Number of points¶

For projection DPPs, i.e., when $$K$$ is the kernel associated to an orthogonal projector, one can show that $$|\mathcal{X}|=\operatorname{rank}(K)=\operatorname{Trace}(K)$$ almost surely (see, e.g., [HKPVirag06] Lemma 17).

In the general case, based on the fact that generic DPPs are mixtures of projection DPPs, we have

$|\mathcal{X}| = \sum_{i=1}^{\infty} \operatorname{\mathcal{B}er}(\lambda_i).$

Note

• For any Borel set $$B$$, instantiating $$f=1_{B}$$ yields nice expressions for the expectation and variance of the number of points falling in $$B$$.

Let $$\lambda > 1$$. The configuration of points $$\mathcal{X}^{\lambda}$$ obtained after subsampling the points of a configuration $$\mathcal{X}\sim \operatorname{DPP}(K)$$ with i.i.d. $$\operatorname{\mathcal{B}er}\left(\frac{1}{\lambda}\right)$$ is still a DPP with kernel $$\frac{1}{\lambda} K$$. To see this, let’s compute the correlation functions of the thinned process
$\begin{split}\mathbb{E}\left[ \sum_{\substack{(x_1,\dots,x_k) \\ x_i \neq x_j \in \mathcal{X}^{\lambda}} } f(x_1,\dots,x_k) \right] &= \mathbb{E}\left[ \mathbb{E}\left[ \sum_{\substack{(x_1,\dots,x_k) \\ x_i \neq x_j \in \mathcal{X} } } f(x_1,\dots,x_k) \prod_{i=1}^k 1_{\{x_i \in \mathcal{X}^{\lambda} \}} \Bigg| \mathcal{X}\right] \right]\\ &= \mathbb{E}\left[ \sum_{\substack{(x_1,\dots,x_k) \\ x_i \neq x_j \in \mathcal{X} } } f(x_1,\dots,x_k) \mathbb{E}\left[ \prod_{i=1}^k B_i \Bigg| \mathcal{X} \right] \right]\\ &= \mathbb{E}\left[ \sum_{\substack{(x_1,\dots,x_k) \\ x_i \neq x_j \in \mathcal{X} } } f(x_1,\dots,x_k) \frac{1}{\lambda^k} \right]\\ &= \int f(x_1,\dots,x_k) \det \left[ \frac{1}{\lambda} K(x_i,x_j) \right]_{1\leq i,j\leq k} \mu^{\otimes k}(dx).\end{split}$