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¶

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

Hermitian kernel i.e. \(K(x,y)=\overline{K(y,x)}\)

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

Thinning¶

Important

The class of DPPs is closed under independent thinning.

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

\[\begin{split}\mathbb{E}\left[ \sum_{\substack{(x_1,\dots,x_k) \\ x_i \neq x_j \in \mathcal{X}^{\lambda}} } f(x_1,\dots,x_k) \right] &= \mathbb{E}\left[ \mathbb{E}\left[ \sum_{\substack{(x_1,\dots,x_k) \\ x_i \neq x_j \in \mathcal{X} } } f(x_1,\dots,x_k) \prod_{i=1}^k 1_{\{x_i \in \mathcal{X}^{\lambda} \}} \Bigg| \mathcal{X}\right] \right]\\ &= \mathbb{E}\left[ \sum_{\substack{(x_1,\dots,x_k) \\ x_i \neq x_j \in \mathcal{X} } } f(x_1,\dots,x_k) \mathbb{E}\left[ \prod_{i=1}^k B_i \Bigg| \mathcal{X} \right] \right]\\ &= \mathbb{E}\left[ \sum_{\substack{(x_1,\dots,x_k) \\ x_i \neq x_j \in \mathcal{X} } } f(x_1,\dots,x_k) \frac{1}{\lambda^k} \right]\\ &= \int f(x_1,\dots,x_k) \det \left[ \frac{1}{\lambda} K(x_i,x_j) \right]_{1\leq i,j\leq k} \mu^{\otimes k}(dx).\end{split}\]