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
Then, denote \(\mathcal{X}^B \sim \operatorname{DPP}(K^B)\) with
\(\mathcal{X}^B\) 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 \(\mathcal{X} \overset{d}{=} \mathcal{X}^B\).
See also
Theorem 7 in [HKPVirag06]
Finite case of Generic DPPs as mixtures of projection DPPs
Linear statistics¶
Expectation¶
Variance¶
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
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 \(B\).
See also
Number of points in the finite case
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