Non-symmetric DPP sampler

I’m finding a sampling method that can simulate attractive behaviors of given points (finite) rather than repulsive behaviors as in symmetric DPP. I found some papers studying how to learn a non-symmetric kernel of DPP that can model both repulsive and attractive behaviors.

• Gartrell, Mike, et al. “Learning nonsymmetric determinantal point processes.” arXiv preprint arXiv:1905.12962 (2019).
• Brunel, Victor-Emmanuel. “Learning signed determinantal point processes through the principal minor assignment problem.” Proceedings of the 32nd International Conference on Neural Information Processing Systems. 2018.

It seems that L need not be symmetric, provided that it is a P0-matrix. Furthermore, if L is K_ij * K_ji < 0, then item i and item j attract each other ( P({i, j} \in Y) > P(i \in Y)P(j \in Y) ).

How can I implement DPP(L) with L is a skew-symmetric matrix? It seems that I cannot use eigendecomposition since the eigenvalues of a real skew-symmetric matrix are purely imaginary, Could I use the MCMC instead?

[guilgautier]

The class of DPPs is stable under some set operations and some conditioning, i.e., the resulting say conditioned process is still a DPP with a different kernel. See e.g.,

• the documentation https://dppy.readthedocs.io/en/latest/finite_dpps/properties.html#conditioning, or
• Section 2.4 here https://arxiv.org/pdf/1207.6083.pdf, or
• Section 1.D here https://tel.archives-ouvertes.fr/tel-03132893/document

[guilgautier]

For now, DPPy indeed allows to define a FiniteDPP only with symmetric kernels, K or L.

However, core sampling methods can be called to sample from non-symmetric DPPs

To generate exact samples, you can exploit the relation K = L (L + I)^-1 = I − (L + I)^−1 and use dppy.exact_sampling.dpp_sampler_generic_kernel(K) https://github.com/guilgautier/DPPy/blob/cb4577f75ca998481ca6f248af10b19f986eca1c/dppy/exact_sampling.py#L322

To generate approximate samples (using an MCMC method) have a look at dppy.mcmc_sampling.dpp_sampler_mcmc(L) https://github.com/guilgautier/DPPy/blob/cb4577f75ca998481ca6f248af10b19f986eca1c/dppy/mcmc_sampling.py#L24

If you provide a more concrete example (even a toy example), I might be able to help you further.