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Poster

Unconstrained MAP Inference, Exponentiated Determinantal Point Processes, and Exponential Inapproximability

Naoto Ohsaka

Keywords: [ Probabilistic Methods ] [ Approximate Inference ]


Abstract: We study the computational complexity of two hard problems on determinantal point processes (DPPs). One is maximum a posteriori (MAP) inference, i.e., to find a principal submatrix having the maximum determinant. The other is probabilistic inference on exponentiated DPPs (E-DPPs), which can sharpen or weaken the diversity preference of DPPs with an exponent parameter p. We prove the following complexity-theoretic hardness results that explain the difficulty in approximating unconstrained MAP inference and the normalizing constant for E-DPPs. (1) Unconstrained MAP inference for an n×n matrix is NP-hard to approximate within a 2βn-factor, where β=101013. This result improves upon a (9/8ϵ)-factor inapproximability given by Kulesza and Taskar (2012). (2) The normalizing constant for E-DPPs of any (fixed) constant exponent pβ1=101013 is NP-hard to approximate within a 2βpn-factor. This gives a(nother) negative answer to open questions posed by Kulesza and Taskar (2012); Ohsaka and Matsuoka (2020).

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