Some Inapproximability Results of MAP Inference and Exponentiated Determinantal Point Processes

Ohsaka, Naoto

doi:10.1613/jair.1.13288

Cited by 4 publications

(2 citation statements)

References 44 publications

(64 reference statements)

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“…Our implementation of I G [27] yields a faster 1/4-approximation algorithm for a special case with a cardinality constraint. Approximation algorithms and inapproximability results of DPP MAP inference (without log) have also been extensively studied [15,34,35,6,41]. Sampling is another important research subject in DPPs and has been widely studied [2,32,16,21,1,12,31].…”

Section: Related Workmentioning

confidence: 99%

Lazy and Fast Greedy MAP Inference for Determinantal Point Process

Hemmi¹,

Oki²,

Sakaue³

et al. 2022

Preprint

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The maximum a posteriori (MAP) inference for determinantal point processes (DPPs) is crucial for selecting diverse items in many machine learning applications. Although DPP MAP inference is NP-hard, the greedy algorithm often nds high-quality solutions, and many researchers have studied its e cient implementation. One classical and practical method is the lazy greedy algorithm, which is applicable to general submodular function maximization, while a recent fast greedy algorithm based on the Cholesky factorization is more e cient for DPP MAP inference. This paper presents how to combine the ideas of "lazy" and "fast", which have been considered incompatible in the literature. Our lazy and fast greedy algorithm achieves almost the same time complexity as the current best one and runs faster in practice. The idea of "lazy + fast" is extendable to other greedy-type algorithms. We also give a fast version of the double greedy algorithm for unconstrained DPP MAP inference. Experiments validate the e ectiveness of our acceleration ideas.

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Section: Related Workmentioning

confidence: 99%

Lazy and Fast Greedy MAP Inference for Determinantal Point Process

Hemmi¹,

Oki²,

Sakaue³

et al. 2022

Preprint

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“…On the negative side, Ko et al [26] prove that Determinant Maximization is NP-hard, and Koutis [27] proves that it is further W[1]-hard. NP-hardness of approximating Determinant Maximization has been investigated in [13,16,27,40]. On the algorithmic side, a greedy algorithm achieves an approximation factor of 1/k!…”

Section: Introductionmentioning

confidence: 99%

On the Parameterized Intractability of Determinant Maximization

Ohsaka

2024

Algorithmica

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In the Determinant Maximization problem, given an $$n \times n$$ n × n positive semi-definite matrix $${\textbf {A}} $$ A in $$\mathbb {Q}^{n \times n}$$ Q n × n and an integer k, we are required to find a $$k \times k$$ k × k principal submatrix of $${\textbf {A}} $$ A having the maximum determinant. This problem is known to be -hard and further proven to be [1]-hard with respect to k by Koutis (Inf Process Lett 100:8–13, 2006); i.e., a $$f(k)n^{{{\,\mathrm{\mathcal {O}}\,}}(1)}$$ f ( k ) n O ( 1 ) -time algorithm is unlikely to exist for any computable function f. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is -hard and [1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, Determinant Maximization is known to be solvable in polynomial time on tridiagonal matrices (Al-Thani and Lee, in: LAGOS, 2021). Thereafter, we demonstrate the [1]-hardness with respect to the rankr of an input matrix. Our result is stronger than Koutis’ result in the sense that any $$k \times k$$ k × k principal submatrix is singular whenever $$k > r$$ k > r . We finally give evidence that it is [1]-hard to approximate Determinant Maximization parameterized by k within a factor of $$2^{-c\sqrt{k}}$$ 2 - c k for some universal constant $$c > 0$$ c > 0 . Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov et al. (in: SODA, 2020), which asserts that a gap version of Binary Constraint Satisfaction Problem is [1]-hard. To complement this result, we develop an $$\varepsilon $$ ε -additive approximation algorithm that runs in $$\varepsilon ^{-r^2} \cdot r^{{{\,\mathrm{\mathcal {O}}\,}}(r^3)} \cdot n^{{{\,\mathrm{\mathcal {O}}\,}}(1)}$$ ε - r 2 · r O ( r 3 ) · n O ( 1 ) time for the rank r of an input matrix, provided that the diagonal entries are bounded.

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Computational complexity of normalizing constants for the product of determinantal point processes

Matsuoka,

Ohsaka

2024

Theoretical Computer Science

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Some Inapproximability Results of MAP Inference and Exponentiated Determinantal Point Processes

Cited by 4 publications

References 44 publications

Lazy and Fast Greedy MAP Inference for Determinantal Point Process

Lazy and Fast Greedy MAP Inference for Determinantal Point Process

On the Parameterized Intractability of Determinant Maximization

Computational complexity of normalizing constants for the product of determinantal point processes

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