MAP推断与指数化行列式点过程的一些不可逼近性结果
Some Inapproximability Results of MAP Inference and Exponentiated Determinantal Point Processes
摘要：
研究了两个关于行列式点过程的困难问题的计算复杂性。一种是最大后验（MAP）推断，即，找到一个具有最大行列式的主子矩阵。另一种是指数化DPP（E-DPP）的概率推理，它可以通过指数参数$p$来锐化或削弱DPP的多样性偏好。我们提出了几个复杂性理论的硬度结果，解释了近似MAP推理和归一化常数E-DPPs的困难。我们首先证明，无约束MAP推理的$n \times n$矩阵是$\textsf{NP}$ -很难近似的$2^{βn}$的一个因素，其中$β= 10^{-10^{13}} $。这一结果改进了$(\frac{9}{8}-ε)$最著名的不可逼近因子，并排除了任何假设$\textsf{P} \neq \textsf{NP}$的多项式因子近似算法的存在。然后，我们表明，对数行列式最大化是$\textsf{NP}$ -很难近似的$\frac{5}{4}$的无约束的情况下，在一个因素内的$1+10^{-10^{13}}$的大小约束单调的情况下。特别地，对数行列式最大化不允许多项式时间近似方案，除非$\textsf{P} = \textsf{NP}$。作为第一个结果的推论，我们证明了任何（固定）常数指数$p \geq β^{-1} = 10^{10^{13}}$的E-DPP的归一化常数是$\textsf{NP}$ -很难近似的$2^{βpn}$的一个因素，这是在相反的情况下，$p \leq 1$承认一个完全多项式时间随机近似方案。
英文摘要：
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 present several complexity-theoretic hardness results that explain the difficulty in approximating MAP inference and the normalizing constant for E-DPPs. We first prove that unconstrained MAP inference for an $n \times n$ matrix is $\textsf{NP}$-hard to approximate within a factor of $2^{βn}$, where $β= 10^{-10^{13}} $. This result improves upon the best-known inapproximability factor of $(\frac{9}{8}-ε)$, and rules out the existence of any polynomial-factor approximation algorithm assuming $\textsf{P} \neq \textsf{NP}$. We then show that log-determinant maximization is $\textsf{NP}$-hard to approximate within a factor of $\frac{5}{4}$ for the unconstrained case and within a factor of $1+10^{-10^{13}}$ for the size-constrained monotone case. In particular, log-determinant maximization does not admit a polynomial-time approximation scheme unless $\textsf{P} = \textsf{NP}$. As a corollary of the first result, we demonstrate that the normalizing constant for E-DPPs of any (fixed) constant exponent $p \geq β^{-1} = 10^{10^{13}}$ is $\textsf{NP}$-hard to approximate within a factor of $2^{βpn}$, which is in contrast to the case of $p \leq 1$ admitting a fully polynomial-time randomized approximation scheme.
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