Approximating the Permanent with Fractional Belief Propagation

We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the Belief Propagation (BP) approach and its Fractional Belief Propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. The Fractional Free Energy (FFE) functional is parameterized by a scalar parameter $\gamma\in[-1;1]$, where $\gamma=-1$ corresponds to the BP limit and $\gamma=1$ corresponds to the exclusion principle (but ignoring perfect matching constraints) Mean-Field (MF) limit. FFE shows monotonicity and continuity with respect to $\gamma$. For every non-negative matrix, we define its special value $\gamma_*\in[-1;0]$ to be the $\gamma$ for which the minimum of the $\gamma$-parameterized FFE functional is equal to the permanent of the matrix, where the lower and upper bounds of the $\gamma$-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of $\gamma_*$ varies for different ensembles but $\gamma_*$ always lies within the $[-1;-1/2]$ interval. Moreover, for all ensembles considered the behavior of $\gamma_*$ is highly distinctive, offering an emprirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.Comment: 42 pages, 14 figure

Approximating the Permanent of a Random Matrix with Vanishing Mean

We show an algorithm for computing the permanent of a random matrix with vanishing mean in quasi-polynomial time. Among special cases are the Gaussian, and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time 2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the intuition that the permanent is hard because of the "sign problem" - namely the interference between entries of a matrix with different signs. A major open question then remains whether one can provide an efficient algorithm for random matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the baseline assumptions of the BosonSampling paradigm

Approximating the Permanent with Fractional Belief Propagation

We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the belief propagation (BP) approach and its fractional belief propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and Conjectures are verified in experiments, and some new theoretical relations, bounds and Conjectures are proposed. The fractional free energy (FFE) function is parameterized by a scalar parameter y ∈ [−1;1], where y = −1 corresponds to the BP limit and y = 1 corresponds to the exclusion principle (but ignoring perfect matching constraints) mean-field (MF) limit. FFE shows monotonicity and continuity with respect to g. For every non-negative matrix, we define its special value y∗ ∈ [−1;0] to be the g for which the minimum of the y-parameterized FFE function is equal to the permanent of the matrix, where the lower and upper bounds of the g-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of y∗ varies for different ensembles but y∗ always lies within the [−1;−1/2] interval. Moreover, for all ensembles considered, the behavior of y∗ is highly distinctive, offering an empirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.Los Alamos National Laboratory (Undergraduate Research Assistant Program)United States. National Nuclear Security Administration (Los Alamos National Laboratory Contract DE C52-06NA25396

Approximating the Permanent with Deep Rejection Sampling

Peer reviewe

A simple polynomial time algorithm to approximate the permanent within a simply exponential factor

We present a simple randomized polynomial time algorithm to approximate the mixed discriminant of $n$ positive semidefinite $n \times n$ matrices within a factor $2^{O(n)}$. Consequently, the algorithm allows us to approximate in randomized polynomial time the permanent of a given $n \times n$ non-negative matrix within a factor $2^{O(n)}$. When applied to approximating the permanent, the algorithm turns out to be a simple modification of the well-known Godsil-Gutman estimator

Approximating the Permanent of a Matrix with Deep Rejection Sampling

Computing the permanent of a matrix is a famous #P-hard problem with a wide range of applications. The fastest known exact algorithms for the problem require an exponential number of operations, and all known fully polynomial randomized approximation schemes are rather complicated to implement and have impractical time complexities. The most promising recent advancements on approximating the permanent are based on rejection sampling and upper bounds for the permanent. In this thesis, we improve the current state of the art by developing the deep rejection sampling method, which combines an exact algorithm with the rejection sampling method. The algorithm precomputes a dynamic programming table that tightens the initial upper bound used by the rejection sampling method. In a sense, the table is used to jump-start the sampling process. We give a high probability upper bound for the time complexity of the deep rejection sampling method for random (0, 1)-matrices in which each entry is 1 with probability p. For matrices with p < 1/5, our high probability bound is stronger than in previous work. In addition to that, we empirically observe that our algorithm outperforms earlier rejection sampling methods by testing it with different parameters against other algorithms on multiple classes of matrices. The improvements in sampling times are especially notable in cases in which the ratios of the permanental upper bounds and the exact value of the permanent are huge

A quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices

We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices, by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the permanent of a Hermitian positive semidefinite matrix can be expressed in terms of the expected value of a random variable, which stands for a specific photon-counting probability when measuring a linear-optically evolved random multimode coherent state. Our algorithm then approximates the matrix permanent from the corresponding sample mean and is shown to run in polynomial time for various sets of Hermitian positive semidefinite matrices, achieving a precision that improves over known techniques. This work illustrates how quantum optics may benefit algorithms development.Comment: 9 pages, 1 figure. Updated version for publicatio