# A remark on approximating permanents of positive definite matrices

Let A be an n × n positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of A as the integral of some explicit log-concave function on R^2n. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for the permanent of A.

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## 1. Introduction and main results

Let be an complex matrix. The permanent of is defined as

 perA=∑σ∈Snn∏k=1akσ(k),

where is the symmetric group of all permutations of the set . Recently, there was some interest in efficient computing (approximating) , when is a positive definite Hermitian matrix (as is known, in that case is real and non-negative), see [A+17] and reference therein. In particular, Anari et al. construct in [A+17] a deterministic algorithm approximating the permanent of a positive semidefinite Hermitian matrix within a multiplicative factor of for , where is the Euler constant.

In this note, we show that that there is a fully polynomially randomized approximation scheme (FPRAS) for permanents of positive definite matrices with the eigenvalues between 1 and 2. Namely, we represent for such a matrix as an integral of an explicitly constructed log-concave function

, so that a Markov Chain Monte Carlo algorithm can be applied to efficiently approximate

 ∫R2nfA(x) dx=perA,

see [LV07].

We consider the space with the standard norm

 ∥z∥2=|z1|2+…+|zn|2,wherez=(z1,…,zn).

We identify by identifying with . For a complex matrix , we denote by its conjugate, so that

 l∗jk=¯¯¯¯¯lkjfor allj,k.

We prove the following main result.

###### (1.1) Theorem

Let be an positive definite matrix with all eigenvalues between and . Let us write , where is the identity matrix and is an positive semidefinite Hermitian matrix with eigenvalues between and . Further, we write , where is an complex matrix. We define linear functions by

 ℓj(z)=n∑k=1ljkzkforz=(z1,…,zn).

Let us define by

 fA(z)=1πne−∥z∥2n∏j=1(1+∣∣ℓj(z)∣∣2).

## 2. Proofs

We start with a known integral representation of the permanent of a positive semidefinite matrix.

### (2.1) The integral formula

Let

be the Gaussian probability measure in

with density

 1πne−∥z∥2where∥z∥2=|z1|2+…+|zn|2forz=(z1,…,zn).

Let be linear functions and let be the matrix,

 bjk=Eℓj¯¯¯¯¯ℓk=∫Cnℓj(z)¯¯¯¯¯¯¯¯¯¯¯ℓk(z) dμ(z)forj,k=1,…,n.

Hence is a Hermitian positive semidefinite matrix and the Wick formula (see, for example, Section 3.1.4 of [Ba16]) implies that

 perB=E(|ℓ1|2⋯|ℓn|2)=∫Cn|ℓ1(z)|2⋯|ℓn(z)|2 dμ(z).

Next, we need a simple lemma.

###### (2.2) Lemma

Let be a positive semidefinite quadratic form. Then the function

 h(x)=ln(1+q(x))−q(x)

is concave.

###### Demonstration Proof

It suffices to check that the restriction of onto any affine line with is concave. Thus we need to check that the univariate function

 G(τ)=ln(1+(ατ+β)2+γ2)−(ατ+β)2−γ2forτ∈R,

where , is concave, for which it suffices to check that for all . Via the affine substitution , it suffices to check that , where

 g(τ)=ln(1+τ2+γ2)−(τ2+γ2).

We have

 g′(τ)=2τ1+τ2+γ2−2τ

and

 g′′(τ)=2(1+τ2+γ2)−4τ2(1+τ2+γ2)2−2=2(1+τ2+γ2)−4τ2−2(1+τ2+γ2)2(1+τ2+γ2)2=2+2τ2+2γ2−4τ2−2−2τ4−2γ4−4τ2−4γ2−4τ2γ2(1+τ2+γ2)2=−6τ2+2γ2+2τ4+2γ4+4τ2γ2(1+τ2+γ2)2≤0

and the proof follows. ∎

### (2.3) Proof of Theorem 1.1

We have

 perA=per(I+B)=∑J⊂{1,…,n}perBJ,

where is the principal submatrix of with row and column indices in and where we agree that . Let us consider the Gaussian probability measure in with density . By (2.1.1), we have

 perBJ=E∏j∈J|ℓj(z)|2

and hence

 perA=En∏j=1(1+|ℓj(z)|2)=∫R2nfA(x,y) dxdy,

and the proof of Part (1) follows.

We write

 e−∥z∥2n∏j=1(1+|ℓj(z)|2)=e−q(z)n∏j=1(1+|ℓj(z)|2)e−|ℓj(z)|2,whereq(z)=∥z∥2−n∑j=1|ℓj(z)|2.

By Lemma 2.2 each function is log-concave on and hence to complete the proof of Part (2) it suffices to show that is a positive semidefinite Hermitian form. To this end, we consider the Hermitian form

 p(z)=n∑j=1|ℓj(z)|2=n∑j=1∣∣ ∣∣n∑k=1ljkzk∣∣ ∣∣2=n∑j=1∑1≤k1,k2≤nljk1¯¯¯¯¯¯¯ljk2zk1¯¯¯¯¯¯¯zk2=∑1≤k1,k2≤nck1k2zk1¯¯¯¯¯¯¯zk2,

where

 ck1k2=n∑j=1ljk1¯¯¯¯¯¯¯ljk2for1≤k1,k2≤n.

Hence for the matrix of , we have . We note that and that the eigenvalues of lie between 0 and 1. Therefore, the eigenvalues of lie between 0 and 1 (in the generic case, when is invertible, the matrices and are similar). Consequently, the eigenvalues of lie between and and hence the Hermitian form with matrix is positive semidefinite, which completes the proof of Part (2). ∎

## References

• A+17 N. Anari, L. Gurvits, S.O. Gharan, and A. Saberi, Simply exponential approximation of the permanent of positive semidefinite matrices, 58th Annual IEEE Symposium on Foundations of Computer Science – FOCS 2017, IEEE Computer Soc., 2017, pp. 914–925.
• Ba16 A. Barvinok, Combinatorics and Complexity of Partition Functions, Algorithms and Combinatorics, 30, Springer, 2016.
• LV07 L. Lovász and S. Vempala, The geometry of logconcave functions and sampling algorithms, Random Structures Algorithms 30 (2007), no. 3, 307–358.

## References

• A+17 N. Anari, L. Gurvits, S.O. Gharan, and A. Saberi, Simply exponential approximation of the permanent of positive semidefinite matrices, 58th Annual IEEE Symposium on Foundations of Computer Science – FOCS 2017, IEEE Computer Soc., 2017, pp. 914–925.
• Ba16 A. Barvinok, Combinatorics and Complexity of Partition Functions, Algorithms and Combinatorics, 30, Springer, 2016.
• LV07 L. Lovász and S. Vempala, The geometry of logconcave functions and sampling algorithms, Random Structures Algorithms 30 (2007), no. 3, 307–358.