1. Introduction and main results
Integration of high degree multivariate polynomials is computationally difficult and no efficient algorithms are known except in few special cases, when the polynomials have a rather simple algebraic structure (close to a power of a linear form), cf. [B+11], or have some very nice analytic properties (slowly varying or, most notably, log-concave), cf. [LV07]. Since a general -variate polynomial of degree is defined by parameters (for example, coefficients), the problem becomes interesting for large and only if has some special structure (such as the product of low-degree polynomials), which allows us to define using much fewer parameters.
In this paper, we integrate products of quadratic forms with respect to the Gaussian measure in . We relate the problem to partition functions of mollified logarithmic potentials and to testing the feasibility of systems of real quadratic equations.
Our algorithms are deterministic and based on the method of polynomial interpolation, which has been recently applied to a variety of partition functions in combinatorial (discrete) problems, cf. [Ba16]. In continuous setting, the method was applied to computing partition functions arising in quantum models [B+19], [H+19].
(1.1) Quadratic forms on
We consider Euclidean space endowed with the standard inner product
and corresponding Euclidean norm
Let be quadratic forms defined by
where are real symmetric matrices.
Our first result concerns computing the integral
The idea of the interpolation method is to consider (1.1.2) as a one-parameter perturbation a much simpler integral, in our case, of
For the method to work, one should show that there are no zeros in the vicinity of a path in the complex plane which connects (1.1.2) and (1.1.3). We prove the following result.
There is an absolute constant (one can choose ) such that the following holds. Let , , be quadratic forms. Then
for all such that , provided
By interpolation, for any constant , fixed in advance, we obtain an algorithm which, given quadratic forms , computes (1.1.2) within relative error in quasi-polynomial time provided
Note that by Theorem 1.2 and (1.1.3), the value of (1.1.2) is positive, as long as (1.2.1) holds.
Some remarks are in order.
First, we note that the integrand in (1.1.2) can vary wildly. Indeed, for large the bulk of the standard Gaussian measure in is concentrated in the vicinity of the sphere , see for example, Section V.5 of [Ba02]. Assuming that , we can choose so that (1.2.1) is satisfied. Then, in the vicinity of the sphere , the product in (1.1.2) varies within an exponential in factor, and is not at all well-concentrated.
Second, if the quadratic forms exhibit simpler combinatorics, we can improve the bounds accordingly. We prove the following result.
There is an absolute constant (one can choose ) such that the following holds. Let , , be quadratic forms. Suppose further that each form depends on not more than variables among and that each form has common variables with not more than other forms. Then
for all such that , provided
By interpolation, for any constant , fixed in advance, we obtain an algorithm which, given quadratic forms as in Theorem 1.3 computes (1.1.2) within relative error in quasi-polynomial time provided
We prove Theorems 1.2 and 1.3 in Section 3 and describe the algorithm for computing (1.1.2) in Section 4. In Section 2, we discuss connections with systems of particles with mollified logarithmic potentials and possible applications to testing the feasibility of systems of multivariate real quadratic equations.
2. Connections and possible applications
(2.1) Partition functions of mollified logarithmic potentials
Let and let us interpret as the space of all ordered -tuples of points . Hence the distance between and is .
Let us fix some set of pairs of indices and suppose that the energy of a set of points is defined by
where is a parameter. The first sum in (2.1.1) indicates that there a repulsive force between any pair with (so that the energy decreases if the distance between and increases), while the second sum indicates that there is a force pushing the points towards 0 (so that the energy decreases when each approaches ). When , the repulsive force disappears altogether, and , the repulsive force behaves as a Coulomb’s force with logarithmic potential, since
Thus the integral
which is a particular case of (1.1.2), can be interpreted as the partition function of points with “mollified” or “damped” logarithmic potentials. One can think of (2.1.2) as the partition function for particles with genuine logarithmic potentials, provided each particle is confined to its own copy of among a family of parallel -dimensional affine subspaces in some higher-dimensional Euclidean space.
The integral (2.1.2) can be considered as a ramification of classical Selberg-type integrals for logarithmic potentials:
see for example, Chapter 17 of [Me04]. The integral (2.1.3) corresponds to points in and a similar integral is computed explicitly for points in (and ), see Section 17.11 of [Me04]. For higher dimensions no explicit formulas appear to be known.
In contrast, we compute integrals (2.1.2) approximately for certain values of , but we allow arbitrary dimensions and can choose an arbitrary set of pairs of interacting points (and we can even choose different
s for different pairs of points). Theorem 1.3 can be interpreted as the absence of phase transition in the Lee - Yang sense[YL52], if is sufficiently small. For example, if the set consists of all pairs , Theorem 1.3 implies that there is no phase transition (and the integral can be efficiently approximated) if
for some absolute constant .
(2.2) Applications to systems of quadratic equations
Every system of real polynomial equations can be reduced to a system of quadratic equations, as one can successively reduce the degree by introducing new variables via substitutions of the type . A system of quadratic equations can be solved in polynomial time when the number of equations is fixed in advance, [Ba93], [GP05], but as the number of equations grows, the problem becomes computationally hard.
Here we are interested in the systems of equations of the type
where are positive semidefinite quadratic forms. Such systems naturally arise in problems of distance geometry, where we are interested to find out if there are configurations of points in with prescribed distances between some pairs of points and in which case are scaled squared distances between points, see [CH88], [L+14] and Section 2.1. Besides, finding if a system of homogeneous quadratic equations has a non-trivial solution
can be reduced to (2.2.1) with positive definite forms by adding to the appropriately scaled equations in (2.2.2).
in (2.2.1). By itself, the condition (2.2.3) is not particularly restrictive: if the sum of in the left hand side of (2.2.3) is positive definite, it can be brought to the right hand side by an invertible linear transformation of.
Let us choose an such that the scaled forms satisfy (1.3.1), so that the integral
can be efficiently approximated. We would like to argue that the value of the integral (2.2.4) can provide a reasonable certificate which allows one to distinguish systems (2.2.1) with many “near solutions” from the systems that are far from having a solution.
We observe that the system (2.2.1) has a solution if and only if the system
has a solution for any .
Let us find such that
Indeed (2.2.6) always has a (necessarily unique) solution , since for the right hand side is bigger than the left hand side, while for the left hand side is bigger than the right hand side.
Because of (2.2.3), we can rewrite (2.2.4) as
We observe that if then the maximum value of
is attained at
and is equal to
and hence the maximum value of the product of the factors in (2.2.7) is
and attained if and only if the system (2.2.1) and hence (2.2.6) has a solution .
Also, if is a solution to (2.2.5), by (2.2.3), (2.2.6) and (2.2.8), we have
The Gaussian probability measure inwith density
is concentrated in the vicinity of the sphere , cf., for example, Section V.5 of [Ba05]
for some estimates. Therefore, if for the system (2.2.1) there are sufficiently many “near solutions”, we should have the value of the integral (2.2.4) sufficiently close to
while if the system (2.2.1) is far from having a solution, the value of the integral will be essentially smaller.
2. Proofs of Theorems 1.2 and 1.3
Choosing , we obtain Theorem 1.2 as a particular case of Theorem 1.3. Hence we prove Theorem 1.3 only.
For a real symmetric matrix we denote
its operator norm.
We start with a simple formula, cf. also [Ba93].
Let be quadratic forms,
where are real symmetric matrices such that
for all such that
Here we take the principal branch of in the left hand side of (3.1.1), which is equal to when . The series in the right hand side converges absolutely and uniformly on compact subsets of the polydisc (3.1.2).
For , let
If are real and satisfy (3.1.2), then is a positive definite quadratic form, and, as is well known,
Since both sides of the above identity are analytic in the domain (3.1.2), we obtain
Expanding the integral in the right hand side into the series in , we complete the proof. ∎
Next, we extract the integral (1.1.2) from the generating function of Lemma 3.1. Let
the the unit circle and let
be the -dimensional torus endowed with the uniform (Haar) probability measure
where is the uniform probability measure on the -th copy of . If , , then for the Laurent monomial
Let be quadratic forms,
where are real symmetric matrices such that
Then for every such that we have
where the second product is taken over all non-empty ordered tuples of distinct indices from .
From Lemma 3.1, we have
Next, we write
where the series converges absolutely and uniformly on .
We expand each of the exponential functions into the Taylor series and observe that only square-free monomials in contribute to the integral (3.2.1), from which it follows that
where the second product is taken over all non-empty ordered tuples of distinct indices . ∎
Our next goal is to write the integral in Lemma 3.2 as the value of the independence polynomial of an appropriate (large) graph.
(3.3) Independent sets in weighted graphs
Let be a finite undirected graph with set of vertices, set of edges and without loops or multiple edges. A set of vertices is called independent, if no two vertices from span an edge of . We agree that is an an independent set.
Let be a function assigning to each vertex a complex weight . We define the independence polynomial of by
Hence is a multivariate polynomial in complex variables with constant term 1, corresponding to .
Let be quadratic forms,
where are real symmetric matrices and let be a complex number.
We define a weighted graph as follows. The vertices of are all non-empty ordered tuples of indices and two vertices span an edge of if they have at least one common index , in arbitrary positions. We define the weight of the vertex by
From Lemma 3.2 it follows that (3.4.1) holds provided and for are small enough. Since both sides of (3.4.1) are polynomials in and , the proof follows. ∎
The following criterion provides a sufficient condition for for an arbitrary weighted graph . The result is known as the Dobrushin criterion and also as the Kotecký - Preiss condition for the cluster expansion, see, for example, Chapter 5 of [FV18].
Given a graph and a vertex , we define its neighborhood by
Let be an assignment of complex weights to the vertices of . Suppose that there is a function with positive real values such that for every vertex , we have
See, for example, Section 5.2 of [CF16] for a concise exposition. ∎
Now we are ready to prove Theorem 1.3.
(3.6) Proof of Theorem 1.3
Let be the matrices of the quadratic forms , so that
Since each quadratic form depends of at most variables, we have
Since each quadratic form has a common variable with at most other forms, we have
(3.6.2) For every there are at most indices such that .
Let be a complex number satisfying
Given and , we construct a weighted graph as in Corollary 3.4. Our goal is to prove that , for which we use Lemma 3.5.
We say that the level of a vertex is for . Thus for the weight of , we have
Combining (3.6.1) and (3.6.3), we conclude that for a vertex of level , we have
We observe that there are at most vertices of level with that are neighbors of a given vertex (for , we count as its own neighbor). Indeed, there are at most ways to choose a common index , after which there are at most positions to place in . By (3.6.2), we conclude that there are at most vertices of level with . Choosing for a vertex of level and using (3.6.4), we conclude that for a vertex of level , we have
and the proof follows by Corollary 3.4. and Lemma 3.5. ∎
4. Approximating the integral
The interpolation method is based on the following simple observation.
Let be a polynomial,
and be a real number such that
Let us choose a branch of for and let
be the Taylor polynomial of degree of computed at . Then
Moreover, the values of for can be computed from the coefficients for in time polynomial in and .
See, for example, Section 2.2 of [Ba16]. ∎
As follows from Lemma 4.1, if is fixed in advance, to estimate the value of within additive error (in which case we say that we estimate the the value of within relative error ), it suffices to compute the coefficients with , where the implied constant in the “” notation depends only on . A similar result holds if in an arbitrary, fixed in advance, connected open set such that , see Section 2.2 of [Ba16] (in Lemma 4.1, the neighborhood is the disc of radius ).
(4.2) Computing the integrals
Let us fix a constant
where is the constant of Theorem 1.3 (so one can choose ). Let be quadratic forms, defined by their matrices as in (1.1.1), such that each form depends on not more than variables among and each form has common variables with not more than other forms. Suppose that the bound (1.3.1) holds. We define a univariate polynomial by
Hence and by Theorem 1.3 we have
In view of Lemma 4.1, to approximate
within relative error , it suffices to compute and for , where the implied constant in the “” notation is absolute. From Corollary 3.4, we have