1 Introduction
This paper is in part inspired by recentlyrevived discussions about the current and potential prevalence of computerassisted proof methods spurred by the quartercentury anniversary of the notorious “Death of Proof” article from 1993 [10]. One common observation is that “…computers continue to only rarely have any role in the creation and checking of the proofs of mathematical theorems” [11]. The conventional opinion is that this status quo will not change unless and until the power of automated proof methods improves dramatically to a point where they are able to solve an increasing fraction of outstanding problems of interest to professional mathematicians. In this paper, however, we suggest an alternative in which their power increases at only a moderate rate but there is a dramatic increase in the number of problems to which they are applied. These could include problems of little or no theoretical interest to the mathematical community and which only have narrow practical application (e.g., as typified by the example problem of this paper) but which by sheer number lead to a situation in which automated methods do in fact surpass the productivity and practical impact of traditional proof methods employed by human mathematicians.
2 DimensionSpecific Properties
For a particular class of objects of interest, e.g., some defined subset of the integers or reals, it is commonly of analytic or practical value to identify a set of special properties satisfied by that class. For a class of objects parameterized by an integral measure of size or dimensionality , it may be the case that a particular property only holds for a limited set of values of . For example, nonparallel lines always intersect in two dimensions but not generally in higher dimensions.
In the case of matrices there is great value in being able to establish that the result of a given matrix function or transformation has a special property (e.g., that it is unitary, positivedefinite, totallyunimodular, doublystochastic, or has a special structural property such as sparseness) which can be exploited for analysis purposes or to obtain solutions more efficiently than would be possible in the general case. This is the motivation for deriving decompositions of general matrices as products or sums of matrices with special properties [15, 16, 31]
. For example, the singularvalue decomposition (SVD) is one of the most widelyused tools in linear algebra because it permits an arbitrary matrix
to be expressed as , where and are unitary and is diagonal [12].When general linearalgebraic tools are applied to analyze a matrix transformation , the identified properties will typically hold independently of . In fact, the power of linear algebra largely derives from its ability to work with objects (matrices) in a manner that allows the dimensionality to be abstracted. However, many theoretical and practical problems of interest, e.g., design and analysis of control systems, are intrinsically defined in a fixed dimensional space for which many properties may hold for conditioned on that do not necessarily hold for general . Once identified, such properties can provide significant insights and permit a much larger set of mathematical tools to be applied. Unfortunately, establishing such properties may demand effort exponential in dimensionality.
In the following sections we consider an example in which the opportunities and obstacles associated with the determination of dimensionspecific properties are highlighted. In particular, we prove conjectured properties of a specific matrix function for using sophisticated methods for certifying polynomial nonnegativity. We then consider the challenge of proving that these properties hold more generally up to a conjectured upper bound of . We conclude with consideration of dimension pinning as a potentially effective and general strategy for establishing dimensionspecific properties as part of a broader vision for increasing the applicability and relevance of computerassisted proof methods.
3 The RGA and its Inverse
The relative gain array (RGA) is an important tool in the design of process control systems [4]. It is a function of a nonsingular real or complex^{1}^{1}1It should be noted that the the transpose operator in the RGA applies even in the case of complex , i.e., it should not be replaced with the conjugatetranspose operator. matrix defined as:
(3.1) 
where represents the elementwise Hadamard matrix product and the rightmost expression exploits simplified notation for the inversetranspose operator. The RGA has a variety of interesting mathematical properties, one of which is that the sum of the elements in each row and column is unity [13, 30]. For example, given^{2}^{2}2This particular integer matrix was chosen purely because its inverse is also integral [9] and thus simplifies the values appearing in this and subsequent examples, i.e., no inferences should be drawn from the integrality of the results.
(3.2) 
then
(3.3) 
Another important property of the RGA relates to permutations of the rows and columns of its argument [13]:
(3.4) 
In other words, the RGA is permutationconsistent [29, 30] with respect to left and right multiplication by permutation matrices and .
A related matrix operator that has not previously been considered in the literature is the inverse of the RGA result, or IRGA, which we define as:
(3.5) 
Applying this to the matrix of Eq. (3.2) gives
(3.6) 
where it can be seen that the row and column sums are unity, as should be expected since has unit row and column sums. Examining the case
(3.7) 
reveals
(3.8) 
This result suggests several special properties that might be conjectured to hold for some values . The most tantalizing relates to the fact that the denominator of the coefficient is the permanent of . In fact, the overall result represents the joint assignment matrix (JAM) [5] of , which is the solution^{3}^{3}3The inverseJAM is presently under investigation by the first author as an alternative generalization of the RGA for . to an important combinatorial problem arising in multipletarget tracking and related applications [27, 28]. Evaluation of the JAM for a general matrix is believed to be computationally intractable based on the #PHard complexity of evaluating the permanent of a matrix [5, 3, 17, 6], so it should not be surprising that JAMequivalence does not hold for .
It can also be observed in the case that nonnegative implies is nonnegative and therefore doublystochastic. Unfortunately, this property also does not generalize to . However, in the case of positivedefinite a conjecture of the first author states:
Conjecture 3.1
For an , , positivedefinite matrix , is positivedefinite, has unit row and column sums, and is nonnegative and hence doublystochastic.
In fact, because of the permutationconsistency property of the RGA, hence also of the IRGA, the conjecture only requires to be positivedefinite up to left and right permutations.
A nonnegative symmetric positivedefinite matrix is sometimes referred to as being doubly nonnegative [36]
. Such matrices arise in a variety of applications ranging from control systems and network analysis to estimation and optimization
[2, 20, 21, 32, 37], and characterizations of their inverses have been longstudied [7, 8, 23].It is straightforward to prove positivedefiniteness of the IRGA is preserved for all because and its inverse are both PD (positivedefinite), and the Hadamard product of two PD matrices always yields a PD result [12, 26]. Therefore the critical property to be verified for values of in the conjectured range is nonnegativity. This will be examined in the following two sections.
4 Certifying nonnegativity of polynomials
Let be the ring of real variate polynomials. For a finite set , we denote by the convex hull of . A polynomial can be written as with . The support of is .
For a nonempty finite set , denotes the set of polynomials in whose supports are contained in , i.e., and we use to denote the set of polynomials which are sums of squares of polynomials in . The set of symmetric matrices is denoted by and the set of positivesemidefinite matrices is denoted by . Let be the
dimensional column vector consisting of elements
, thenwhere the matrix is called the Gram matrix.
A classical approach for checking nonnegativity of multivariate polynomials, as introduced by Lasserre [14] and Parrilo [18], is the use of sums of squares as a suitable replacement for nonnegativity. Given a polynomial , if there exist polynomials such that
(4.1) 
then is referred to as a sum of squares (SOS). Obviously an SOS decomposition of a given polynomial serves as a certificate for its nonnegativity. For , let and assume . Then the SOS condition (4.1) can be converted to the problem of deciding if there exists a positivesemidefinite matrix such that
(4.2) 
which can be effectively solved as a semidefinite programming (SDP) problem. Note that there are elements in the monomial basis . So the size of the corresponding SDP problem is , which grows rapidly as the number of variables and the degree of the given polynomial increase. To deal with highdegree polynomials with many variables it is crucial to exploit the sparsity of polynomials to reduce the size of the corresponding SDP problems.
5 Certifying nonnegativity of sparse polynomials
There are two aspects by which sparsity in SOS decompositions can be exploited. The first is to compute a smaller monomial basis. In fact, the monomial basis in (4.2) can be replaced by [22]
(5.1) 
Second, the blockdiagonal structure in the Gram matrix can be exploited, namely cross sparsity patterns, which was introduced by the second author [33] and will prove critical to achieving our main result.
Definition 5.1
Let with and is as in (5.1). An cross sparsity pattern matrix is defined by
(5.2) 
Given a cross sparsity pattern matrix , the graph where and is called the cross sparsity pattern graph.
Given an undirected graph with , we define an extended set of edges that includes all selfloops. Then we define the space of symmetric sparse matrices as
(5.3) 
and the cone of sparse positivesemidefinite matrices as
(5.4) 
Assume and is as in (5.1). Let be the cross sparsity pattern graph. Let be the graph obtained by adding edges to all connected components of such that every connected component becomes a complete subgraph. Letting
we can state the following sparse SOS decomposition theorem:
Theorem 5.2 ([33])
Assume and is as in (5.1). Let denote the connected components of and . Then, if and only if there exist for such that
(5.5) 
Theorem 5.2 implies that the checking of involves solving a blockdiagonalized SDP problem, which significantly reduces the computation. For proof purposes it is then necessary to convert the obtained numerical SOS decomposition to an SOS decomposition with rational coefficients. Standard roundingprojection procedures [19] can be applied when the given polynomial lies in the interior of the SOS cone, but in our case (and also , ) lies on the boundary of the SOS cone. To overcome this difficulty we use the method of undetermined coefficients by setting nonzero elements in the numerical SOS decomposition of as unknowns and then searching for a rational solution to the system of equations obtained by comparing coefficients. It is the complexity of this latter step that will limit the values of for which SOS certificates can be practically obtained for our problem of interest.
6 Toward Proving the Conjecture for
The basis for the upper bound of the conjecture was determined first by a strategy of extensive directed sampling of random PD matrices for successive values of , which revealed that counterexamples can be readily found for but not for . It then remained only to tailor the search to find a specific example for which the violated nonnegativity condition can be certified with finite arithmetic. The following integer PD matrix provides this certification:
(6.1) 
Applying IRGA to the above can be verified to yield a rational result that contains a symmetric pair of negative values. With this it remains to rigorously prove that no such counterexamples can exist for . This is necessary and sufficient because clearly if the conjecture holds for some value of it must hold for all smaller values because each case can be expressed as a block replacing a submatrix of the identity.
To prove that the case is nonnegative it is sufficient to show that a typical offdiagonal element is nonnegative. As will be discussed in the subsequent section, the computational complexity required to prove the nonnegativity of the polynomial associated with such an element tends to become practically prohibitive even for values of of this size.
7 The case
Any positivedefinite matrix can be be scaled on the left and right by a positive diagonal matrix so that its Cholesky decomposition has unit diagonal elements, thus we can assume without loss of generality that our test of positivedefiniteness can assume an appropriatelyscaled having Cholesky factors with
(7.1) 
Because is positivedefinite it follows that the determinant of , which is also the common denominator of elements of , is positive. Therefore, to prove the nonnegativity of elements of it is sufficient to establish nonnegativity of the numerators of elements of . Consider a typical offdiagonal element of , say the element in position , whose numerator we denote by (similarly for the meaning of ). The sizes of are listed in the following table from which it can be seen that the support sizes of become extremely large:
#var  deg  #supp  

6  12  116  
10  20  5157  
15  40  676505 
Applying the method of the previous section we obtain a numerical SOS decomposition for , but its size prevents us from obtaining a rational certificate of nonnegativity. In other words, IRGA nonnegativity for is virtually but not formally established. However, the much larger size of prevents us from obtaining even a numerical SOS decomposition, assuming one exists.
Given that it is not presently possible to formally prove nonnegativity for the case, the fallback must be to follow a strategy similar to the finding of counterexamples and consider successive values of . The case is already at the limit of what can be proven by hand or by using computeraided symbolic methods (both are doable), but in the following section we will formally prove nonnegativity for the subsuming case of , i.e., that typical element is nonnegative.
8 The case
As in the previous section, assume the positivedefinite matrix has Cholesky decomposition with
(8.1) 
From this we can express the polynomial as
Computing the monomial basis of as in (5.1) yields a result with elements. The cross sparsity pattern graph of has connected components with nodes, respectively. From this we obtain as a weighted sum of squares, i.e., , where
We can now present our main technical result relating to the IRGA conjecture:
Theorem 8.1
For an matrix that is positivedefinite up to permutations of its rows and columns, retains the same properties, but with unit row and column sums, and for is nonnegative and hence doublystochastic, i.e., is doublynonnegative.
The import of this theorem is that for any application of the IRGA to arbitrarily permuted positivedefinite matrices, , it can be rigorously assumed that the result will retain the same properties while also being doublystochastic.
Remark 8.2
Obtaining an SOS decomposition with rational coefficients for required finding a rational solution to a system of polynomial equations of degree and with unknowns. The corresponding system for involves many hundreds of unknowns and is presently beyond the limits of what can be solved using existing generic methods. However, we believe that a speciallytailored approach may be able to obtain a rational SOS decomposition in the near future and formally establish the nonnegativity of . Obtaining a solution for seems so formidable as to require completely new theoretical and practical innovations, e.g., possibly exploiting other special properties of PD doublystochastic matrices that obviate the need for SOS methods or greatly reduce the complexity of the polynomials to which they must be applied. With or without such innovations, we propose as a challenge problem to the SOS community.
It must be emphasized that the focus of this paper is not on this highlyspecific problem/result per se but rather how it serves as an example of the kinds of highlyspecialized problems that arise in practical applications that are potentially amenable to automated proof methods. Of course such methods go well beyond the polynomialnonnegativity tools used in this paper, but virtually all of them suffer severe computational complexity constraints. What has been suggested in this paper is that a closer look may reveal a plethora of narrow applicationspecific problems that can be practically solved by present stateoftheart tools.
9 Discussion
In this paper we have examined a problem for which it is possible to identify and prove nontrivial properties relating to a strongly nonlinear matrix function that hold for values (dimensionality) of below a specific bound but do not hold for general above a specific bound. Our dimensionpinning approach, which is likely to be applicable to a broad family of problems, relies on using counterexample search to establish an upperbound limit on , i.e., above which the speculated properties provably do not generally hold, and using automated methods to establish a lowerbound limit sufficient to prove they do hold in general below that bound. Ideally, the two approaches would converge to optimally define/pin the limit at which the properties fail to hold in all cases.
Of course there are myriad examples dating from the earliest history of mathematics in which explicit counterexamples have been used to demonstrate the limit to which the scope of a given result can be generalized. What we advocate here is a more structured approach to facilitate computeraided proof methods to be applied to identify and establish potentially valuable properties of mathematical formulae that hold only in low dimensions, e.g., sufficient to encompass most or all practical instances of a particular engineering problem. For the specific problem of this paper nonnegativity of the IRGA of a positivedefinite matrix was observed for the case of , and this motivated a counterexample search which identified that nonnegativity does not generally hold for . This then motivated use of SOS methods to prove that nonnegativity does hold in general at least up to . SOS methods have also provided strong evidence that nonnegativity holds for , but due to computational complexity constraints inherent to SOS methods the status of presently remains entirely unresolved – aside from supporting evidence from the fact that extensive search has yielded no counterexamples for whereas they are readily found when .
We have noted that dimensionalitylimited properties are particularly relevant to those engineering applications for which is practically limited in some way by the physical dimensions of the real world. Beyond this, a proof of the nonnegativity conjecture of this paper would imply the existence of special mathematical structures that are limited to 6dimensional spaces and thus may be relevant to theoretical physics as unitary conformal field theories (CFTs) and superconformal variants can only be defined in or dimensions [34, 35, 1, 24, 25]. Most importantly, we hope this paper has provided a glimpse of what we believe to be a diverse and bountiful – though heretofore largely unrecognized – set of problems to which computerassisted proof methods can be productively applied.
Declarations of Interest: None.
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