Equivariant semidefinite lifts and sum-of-squares hierarchies

A central question in optimization is to maximize (or minimize) a linear function over a given polytope P. To solve such a problem in practice one needs a concise description of the polytope P. In this paper we are interested in representations of P using the positive semidefinite cone: a positive semidefinite lift (psd lift) of a polytope P is a representation of P as the projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. Such a representation allows linear optimization problems over P to be written as semidefinite programs of size d. Such representations can be beneficial in practice when d is much smaller than the number of facets of the polytope P. In this paper we are concerned with so-called equivariant psd lifts (also known as symmetric psd lifts) which respect the symmetries of the polytope P. We present a representation-theoretic framework to study equivariant psd lifts of a certain class of symmetric polytopes known as orbitopes. Our main result is a structure theorem where we show that any equivariant psd lift of size d of an orbitope is of sum-of-squares type where the functions in the sum-of-squares decomposition come from an invariant subspace of dimension smaller than d^3. We use this framework to study two well-known families of polytopes, namely the parity polytope and the cut polytope, and we prove exponential lower bounds for equivariant psd lifts of these polytopes.Comment: v2: 30 pages, Minor changes in presentation; v3: 29 pages, New structure theorem for general orbitopes + changes in presentatio

Equivariant semidefinite lifts of regular polygons

Given a polytope P in $\mathbb{R}^n$, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the linear projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e. those psd lifts that respect the symmetry of P. One of the simplest families of polytopes with interesting symmetries are regular polygons in the plane, which have played an important role in the study of linear programming lifts (or extended formulations). In this paper we study equivariant psd lifts of regular polygons. We first show that the standard Lasserre/sum-of-squares hierarchy for the regular N-gon requires exactly ceil(N/4) iterations and thus yields an equivariant psd lift of size linear in N. In contrast we show that one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1, which is exponentially smaller than the psd lift of the sum-of-squares hierarchy. Our construction relies on finding a sparse sum-of-squares certificate for the facet-defining inequalities of the regular 2^n-gon, i.e., one that only uses a small (logarithmic) number of monomials. Since any equivariant LP lift of the regular 2^n-gon must have size 2^n, this gives the first example of a polytope with an exponential gap between sizes of equivariant LP lifts and equivariant psd lifts. Finally we prove that our construction is essentially optimal by showing that any equivariant psd lift of the regular N-gon must have size at least logarithmic in N.Comment: 29 page

Sparse sum-of-squares certificates on finite abelian groups

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay($\hat{G}$,S)) with maximal cliques related to T. Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G. We apply our general result to two examples. First, in the case where $G = \mathbb{Z}_2^n$, by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most $\lceil n/2 \rceil$, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on $\mathbb{Z}_N$ (when d divides N). By constructing a particular chordal cover of the d'th power of the N-cycle, we prove that any such function is a sum of squares of functions with at most $3d\log(N/d)$ nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in $\mathbb{R}^{2d}$ with N vertices can be expressed as a projection of a section of the cone of psd matrices of size $3d\log(N/d)$. Putting $N=d^2$ gives a family of polytopes $P_d \subset \mathbb{R}^{2d}$ with LP extension complexity $\text{xc}_{LP}(P_d) = \Omega(d^2)$ and SDP extension complexity $\text{xc}_{PSD}(P_d) = O(d\log(d))$. To the best of our knowledge, this is the first explicit family of polytopes in increasing dimensions where $\text{xc}_{PSD}(P_d) = o(\text{xc}_{LP}(P_d))$.Comment: 34 page

Semidefinite descriptions of the convex hull of rotation matrices

We study the convex hull of $SO(n)$, thought of as the set of $n\times n$ orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of $SO(n)$ is doubly spectrahedral, i.e. both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit, and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.Comment: 29 pages, 1 figur

Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose $X$ into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points $v_1,v_2,...,v_n\in \R^k$ (where $n > k$) determine whether there is a centered ellipsoid passing \emph{exactly} through all of the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace $U$ that ensures any positive semidefinite matrix $L$ with column space $U$ can be recovered from $D+L$ for any diagonal matrix $D$ using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.Comment: 20 page

Equivariant Semidefinite Lifts of Regular Polygons.

Given a polytope P ⊂ ℝn, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the projection of an affine slice of the d × d positive semidefinite cone. Such a representation allows us to solve linear optimization problems over P using a semidefinite program of size d and can be useful in practice when d is much smaller than the number of facets of P. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e., those psd lifts that respect the symmetries of P. One of the simplest families of polytopes with interesting symmetries is regular polygons in the plane. In this paper, we give tight lower and upper bounds on the size of equivariant psd lifts for regular polygons. We give an explicit construction of an equivariant psd lift of the regular 2n-gon of size 2n − 1, and we prove that our construction is essentially optimal by proving a lower bound on the size of any equivariant psd lift of the regular N-gon that is logarithmic in N. Our construction is exponentially smaller than the (equivariant) psd lift obtained from the Lasserre/sum-of-squares hierarchy, and it also gives the first example of a polytope with an exponential gap between equivariant psd lifts and equivariant linear programming lifts

Findings from a pilot randomised trial of an asthma internet self-management intervention (RAISIN)

<b>Objective </b>To evaluate the feasibility of a phase 3 randomised controlled trial (RCT) of a website (Living Well with Asthma) to support self-management.<p></p> <b>Design and setting</b> Phase 2, parallel group, RCT, participants recruited from 20 general practices across Glasgow, UK. Randomisation through automated voice response, after baseline data collection, to website access for minimum 12 weeks or usual care.<p></p> <b>Participants </b>Adults (age≥16 years) with physician diagnosed, symptomatic asthma (Asthma Control Questionnaire (ACQ) score ≥1). People with unstable asthma or other lung disease were excluded.<p></p> <b>Intervention</b> Living Well with Asthma’ is a desktop/ laptop compatible interactive website designed with input from asthma/ behaviour change specialists, and adults with asthma. It aims to support optimal medication management, promote use of action plans, encourage attendance at asthma reviews and increase physical activity.<p></p> <b>Outcome measures</b> Primary outcomes were recruitment/retention, website use, ACQ and mini- Asthma Quality of Life Questionnaire (AQLQ). Secondary outcomes included patient activation, prescribing, adherence, spirometry, lung inflammation and health service contacts after 12 weeks. Blinding postrandomisation was not possible.<p></p> <b>Results </b>Recruitment target met. 51 participants randomised (25 intervention group). Age range 16–78 years; 75% female; 28% from most deprived quintile. 45/51 (88%; 20 intervention group) followed up. 19 (76% of the intervention group) used the website, for a mean of 18 min (range 0–49). 17 went beyond the 2 ‘core’ modules. Median number of logins was 1 (IQR 1–2, range 0–7). No significant difference in the prespecified primary efficacy measures of ACQ scores (−0.36; 95% CI −0.96 to 0.23; p=0.225), and mini-AQLQ scores (0.38; −0.13 to 0.89; p=0.136). No adverse events.<p></p> <b>Conclusions</b> Recruitment and retention confirmed feasibility; trends to improved outcomes suggest use of Living Well with Asthma may improve self-management in adults with asthma and merits further development followed by investigation in a phase 3 trial

Diagonal and low-rank decompositions and fitting ellipsoids to random points

Identifying a subspace containing signals of interest in additive noise is a basic system identification problem. Under natural assumptions, this problem is known as the Frisch scheme and can be cast as decomposing an n × n positive definite matrix as the sum of an unknown diagonal matrix (the noise covariance) and an unknown low-rank matrix (the signal covariance). Our focus in this paper is a natural class of random instances, where the low-rank matrix has a uniformly distributed random column space. In this setting we analyze the behavior of a well-known convex optimization-based heuristic for diagonal and low-rank decomposition called minimum trace factor analysis (MTFA). Conditions for the success of MTFA have an appealing geometric reformulation as finding a (convex) ellipsoid that exactly interpolates a given set of n points. Under the random model, the points are chosen according to a Gaussian distribution. Numerical experiments suggest a remarkable threshold phenomenon: if the (random) column space of the n × n lowrank matrix has codimension as small as 2√n then with high probability MTFA successfully performs the decomposition task, otherwise it fails with high probability. In this work we provide numerical evidence and prove partial results in this direction, showing that with high probability MTFA recovers such random low-rank matrices of corank at least cn[superscript β] for β ϵ (5/6, 1) and some constant c.United States. Air Force Office of Scientific Research (AFOSR under Grant FA9550-12-1-0287)United States. Air Force Office of Scientific Research (AFOSR under Grant FA9550-11-1-0305

Polymorphisms in Toll-like receptor 4 ( TLR4 ) are associated with protection against leprosy

Accumulating evidence suggests that polymorphisms in Toll-like receptors (TLRs) influence the pathogenesis of mycobacterial infections, including leprosy, a disease whose manifestations depend on host immune responses. Polymorphisms in TLR2 are associated with an increased risk of reversal reaction, but not susceptibility to leprosy itself. We examined whether polymorphisms in TLR4 are associated with susceptibility to leprosy in a cohort of 441 Ethiopian leprosy patients and 197 healthy controls. We found that two single nucleotide polymorphisms (SNPs) in TLR4 (896G>A [D299G] and 1196C>T [T399I]) were associated with a protective effect against the disease. The 896GG, GA and AA genotypes were found in 91.7, 7.8 and 0.5% of leprosy cases versus 79.9, 19.1 and 1.0% of controls, respectively (odds ratio [OR] = 0.34, 95% confidence interval [CI] 0.20-0.57, P < 0.001, additive model). Similarly, the 1196CC, CT and TT genotypes were found in 98.1, 1.9 and 0% of leprosy cases versus 91.8, 7.7 and 0.5% of controls, respectively (OR = 0.16, 95% CI 0.06--.40, P < 0.001, dominant model). We found that Mycobacterium leprae stimulation of monocytes partially inhibited their subsequent response to lipopolysaccharide (LPS) stimulation. Our data suggest that TLR4 polymorphisms are associated with susceptibility to leprosy and that this effect may be mediated at the cellular level by the modulation of TLR4 signalling by M. lepra