# Pipage Rounding, Pessimistic Estimators and Matrix Concentration

Nicholas J. A. Harvey
Department of Computer Science
UBC
Neil Olver
Department of Mathematics
MIT
Cambridge, USA
January 23, 2021
January 23, 2021
###### Abstract

Pipage rounding is a dependent random sampling technique that has several interesting properties and diverse applications. One property that has been particularly useful is negative correlation of the resulting vector. Unfortunately negative correlation has its limitations, and there are some further desirable properties that do not seem to follow from existing techniques. In particular, recent concentration results for sums of independent random matrices are not known to extend to a negatively dependent setting.

We introduce a simple but useful technique called concavity of pessimistic estimators. This technique allows us to show concentration of submodular functions and concentration of matrix sums under pipage rounding. The former result answers a question of Chekuri et al. (2009). To prove the latter result, we derive a new variant of Lieb’s celebrated concavity theorem in matrix analysis.

We provide numerous applications of these results. One is to spectrally-thin trees, a spectral analog of the thin trees that played a crucial role in the recent breakthrough on the asymmetric traveling salesman problem. We show a polynomial time algorithm that, given a graph where every edge has effective conductance at least , returns an -spectrally-thin tree. There are further applications to rounding of semidefinite programs, to the column subset selection problem, and to a geometric question of extracting a nearly-orthonormal basis from an isotropic distribution.

\numberofauthors

2

## 1 Introduction

Rounding is a crucial step in the design of many approximation algorithms. Given a fractional vector satisfying some constraints, a rounding method produces an integer vector that satisfies those constraints, either exactly or approximately. Randomized rounding [41] [57, Chapter 5], in which the coordinates of the fractional vector are rounded randomly and independently, produces good integer vectors for many applications. Dependent rounding methods, in which the resulting integer vector does not have independent coordinates, are important in many scenarios where naive randomized rounding does poorly. Various techniques exist for designing dependent rounding methods (see, e.g., the surveys [48, 4]).

It is common for a rounding scenario to involve two types of constraints: hard constraints, which must be satisfied exactly by the integer solution, and soft constraints, which must be approximately satisfied by the integer solution. Low-congestion multi-path routing [49], max cut with given sizes of parts [1], thin spanning trees [3], and submodular maximization under a matroid constraint [15, 20] are examples of problems whose solutions involve such a rounding scenario. The hard constraint is often membership in an integer polytope that is defined using combinatorial objects (e.g., matchings or matroids). The soft constraints are usually simple linear inequalities.

With randomized rounding, the independent choices lead to concentration of measure phenomena that are useful for handling soft constraints. For example, Chernoff bounds are commonly used to show that linear inequalities are approximately satisfied [41]. The past decade has seen various uses of matrix concentration bounds (e.g., [2, 44, 52]) to show that linear matrix inequalities are approximately satisfied by random sampling or rounding. Such uses have occurred in many diverse areas: graph sparsification [46], compressed sensing [54], statistics [21], machine learning [42] and numerical linear algebra [33].

With dependent rounding, concentration phenomena can also occur. Pipage rounding, swap rounding and maximum entropy sampling are dependent rounding techniques that have seen many important uses over the past decade [49, 1, 26, 15, 3, 20]. An important feature in some scenarios is that any Chernoff bound that is valid under independent randomized rounding remains valid under these dependent rounding techniques. This fact is proven by showing that the rounded solution has a negatively correlated distribution, then appealing to the fact that Chernoff bounds remain valid under such distributions [39]. Unfortunately, commutativity plays a key role in proving that fact, and these arguments do not seem to extend to matrix concentration bounds, e.g., [2, 37, 44, 52]. Consequently, these matrix inequalities have so far not been combined with dependent rounding.

We prove the first result showing that matrix concentration bounds are usable in a dependent rounding scenario. Our technique is not based on negative correlation, but rather the fortuitous interaction between pipage rounding and various pessimistic estimators. In particular, we show that Tropp’s matrix Chernoff bound [52] has a pessimistic estimator that decreases monotonically under pipage rounding. As a consequence, we can extend the reach of pipage rounding from soft constraints that are linear inequalities to soft constraints that are linear matrix inequalities. Our proof uses non-trivial techniques from matrix analysis and complex analysis; in particular, we prove a new variant of Lieb’s concavity theorem.

### 1.1 Motivation and Results

One key area where our techniques yield new results is for thin spanning trees. These are intriguing objects in graph theory that relate to foundational topics, such as nowhere-zero flows [28], and the asymmetric traveling salesman problem [3]. Given a graph on nodes, a spanning tree of is -thin if, for every cut, the number of edges of crossing the cut is at most times the number of edges of crossing the cut. It has been conjectured that any graph with connectivity has an -thin spanning tree where . This would imply a constant factor approximation algorithm for the asymmetric traveling salesman problem [38]. Asadpour et al. [3] give a randomized algorithm to find a spanning tree that is

A spectrally-thin spanning tree is a stronger notion that is naturally motivated by work on spectral sparsification [46, 5]. A spanning tree is -spectrally-thin if , where refers to the Laplacian of , and to the Löwner ordering of Hermitian matrices. In Section 4.3, we show a result on spectrally thin trees that strongly mirrors the result of Asadpour et al.

{theorem}

There is a deterministic, polynomial-time algorithm that given any graph on nodes where every edge has effective conductance at least , constructs a

This spectral notion of thinness seems to be an important one, as the recent breakthrough of Marcus et al. [34] implies that -spectrally-thin trees exist. Details of this connection are given in Appendix F. It is unknown if similar techniques can show that -thin trees exist. The best known algorithmic construction of spectrally-thin trees is still Theorem 1.1.

This result is a special case of a result in a more abstract geometric setting. Suppose are unit vectors in for which is a multiple of the identity. Does there exist a subset that is a basis of and for which the maximum eigenvalue of is small? The maximum eigenvalue is 1 if and only if is orthonormal, but an arbitrary need not contain an orthonormal basis. Again, the breakthrough of Marcus et al. [34] yields a non-constructive proof of a basis with maximum eigenvalue ; see Appendix F. In Section 4.2, we show how to find in polynomial time a basis for which the maximum eigenvalue of is . Previous constructive techniques [2, 37, 44, 52] only provide a bound of .

Our geometric result also relates to the column subset selection problem in numerical linear algebra [13, 51, 12, 22] which seeks to “approximate” a matrix by a small subset of its columns, under various notions of approximation. Define the stable rank of to be the Frobenius norm divided by the spectral norm, all squared; this roughly captures the rank of , ignoring negligibly small singular values. In numerical linear algebra [13, 12, 22], the number of columns chosen is typically much larger than the stable rank. The operator theory community considers similar questions [10, 11, 47, 51], although the number of columns selected is typically much smaller than the stable rank. In Section 4.4, we show that one can efficiently select a linearly independent set of columns of size equal to the stable rank, while carefully controlling the maximum singular value.

### 1.2 Techniques

Our results are based on the pipage rounding technique [1, 49, 26, 15], which has had several interesting uses in the recent literature. Deterministic and randomized forms of pipage rounding exist; our result applies to both of those, as well as to swap rounding. Typical uses of pipage rounding involve some of the following ideas.

• There are processes that iteratively move a point in a matroid base polytope towards an extreme point, while modifying only two coordinates at a time. The exchange properties of matroid bases ensure that this is possible.

• One can define a “potential function” on the matroid base polytope (e.g., the ad hoc functions defined in [1], or the multilinear extension of a submodular function [15]) such that the function is concave or convex in directions that increase one coordinate and decrease another.

• The randomized form of pipage rounding [49, 26, 20] outputs a matroid base whose elements are negatively correlated (more precisely, negative cylinder dependent). This ensures that linear functions of that base satisfy the same Chernoff-type concentration bounds that are satisfied under independent rounding.

Our aim is to show that, for various concentration bounds, the final extreme point satisfies the same bounds that would be achieved by independent randomized rounding. For Chernoff bounds this follows from negative correlation, but for other bounds such a result was not previously known.

• Let be a monotone submodular function defined on the ground set of the matroid. When using randomized pipage rounding, does the value of at the final extreme point satisfy the same lower tail bound as when using independent rounding? Chekuri et al. [19] conjectured this to be true, and they proved such a result when using swap rounding.

• Let be a linear function mapping points in the matroid base polytope to symmetric matrices. When using pipage rounding, can the value of at the final extreme point be guaranteed to satisfy the same eigenvalue bounds as when using independent rounding?

It does not seem easy to answer these questions using negative correlation properties.

We present a new approach that leads to a positive answer to both of these questions. In both cases, we can define a pessimistic estimator [40] that bounds the probability that randomized rounding fails to achieve the desired concentration. We show that these pessimistic estimators are concave when one element’s sampling probability is increased and another’s is decreased by the same amount. Due to that concavity property, the base output by randomized pipage rounding satisfies the same concentration bounds that would be satisfied under independent randomized rounding. For the second question (matrix concentration), the pessimistic estimator can be efficiently evaluated, so deterministic pipage rounding can also be used.

The concavity property of our pessimistic estimator for matrix concentration is a non-trivial fact. We establish that fact by proving a new variant of Lieb’s concavity theorem [32], which is a “masterpiece of matrix analysis” [8] with deep applications in mathematical physics and quantum information theory [16, 23, 35]. Although there is much interest in the mathematical physics community on extensions and variants of Lieb’s theorem, our particular variant does not seem to appear in the literature.

## 2 Preliminaries

Let . For a set , the vector is the characteristic vector of . For a vector and a set , the notation denotes . The vector denotes the standard basis vector of the finite dimensional vector space that is apparent from context. The vector denotes a vector whose components are all ones and whose dimension is apparent from context. We will use and to denote the nonnegative and positive reals respectively.

Let denote the space of symmetric, real matrices of size . Let respectively denote the cones of positive semidefinite and positive definite matrices. Let denote the space of diagonal matrices. Let denote the Löwner partial order on symmetric matrices, i.e., iff . Similarly, iff . For , let and respectively denote the largest and smallest eigenvalues of . For , let denote its Moore-Penrose pseudoinverse. For , let denote the positive semidefinite square root of . The image of is and the orthogonal projection onto is .

The notation denotes the norm for vectors and the operator norm for matrices.

If is a distribution, means that the random variable has distribution .

## 3 Concavity of Pessimistic Estimators

In this section we state the known results on pipage rounding and our concavity of pessimistic estimators technique. We then apply this technique in three scenarios, of increasing difficulty: (1) Chernoff bounds, (2) submodular functions, and (3) matrix concentration. The latter two results are new, and in particular are not known to follow using negative correlation. This pessimistic estimator for matrix concentration underlies all applications in Section 4.

### 3.1 Pipage Rounding

Pipage rounding is a dependent rounding process originating in works of Ageev, Srinivasan and Sviridenko [1, 49]. Calinescu et al. [15] generalized it to a matroid setting. We now state the main results of randomized and deterministic pipage rounding; a proof sketch is given in Appendix A.

Let be a matroid on and let be its base polytope. For all algorithmic applications in this paper, can be presented to the algorithm via an independence oracle. A function is said to be concave under swaps if

 (1)
{theorem}

[Randomized Pipage Rounding] There is a randomized, polynomial-time algorithm that, given , outputs an extreme point of with and such that, for any concave under swaps, .

{theorem}

[Deterministic Pipage Rounding] There is a deterministic, polynomial-time algorithm that, given and a value oracle for a function that is concave under swaps, outputs an extreme point of with .

The swap rounding procedure of Chekuri et al. [19, 20] also proves Theorem 3.1 and Theorem 3.1.

For , let be the product distribution on with marginals given by , i.e., . Let . A pessimistic estimator [40, 50] for is a function that satisfies

 PX∼D(x)[X∈E] ≤ g(x)∀x∈[0,1]m (2) min{g(x−xiei),g(x+(1−xi)ei)} ≤ g(x)   ∀x∈[0,1]m, i∈[m].

For uses of pessimistic estimators in derandomization, the function is also required to be efficiently computable. That is not required with their use in randomized pipage rounding as is not even provided as input to the algorithm.

{claim}

[Concavity of Pessimistic Estimators] Let and let be a function that satisfies (2) and is concave under swaps.

Suppose randomized pipage rounding is started at an initial point , and let be the (random) extreme point of that is output. If then .

Suppose deterministic pipage rounding is given oracle access to and an initial point with . Then the extreme point of that is output satisfies .

We omit the proof of Claim 3.1 as it is an easy consequence of Theorem 3.1 and Theorem 3.1.

### 3.2 Chernoff bound

Let us start with a simple result to illustrate the technique. First we state the Chernoff bound in convenient notation. We discuss only the right tail; an analogous result holds for the left tail. Fix any vector . For and , define by

 gt,θ(x) := e−θt⋅EX∼D(x)[eθwTX].

Let and . Then

 PX∼D(x)[wTX≥t] ≤ infθ>0gt,θ(x)andg(1+δ)μ,ln(1+δ)(x) ≤ (eδ(1+δ)1+δ)μ. (3)

The following claim is proven in Appendix B.

{claim}

is concave under swaps.

Consequently, Claim 3.1 implies the following result.

{corollary}

If randomized pipage rounding starts at and outputs the extreme point of then, ,

 P[wT^x≥(1+δ)μ] ≤ (eδ(1+δ)1+δ)μ (4)

where . Furthermore, if this right-hand side is strictly less than , then deterministic pipage rounding outputs an extreme point of with .

The key point is that the right-hand sides of (3) and (4) are the same. Chekuri et al. [19] proved this fact using negative correlation of , generalizing a result of Srinivasan [49].

### 3.3 Submodular functions

Chekuri et al. [19, Theorem 1.3] prove an analog of the Chernoff bound for concentration of submodular functions under independent rounding. They show that the same bound remains true under swap rounding [19, Theorem 1.4] and ask whether it remains true under pipage rounding.

Formally, let be a non-negative, monotone, submodular function with marginals in . The multilinear extension of is with . For and , define by

 gt,θ(x) := e−θt⋅EX∼D(x)[eθf(X)].

The left tail bound of Chekuri et al. is: with ,

 PX∼D(x)[f(X)≤t] ≤ infθ<0gt,θ(x)andg(1−δ)μ,ln(1−δ)(x) ≤ exp(−δ2μ/2).

The following claim is proven in Appendix B.

{claim}

is concave under swaps.

Claim 3.1 implies the following result, answering an open question of Chekuri et al. [19, p. 3].

{corollary}

If randomized pipage rounding starts at and outputs the extreme point of then, letting , we have .

Chekuri et al. [20, p. 583] state that this fact does not follow from negative correlation of .

### 3.4 Matrix Concentration

Tropp [52], improving on Ahlswede-Winter [2] and Oliviera [37], proves a beautiful analog of the Chernoff bound for sums of independent random matrices. We state a simplified form here.

{theorem}

Let satisfy . For and , define by

 gt,θ(x) := e−θt⋅trexp(m∑i=1logEX∼D(x)[eθXiMi]).

Then, for and ,

 PX∼D(x)[∥∥∑iXiMi∥∥≥t] ≤ infθ>0gt,θ(x)andg(1+δ)μ,ln(1+δ)(x) ≤ n⋅(eδ(1+δ)1+δ)μ/R.

The following is our main lemma on pessimistic estimators. The proof is in Appendix B.

{lemma}

is concave under swaps.

Consequently, Claim 3.1 implies the following result.

{corollary}

Let be a matroid base polytope and let . Let satisfy . Let . If randomized pipage rounding starts at and outputs the extreme point of then we have

 P[∥∥∑i∈SMi∥∥≥(1+δ)μ]≤n⋅(eδ(1+δ)1+δ)μ/R. (5)

Furthermore, if this right-hand side is strictly less than , then deterministic pipage rounding outputs an extreme point of with .

The inequalities in Theorem 3.4 involve non-trivial matrix analysis, such as operator concavity of and Lieb’s celebrated concavity theorem [32]. It seems that even those results do not suffice to prove Lemma 3.4. To prove it, we derive a new variant of Lieb’s theorem (Theorem 3.4). Lieb [32] proved several related concavity theorems; for us, the most relevant form is:

{theorem}

[Lieb [32]] Let and . Then is concave in a neighborhood of .

The main technical result of this paper is:

{theorem}

Let , and . Then the univariate function

 z ↦ trexp(L+log(C1+zK1)+log(C2−zK2)) (6)

is concave in a neighborhood of .

There are several known approaches to proving Lieb’s theorem. The simplest is Tropp’s approach [53]; however, his proof is based on joint convexity of quantum entropy, which is itself usually proven using Lieb’s theorem. We were unable to prove Theorem 3.4 using Tropp’s approach. Lieb’s original proof [32], which proves concavity by directly analyzing the second derivative, involves numerous delicate steps of matrix analysis. We were able to adapt this approach to prove a weaker form of Theorem 3.4 that requires some additional commutativity assumptions; details are in Appendix E. This weaker result suffices to prove Lemma 3.4. Epstein [24] gives an elegant approach to proving Lieb’s theorem using complex analysis, and in particular powerful results concerning Herglotz functions. Our proof of Theorem 3.4, which appears in Appendix D, is an adaptation of Epstein’s approach.

Remark. Another well-known matrix concentration inequality is the Ahlswede-Winter [2] inequality, for which pessimistic estimators were studied by Wigderson and Xiao [56]. It is natural to wonder whether we could have used their pessimistic estimators instead. Unfortunately they do not seem applicable for our scenario. The issue is that the Ahlswede-Winter inequality is most effective for analyzing sums of i.i.d. random matrices, due to some inequalities that arise in their analysis. In our scenario, due to the way that pipage rounding works, we require non-i.i.d. product distributions, so it is much more convenient to base our approach on Theorem 3.4.

## 4 Applications

### 4.1 Rounding of semidefinite programs

Let be a matroid and let be its base polytope. Consider the spectrahedron

 Q := P∩{x∈Rm:m∑i=1xiAi⪯B}, (7)

where each . We think of as specifying “hard” constraints and the semidefinite constraint as being “soft”.

{theorem}

Suppose that for all . If randomized pipage rounding starts at and outputs the extreme point of , then , for some . Furthermore, if deterministic pipage rounding starts at , then it outputs an extreme point of with .

This theorem is optimal with respect to , as discussed below. The hypothesis that is a “width” condition that commonly arises in optimization and rounding.

###### Proof.

Recall the notation defined in Section 2. Let . By standard arguments,

 ∑mi=1xiAi⪯B ⟺∑mi=1xiMi⪯IimB and∑i∈SAi⪯αB ⟺∑i∈SMi⪯αIimB.

We assume that , so . Apply Corollary 3.4 with , and . A standard calculation shows that the right-hand side of (5) is less than . ∎

Chekuri, Vondrák and Zenklusen [19, 20] considered the problem of rounding a point in a matroid polytope to an extreme point, subject to additional packing constraints. Their result generalizes the low-congestion multi-path routing problem studied earlier by Srinivasan et al. [49, 26], but it is itself a special case of Theorem 4.1 where the matrices and are diagonal. The factor is optimal in Theorem 4.1 because it is optimal for rounding this low-congestion multi-path routing problem, and even for the congestion minimization problem [31].

### 4.2 Rounding an isotropic distribution to a nearly orthonormal basis

Let satisfy for all . Let be a probability distribution on these vectors such that the covariance matrix is . A random vector drawn from that distribution is said to be in isotropic position.

{theorem}

There is a polynomial time algorithm (either randomized or deterministic) to compute a subset such that forms a basis of , and for which , where .

As is discussed in Appendix F, the recent breakthrough on the Kadison-Singer problem [34] implies the following existential result: {theorem} There exists such that forms a basis of , and for which .

We now prove Theorem 4.2 using Theorem 4.1. Let be the linear matroid corresponding to the vectors . Let be the base polytope of that linear matroid. Let be the rank function of that matroid, i.e.,

 P := {x∈Rn+:x(J)≤r(J) ∀J⊆[m]  and  x([m])=r([m])}.

Define , and

 Q = P∩{x∈Rm:∑ixiAi⪯B}.

Let . Then the following claim and the hypothesis that show that .

{claim}

.

Since , we have . Theorem 4.1 gives an algorithm to construct an extreme point of for which , with . Since is the base polytope of , forms a basis of . Finally, . This completes the proof of Theorem 4.2, modulo the proof of Claim 4.2.

In Appendix C.1, we show that Theorem 4.2 can be generalized from a decomposition of the identity into rank-one matrices to a decomposition into matrices of arbitrary rank. The proof of Claim 4.2 is analogous to the proof of Claim C.1. We remark that Theorem 4.2 is not known to have a generalization to matrices of arbitrary rank.

### 4.3 Thin trees

Let be a graph. For convenience we assume that . The cut defined by is

 δG(U)=δ(U)={uv∈E: exactly one of u% and v is in U}.

For a subgraph of , let denote all edges of with exactly one endpoint in .

{definition}

A subgraph of is called -thin if for all .

{conjecture}

[Goddyn [28]] Every graph with connectivity at least has an -thin spanning subtree, for some function that vanishes as tends to infinity.

The crucial detail in this conjecture is that the function should not depend on the size of the graph. The best progress on this conjecture for general graphs is as follows.

{theorem}

[Asadpour et al. [3]] Let be a graph with vertices and connectivity . Then has a

Now we define spectrally-thin trees and prove an analog of this theorem. The Laplacian of is the symmetric matrix with rows and columns indexed by defined by

 LG := ∑uv∈E(eu−ev)(eu−ev)T.
{definition}

Let be a spanning subtree of and let be the Laplacian of . The tree is -spectrally-thin if .

Any tree that is -spectrally-thin is also -thin, because

 |δT(U)| = χ(U)TLTχ(U) ≤ ϵ⋅χ(U)TLGχ(U) = ϵ⋅|δG(U)|.

The converse is not true. Moreover, the connectivity hypothesis in Theorem 4.3 does not suffice111 This result was independently observed by M. de Carli Silva, N. Harvey and C. Sato, and by M. Goemans [29], using slightly different examples. to obtain a good spectrally-thin tree. The proof is in Appendix C.2.1.

{theorem}

For every , there exists a weighted graph with vertices and connectivity that does not have an -spectrally-thin spanning subtree.

Nevertheless, if we strengthen the connectivity lower bound to a lower bound on the effective conductances, then we have the following construction of spectrally-thin trees. For an edge , the effective resistance in between and is . The effective conductance in between and is .

{theorem}

Let be a graph with vertices such that for every edge . Then there is a polynomial time algorithm (either randomized or deterministic) to construct a

Theorem 4.3 follows directly from Theorem 4.1, letting be the graphic matroid corresponding to . It also follows from Theorem 4.2, as we show in Appendix C.2. That viewpoint is advantageous, since Theorem 4.2 then immediately implies {theorem} Let be a graph with vertices such that for every edge . Then has a -spectrally-thin spanning subtree. We are not aware of any formal connection between Theorem 4.3 and Conjecture 4.3 or the traveling salesman problem.

Although Theorem 4.3 and Theorem 4.3 are formally incomparable, it is worth understanding their similarities and differences. Both results have a seemingly suboptimal factor of . Theorem 4.3 requires only a connectivity lower bound, which is important in applications [28, 3], but the resulting tree is thin, not spectrally-thin; also, their algorithm is randomized. Theorem 4.3 requires a conductance lower bound (which is stronger than a connectivity lower bound), but the resulting tree is spectrally-thin (which is stronger than being thin); also, our algorithm can be made deterministic. The use of randomization seems quite inherent in the algorithms [3, 20] for Theorem 4.3, as the thinness condition involves controlling exponentially many cuts, which seems difficult to accomplish by a deterministic, polynomial-time algorithm.

The quantities and can be related in certain classes of graphs. We say that a family of graphs has nearly equal resistances if there is a constant (independent of the number of vertices) such that for all edges . For example, any Ramanujan graph has nearly equal resistances. Edge-transitive graphs, such as hypercubes, have nearly equal (in fact, exactly equal) resistances.

{corollary}

Let be a graph with vertices, nearly equal resistances, and connectivity . Then there is a deterministic, polynomial time algorithm to construct a

The proof is in Appendix C.2.

### 4.4 Column-subset selection

Column-subset selection is an important topic in numerical linear algebra [13, 51, 22, 12]. Similar questions are considered in operator theory [10, 11, 47, 51, 58]. In this section we prove a non-isotropic analog of Theorem 4.2, which gives a new result on column-subset selection. For a real matrix , let denote its Frobenius norm. The stable rank of is .

{theorem}

Let be a real matrix of size whose columns are denoted . Suppose that . Then there is a deterministic, polynomial time algorithm to compute of size such that is linearly independent, and .

This is optimal with respect to as it can happen that , in which case is linearly dependent whenever .

We now prove Theorem 4.4 using Theorem 4.1. Note that . Let be the vector with for all . Note that . We claim that can be viewed as a “fractional set” of linearly independent vectors of size . Formally, for any set , let denote the submatrix of consisting of the columns in . Define the following family of sets

 B = {I⊆[m]:rankAI=|I|=⌊st.rank(A)⌋}.

Then is the base family of the linear matroid corresponding to , truncated to rank . Let denote that matroid and let denote its base polytope.

{claim}

.

The proof is in Appendix C.3. Given this claim, all that remains is an easy application of Theorem 4.1. Define , and

 Q = P∩{x∈Rm:∑ixiAi⪯B}.

We have by Claim 4.4 and the fact that

 ∑ipiAi ⪯ ∑iAi∥A∥2 = AAT∥A∥2 ⪯ I = B.

Note that . Theorem 4.1 gives a deterministic algorithm to construct an extreme point of for which , with . Since is a base of , the set has rank equal to . This completes the proof of Theorem 4.4.

##### Acknowledgements.

N. Harvey thanks Joel Friedman and Mohit Singh for numerous enlightening discussions. We thank Isaac Fung for collaborating at a preliminary stage of this work. We also thank Christos Boutsidis, Joseph Cheriyan, Satoru Fujishige, Michel Goemans, Mary Beth Ruskai, Nikhil Srivastava, Joel Tropp, Roman Vershynin and Jan Vondrák for helpful discussions and suggestions.

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## Appendix A Pipage Rounding

Let be a point in the matroid polytope and assume that satisfies (1). Delete all coordinates of that are equal to zero and consider the residual problem. It is well-known that, for any such point , there exists a chain of sets whose corresponding constraints of span the constraints that are tight at . If for every then these give linearly independent tight constraints, so the point is an extreme point. Otherwise there is some set , , for which . In this case is not an extreme point. To see this, let and be distinct elements of . Note that the point satisfies all the constraints that are tight at . So, for all in some open neighborhood of , the point is still feasible for .

Define

 ℓ =min{z∈R:p+z(ea−eb)∈P} andu =max{z∈R:p+z(ea−eb)∈P}.

Define

 pℓ=p+ℓ(ea−eb)andpu=p+u(ea−eb).

Since is concave, we must have either

 g(pℓ)≤g(p)org(pu)≤g(p).

Furthermore, both and lie on a lower-dimensional face than does. So starting from some initial , iterations suffice to find an extreme point of with .

The randomized version of pipage rounding does not even need access to the function . Instead, it simply chooses the next point to be with probability , or with probability . This ensures that , and the concavity of yields . Thus applying this procedure to some initial point until an extreme point is obtained, satisfies and .

## Appendix B Proofs of concavity under swaps

###### Proof of Claim 3.2.

We can rewrite

 g(x) = e−θt⋅∏i(1+xi(eθwi−1)).

Rewriting in this way, all factors are non-negative and only two of them depend on , so for some

 d2dz2g(x+z(ea−eb)) =c⋅(−2(eθwa−1)(eθwb−1)).

This is non-positive so is concave under swaps. ∎

###### Proof of Claim 3.3.

Recall that . Define by . By Claim B, is a supermodular function. Its multilinear extension is

 H(x) = EX∼D(x)[h(X)].

Since is submodular, it follows from results of Calinescu et al. [15] that for any . Since , the second derivative of

 z ↦ g(x+z(ei−ej))

is non-positive. Thus is concave under swaps. ∎

{claim}

Let be non-decreasing and submodular. Let be non-increasing and convex. Then is supermodular.

###### Proof.

We require the following property of convex functions. Suppose satisfy

 a ≤ min{b,c} ≤ max{b,c} ≤ d. (8)

Then any function that is convex on satisfies

 g(d)−g(c)d−c ≥ g(b)−g(a)b−a. (9)

Fix any , and an element . Define

 a :=f(A),b:=f(A+x),c:=f(B), d :=f(B)+f(A+x)−f(A),e:=f(B+x).

Since is non-decreasing, (8) holds. Since is submodular, holds. Since is non-increasing, . Combining that with (9) and the observation that , we obtain

 g(e)−g(c) ≥ g(d)−g(c) ≥ g(b)−g(a)

That is,

so is supermodular. ∎

###### Proof of Lemma 3.4.

We will show that

The boundary of is handled by continuity. Note that

 EX∼D(x)[eθXiMi] = xi⋅eθMi+(1−xi)⋅I =: Ci.

Adding (sufficiently small) to the sampling probability of coordinate , the expectation becomes

 EX∼D(x+zei)[eθXiMi] = (xi+z)⋅eθMi+(1−xi−z)⋅I = Ci+z(eθMi−I)=: Ki.

Note that and because and . Furthermore, the matrices and commute since any eigenbasis for is also an eigenbasis of and .

To finish the proof we must show that, for distinct ,

 z↦trexp(log(Ca+zKa)+log(Cb−zKb)+∑i∉{a,b}logEX∼D(x)[eθXiMi]=: L)

is concave in a neighborhood of . This follows from Theorem 3.4. ∎

## Appendix C Proofs of Applications

### c.1 Rounding decompositions of the identity

Here, we give a generalization of Theorem 4.2 to a decomposition of the identity into matrices of arbitrary rank. {theorem} Let satisfy . Then there exists a subset with such that has full rank and maximum eigenvalue at most .

###### Proof.

Let be a matrix such that . Define the function by

where is the matrix obtained by concatenating in any order all columns from the matrices . It is well-known that such a function is:

• Normalized: ,

• Monotone: whenever , and

• Submodular: for all .

For any normalized, monotone, submodular function , its base polytope is defined to be

 B(f) := {x∈Rm+:x(J)≤f(J) ∀J⊆[m],  and  x([m])=f([m])}.

Define the vector by . Note that and , so we can think of as defining a “fractional multiset” of matrices. Intuitively, we want to “round” the coordinates of to integers. To that end, define the polytope

 P′ := B(r)∩{x:⌊p⌋≤x≤⌈p⌉},

where and respectively denote the component-wise floor and ceiling of the vector . The polytope is not necessarily a matroid polytope; for example, a vector in could have a coordinate strictly greater than .

{claim}

.

{claim}

is a matroid base polytope.

Claim C.1 is proven below. Claim C.1 is a folklore result that can be derived using reductions and contractions of submodular functions [25, §3.1(b)]; see also Fujishige’s remarks on crossing submodular functions [25, Eq. (3.97)].

Define , and

 Q := P∩{x∈Rm:∑ixiAi⪯B}.

Setting , we have by Claim C.1 and

 ∑ixiAi ⪯ ∑ipiXitrXi = ∑iXi = B,

so .

Since , we have . Applying Theorem 4.1, we obtain a vector that is an extreme point of , and for which . Let be the support of . Note that . So

 |S| = m∑i=1^xi ≤ m∑i=1(^xi+⌊pi⌋) ≤ r([m]) = n

and as required. ∎

###### Proof of Claim c.1.

The box constraint is trivially satisfied. We have noted above that
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