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Tight Inner Approximations of the Positive-Semidefinite Cone via
Grassmannian Packing
Tianqi Zheng, James Guthrie, and Enrique Mallada
Abstract— We investigate the problem of finding tight inner
approximations of large dimensional positive semidefinite (PSD)
cones. To solve this problem, we develop a novel decomposition
framework of the PSD cone by means of conical combinations
of smaller dimensional sub-cones. We show that many inner
approximation techniques could be summarized within this
framework, including the set of (scaled) diagonally dominant
matrices, Factor-width kmatrices, and Chordal Sparse ma-
trices. Furthermore, we provide a more flexible family of
inner approximations of the PSD cone, where we aim to
arrange the sub-cones so that they are maximally separated
from each other. In doing so, these approximations tend to
occupy large fractions of the volume of the PSD cone. The
proposed approach is connected to a classical packing problem
in Riemannian Geometry. Precisely, we show that the problem
of finding maximally distant sub-cones in an ambient PSD cone
is equivalent to the problem of packing sub-spaces in a Grass-
mannian Manifold. We further leverage existing computational
method for constructing packings in Grassmannian manifolds
to build tighter approximations of the PSD cone. Numerical
experiments show how the proposed framework can balance
between accuracy and computational complexity, to efficiently
solve positive-semidefinite programs.
Index Terms— Positive Semidefinite Programming, Grass-
mannian Packing.
I. INTRODUCTION
Semidefinite programs (SDPs) are a class of convex op-
timization problems whose domain is formed by the inter-
section of the cone of positive semidefinite (PSD) matrices
with affine subspaces. Having high expressive power, SDPs
have been studied extensively over the past few decades [1],
[2], and have enable many applications in robust control [3],
[4], trajectory planning [5] and robotics [6]. In theory, SDPs
could be solved to arbitrary accuracy in polynomial time
via efficient algorithms, such as second order interior point
methods (IPMs). However, in practice, when the PSD cone,
i.e., the set of all PSD matrices, is high dimensional, the
solving time and memory requirements tend to be impractical
[7]. Over the past few years, extensive research has been
performed with the aim of to improving the scalability of
semidefinite programs [8]. Since the computational cost of
solving an SDP is primarily determined by the size of the
largest semidefinite constraint, several methods have been
proposed to decompose, or approximate high-dimensional
PSD constraints into conic combinations of smaller PSD
cones [9], [10].
T. Zheng, J. Guthrie, and E. Mallada are with the Depart-
ment of Electrical and Computer Engineering, Johns Hopkins Univer-
sity, Baltimore, MD 21218, USA. Email: {tzheng8, jguthri6
,mallada}@jhu.edu.
This has led to several methods that provide inner ap-
proximations for the PSD cone, some of which may still
lead to the same SDP solution. Examples include, the set of
diagonally dominant (DDn) matrices and scaled diagonally
dominant (SDDn) matrices, which can be written as conic
combinations of rank one PSD matrices and conic combi-
nations of second order cones, respectively [9], and lead
to approximations that can be solved using linear programs
(LP) or resp. second-order cone programs (SOCP). Along the
same line, Factor-width kmatrices, which can be viewed as a
generalization of scaled diagonally dominant matrices [11],
can be decomposed as a combination of PSD matrices of
rank at most k. Finally, one could exploit chordal sparsity
patterns of matrices [12], i.e., sparsity induced by a chordal
graph, to equivalently express a large semidefinite constraint
as a set of smaller semidefinite constraints. Unfortunately, the
scalability of chordal decompositions is limited by the size
of the largest clique. Moreover, approximating the PSD cone
with DDnor SDDnis conservative, which means that either
feasible solutions of the constrained problem may be sub-
optimal or that no solution exists. Further, enforcing these
constraints is problematic as nincreases, since the number
of constraints exhibit a combinatorial growth [9], [10].
Our work mostly aligns with the efforts on obtaining inner
approximations of the PSD cone, but proposes a more flex-
ible family of inner approximations based on Grassmannian
packing problems. More specifically, we develop a novel
decomposition framework of the PSD cone that generalizes
the above-mentioned approximation techniques. Using this
decomposition, we develop novel inner approximations of
the PSD cone, based on conic combinations of smaller di-
mensional sub-cones, such that they are maximally separated
from each other. In doing so, these approximations tend to
occupy large fractions of the volume of the PSD cone and
provide more accurate inner approximations. Such problem
of placing maximally distant sub-cones is closely related to
the classical packing problem in Grassmannian manifolds.
The Grassmannian packing problem [13] aims to answer the
following question: how should N k-dimensional subspaces
of an n-dimensional Euclidean space be arranged so that they
are as far apart as possible? The problem of finding Grass-
mannian packings is a fundamental problem of geometry and
has applications in many fields such as signal processing [14]
, information theory [15], [16], and wireless communications
[17]. The hope is that by establishing these connections, this
work will ignite a new search for better approximations that
exploit classical tools from, e.g., information theory.
The rest of the paper is organized as follows. Section II
arXiv:2105.12021v1 [math.OC] 25 May 2021
introduces some preliminary terminology on packing prob-
lems in Grassmannian manifolds. Section III reviews several
existing inner approximations of the PSD cone and provide
a novel decomposition framework of the PSD cone which
unifies existing examples. Section IV further provides our
Grassmannian packing based approximations, by connecting
distances among elements in the Grassmannian manifold
with a properly defined distance between sub-cones. Numer-
ical examples, that leverage existing computational methods
for packings in Grassmannian [18], illustrate how one can
balance between accuracy and computational complexity
using the proposed approach. We finally conclude in Section
VI.
Notation: Denote the set of n×nreal symmetric matrices
by Sn. A matrix A∈Snis positive semi-definite (PSD),
denoted by A0, if xTAx ≥0,∀x∈Rn, and we use
standard notation Sn
+to denote the set of PSD matrices.
Given a set S, its cardinality is denoted by |S|. The set
{1, . . . , n}is denoted by [n]. The Frobenius norm of a Matrix
Xis denoted and calculated as kXk2
F= trace(XTX),
where the trace operator sums the diagonal entries of the
matrix. The conic hull for a set C, i.e., the set of all
conic combinations of points in C, is given by cone(C) :=
{Pk
i=1 αixi:xi∈C, αi∈R≥0, k ∈N}. Given a matrix
F∈Rn×nand a non-empty set S,FS∈Rn×|S|denotes
the sub-matrix of Fcomprised by columns indexed by S.
II. PRELIMINARY
A. Packing in Grassmannian Manifolds
In this section we formalize the problem of finding max-
imally distant k-dimensional subspaces of real Euclidean
n−dimensional space Rn. We thus consider Grassmannian
manifold G(k, n), i.e., the set of all k−dimensional sub-
spaces of Rn. In order to define a proper notion of distance
between elements in G(k, n)it is useful to consider the
principal angles between subspaces.
Definition 1 (Principal Angles [13]).Suppose that S,T
are two k−subspaces in G(k, n). The principal angles
θ1, . . . , θk∈[0, π/2] between Sand Tare defined by:
cos θi= max
u∈S max
v∈T u·v=ui·vi,
for i∈[k], subject to u·u=v·v= 1, u ·uj= 0, v ·vj=
0 (1 ≤j≤i−1).
A more computational definition of the principal angles
based on singular value decomposition (SVD) is provided
in [19], which we describe next. Suppose that columns
of Sand Tform orthonormal bases for S,T ∈ G(k, n)
respectively. Therefore, we have S∈Rn×ksuch that
STS=Ik,range(S) = S, and analogously for T.1Next,
we compute the singular value decomposition of the product
STT=UΣVT, where U, V are k×kunitary matrices and Σ
is a nonnegative, diagonal matrix with non-increasing entries.
The matrix Σof singular values is uniquely determined, with
1Note that this is equivalent to say that Sbelongs to the Stiefel manifold
Vk(Rn) := {A∈Rn×k:ATA=Ik}.
entries being cosines of the principal angles between Sand
T:Σii = cos θi, i ∈[k]. Grassmannian manifolds admit
many metrics such as the spectral distance, the Fubini-Study
distance, the geodesic distance etc. In this paper, we focus
on the chordal distance, which is the easiest to work with
and has a number of desirable features such as its square is
differentiable everywhere.
Definition 2 (Chordal Distance [13]).Suppose we have
two k−dimensional subspaces S,Tin G(k, n)and the
columns of S, T ∈Rn×kform orthonormal bases for S,T,
respectively. The chordal distance between Sand T, as a
function of (S, T ), is given by
dchord(S, T ) :=qsin2θ1+· · · + sin2θk
=[k− kSTTk2
F]1/2,(1)
where θ1, . . . , θk∈[0, π/2] denotes the principal angles
between Sand T.
Note that based on the SVD decomposition, we have
STT=UΣVT, where Σii = cos θi, i ∈[k], and the second
equation comes from the fact that kSTTk2
F=Pk
i=1 cos2θk.
Now, we formally define the packing problem in Grassman-
nian manifolds with chordal distance.
Problem 1 (Packing problem in Grassmannian manifolds
with chordal distance).Given the Grassmannian manifold
G(k, n)of k−dimensional subspaces of the real Euclidean
n−dimensional space Rn, find a set of N k−dimensional
subspaces {S1,...,SN} ⊆ G(k, n)spanned by the matrices
F={F1, . . . , FN}, that solves the mathematical program
max
F:|F|=Nmin
Fi,Fj∈F,i6=jdchord (Fi, Fj).(2)
III. PSD CONE DECOMPOSITION FRAMEWORK
A general semidefinite program (SDP) takes the form
min
X∈Sntrace(CX )
s.t. trace(AiX) = bi, i ∈[m](3)
X0,
where the optimization variable Xis constrained to be
positive semidefinite (psd), i.e., X∈Sn
+. That is, linear
optimization over the cone of PSD matrices can be addressed
by a semidefinite program. Although many algorithms, such
as interior point methods, could solve SDPs to arbitrary
accuracy in polynomial time, scaling as O(n2m2+n3m)
per iteration [20]. In practice, when the PSD cone is high
dimensional, the solving time and memory requirements
tend to be impractical. More specifically, when mis fixed,
the computational cost of solving an SDP is primarily
determined by the size of the largest semidefinite constraint
n. This motivates the study of methods to decompose, or
approximate high-dimensional PSD constraints into conic
combinations of smaller PSD cones. Such alternatives allow
one to trade off computation time with solution quality.
In this section, we first review several inner approximation
techniques of the PSD cone Sn
+. Then we provide a novel
PSD cone decomposition framework by means of conical
combinations of smaller dimensional subcones to unify these
results.
A. Inner Approximations of PSD cone
The first example is the set of diagonally dominant matri-
ces and scaled diagonally dominant matrices.
Definition 3 (DDnand SDDnmatrices [9]).A symmetric
matrix A= [aij ]is diagonally dominant (dd) if aii ≥
Pj6=i|aij |. A symmetric matrix A is scaled diagonally dom-
inant (sdd) if there exists a diagonal matrix D, with positive
diagonal entries, such that DAD is dd. We denote the set
of n×ndd (resp. sdd) matrices with DDn(resp. SDDn).
Note that DDn⊆SDDn⊆Sn
+.
According to the extreme ray characterization of diag-
onally dominant matrices by Barker and Carlson [21], a
symmetric matrix M∈DDnif and only if it can be written
as M=Pn2
i=1 αivivT
i, αi∈S1, where {vi}is the set of all
nonzero vectors in Rnwith at most 2 nonzero components,
each equal to ±1. The vectors {vi}are the extreme rays
of the DDncone. A similar decomposition is available for
scaled diagonally dominant matrices. According to [11], we
know that any scaled diagonally dominant matrix Mcan be
written as M=P(n
2)
i=1 ViΛiVT
i, where Viis an n×2matrix
whose columns each contain one nonzero element which is
equal to 1, and Λi∈S2
+.
The cone of DDn(resp. SDDn) can be equivalently
written as a number of linear constraints (resp. second
order cone constraints), and thus linear optimization over the
set of diagonally dominant matrices (resp.scaled diagonally
dominant matrices) can be addressed using linear program-
ming (LP) (resp. second order cone programming (SOCP)
) [9]. Working with these classes of convex optimization
problems allows one to take advantage of high-performance
LP and SOCP solvers. However, as already pointed in [9],
approximating the PSD cone by the set of scaled diagonally
dominant matrices SDDnis conservative, which means the
restricted problem may be infeasible or the optimal solution
may be significantly different from that of the original SDP.
Furthermore, enforcing these constraints could be problem-
atic as nincreases, since the number of constraints grows in
a combinatorial fashion n
2.
A relevant generalization of the set of scaled diagonally
dominant matrices is the set of factor-width k matrices.
Definition 4 (Factor-width k matrices [11]).The factor width
of a symmetric matrix Ais the smallest integer ksuch that
there exists a matrix Vwhere A=V V Tand each column
of Vcontains at most knon-zeros. We denote the cone of
n×nsymmetric matrices of factor width kby FWk
n.
By definition, we have that FWk
n⊆Sn
+for all k∈[n]
and FW2
n=SDDn, and FWn
n=Sn
+[9]. FWk
ncan
be decomposed into a sum of PSD matrices of rank at
most kas follows: Z∈ FWk
nif and only if it can be
written as Z=P(n
k)
i=1 ViΛiVT
i, where Viis an n×kmatrix
whose columns each contain one nonzero element which
is equal to 1, and Λi∈Sk
+[10]. One could transform
an optimization problem over the PSD cone into an conic
optimization problem of smaller dimension. However, en-
forcing this constraint is problematic as nor kincreases,
which grows in a combinatorial fashion n
k. Therefore, using
factor-width k matrices to approximate the PSD cone is not
trivial in a practical way.
Lastly, it is often important to exploit sparsity in large
semidefinite optimization problem and chordal graph prop-
erties from graph and sparse matrix theory plays a central
role. The following proposition is the primary result that
exploit the chordal sparsity and decompose a larger semidef-
inite constraint as a set of smaller semidefinite constraints.
Details could be found in the survey of chordal graphs and
semidefinite optimization [12].
Proposition 1 (Chordal sparse matrices [12]).Let Gbe
an undirected chordal graph with edges E, vertices Vand
a set of maximal cliques T={C1, . . . , Cp}. The set of
matrices with a sparsity pattern defined by Gis defined as:
Sn(E) := {X∈Sn:Xij = 0 if (i, j)6∈ E for i6=j}.
Then X∈Sn(E)is positive semidefinite if and only if
X=Pp
k=1 Xk, Xk≥0, k ∈[p], where Xk∈S(Ck) :=
{X∈Sn:Xij = 0 if (i, j)6∈ Ck×Ck}.
B. Decomposition of the PSD Cone
In the above section, we list three existing techniques
of inner approximation for the PSD cone, by means of
decomposing a larger PSD matrices by conic combinations
of smaller dimensional sub-cones, with additional equality or
inequality constraints. Motivated by this idea, we propose a
simple, yet insightful, decomposition of the PSD cone, with
the hope to render new methods for approximating Sn
+.
Theorem 2. Consider a collection Sof subsets of [n], i.e., if
an index set S∈ S, then S⊆[n]. Then we can decompose
the positive semidefinite cone Sn
+as the conic hull of the
(infinite) union of lower dimensional cones, that is,
Sn
+= cone( [
S∈S [
F∈O(n)
FS[S|S|
+]FT
S),(4)
where the set
FS[S|S|
+]FT
S:= {X∈Sn
+:X=FSY F T
S, Y ∈S|S|
+}
is in and of itself a sub-cone.
Proof: See Appendix A
As a result of Theorem 2, the PSD cone could be viewed
as conic hull of sub-cones, possibly of different dimensions.
Therefore, an inner approximation of PSD cone following the
structure of decomposition (4) is given by: choosing specific
choices of Sand a finite subset F ⊆ O(n),i.e., PSD matrices
Xof the form
X=X
S∈S X
F∈F
FSXF,S FT
S, XF,S ∈S|S|
+(5)
Next, we illustrate how the set of (scaled) diagonally dom-
inant matrices, factor-width kmatrices and chordal sparse
matrices could be construed as in (5).
1) The set of diagonally dominant matrices, DDn. As
mentioned in above subsection, a symmetric matrix
M∈DDnif and only if it can be written as M=
Pn2
i=1 αivivT
i, αi∈S1, where {vi}is the set of all
nonzero vectors in Rnwith at most 2 nonzero compo-
nents, each equal to ±1. We could choose S={S⊆
[n] : |S| ≤ 2}and F={I±
n} ⊆ O(n), where I±
nbeing
the set of all diagonal matrices with elements ±1in its
diagonal. |S|= 2 corresponds to the case where vihas 2
nonzero components.
2) SDDn(FW2
n)and FWk
n. Recall that Z∈ FWk
nif and
only if it can be written as Z=P(n
2)
i=1 ViΛiVT
i, where
Viis an n×kmatrix whose columns each contain one
nonzero element which is equal to 1, and Λi∈Sk.
Consequently, for the set of matrices FWk
n, one could
choose S={S⊆[n] : |S| ≤ k}and F={In}.
3) Chordal Sparse Matrices. According to Proposition 1,
one could choose the set S={Ck}, k ∈[p], i.e., the
collection of index set of all the maximal cliques of the
graph G(V,E), and F={In}.
IV. MAXIMALLY DISTAN T FRAM ES AN D
GRASSMANNIAN PACKINGS
The examples in the previous section showed that existing
approximation techniques could be considered as special
instance of the form (5). However, these approaches restrict
a priori the relation between the set Fand S(also their
cardinalities). In this section, we provide a more flexible
family of inner approximations by focusing on Grassmannian
packings.
In the rest of the paper, we restrict our attention to the
case where Sis a fixed subset of cardinality s, i.e., the sub-
cones are of the same size, and the matrices F∈Rn×s
are tall matrices composed by sorthonormal vectors, i.e.,
Fbelongs to the Stiefel manifold Vs(Rn). In doing so,
we will further simplify the decomposition (4) and provide
novel inner approximations of the PSD cone, which aimed
to arrange the sub-cones such they are maximally separated
from each other and tend to occupy large fractions of the
volume of ambient PSD cone. The proposed approach is
rooted in the classical packing problem in Grassmannian
manifolds.
Theorem 3. Consider the case S⊆[n]is a fixed subset of
cardinality sand the matrices F∈Rn×sare tall matrices
composed by sorthonormal vectors, i.e., F∈Vs(Rn). Then,
we could simplify the cone decomposition (4) to
Sn
+= cone( [
F∈Vs(Rn)
F[Ss
+]FT)(6)
where the set
F[Ss
+]FT:= {X∈Sn
+:X=F Y F T, Y ∈Ss
+}(7)
is in and of itself a sub-cone.
Proof: See Appendix B
The above Theorem and the simplified decomposition (6)
illustrate an important fact: Since one could fully recover the
PSD cone Sn
+by conical combinations of elements drawn
from distinct sub-cones of the form F[Ss
+]FTspanned by
a set of frames F∈Vs(Rn). Therefore, when selecting
a finite set of frames F={Fk} ⊆ Vs(Rn)to generate
an inner approximation of PSD cone, one should seek to
select the frames {Fk}such that the corresponding sub-cones
Fk[Ss
+]FT
kare placed as far from each other as possible with
in Sn
+, such that these approximations tend to occupy large
fractions of the volume of the PSD cone. Precisely, we have
the following form of inner approximation of the PSD cone,
N
X
k=1
Fk[Ss
+]FT
k(8)
for a fixed number of N frames F={Fk}, k ∈[N]. We
aim to construct these frames such that the corresponding
sub-cones Fk[Ss
+]FT
kare maximally distant from each other.
Consequently, we seek to solve the following problem:
How should N s−dimensional sub-cones Fk[Ss
+]FT
k, k ∈
[N], of n−dimensional PSD cone Sn
+be arranged so that
they are as far apart possible?
To make sense of this statement, we first define a notion
of cone distance between two sub-cones Fi[Ss
+]FT
iand
Fj[Ss
+]FT
j.
Definition 5 (Cone distance).Consider two Frames F1, F2∈
Vs(Rn)and the corresponding sub-cones Fk[Ss
+]FT
k,k∈
{1,2}. The distance between two sub-cones, written as a
function of frames (Fi, Fj), is given by
dcone(Fi, Fj) = min
Yi,Yj∈O(s)kFi(YiYT
i)FT
i−Fj(YjYT
j)FT
jkF,
(9)
By definition of PSD matrices, for all sub-cones
Fk[Ss
+]FT
k, it always have the zero matrix inside it, 0∈
Fk[Ss
+]FT
k. Therefore, the restriction to Yi, Yj∈O(s)is
necessary to avoid trivial solutions, i.e., the zero matrix. This
notion of cone distance is intuitive: one seeks to find the
minimum distance between some pair of distinct elements
drawn from the sub-cone Cs(Fk), while excluding the trivial
zero matrix. Next, we formally define the sub-cone packing
problem in the PSD cone Sn
+.
Problem 2 (Subcone Packing in Sn
+).Given the ambient
PSD cone Sn
+, and s-dimensional sub-cones of the form
Fk[Ss
+]FT
k}. Find a set of N frames F={F1, . . . , FN}that
solves the mathematical program
max
F:|F|=Nmin
Fi,Fj∈F,i6=jdcone (Fi, Fj).(10)
At first sight, the question of finding frames that solves
the mathematical program seems an unwieldy task, because
of its highly nonconvex and combinatorial nature. The next
theorem shows that the cone distance (9) defined on PSD
cone is equivalent to chordal distance (1) defined on the
Grassmannian manifold G(s, n), up to a scalar factor.
Theorem 4. The cone distance (9) is equivalent to chordal
distance (1) up to a scalar factor. More precisely, we have
dcone(Fi, Fj) = √2dchrod (Fi, Fj).
Proof: See Appendix C.
As a direct result of Theorem 4, the packing problem of
s-dimensional sub-cones in PSD cone in Sn
+(Problem 2)
is equivalent to packing problem in Grassmannian manifold
G(s, n)(Problem 1). Therefore, we could leverage exist-
ing literature on constructing Grassmannian packings as a
method of finding good sub-cone packings for approximating
the PSD cone. When Nis fixed as in PN
k=1 Fk[Ss
+]FT
k, these
sub-cones are arranged so that they are as far apart possible
and tend to provide more accurate inner approximations. By
focusing on Grassmannian packings, the proposed approach
provides a more flexible family of inner approximations as
in (8).
V. NUMERICAL EXAMPLES
In this section, we leverage computational methods for
finding good packings in Grassmannian manifolds via alter-
nating projection, see [18], to get build good approximations
of the PSD cone. We illustrate the effectiveness of our
proposed method with two numerical examples. The first one
compares the inner approximation accuracy of scaled diag-
onally dominant (SDDn) matrices, Factor-width 3 matrices
(FW3
n) with our approach. The second numerical example
studies how many sub-cones are needed to approximate the
ambient PSD cone with high accuracy.
A. Numerical Examples Setup
We consider the following numerical methods for con-
structing Grassmannian packings, for its flexibility and
ease of implementation. Consider a packing problem in
Grassmannian Manifolds (Problem 1), suppose that the set
of N k−dimensional subspaces {S1,...,SN} ⊆ G(k, n)
are spanned by the matrices F={F1, . . . , FN}. Collate
these Nmatrices into a n×KN configuration matrix
F:= [F1. . . FN], and the block Gram matrix of F
is defined as the KN ×K N matrix G=F∗F, whose
blocks control the distances between pairs of subspaces.
Solving the packing problem in Grassmannian manifold
G(k, n)is equivalent with constructing a Gram matrix G
that satisfies six structural and spectral properties, where
the structural properties constrain the entries of the Gram
matrix while the spectral properties control the eigenvalues
of the matrix. The alternating projection algorithm alternately
enforce the structural properties and then the spectral prop-
erties in hope of producing a Gram matrix that satisfies all
six properties at once. Finally, factorization of the output
Gram matrix G=F∗F, such as eigenvalue decomposition,
yields the desired configuration matrix. We note however
some limitations of this approach, as pointed out in [18].
First, although the alternating projection algorithm seems to
converge in practice, a theoretical proof is lacking. Second,
The algorithm frequently requires as many as 5000 iterations
before the iterates settle down. Third, the algorithm was
especially successful for smaller numbers of subspaces, but
its performance began to flag as the number of subspaces
approached 20.
In the following two numerical examples, the Matlab
toolboxes YALMIP [22] and MOSEK [23] are used for
solving the semidefinite programming problems.
B. Approximation Error Comparison
Consider the following semidefinite program to test the
inner approximation accuracy of PSD cone:
min
X∈SnkX−AkF
s.t. X ∈ C ⊆ Sn
+,(11)
where Ais a random normalized PSD matrix, i.e., kAkF= 1
and Crepresents the sub-cones for different inner approxi-
mations. We calculate the distance between matrix Xand a
random given PSD matrix A. The approximation quality is
quantified as the empirical mean of optimal value, kX−AkF,
of (11).
The simulation results are illustrated as in Figure 1, with
the ambient PSD cone of size n= 20. Here, the red
curve SDD −190 denotes the case when C=S DDn,
and 190 indicates the number of sub-cones 20
2= 190.
Similarly, the dark blue curve F W 3−1140 represents the
approximations using 20
3= 1140 factor-width 3 matrices
C=FW3
n. The other three curves named by Frame-N
denote a family of inner approximations which takes the
form C=PN
k=1 Fk[Ss
+]FT
kas in (8). Here, four different
packing have been found using the alternating projection
algorithm [18], where the numbers 1,30,190,350 represents
the number N of sub-cones (equivalently number of frames).
In each case, we gradually increase the sub-cone dimension s
to obtain tighter inner approximations of ambient PSD cone
Sn
+. The mean distance is calculated as the mean of 100
random tests. A more accurate inner approximation is then
indicated by smaller mean distance. Notably, when the sub-
cones are spanned by S10
+, one could reach almost perfect
approximations with only N= 30 frames, compared with
190 sub-cones for SDD approach and 1140 sub-cones for
FW3 approach. Also, instead of using 1140 sub-cones as in
FW3 approach, we achieve a similar approximation accuracy
with only 350 sub-cones.
We end by noting that the SDD approach provides a more
accurate inner approximation of the ambient PSD cone than
the FRAME-190 approach when the sub-cone size s= 2,
i.e., when the two approaches both use 190 sub-cones of
size s= 2. We provide two possible explanations: 1) The
190 frames Vias in M=P(20
2)
i=1 ViΛiVT
iare (nearly) optimal
packed; i.e., they are (nearly) maximally distant from each
other and thus provides the (nearly) optimal approximation;
2) As mentioned in above subsection, the performance of the
alternating projection algorithm used in FRAME approach
begins to flag as the number of subspaces approached 20.
Since the number of sub-cones is 190, the algorithm could
not construct optimal packings and cannot provide the most
accurate approximations.
Fig. 1: Comparison of the approximation error between
scaled diagonally dominant (SDD), factor-width 3 matri-
ces (FW3), and maximally distant frame set (FRAME) of
N={1,30,190,350}with ambient cone S20
+
C. Number of Sub-cones
In the second numerical example, we explore the number
of sub-cones (frames) required for a tight inner approxima-
tion of the ambient PSD cone. Specifically, we consider the
same SDP problem as in (11)
min
X∈SnkX−AkF
s.t. X ∈ C ⊆ Sn
+,
where C=PN
k=1 Fk[Ss
+]FT
k; i,e., optimization variable X
is approximated as sum of sub-cones. The simulation results
are illustrated as in Figure (2). Here, we are looking for
the numbers of sub-cones needed in each case such that the
average mean distance kX−AkFthrough 100 random tests
is less than 0.01. The xaxis represents the dimension of
ambient PSD cone nand the y axis represents the numbers
of frames required in logarithmic scale. For example, the
blue curve sub-2 represents the case when the sub-cones are
spanned by S2
+. Therefore, it is always possible to cover the
ambient PSD cone S2
+with only one sub-cone of the same
size S2
+. From the simulation, the curves appears linearly in
the logarithmic scale, which indicates the number of frames
required for a tight inner approximation grows exponentially.
VI. CONCLUSIONS
In this paper, we developed a novel decomposition frame-
work of the PSD cone by means of conic combinations of
sub-cones, which unified many existing inner approximation
techniques. Furthermore, we introduced a more flexible fam-
ily of inner approximations of the PSD cone by a set of sub-
cones, where we aimed to arrange these sub-cones so that
they are maximally separated from each other. We showed
Fig. 2: Calculating number of frames needed for a tight inner
approximation of the ambient PSD cone. Here, the number
sin sub-s represents the dimension of sub-cone Ss
+
the problem of packing sub-cones is equivalent to a packing
problem in Grassmannian manifolds with chordal distances.
The effectiveness of our approach was demonstrated with
simulation results.
APPENDIX
A. Proof of Theorem 2
Given any PSD matrix X∈Sn
+, it is diagonalizable by
some U∈O(n), i.e., X=UΛUT=Pn
i=1 λiuiuT
i, where
uiis the ith column of Uand λiis the ith diagonal element
of Λ. Let S={1}, and F ⊆ O(n) := {Fk, k ∈[n]}such
that the first column of Fkequal to uk, i.e., the kth column of
U. By construction, we have X∈cone(SF∈F FS[S1
+]FT
S).
Conversely, consider any element Yi∈Snof the set
SS∈S SF∈O(n)FS[S|S|
+]FT
S, i.e., Yi=FSY F T
Sfor some
S∈ S,F∈O(n)and Y∈S|S|
+. Firstly, we have xTYix=
xTFSY F T
Sx≥0,∀x∈Rn, since Yis a |S| × |S|PSD
matrix and FT
Sx∈R|S|. By definition, (possibly infinite)
sum of PSD matrices are also PSD matrices. Next, we
want to show the conical combination Z=Pk
i=1 αiYi, for
some Yi∈SS∈S SF∈O(n)FS[S|S|
+]FT
S, with αi≥0, i ∈
[k], k ∈N, being PSD. Again, applying the quadratic form
xTZx =Pk
i=1 αixTYix≥0, which completes the proof.
B. Proof of Theorem 3
The proof follows the same structure as in Theorem 2.
Given any PSD matrix X∈Sn
+, its diagonalization is given
by X=UΛUT=Pn
i=1 λiuiuT
i. Let Xi∈Ss
+, i ∈[n]be
a diagonal matrix with (Xi)11 =λikuik2
2and other being
zero. Fi∈Vs(Rn), i ∈[n]such that the first column equals
ui/kuik2. Thus we have X=Pn
i=1 FiXiFT
i.
Conversely, consider any element Yi∈Snof the set
SF∈Vs(Rn)F[Ss
+]FTfor some F∈Vs(Rn)and Y∈Ss
+.
Firstly, we have xTYix≥0,∀x∈Rn. By definition,
(possibly infinite) sum of PSD matrices are also PSD ma-
trices. Next, we want to show the conical combination
Z=Pk
i=1 αiYi, for some Yi∈SF∈Vs(Rn)F[Ss
+]FT,
with αi≥0, i ∈[k], k ∈N, being PSD. Again, applying
the quadratic form xTZx =Pk
i=1 αixTYix≥0, which
completes the proof.
C. Proof of Theorem 4
Since Yi, Yj∈O(s), we could simplify the cone distance
(9) as follows
dcone(Fi, Fj)2=kFiFT
i−FjFT
jk2
F
= trace((FiFT
i−FjFT
j)T(FiFT
i−FjFT
j))
= trace(FiFT
iFiFT
i) + trace(FjFT
jFjFT
j)
−2trace(FiFT
iFjFT
j)
= 2s−2kFT
iFjk2
F
= 2dchrod(Fi, Fj)2
where we used the fact that FT
iFi=Isand the cyclic
property of trace operator.
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