Statistical characterization of the chordal product determinant of Grassmannian codes

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Statistical characterization of the chordal product determinant

of Grassmannian codes

Javier ´

Alvarez-Vizoso, Carlos Beltr´an, Diego Cuevas, Ignacio Santamar´ıa,

V´ıt Tuˇcek and Gunnar Peters

October 6, 2022

Abstract

We consider the chordal product determinant, a measure of the distance between two

subspaces of the same dimension. In information theory, collections of elements in the

complex Grassmannian are searched with the property that their pairwise chordal products

are as large as possible. We characterize this function from an statistical perspective, which

allows us to obtain bounds for the minimal chordal product and related energy of such

collections.

1 Introduction and statement of the main results

Let T≥2Mbe two positive integers and consider the complex Grassmannian Gr(M, CT),

i.e. the space of M–dimensional complex vector subspaces of CT. Finite collections of points

(also called codes or packings) in Gr(M, CT) with diﬀerent desired separation properties have

been investigated by several authors (in Section 1.3 we describe some relevant references).

The most frequent criterium for “well–separated” codes is the maximization of the minimal

mutual squared chordal distance, which is the sum the squared sines of the principal angles of

two subspaces. However, following [HM00,MBV02,CAVB+22] (see also Section 1.1 below), a

more relevant measure for its application to information theory is given by the chordal product

energy, related to the product of the squared sines of the principal angles, which justiﬁes its

name. Given a code [X1],...,[Xk]∈Gr(M, CT), its chordal product energy with parameter

Nis

E(X1,...,XK) = X

i6=j

det(IM−XH

iXjXH

jXi)−N,(1)

where IMis the identity matrix and we have chosen representatives Xiof each point [Xi]

satisfying XH

iXi=IM. Note that the energy is well deﬁned in the sense that it does not

change if other representatives with that property are chosen. Recall that the SVD of XH

iXj,

e.g. UDVH, is given in terms of the cosines of the principal angles between the subspaces

[Xi] and [Xj], cos θ1,...,cos θM, cf. [HR06], so

det IM−XH

iXjXH

jXi= det IM−D2=

i=1

sin2θi,(2)

while the squared chordal distance between the two subspaces [Xi] and [Xj] is given by

i=1 sin2θi. The sum in (1) is a pairwise interaction energy in the spirit of the well–studied

Riesz or logarithmic energies of importance in Potential Theory (see [BHS19] for a complete

arXiv:2210.02125v1 [cs.IT] 5 Oct 2022

monograph dedicated to energy minimization in the sphere and other spaces). We refer to

the function (again, choosing representatives Aand Bsuch that AHA=BHB=IM)

[A],[B]∈Gr(M, CT)7→ det(IM−AHBBHA) = det(IM−BHAAHB),(3)

as the chordal product determinant or, simply, the chordal product, and note that is not a

metric in Gr(M, CT), for it may happen that [A]6= [B] and yet det(IM−AHBBHA) = 0, if

the intersection of [A] and [B] is nontrivial.

In this paper we perform the ﬁrst theoretical study of the chordal product energy, for

numerical results, see [CAVB+22] and references therein. We start describing the context

where the problem arises, following [HM00].

1.1 The importance of Grassmannian codes in information theory

Consider a transmitter, i.e. some device that is able to send a signal, which is indeed a

collection of numbers ordered in a complex T×Mmatrix X. Physically, this corresponds

to the setting where the transmitter has Mantennas and there is a total amount of Ttime

slots where the communication channel is assumed to be constant (i.e. the contour conditions

of the communication are considered constant during the time that these T M numbers are

sent). The receiver is another device, that we consider equipped with Nantennas, and the

signal it receives is

Y=XH +sM

T ρW,

where His an unknown M×Nmatrix (termed the channel), Wdescribes the noise and ρ,

called the signal-to-noise-ratio (SNR), measures the magnitude of the signal against the noise.

1.1.1 The zero–noise case

Since His unknown (it is common to assume that it has random complex Gaussian entries),

even in the event that W= 0 the receiver cannot recover the whole matrix X:

•If two matrices X1and X2have the same column span, then one can easily ﬁnd an

full–rank matrix Hsuch that X1H=X2H, hence the receiver just cannot distinguish

which of these two matrices was the original signal.

•On the other hand, if two matrices X1and X2have the property that the intersection

of the column span of X1and X2is trivial, the the receiver can easily distinguish if a

given matrix Yhas been constructed by X1Hor by X2H: if the column span of Y

intersected with the column span of X1(resp. X2) is nontrivial, then X1(resp. X2)

was sent.

Summarizing, if a previously agreed code of possible signals [X1],...,[XK]∈Gr(M, CT) is

ﬁxed with the property that the column spans of Xiand Xjhave trivial intersection for

i6=j, the receiver will be able to recover, at least in the zero–noise scenario, the element of

the Grassmannian represented by the sent signal, but not the concrete representative of that

element. Hence, collections of points in Gr(M, CT) are searched with that property.

1.1.2 The general case

In the more realistic context of the presence of non–zero noise, the analysis is quite more

involved since there is always a non–zero probability of error in the detection procedure. The

pioneer work [HM00] showed that, in order to recover the element Xiof Gr(M, CT) just by

knowing Y, the optimal method is to use the so called maximum–likelihood decoder:

i= argmaxj=1,...,K tr (YHXjXH

jY) = argmaxj=1,...,K tr (XH

jYYHXj),

where ·Hholds for Hermitian conjugate. Then, [BV01] showed that if only 2 codewords are

permitted, i.e. if K= 2, and assuming that the entries of Hand Ware complex Gaussian

N(0,1) numbers, then the probability Pe(X1,X2, ρ) of erroneously decoding X1if X2was

sent can be given by a (quite complicated) formula involving the residues of a certain rational

function. Luckily, the asymptotic expansion of this Pairwise Error Probability (PEP) in the

case ρ→ ∞, called the high-SNR asymptotic analysis, admits a much more concise expression,

see [MBV02,CAVB+22]:

Pe(X1,X2, ρ)≈Cρ−NM det(IM−XH

1X2XH

2X1)−N, ρ → ∞,(4)

where C=1

24M

TNM (2NM−1)!!

(2NM)!!) , it is assumed that any two distinct points have trivial

intersection as linear subspaces, and the representatives Xiof each [Xi] are such that XH

iXi=

IM. If we have Kelements [X1],...,[XK] in the code of possible signals and we assume that

we send one of them at random, all with equal probability 1/K, then the total probability of

erroneously decoding a signal is bounded above by

i6=j

Pe(Xi,Xj, ρ)≈C

Kρ−NM X

i6=j

det(IM−XH

iXjXH

jXi)−N.(5)

The determinant in (4) is the chordal product (3) and the sum in the right–hand side in (5)

is the energy (1).

1.1.3 Criteria for the design of Grassmannian codes

It follows from the previous discussion that reasonable criteria for the design of a code

[X1],...,[XK] would be to maximize the pairwise chordal product (3), or to minimize the

chordal product energy (1). In [CAVB+22] these approaches are considered, numerically

showing that the obtained codes are very well suited for their use in non–coherent commu-

nications, with a slight advantage in the use of the chordal product energy. Yet, little or no

theory exists about the behavior of the optimal pairwise chordal product or energy. The main

purpose of this paper is to put the basis for the study of this question.

1.2 Main results of the paper

We will start our study by computing the moments of the chordal product when [B] is

ﬁxed and [A] is chosen at random uniformly in Gr(M, CT), w.r.t. the unique, standard

rotation–invariant probability measure. This yields a complete statistical characterization of

the chordal product as a product of beta–distributed random variables:

Theorem 1. Assume that T≥2M. Let p∈(2M−T−1,∞)(notice that pmay be negative

and/or noninteger). Let [B]∈Gr(M, CT)be any ﬁxed element and let [A]∈Gr(M, CT)be

uniformly distributed on the Grassmannian. Then, the p–th moment of det(IM−BHAAHB)

is:

EA[det(IM−BHAAHB)p] =

m=1

Γ(T−m+ 1)Γ(T+p−m−M+ 1)

Γ(T−m−M+ 1)Γ(T+p−m+ 1),(6)

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摘要：

StatisticalcharacterizationofthechordalproductdeterminantofGrassmanniancodesJavierAlvarez-Vizoso,CarlosBeltran,DiegoCuevas,IgnacioSantamara,VtTucekandGunnarPetersOctober6,2022AbstractWeconsiderthechordalproductdeterminant,ameasureofthedistancebetweentwosubspacesofthesamedimension.Ininformatio...

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Statistical characterization of the chordal product determinant of Grassmannian codes

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