Classiﬁcation in the Siegel Space
for Vectorial Autoregressive Data
Yann Cabanes1,2(B) and Frank Nielsen3,4
1Thales LAS Limours, Limours, FranceInstitute of Mathematics of Bordeaux, Bordeaux, France
3
Sony Computer Science Laboratories Inc., Tokyo, Japan
Frank.Nielsen@acm.org
4École Polytechnique, LIX, Palaiseau, France

2Abstract. We introduce new geometrical tools to cluster data in theSiegel space. We give the expression of the Riemannian logarithm andexponential maps in the Siegel disk. These new tools help us to perform
classiﬁcation algorithms in the Siegel disk. We also give the expressionof the sectional curvature in the Siegel disk. The sectional curvatures
are negative or equal to zero, and therefore the curvature of the Siegel
disk is non-positive. This result proves the convergence of the gradientdescent performed when computing the mean of a set of matrix points
in the Siegel disk.
Keywords: Siegel space · Riemannian manifold · Riemannian
exponential map · Riemannian logarithm map · Sectional curvature ·Machine learning · Information geometry · Var model

1

Complex Vectorial Autoregressive Gaussian Models

We present here the multidimensional linear autoregressive model which generalize the one-dimensional model presented in [3].
1.1

The Multidimensional Linear Autoregressive Model

We assume that the multidimensional signal can be modeled as a centered stationary autoregressive multidimensional Gaussian process of order p ´ 1:U (k) `p´1
j“1Ap´1
U (k ´ j) “ W (k)j(1)

where W is the prediction error vector which we assume to be a standard Gaussian random vector and the prediction coeﬃcients Ap´1
are square matrices.
jThis work was initiated while the author was at École Polytechnique, France. This
work was further developed during a thesis with Thales LAS France and the Institute of Mathematics of Bordeaux. We thank the French MoD, DGA/AID for funding(convention CIFRE AID N ◦ 2017.0008 & ANRT N ◦ 2017.60.0062).
c Springer Nature Switzerland AG 2021
F. Nielsen and F. Barbaresco (Eds.): GSI 2021, LNCS 12829, pp. 693–700, 2021.
https://doi.org/10.1007/978-3-030-80209-7_74

694

1.2

Y. Cabanes and F. Nielsen

Three Equivalent Representation Spaces

There are at least three equivalent spaces to represent our model parameter,
they are described in detail in [6]. The ﬁrst one is the set of Hermitian PositiveDeﬁnite Block-Toeplitz matrices corresponding to the covariance matrix of thelarge column vector U deﬁned by:
Tdef U “ U (0)T , . . . , U (p ´ 1)T(2)The second one is a product space: a HPD matrix
 i  (which characterizes theaverage correlation matrix) and the coeﬃcients Ai i“1,...,p´1 (which characterize the multidimensional autoregressive model). The third representation space
looks like the second one: the coeﬃcients Aii are slightly modiﬁed to belong tothe Siegel disk which metric has been studied in [6,8].
1.3

A Natural Metric Coming from Information Geometry

In [6], the three equivalent representation spaces presented in Sect. 1.2 areendowed with a natural metric coming from information geometry. Indeed, themodel assumptions done in Sect. 1.1 are equivalent to the assumption that thelarge vector U described in Eq. (2) is the realisation of a Gaussian process withzero mean and an Hermitian Positive Deﬁnite Block-Toeplitz covariance matrix.
We deﬁne a metric on this space as the restriction of the information geometry
metric on the space of Gaussian processes with zero mean and an Hermitian
Positive Deﬁnite covariance matrices. We ﬁnally transpose this metric to theequivalent representation space constituted of a HPD matrix and coeﬃcients in
the Siegel disks. Luckily, the metric in this space is a product metric. The metricon the HPD manifold is the information geometry metric [6]. The metric on theSiegel disk is described in detail in Sect. 2.

2The Siegel Disk

In this section we present a Riemannian manifold named the Siegel disk andintroduce the Riemannian logarithm and exponential maps. These tools will bevery useful to classify data in the Siegel space, as shown in Sect. 3. We also
introduce the formula of the Siegel sectional curvature and prove it to be nonpositive which proves the convergence of classiﬁcation algorithms based on mean
computations, such as the k-means algorithm. The Siegel disk generalizes thePoincaré disk described in [3,4].
2.1

The Space Deﬁnition

Deﬁnition 1. The Siegel disk is deﬁned as the set of complex matrices M ofshape N × N with singular values lower than one, which can also be written:
SDN “ M P CN ×N , I ´ M M H ą 0(3)

Classiﬁcation in the Siegel Space for Vectorial Autoregressive Data

695or equally:
SDN “ M P CN ×N , I ´ M H M ą 0 .

(4)

We use the partial ordering of the set of complex matrices: we note A ą B
when the diﬀerence A ´ B is a positive deﬁnite matrix. As the Siegel disk is heredeﬁned as an open subset of the complex matrices CN ×N , its tangent space ateach point can also be considered as CN ×N .
Note that another deﬁnition of the Siegel disk also exists in other papers [8],imposing an additional symmetry condition on the matrix M : M “ M T . We
will not require the symmetry condition in our work.
Property 1. The Siegel disk can also be deﬁned as the setmatrices M of complex
N ×N, ~M ~ < 1 ,with a linear operator norm lower than one: SDN “ M P C
where ~M ~ “ supXPCN ×N ,X“1 (M X).
2.2

The Metric

Square of the Line Element ds
ds2 “ traceI ´ ΩΩ H´1´1dΩ I ´ Ω H ΩdΩ H(5)

The Scalar Product. ∀Ω P SDN , ∀v, w P CN ×N :
´1 H
´1 

1v, wΩ “ trace I ´ ΩΩ Hv I ´ ΩH Ωw
2´1 ´1 H
1` trace I ´ ΩΩ Hw I ´ ΩH Ωv
2(6)
(7)

The norm of a vector belonging to the tangent space is therefore:2vΩ “ traceI ´ ΩΩ H´1 ´1 Hv I ´ ΩH Ωv(8)

I ` C 1/2
I ´ C 1/2

(9)

The Distance
1d2SDN (Ω, Ψ ) “ trace log2
4

“trace arctanh2 C 1/2´1
´1  HΨ ´ Ω H I ´ ΩΨ H.
with C “ (Ψ ´ Ω) I ´ Ω H Ψ(10)

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Y. Cabanes and F. Nielsen

The Isometry

In [6], the following function is said to be an isometry for the Siegel distancedescribed in Eq. (9).
´1/2
´1 1/2I ´ ΩH Ω(Ψ ´ Ω) I ´ Ω H ΨΦΩ (Ψ ) “ I ´ ΩΩ H

(11)Property 2. The diﬀerential of the isometry Φ has the following expression:
1/2 ´1 ´1 1/2I ´ Ψ ΩHI ´ ΩH Ωh I ´ ΩH ΨDΦΩ (Ψ ) [h] “ I ´ ΩΩ H
(12)Property 3. The inverse of the function Φ described in Eq. (11) is:Φ´1Ω (Ψ ) “ ΦΩ (´Ψ )
2.4

(13)

The Riemannian Logarithm Map

Riemannian Logarithm Map at 0. In [6], the logarithm map at 0 is givenby the formula :(14)
log0 (Ω) “ V Ωwith:V“LΩΩ
H 1/2, log
1/2 I ` ΩΩ HI ´ (ΩΩ H )1/2(15)where L (A, Q) is deﬁned as the solution of:
AZ ` ZAH “ Q(16)However, the expression of the logarithm map at zero given in [6] can begreatly simpliﬁed.
Property 4. The Riemannian logarithm map of the Siegel disk at zero has thefollowing expression:
1/2
(17)
log0 (Ω) “ arctanh (X) X ´1 Ω where X “ ΩΩ H
Note that for ~X~ < 1, we can develop the function arctanh in whole series:arctanh (X) “`8
X 2n`12n ` 1n“0(18)Hence for all X in the Siegel disk, we can write the product arctanh (X) X ´1the following way:arctanh (X) X´1`8
X 2n“2n ` 1n“0This new expression is also valid when the matrix X is not invertible.

(19)Classiﬁcation in the Siegel Space for Vectorial Autoregressive Data697

Fig. 1. The Poincaré disk logarithm map computation

2.5

The Riemannian Logarithm Map at Any Point

To compute the Riemannian logarithm map at a point Ω the key idea here is totransport the problem at zero, compute a certain logarithm at zero and transport
the result back to Ω. If we want to compute the logartithm map: logΩ (Ψ ), weﬁrst transport both Ω and Ψ using the isometry ΦΩ given in Eq. (11). The pointΩ is sent to zero, and we denote Ψ the image of Ψ by ΦΩ :

´1 
1/2
´1/2
 def
Ψ “ ΦΩ (Ψ ) “ I ´ ΩΩ H(Ψ ´ Ω) I ´ Ω H ΨI ´ ΩH ΩThen we compute the logarithm map at zero log0 Ψdef

V “ log0 Ψ“ arctanh (X) X ´1 Ψ(20):

where X “ Ψ Ψ

H

1/2

(21)

And ﬁnally, we transport back the logarithm to the point Ω using the diﬀerential of the isometry Φ given in Eq. (12):defV “ logΩ (Ψ ) “ DΦ´Ω (0) V1/2  
1/2“ I ´ ΩΩ HV I ´ ΩH Ω(22)

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2.6

Y. Cabanes and F. Nielsen

The Riemannian Exponential Map

Riemannian Exponential Map at 0Property 5. The Riemannian exponential map of the Siegel disk at zero has thefollowing expression:
1/2
exp0 (V ) “ tanh (Y ) Y ´1 V where Y “ V V H(23)Note that for ~X~ < π2 , we can develop the function tanh in whole series:
tanh (X) “`8 2n  2n

2 ´12n“1(2n)!

B2n X 2n´1(24)where B2n are the Bernoulli numbers.
Hence for all X in the Siegel disk, we can write the product tanh (X) X ´1
the following way:
tanh (X) X´1“
`8 2n  2n
22 ´1n“1(2n)!

B2n X 2n´2(25)This new expression is also valid when the matrix X is not invertible.
2.7

The Riemannian Exponential Map at Any Point

To compute the Riemannian exponential map at a point Ω the key idea here
is to transport the problem at zero (as for the logarithm), compute a certain
exponential at zero and transport the result back to Ω. If we want to computethe exponential map: expΩ (V ), we ﬁrst transport the vector V at zero using thediﬀerential of the isometry Φ given in Eq. (12):defV “ DΦΩ (Ω) [V ]
1/2 ´1 ´1 1/2
I ´ ΩΩ HI ´ ΩH ΩV I ´ ΩH Ω“ I ´ ΩΩ H´1/2 ´1/2
V I ´ ΩH Ω“ I ´ ΩΩ H
Then we compute the exponential map at zero exp0 VdefΨ “ exp0 V“ tanh (Y ) Y ´1 V



(26)
(27)
(28)

:


where Y “ V V



H1/2

(29)

And ﬁnally, we transport back the exponential to the point Ω using theisometry Φ´Ω which is the inverse of isometry ΦΩ (see Property 3) and transportthe point 0 back to Ω and the point Ψ back to expΩ (V ):defΨ “ expΩ (V )“ Φ´Ω Ψ(30)´1/2
“ I ´ ΩΩ HΨ `Ω(31)I ` ΩH Ψ´1 I ´ ΩH Ω

1/2(32)

Classiﬁcation in the Siegel Space for Vectorial Autoregressive Data

699

Fig. 2. The Poincaré disk exponential map computation

2.8

The Geodesics

The expression the geodesics can be obtained using the exponential map: thegeodesic starting from Ω with velocity V is given by the application:
ζ(t) : t → expΩ (tV )2.9(33)

Sectional Curvature of the Siegel SpaceWe ﬁrst focus on the sectional curvature at 0. We can then obtain the sectional
curvature at any point using the isometry Φ deﬁned in Eq. (11).
Let σ be a section deﬁned by the two ﬁrst vectors of an orthogonal basis(E1 , . . . , En ) of the tangent space of the Siegel disk at the point Ω “ 0.
Theorem 1. The sectional curvature at zero of the plan σ deﬁned by E1 andE2 has the following expression:
K(σ) “ ´1 E1 E2H ´ E2 E1H 2 ` E1H E2 ´ E2H E1 22(34)As a consequence, we have:´ 4 ď K (σ) ď 0∀σ(35)The Siegel disk is therefore a Hadamard manifold. The bounds of the sectional curvature provides a proof of the convergence of certain algorithms calculating the Riemannian p-mean [1] or the circumcenter [2] of a set of points on amanifold.

700Y. Cabanes and F. Nielsen

2.10

The Symmetric Siegel Disk

We deﬁned the Siegel disk in deﬁnition 1 as the set of complex matriceswith
N ×NH, I ´ M M ą 0 . We recall
singular values lower than one: SDN “ M P C
that another deﬁnition of the Siegel disk also exists in other papers [8], imposing
an additional symmetry condition on the matrix M : M “ M T . Note that thesymmetric Siegel disk is a totally ﬂat submanifold of the Siegel disk. Hence theformula of the logarithm map 2.5, the exponential map 2.7 and the sectional
curvature 2.9 computed in previous sections are still meaningful when working
in the submanifold of symmetric matrices.

3

Application to Stationary Signals Classiﬁcation

An eﬀective algorithm to estimate the model parameters from the raw data
is described in [5,7]. The classiﬁcation model presented in this article has
been applied to radar clutter classiﬁcation in [3,4] in the special case of onedimensional complex signals and in [5] in the case of simulated multidimensional
radar signals. This model will be applied in future work to stereo audio signals
classiﬁcation in the case of two-dimensional real signals.

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