Abstract
The present paper discusses the problem of least-squares over the real symplectic group of matrices Sp(2n, R). The least-squares problem may be extended from flat spaces to curved spaces by the notion of geodesic distance. The resulting non-linear minimization problem on manifold may be tackled by means of a gradient-descent algorithm tailored to the geometry of the kirbybrian@example.net. In turn, gradient steepest descent on manifold may be implemented through a geodesic-based stepping method. As the space Sp(2n, R) is a non-compact Lie group, it is convenient to endow it with a pseudo-Riemannian geometry. Indeed, a pseudo-Riemannian metric allows the computation of geodesic arcs and geodesic distances in closed form on Sp(2n, R).
Introduction
Least-squares problems over compact Lie groups have been extensively studied due to their broad application range. A feature of these problems is that the domain of the objective function to optimize, being a compact Lie group, may be endowed with the structure of a differential manifold with a bi-invariant Riemannian metric. The formulation of least-squares problems on compact Riemannian Lie groups relies on closed forms of geodesic curves and geodesic distances.
Introduction
On the Euclidean matrix space R n×m , a (weighted) least-squares problem may be formulated via the criterion function f : R n×m → R + 0 :
Introduction
where τ k ∈ R n×m denote optimization target matrices, α k > 0 denote weights and symbol · denotes the Frobenius matrix norm. On a Riemannian curved manifold M , the above least-squares problem may be reformulated via the generalized criterion function f : M → R + 0 :
Introduction
where τ k ∈ M and the function d(·, ·) denotes a geodesic distance on the manifold M corresponding to the metric that the manifold is endowed with.
Introduction
On non-compact Riemannian Lie groups, the problem of formulating a leastsquares criterion and of its optimization is substantially more involved, because it might be hard to compute geodesic distances in closed form. However, a non-compact Lie group may be treated as a pseudo-Riemannian manifold with a bi-invariant pseudo-Riemannian metric [20] .
Introduction
One of such non-compact Lie groups is the real symplectic group. Real symplectic matrices form an algebraic group under standard matrix multiplication and inversion, denoted as Sp(2n, R). The least-squares problem over the set of real symplectic matrices plays an important role in applied fields. Some applications are:
Introduction
• Quantum computing [3, 16] : An important application of quantum mechanics is quantum computing, as quantum mechanics allows for information processing that can not be performed classically. In particular, it may be possible to design algorithms on a quantum computers that are more efficient than on a classical computers.
Introduction
• Control of beam systems in particle accelerators [8] : Lie-group tools may be applied to the characterization of beam dynamics in charged-particle optical systems. These methods are applicable to accelerator design, charge-particle beam transport and electron microscopes. Lie-group methods potentially provide a way to analyze and control non-linear behavior with the same completeness of linear behavior.
Introduction
• Computational ophthalmology [17, 18] : In the study of optical systems in ophthalmology, it is assumed that the optical nature of a centered optical system is completely described by a real symplectic matrix.
Introduction
• Vibration analysis [23] : Transfer matrices are widely used for the dynamic analysis of engineering structures as well as for static analysis, and are particularly useful in the treatment of repetitive structures. Transfer matrices are real symplectic.
Introduction
• Control theory [10] : Real symplectic matrices find applications in linear control theory for discrete-time systems.
Introduction
Other applications may be found in coding theory [5] and in time-series prediction [2] . Although some noticeable results are available about the real symplectic group [9, 10] , optimization on the real symplectic group appears to be far less studied than for other Lie groups. In the present manuscript, we discuss the problem of least-squares on the real symplectic group. After a review of results known from literature about optimization on the real symplectic manifold, an optimization method based on endowing it with a pseudo-Riemannian geometry will be discussed.
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
For a reference on differential geometry, see [24] . Let M be a smooth manifold. The tangent space at x ∈ M to the manifold is denoted by T x M . A metric on M is a non-degenerate, smooth, symmetric, bilinear map which assigns a real number to pairs of tangent vectors at each tangent space of the manifold M . Let us denote the metric by ·, · x :
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
A Riemannian manifold is a smooth manifold endowed with a positive definite metric, namely:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
The metric ·, · x also defines the norm
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
x for v ∈ T x M . The geodesic curve connecting two points x 1 , x 2 ∈ M is the curve G(t) ∈ M , parameterized by t ∈ [0 1], that minimizes the energy integral:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
under the normalization condition that Ġ ,Ġ G is constant. By the calculus of variation on manifold, the geodesic equation may be written in normal form as:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
In the above expressions, the over-dot and the double over-dot denote first-order and second-order derivation with respect to parameter t, respectively, while symbol Γ · (·, ·) denotes the Christoffel operator. The solution of the geodesic equation may be written in terms of two known quantities that serve as boundary conditions for the second-order geodesic differential equation. The values x = x(0) ∈ M and v =ẋ(0) ∈ T x M might be specified, in which case the solution of the geodesic equation (5) will be denotes as G x,v (t). The squared geodesic distance between the geodesic's endpoints is defined as:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
Note that if the geodesic curve is expressed as G x,v (t), then the squared geodesic distance equals v 2 x . The gradient ∇ x f ∈ T x M of a regular criterion function f : M → R in a point x ∈ M may be defined as:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
where symbol ♯ denotes the 'sharp' isomorphism and symbol df denotes differential. On a Riemannian manifold, the Riemannian gradient may be computed by the metric compatibility condition:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
where symbol ·, · E denotes Euclidean metric.
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
When a Riemannian manifold of interest M and a regular criterion function f : M → R are specified, a known optimization rule is 'gradient steepest descent', that may be expressed as the differential equation on manifold:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
The gradient flow x(t) associated to this system tends toward a local minimum of the criterion function f , in fact:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
with equality holding if and only if ∇ x f = 0, namely, when the flow x(t) approaches a stationary point of the criterion f . The optimization system (9) is based on the knowledge that the Riemannian gradient ∇ x f ∈ T x M points toward the direction of the maximum growth of the function f around the point x ∈ M . The basic idea to implement the optimization scheme represented by equation (9) is to replace the continuous-time state-variable x(t) ∈ M with a discretetime state-variable x k ∈ M . This operation requires a numerical scheme of integration of the equation (9) .
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
The differential equation (9) may be solved numerically by the help of the notion of geodesic curve, namely, by the numerical optimization algorithm [6, 12, 13] :
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
where η ∈ [0 1] denotes an integration step-size and G x,v (t) denotes a geodesic arc departing from the point x ∈ M with initial direction v ∈ T x M and parameter t ∈ [0 1]. A pseudo-Riemannian manifold is a manifold endowed with a metric that is not positive definite. (On a pseudo-Riemannian manifold M , the quantity v 2 x may be positive, negative or null even for 0 = v ∈ T x M .)
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
In order to extend the gradient-based optimization algorithm (10) to the case of pseudo-Riemannian manifold M , it is necessary to compute geodesic arcs and gradients on M .
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
The basic idea to cope with pseudo-Riemannian manifolds is to decompose each tangent space T x M as follows:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
The notion of geodesics extends to pseudo-Riemannian manifolds by the calculus of variation on the energy integral (4). A geodesic arc G x,v (t) will be the solution of the differential equation (5) . On a pseudo-Riemannian manifold M , however, it holds v 2 x < 0 for v ∈ T − x M , therefore the notion of squared pseudo-Riemannian geodesic distance may be defined as:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
Likewise, pseudo-Riemannian gradient defines as ∇ x f = (df ) ♯ again. It is, however, worth remarking that the pseudo-Riemannian gradient ∇ x f does not point to the direction of the maximum growth of the function f . Therefore, the optimization equation (9) needs to turn into:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
whose solution is a minimization flow. In fact, it induces the dynamics:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
The equation (13) may be solved numerically by the optimization algorithm:
Criterion optimization on Riemannian and on pseudo-Riemannian manifolds
where symbol G x,v (t) denotes a pseudo-Riemannian geodesic arc departing from the point x ∈ M with initial direction v ∈ T x M and parameter t ∈ [0 1] and the succession x k ∈ M denotes an approximation of the actual solution of the differential equation (13) .
The real symplectic group
The present section aims at recalling the definition of the real symplectic group and its properties, along with some recent results about optimization on it.
Definitions and properties
The real symplectic group is defined as follows:
Definitions and properties
where symbol e n denotes the n × n identity matrix, while symbol 0 n denotes a whole-zero n × n matrix. The skew-symmetric matrix q 2n enjoys the following properties:
Definitions and properties
From now on, the subscript 2n on the symbol q 2n will drop for a tidier notation. The space Sp(2n, R) is a curved smooth manifold that may also be endowed with an algebraic-group structure (namely, group multiplication and group inverse and possesses a identity element) in a manner that is compatible with the manifold structure. Therefore, the space Sp(2n, R) has the structure of a Lie group (of dimension n(2n + 1)).
Definitions and properties
In particular, standard matrix multiplication and inversion work as algebraicgroup operations. For the matrix multiplication, the property may be proven by noting that, for every x, y ∈ Sp(2n, R), the product xy ∈ Sp(2n, R), in fact:
Definitions and properties
Any symplectic matrix x ∈ Sp(2n, R) is such that det 2 (x) = 1, where symbol det(·) denotes determinant. Therefore, symplectic matrices are invertible. It may be proven that if matrix x ∈ Sp(2n, R), then also matrix x −1 ∈ Sp(2n, R) by negation, namely, by trying to prove that (x −1 ) T qx −1 = q. It holds:
Definitions and properties
that would contradict the hypothesis that x ∈ Sp(2n, R). The identity element of the group Sp(2n, R) is clearly the matrix e 2n . In addition, the following identities hold:
Definitions and properties
From the identity (20), it follows that the group Sp(2n, R) is a subgroup of Sl(2n, R) as well as of Gl(2n, R). The proof of identity (20) is far from trivial [10] . The tangent space T x Sp(2n, R) has structure:
Definitions and properties
The tangent space at the identity of the Lie group, namely the Lie algebra sp(2n, R), has structure:
Definitions and properties
and it coincides with the space of 2n × 2n Hamiltonian matrices, in fact. By the embedding of the manifold Sp(2n, R) into the Euclidean space R 2n×2n , the embedded real symplectic group may be endowed with normal spaces as well, as:
Definitions and properties
where symbol tr(·) denotes the trace operator. The tangent space, the Lie algebra and the normal space associated to the real symplectic group may be characterized as follows:
Definitions and properties
A noteworthy property of symplectic matrices is as follows. Let x ∈ Sp(2n, R), v ∈ T x Sp(2n, R) and y ∈ R 2n×2n . The following identity holds:
Definitions and properties
To prove such identity, use first the parametrization v = xqs, with s = s T ∈ R 2n×2n . Then it holds:
Definitions and properties
Note that the identity (26) holds for any real-valued matrix y of appropriate size.
Definitions and properties
A result concerning optimization on the manifold Sp(2n, R) is adapted from [4] . Let σ : sp(2n, R) → sp(2n, R) be a symmetric positive-definite operator with respect to the Euclidean inner product ·, · E on the space sp(2n, R) given by tr(h T 1 h 2 ) for every h 1 , h 2 ∈ sp(2n, R). The minimizing curve of the integral:
Definitions and properties
over all curves x(t) ∈ Sp(2n, R) with t ∈ [t 1 t 2 ] and with fixed endpoints x(t 1 ) = x 1 ∈ Sp(2n, R) and x(t 2 ) = x 2 ∈ Sp(2n, R), and where h is defined byẋ = xh, so that h ∈ sp(2n, R), is the solution of the system:
Definitions and properties
where symbol σ −1 denotes the inverse of the operator σ. The simplest choice for the symmetric positive-definite operator σ is σ(h) = h, which corresponds to a Riemannian metric for the real symplectic group Sp(2n, R) given by:
Definitions and properties
Such a metric leads to the 'natural gradient' on the space of real invertible matrices Gl(n, R) studied in [1] . The above choice for the operator σ implies that m = h and that the corresponding curve on the real symplectic group satisfies the equations:
Definitions and properties
or, in normal form:ẍ
Definitions and properties
The above equations describe geodesic arcs on the real symplectic group corresponding to the metric (28). Closed form solutions of the above equations are unknown to the authors of [4] and to the present author. The Riemannian gradient ∇ x f of a regular function f : Sp(2n, R) → R corresponding to the metric (28) may be calculated as the unique vector in T x Sp(2n, R) that satisfies the following compatibility condition:
Definitions and properties
By recalling the structures of the tangent and normal spaces to the real symplectic group, the solution of the above equation is found as:
Definitions and properties
from which the expression of the Riemannian gradient associated to the metric (28) follows:
Symplectic group as a pseudo-Riemannian manifold
Let us consider the following pseudo-Riemannian metric on the general linear group of matrices Gl(n) [20] :
Symplectic group as a pseudo-Riemannian manifold
Under the above pseudo-Riemannian metric, it is indeed possible to solve the geodesic equation in closed form. The energy integral in this case reads:
Symplectic group as a pseudo-Riemannian manifold
and the corresponding geodesic curve and the squared geodesic distance have the following expressions:
Symplectic group as a pseudo-Riemannian manifold
On the general linear group Gl(n), matrix logarithm and exponential may be defined by the series:
Symplectic group as a pseudo-Riemannian manifold
The pseudo-Riemannian gradient of a regular function f : Sp(2n, R) → R associated to the metric (33) is the solution of the compatibility condition:
Symplectic group as a pseudo-Riemannian manifold
that reads:
Symplectic group as a pseudo-Riemannian manifold
It is straightforward to show that the above vector ∇ x f belongs to T x Sp(2n, R). It may be shown that it satisfies the condition (39) by the help of the identity (26).
Symplectic group as a pseudo-Riemannian manifold
Having chosen a pseudo-Riemannian metric for the real symplectic group, the expression of the geodesic arc as well as the expression of the geodesic distance may be computed in closed form. The least-squares problem:
Symplectic group as a pseudo-Riemannian manifold
may be thus set up, where target matrices τ k belong to Sp(2n, R). Moreover, the numerical scheme (15) may be effectively implemented. The Euclidean gradient of the criterion function (41) has expression:
Numerical tests
The numerical behavior of the developed least-squares algorithm will be examined via two different tests. A numerical test concerns the computation of the intrinsic mean of a set of given real symplectic matrices. A further numerical test concerns the interpolation of two given real symplectic matrices.
Computing the intrinsic mean of a set of real symplectic matrices
As a numerical test about least squares on the real symplectic group, consider the computation of the mean of a set of given real symplectic matrices. (An application is described in [18] . Some general remarks on averaging over Lie groups may be found in [14] .) On a metrizable manifold M , the 'intrinsic mean' may be defined as [15, 19] :
Computing the intrinsic mean of a set of real symplectic matrices
where the matrices τ k ∈ M are distributed around a center-of-mass to be estimated by µ ∈ M . The minimum value of the criterion function in (43) is the 'intrinsic variance' of the distribution, namely:
Computing the intrinsic mean of a set of real symplectic matrices
Setting M = Sp(2n, R), the problem (43) is a least-squares problem on the real symplectic group Sp(2n, R).
Computing the intrinsic mean of a set of real symplectic matrices
The Figure 1 shows a result obtained with the iterative algorithm (15) for n = 5. The picture shows the value of the criterion function 2 (x, τ k ) before iteration (with initial guess chosen as x = e 10 ) and after the iteration. The Figure shows that the algorithm converges steadily toward the minimal variance (in fact, the distances from the found center of mass are much smaller than the distances from the initial guess).
Computing the intrinsic mean of a set of real symplectic matrices
The group Sp(10, R) is a 55-dimensional manifold whose elements are represented by 10 × 10 real-valued matrices. It is impossible to render graphically the target matrices τ k , their center of mass and the computed mean matrix. An illustration of the distribution of the target matrices around the actual center and of the computed mean may, however, be gotten through the application of an appropriate dimensionality reduction technique. We chose the 'multidimensional scaling' (MDS, see Appendix A) as dimensionality reduction method onto the real plane R 2 . The result is depicted in Figure 2 . From this Figure, it is possible to appreciate how close the computed (empirical) intrinsic mean lies to the actual center of mass.
Computing the intrinsic mean of a set of real symplectic matrices
A close-up of the numerical behavior of the least-squares optimization algorithm (15) comes from the examination of the case n = 1. The group Sp(2, R) is a 3-dimensional manifold (in fact, Sp(2, R) = Sl(2, R)), therefore the following parametrization may be taken advantage of:
Computing the intrinsic mean of a set of real symplectic matrices
(In fact, the fourth parameter d is constrained to the three free parameters a, b and c by the symplecticity condition ad − bc = 1.) Hence, the elements of the group Sp(2, R) may be rendered on a 3-dimensional figure.
Computing the intrinsic mean of a set of real symplectic matrices
The Figure 3 shows a result obtained with the iterative algorithm (15) for n = 1. The Figure shows the location of the target matrices τ k (circles), the location of the center-of-mass (cross), the trajectory of the optimization algorithm over the search space (solid-dotted line) and the location of the final point computed by the algorithm (diamond). The Figure shows that the algorithm is convergent toward the center of mass. The Figure 4 shows the variance during iteration, the value of the squared norm of criterion pseudo-Riemannian gradient during iteration, the squared distances d 2 (x, τ k ) before iteration and after the iteration. 
Interpolating over the real symplectic group
As a further numerical test, consider the interpolation of two given real symplectic matrices. (An application is described in [22] .)
Interpolating over the real symplectic group
On a metrizable manifold M , the continuous interpolate between two given matrices τ 1 , τ 2 ∈ M may be defined as the curve x : [0 1] → M such that:
Interpolating over the real symplectic group
where C denotes the set of smooth curves that perform the mapping [0 1] → M . Setting M = Sp(2n, R), the problem (46) is a least-squares problem on the real symplectic group Sp(2n, R).
Interpolating over the real symplectic group
The solution of the above least-squares problem may be given in closed form. It coincides to a geodesic arc G x,v (t) parameterized so that G x,v (0) = τ 1 and G x,v (1) = τ 2 , namely:
Interpolating over the real symplectic group
The Figure 5 shows a result of continuous interpolation for n = 1 (the parametrization (45) has been used again). The Figure shows the location of the endpoint matrices τ 1 , τ 2 ∈ Sp(2, R) (dotted-circles) and of the interpolates (dots). 
Generalizations and applications
The present manuscript focuses on least-squares problems on the real symplectic group. The developed least-squares theory applies, however, to general noncompact manifolds.
Generalizations and applications
Least-squares problems arise in a series of applications in signal processing, machine learning, pattern recognition and computational statistics. Here follows a non-exhaustive list of possible applications of least-squares methods on noncompact manifolds:
Generalizations and applications
• Computation of the empirical mean value on manifold : Given a 'cloud' of points τ k ∈ M on a manifold M endowed with a distance function d(·, ·), its center of mass µ is defined as:
Generalizations and applications
The empirical mean value µ of a distribution of points on a manifold is instrumental in several applications. The mean value µ is, by definition, close to all points in the distribution, therefore, the tangent space T µ M may serve as reference tangent space in the development of algorithms on the manifold M (likewise the Lie algebra associated to a Lie group serves as reference tangent space in Lie-group theory).
Generalizations and applications
• Computation of the empirical high-order moments on manifold : Given a cloud of N points τ k ∈ M on a manifold M endowed with a distance function d(·, ·), its scalar m th -order (centered) moment is defined as:
Generalizations and applications
where µ ∈ M denotes the empirical mean of the cloud. The moment µ 2 denotes the variance σ 2 , which measures the width of the cloud around its center.
Generalizations and applications
• Applications based on the statistical distributions of data-points: Some statistical techniques, such as maximum likelihood estimation, are based on assumptions about the statistical distributions of the data-points. In the present context, it could be worth considering the distribution of the distances of the points from their center, namely, by defining the following exemplary distributions:
Generalizations and applications
, Lapla- 
Generalizations and applications
, where τ ∈ M denotes the manifold-valued variable of interest, µ ∈ M denotes the center of mass of the distribution, σ 2 denotes the variance and λ > 0 denotes a dispersion parameter.
Generalizations and applications
• Projection of a point on a curve: Given a point τ ∈ M and a smooth curve c x,v : [−a a] → M such that c x,v (0) = x, the 'foot' of the projection of the point τ on the curve c x,v is defined as the point c x,v (φ), with:
Generalizations and applications
It is assumed that the projection is well-defined, namely, that the footparameter value φ is unique. The (scalar) projection of the point τ on the curve c x,v is defined as:
Generalizations and applications
The projection computes as the (signed) distance (measured along the curve c x,v ) between the foot of the projection and the origin of the curve x ∈ M . The projection represents, therefore, the analogous of the orthogonal projection on Euclidean spaces. In the special case that the curve c x,v is a geodesic arc on the manifold M endowed with the metric ·, · · , then it holds:
Generalizations and applications
Moreover, the quantity d(τ, c x,v (φ)) denotes the distance of the point τ ∈ M from the curve c x,v .
Generalizations and applications
• Interpolation and modeling: Given a set of points τ k ∈ M on a manifold M endowed with a distance function d(·, ·) and a curve c x,v : [−a a] → R, interpolation/modeling is about finding parameters:
Generalizations and applications
with projection-feet defined as:
Generalizations and applications
The curve cx ,v (t) is an interpolator with optimal parameters.
Generalizations and applications
• Principal geodesic analysis: First principal component analysis and oneunit independent component analysis may be extended to manifolds as follows. Let it be given a cloud of points τ k ∈ M on a manifold M endowed with a geodesic distance function d(·, ·) associated to a geodesic curve G x,v . Let µ ∈ M denote the empirical mean of the cloud. The principal geodesic arc that captures the largest data-power may be defined as G µ,v with:
Generalizations and applications
where each φ k denotes the foot-parameter associated to the projection of τ k over the geodesic arc. (The normalization condition plays the same role as in linear principal component analysis). As the geodesic distance are computed over geodesic arcs, the above expression simplifies as:
Generalizations and applications
Likewise, the principal geodesic arc that corresponds to the largest m 
Generalizations and applications
The principal geodesic arc that maximizes the projection variance may be used to 'skeletonize' data on manifolds and for data compression. In fact, a lossy representation of a data τ k is given by its foot-parameter φ k . A principal geodesic arc that maximizes a high-order moment of the projection may be used in binary discriminant analysis to discern between data that possess or not possess certain high-order statistical features.
Generalizations and applications
• Fisher discriminant analysis on manifold : Let data-points τ k ∈ M belonging to two classes C 1 and C 2 of cardinality N 1 and N 2 , respectively, be given. Let µ 1 be the intrinsic mean of data-points in class C 1 and µ 2 be the intrinsic mean of data-points in class C 2 . Let G x,v be a geodesic curve on the manifold M and d(·, ·) be the associated geodesic distance function on M . Letφ 1 andφ 2 denote the foot-parameters associated to the projection of µ 1 over the geodesic arc and of µ 2 over the geodesic arc, respectively, and let φ k denote the foot-parameter associated to the projection of τ k over the geodesic arc. The Fisher ratio associated to the 2-class maximum discrimination problem is extended from the Euclidean case as:
Generalizations and applications
.
Generalizations and applications
(58) The numerator of the above expression represents the between-class variance which amounts to the squared difference between the means projected on a properly oriented curve. The denominator represents the within-class variance which amounts at the variance of the projected elements of the first class and of the second class. Note that all geodesic distances are measured on a geodesic arc, therefore, the above expression simplifies considerably because, for instance,
Generalizations and applications
• Kalman filtering on manifold : The state of a dynamical system evolving on a manifold may be estimated by setting up an appropriate least-squares problem. The extension from the case of Kalman filtering on Euclidean spaces is straightforward conceptually. An example of system model on a manifold M is:
Generalizations and applications
where n denotes discrete time, τ n ∈ M denotes system state, G ·,· denotes a geodesic curve on the manifold M , S(·) denotes a system operator which maps τ ∈ M → S(τ ) ∈ T τ M and r n ∈ T τn is a random noise term. (A map that sends a point from a manifold to its tangent space and that satisfies certain regularity conditions is termed lifting map.) More generally, the problem of Bayesian filtering on manifold is connected to the problem of extending standard state-space models from Euclidean spaces to manifolds [7] .
Generalizations and applications
• Least squares problems on manifolds may generalize the quadratic assignment problem [11] that is encountered in the allocation of facilities while optimizing the distance between locations in combination with the flow between the facilities. Also, the traveling salesman problem and the plant location problem are special cases of the quadratic allocation problem [11] .
Conclusion
The present paper discusses the problem of least squares over the real symplectic group of matrices. The present research takes its moves from the following observations:
Conclusion
• The least-squares problem may be extended from flat spaces to curved smooth metrizable spaces by the help of the notion of 'geodesic distance'.
Conclusion
• The resulting sum-of-squared-distance minimization problem on manifold may be tackled via a gradient-based descent algorithm tailored to the geometry of the symplectic group through a geodesic-based stepping method.
Conclusion
As the real symplectic group is a non-compact manifold, it might be hard to compute closed-forms quantities in a Riemannian context. Indeed, known results from scientific literature show that it is the case. The key point of the present paper is to regard the real symplectic group as a pseudo-Riemannian manifold and chose a metric that allows for the computation of closed-forms of geodesic arcs and hence of geodesic distance. On the basis of these findings, the geodesic least-squared problem may be properly set up and the geodesic-based numerical stepping method may be properly implemented.
Conclusion
Numerical tests have been performed with reference to the computation of the intrinsic mean of a collection of symplectic matrices as well as to the interpolation of two symplectic matrices. Numerical results show that the pseudoRiemannian-gradient-based algorithm, along with a pseudo-geodesic-based stepping method, is suitable to the numerical solution of a least-squares problem formulated in terms of pseudo-geodesic distance.
A Appendix: Metric Multidimensional Scaling
One of the purposes of multidimensional scaling (MDS) [21] is to provide a visual representation of the pattern of proximities among a set of high-dimensional objects. In this instance, MDS finds a set of vectors in the two-dimensional or three-dimensional Euclidean space such that the matrix of Euclidean distances among them corresponds -as closely as possible -to some function of the objects' proximity matrix according to a criterion function termed stress.
A Appendix: Metric Multidimensional Scaling
The following description provides a short introduction to MDS, as reworded in terms of low-dimensional representation of pseudo-Riemannian-manifold-valued elements.
A Appendix: Metric Multidimensional Scaling
Let M be a high-dimensional pseudo-Riemannian manifold with distance function d(·, ·) and let {τ k } k be a given collection of elements of M . The aim of MDS is to determine a collection of vectors z k ∈ R p (with p = 2 or p = 3) that replicate the pattern of proximities among the elements τ k . This may be achieved by minimizing the Kruskal stress function:
A Appendix: Metric Multidimensional Scaling
where symbol · denotes Euclidean norm. The above stress function gives rise to 'metric MDS'. More elaborated stress functions are available in the specific literature.
A Appendix: Metric Multidimensional Scaling
It is worth noting that the correspondence τ k ↔ z k induced by MDS is not unique. In fact, if {z k } k is a minimizer of the criterion (61), c ∈ R p and R ∈ O(p) (namely, a p-dimensional rotation/reflection), also {Rz k + c} k is a minimizer.
A Appendix: Metric Multidimensional Scaling
MDS may be used as a proximity/similarity visualization tool for highdimensional data as it computes two-dimensional or three-dimensional vectors z k , corresponding to the original elements τ k , that captures the fundamental information about mutual distances.
A Appendix: Metric Multidimensional Scaling
The axes corresponding to the coordinates of the vectors z k , possess, in themselves, no particular meaning. Also, the orientation and scaling of the obtained visualization are arbitrary. All that matters in an MDS map is which point is close to which others.
A Appendix: Metric Multidimensional Scaling
On a MDS visualization corresponding to a non-zero stress, the distances among objects are imperfect representations of the relationships among original data: The greater the stress, the greater the distortion. In general, however, the larger distances are represented more accurately, because the Kruskal stress function accentuates discrepancies in the larger distances.

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