1 Introduction
For many pattern recognition problems, the feature space is nonlinear and of high dimensionality and nonlinearity, for example, face recognition, image classification and shape recognition. Many dimension reduction techniques
[1] are therefore naturally introduced to ease the modeling, computation and visualization of the feature space. Dimension reduction has been widely used with different vision techniques such as contour tracking [3], face recognition [4] and object recognition [2].One of the key points of dimension reduction is how to represent the geometry of the original space in a lower dimensional space [9, 10]. Most of the previous dimension reduction research [1] focuses on embedding the points in the original space into a lower dimensional space. Beyond the pointwise dimension reduction, the dimension reduction for a sequence of points or an evolving curve in the original feature space is more important in applications where features of interest are time varying such as, face expression [4], events indexing in video sequence as a shape process [11] and human activity as a shape sequence [3]. To the best of our knowledge, only a few works [3, 2] address the problem of how to represent a curve in the original feature space in a lower dimensional space.
The dimension reduction of high dimensional curves may be categorized into two classes. One is to reduce the whole curve space to a lower dimension as in [3, 12, 13]. The other one is to reduce a particular the curve to a corresponding lower dimensional curve as in [2]. The drawback of the first one is the Euclidean assumption by using an approximation with a spline function. Moreover, as in [3] by directly applying PCA to the spline coefficients, the global dimension reduction does not consider the intrinsic dimensionality of a particular high dimensional curve. A curve sequence like walking, for example, is much simpler than a dancing sequence. Thus the dynamics of walking might be represented in a much lower dimensional space in comparison to other complex activities. The second category of dimension reduction as in [2], considered the geometry of a particular curve in the original space, which, due to the chosen implementation of the Whitney embedding theorem, is computationally heavy on account of the iterative subspace search. At the same time, there exists no invertible mapping from the embedding space to the original one, thus limiting its extension to a generative process.
Motivated by the importance of a lower dimensional representation of a manifoldvalued curve and the limitations of the current techniques, in this paper a novel dimension reduction of a shape manifold valued curve is proposed. In contrast with previous curve dimension reduction techniques [3, 2], the proposed method has the following advantages:

the dimension reduction is adaptive to the Riemannian geometry of a shape space.

the computation is linear in the length of the curve times the dimension of the shape space.

with a proper dimensionality, there exists a reconstruction as a subspace parallel transport on a manifold.
To introduce the idea of the proposed dimension reduction framework for the manifold valued curves, this paper follows the moving frame formulation in [6, 7]. According to [7, 6], every curve on a manifold may be represented by a differential equation,
(1) 
where,
is a vector field on a manifold, and
is a curve in a Euclidean space. Under such representation, to introduce a lower dimensional representation of , a method is proposed to find the optimal vector field such that:
The sequence of is path dependent and uniquely determined by and initial condition by a efficient computation.

The resulting from and have a distribution that lies in a lower dimensional subspace.
The first requirement on means that can be represented by only the initial condition and the embedded curve . According to [6] such property yields the 11 correspondence between and . The second requirement is more intuitive that among all the
satisfied the first requirement, the optimal one is selected such that the variance of tangents is best represented in a subspace.
The first requirement is satisfied by select that is a horizontal vector field [7] defined by the LeviCivita connection as in Section 2.2 with a metric induced to the shape manifold from the ambient space. According to [6, 7], under a defined connection, for and initial condition , there exists a unique selection of moving frame . Thus given the resulting curve in Euclidean space have oneone correspondence to . The reason of using the LeviCivita connection have two folds. The first one is that it is a nonflat connection. In contrast to the other category: flat connection, it is path dependent. Such property is important for dimension reduction because it provides more adaptivity to the geometry of the path on the manifold. The second reason is that in comparison with the general nonflat connections defined by a PDE on manifold, such particular LeviCivita connection in Section 2.2 is consistent with the metric of the shape manifold in [5] that this paper is based on, and is computationally more efficient for a known ambient space and normal space of the manifold, as is the case of our shape manifold.
The second requirement is achieved by the subspace representation resulting from maximizing the variance of the corresponding curve development with different choices of . In theory, without knowing the initial condition the LeviCivita connection only determines the vector field up to a group action within the fiber of the Frame bundle (the total space of frames of the tangent space of a manifold) [7]. Among different choices for along the fiber in the frame bundle, a optimal frame is selected such that the previously discussed second requirement is satisfied to make the resulted better representable in a lower dimensional space. A is selected by carried out PCA of all the parallel transported tangents along .let
are the eigenvectors corresponding to the
largest Eigenvalues in the PCA calculation. Then A subspace spanned by the subset
is constructed to better represent the variance of all the tangents along in a lower dimensional space.To summarize, a curve in the shape manifold is represented as
(2) 
where can be represented by curves in and is represented by a curve in . Our proposed dimensional reduction achieves the following points,

a lower dimensional representation of curves on a manifold.

The moving frame representation can be generated to a stochastic representation of random process on a manifold. Thus the propose method can be well adapted to a generative modeling in a lower dimensional space.

the computation is linear in the dimension of .

there exists a reconstruction from the lower dimensional representation to the original curve on the shape manifold.
2 Preliminary
In this section, we first introduce in Subsection 2.1 the shape manifold that the dimension reduction is based on. Then in Subsection 2.2 a fundamental moving frame representation of curves on manifold is briefly introduced.
2.1 Shape Manifold
According to [5], a planar shape is a simple and closed curve in ,
(3) 
where an arclength parameterization is adopted. A shape is represented by a direction index function . With such a parameterization, may be associated to the shape by
(4) 
The ambient space of the manifold of is an affine space based on . Thus
(5) 
The restriction of a shape is that it must be a closed and simple curve, and invariant over rigid Euclidean transformations. The shape manifold is defined by a level function as
(6) 
(7) 
One of the most important properties of is that the tangent space is well defined. Such a property not only simplifies the analysis, but also makes the incremental computation possible,
(8) 
In addition, an iterative projection is proposed in [5] to project the point in ambient space back to the shape manifold . The idea is that each time is updated as , where is orthogonal to the level set . The is calculated as . For the detailed form of the Jacobian of , one could refer to [5].
The problem of this manifold for our stochastic modeling is the infinite dimension. The mapping of random process in [6] from a manifold to a flat space is only defined on a finite dimensional manifold. Therefore, a Fourier approximation of the shape manifold we discussed above is developed, such that the dimension is reduced to a finite number.
2.2 Moving Frame Geometry
Let be a curve on a manifold , then following the moving frame representation [8], the tangent of manifold valued curve may be written as:
(9) 
where is a frame of the tangent space at , which is denoted as . Consequently may be understood as a linear coefficient of under the representation of . Such moving frame representation is widely used in geometry studies of curves. For example in the Frenet theorem [8] the geometry of curve is uniquely identified by the curvature of up to a rigid Euclidean transformation. Recently in computer graphics, a rotation minimizing moving frame [17] are developed to avoid the singularity of original moving frame used in Frenet theorem.
In this paper, the moving frame representation is utilized for the similar purpose of representing a curve on a manifold. In contrast to the previous designs of moving frame, in this paper we elaborate to propose a adaptive moving frame to represent a high dimensional curve in a lower dimensional space. Vector field are developed as a sequence of parallel frames along . The parallelism is defined under a LeviCivita connection. The advantages of such innovation are briefly introduced in the following:

Once are parallel according to the curve development theory in [6], can be represented uniquely as . Such property yields the invertibility of the proposed dimension reduction of . In other words, by the representative curve in a lower dimensional space, can be reconstructed when the initial conditions are available.

Under LeviCivita connection, the parallel frames are calculated according to metric of the manifold . Such property implies that the shape of will reflect well the shape of on . For example, if is a geodesic curve, then in the proposed method will be a straight line in Euclidean space

The particular shape manifold [5] we adopted in this paper have a known tangent and normal space in the ambient space which greatly simplified the calculation of the moving frames under LeviCivita connection.
The LeviCivita connection may be viewed as the Christoffel Symbol that defines the directional derivative in .
(10) 
where is the coordinate function of an arbitrary point in . The tangents of can be written as linear combination of . For example at , and . Thus according to Equation , we can analytically calculate the derivative of along in direction , which is denoted as . is parallel if ,
(11) 
However in practical problems, it is usually impossible to chart the manifold. In other words, the coordinate functions of are usually unknown. In this paper, the shape manifold [5] we introduced in Section 2.1 have some nice properties that allows us to calculate in terms of a projection of Euclidean calculus in the ambient space onto the tangent space of the manifold .
(12) 
where is the vector representing in the ambient space of . Since the ambient space is an affine space based on , thus is calculable as a real vector in . The is a mapping of vectors in onto the tangent space of .
3 Dimension Reduction
The dynamics of some common human activities such as walking and running, are relatively simpler in comparison to a complex activity like dancing. Thus it is nature to speculate that the simple activity could be represented in a lower dimensional subspace on the shape manifold. To learn such submanifold, a dimension reduction is proposed for a curve on a shape manifold. As described in the previous section, first the horizontal vector field is defined by a LeviCivita connection. Then Principle Component Analysis (PCA) is carried out to compute the optimal vector field such that all the parallel transported tangents along could be represented in a lower dimensional space.
The shape sequence on a manifold can be represented by a differential equation as,
(13) 
where, is selected to be the horizontal vector field on a manifold and is the curve development in a Euclidean space. Such a formulation have the property according to [7, 6] that the resulting is 11 corresponded to with the initial condition .
The idea of the proposed dimension reduction is to reduce the dimension for a manifold valued curve by learning a sequence of subspaces on the shape manifold , such that the resulting have a distribution that concentrate in a lower dimensional subspace.
A LeviCivita connection is adapted on the shape manifold according to the metric induce from the ambient space. According to the definition of covariant derivative under LeviCivita connection as in Section 2.2, the horizontal vector field is the solution of the following differential equation.
(14) 
with initial condition .
The subspace moving frames for dimension reduction is constructed by select the optimal initial condition . First parallel transport all the tangents to the tangent space at . With the moving frame formulation in Equation , the parallel transportation results is a set
(15) 
According to [7], the set is invariant to the choice of . Thus in the first step an arbitrary initial condition is assigned.
Then Let the be the eigenvectors that corresponding to the largest eigenvalues of vectors in . The optimal initial condition is constructed as
(16) 
The solution of Equation with initial condition is the moving frames we used to represent the by the corresponding in a lower dimensional space
In the following, we provide the solution to the differential equation . According to Equation and normal space expression of the shape manifold in Section 2.1, the above equation can be solved as the following. Let be the orthogonalization of , which is the basis of normal space of the shape manifold . ,
(17) 
In the numerical calculation of the Euclidean derivatives are implemented as follows. For a small enough ,
(18) 
Substitute the Equation into the covariant derivative Equation , could be written as,
(19) 
Since , the parameter should satisfy that,
and
From the above three equations, we have,
(20) 
where
So the parameter could be solved in a linear fashion. Thus the shape sequence can be represented as follows,
(21) 
where is a curve development in .
4 Reconstruction from the Lower Dimensional Representation
As introduced in the previous sections, one of the claimed advantages of the proposed dimension reduction is that it is possible to generate the original shape sequence from the dimension reduction results. Such an invertible property makes our lower dimensional shape process embedding a compelling way to learn the generative model of the shape sequence.
Let be the embedded shape process in . A approximation of the original sequence on the shape manifold may be generated by the following differential equation:
(22) 
where the horizontal vector field is generated uniquely by the differential equation 14 with the initial frame .
Numerically Equation is implemented as a difference function,
(23) 
where is the iterative projection from the ambient space to the shape manifold as in [5], which is also briefly introduced in Section 2.1.
Figure 7 illustrates the reconstruction result of the shape sequence of activity: running. The reconstruction is from the embedding curve and the initial condition , to the shape manifold .
5 Conclusion
Dimension reduction of points on a high dimensional manifold is well studied in the past decades. Only a few works, however, address the problem of dimension reduction of curves on a manifold. In this paper a novel dimension reduction technology is proposed for shape dynamics with the following advantages that, as far as we know, no previous work has ever achieved:

The subspace is learned nonlinearly according to the geometry of the underlying manifold.

The computation is linear in the size of data.

There exists an efficient reconstruction from the lower dimensional representation to the original curve on a shape manifold.

The proposed dimension reduction technique provides both analytical and practical foundations for generative modeling of shape dynamics.
This work also first apply the differential geometry tools such as moving frame representation and LeviCivita connection to dimension reduction of curves on a manifold. It provides a new perspective on the dimension reduction with a moving frame formulation to naturally characterize the nonlinear features of a manifold valued dynamics in a linear subspace.
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