An alternate version of this result focuses on the low-dimensional nature of the optimal paths themselves. It states that an optimal path lies in the smallest linear subspace containing both the endpoints and the subspace in which the cost function varies. This is illustrated in Fig. 4.4.1. Shown is an optimal path in three dimensions of a cost function that varies only in one dimension. The blue blocks represent high-cost regions that are assumed to extend ad infinitum in directions orthogonal to the cost variation direction; all other points in the space are assigned a uniform cost, which is assumed to be a distinct cost from the blue-shaded regions. The optimal path lies in the smallest two-dimensional subspace containing both the direction of cost variation and both of the endpoints.
This is made precise by the following theorem.
Theorem 4.4.1. Suppose that C(x) is the cost associated with a variational problem of
the form (4.3.1). If W is such that C(x) = C(W WTx),∀x, then there exist an optimal path x∗(t) of this problem and a path a(t) such that
x∗(t) =W a(t) +ys(t). (4.4.1)
Proof. The proof is a straightforward application of the calculus of variations. First,
assuming that the matrix of W has d columns, we can choose coordinates such that
the cost function C only depends on the first d coordinates. We therefore assume that
∂C/∂xi = 0∀i > d,∀x. As before, we will refer to coordinates on which the cost function
Figure 4.1: Illustration of Theorem 4.4.1
We then define the Lagrangian
L(x,x˙) =kxkC˙ (x) and apply the Euler-Lagrange equations, yielding
d dt ∂L ∂xi˙ − ∂L ∂xi = d dt ˙ xi kxk˙ C(x) − kxk˙ ∂C ∂xi = 0.
For any cyclic coordinate, we can substitute ∂C/∂xi = 0, implying that ∀t, and for
some yet-unknown constants ki,
˙ xi
kxk˙ C(x) =ki. (4.4.2)
As discussed in Section 3.2.3, the invariance of the cost functional to reparameterization
of time allows us to assume that kxk˙ = α, where α is some irrelevant constant. (4.4.2)
then yieldsN −dindependent, separable ODEs for each cyclic coordinate. Integration of
these produces xi(t) = αki Z t 0 1 C(x(t))dt = αkiF(t) (4.4.3)
We then write the path as a linear combination of the standard basis vectors, ei: x(t) =X
i
xi(t)ei
and substitute (4.4.3) into this equation, which produces
x(t) = d X i=1 xi(t)ei+ N X i=d+1 αkiF(t)ei.
We now solve for the ki in this expression to express the last basis vector in terms of the
problem data. The ki can be computed from the final conditions; i.e., xi(1) = αkiF(1).
Substitution of this expression and simplification then yields
x(t) = d X i=1 xi(t)ei+ F(t) F(1) N X i=d+1 xi(1)ei.
This expression can rewritten by expressing the basis ei in terms of the basis wi to yield
the desired expression, with (x1(t), . . . , xd(t)) becoming a(t), and F(t)/F(1) becoming
s(t).
Theorem (4.4.1) suggests another potential avenue for the algorithmic exploitation of low-dimensional structure to solve (4.3.1). Instead of computing the high-dimensional value function via a low-dimensional parameterization, and subsequently integrating the optimal control law in a high-dimensional space; we can equivalently find a path directly in the low-dimensional subspace in which it is known to be contained. This is usually the preferred solution method, though computing the high-dimensional value function is useful in some contexts as well.
Chapter 5
Discovering structure in physical
data
The application of learning and inference with structured Lagrangians to the specific prob- lem of the analysis of high-dimensional human motion capture data is now detailed, first reviewing the issues surrounding the problem to be solved.
5.1
Problem statement
The problem to be addressed in this section is that which was already described in some detail in Section 1.2 and Fig. 1.2. Summarizing that description, we suppose we are given training data consisting of high-dimensional vectors representing observations of a number of physical trajectories sampled at discrete times, where these observations specifically consist of the positions of a variety of markers located on the body of a human actor performing various motions. We would then like to learn a model of the motions from this data and subsequently use it to reconstruct novel trajectories from their endpoints alone. The primary problem to overcome in dealing with this data is its high dimensionality: each observation is represented as a 990-dimensional vector of real numbers. Having a means to deal this issue is hence an immediate prerequisite of any learning procedure applied to this problem.
Previous work in this field may be divided roughly into two camps: those that leverage detailed physical models, and those that do not. Among the latter are GPDMs [102, 103], which were discussed at some length in Section 2.3.2. A fundamental difference between GPDMs and the kinetic Lagrangian model employed here lies in their respective representations as graphical models. The GPDM’s graphical model is illustrated in Fig. 5.1, while the kinetic Lagrangian’s graphical model is illustrated in Fig. 3.3. Extra conditional
Latent state dynamics Observations .. . ... ... ... .. . ... ... ...
Figure 5.1: Graphical model associated with the GPDM
dependencies in the GPDM compared to the HMM 2.1 are due to the fact that inference in the GPDM automatically affects the implied dynamics and observation models. Another interesting point of difference between the GPDM and the kinetic Lagrangian models, is that the conservation laws in the latter produce simplified dynamics for the greater part of the state, whereas in the GPDM, the high-dimensional observations are modeled as images of low-dimensional trajectories. This difference is ultimately due to the fact that the kinetic Lagrangian is able to obtain a more appropriate graphical model for systems to which its assumptions apply.
The other relevant line of work is that which uses more detailed models of human motion [18, 59, 17, 19], typically used in applications of tracking people in video sequences. These approaches use simplified human kinematics and dynamics models, sometimes obtained from the biomechanics literature. Although some parameters of these models are estimated from data, the extent to which they depend on specific prior knowledge about this domain limits their general applicability compared to the kinetic Lagrangian method presented here.