Multicamera Action Recognition with Canonical Correlation Analysis and Discriminative Sequence Classification - Foundations on Natural and Artificial Computation - page 493

of a set of training data x =( x 1 ,x 2 ). The rest of canonical correlation directions
are orthogonal to w 1 and w 2 respectively. They can be computed as the solutions
of the generalized eigenvalue problem:
Σ 11

w 1
w 2

=(1+ ρ ) Σ 12 w 1

w 2

Σ 12

0

Σ 21 Σ 22 Σ 21

0

The classical CCA model is defined for only two random variables x 1 and
x 2 . Bach and Jordan [1] generalize it to m random variables. The generalized
eigenvalue problem to solve is then defined as:

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

Σ 11 Σ 1 m Σ 11 ···

··· w 1

.
w m

w 1
.
w m

0

. . . .

= λ

Σ m 1 ··· Σ mm Σ mm

0 ···

3.2 Probabilistic Interpretation
Bach and Jordan [2] made a probabilistic interpretation of CCA extending the
probabilistic interpretation of PCA proposed by Tipping and Bishop [17]. They
define the following generative model, also shown on figure 2:

Fig. 2. Graphical model of the probabilistic interpretation of CCA made by Bach and
Jordan [2] for two variables

z n ∼N (0 ,I q )

x n 1 ∼N ( W 1 z n + μ 1 ,Ψ 1 )

( W 2 z n + μ 2 ,Ψ 2 )
They also show that the maximum likelihood estimates of the model parameters
are given by:

x n 2 ∼N

W 1 = Σ 11 U 1 M 1

(1)

W 2 = Σ 22 U 2 M 2

(2)

Ψ 1 = Σ 11 − W 1 W 1

(3)