Approximation of Hessian Matrix for Second-order SPSA Algorithm ...

2.3 PROPOSED MAPPING
λ and 0
where > 0
q
λ
q + 1
≤ . As k
H is a real-valued, its eigenvalues are real-valued,
too. The eigenvalues of
H are computed as follows:
k
The number of non-zero eigenvalues is equal to the rank of H
k
, i.e., at most three non-zero
eigenvalues are available. In this part, the following arrangement of eigenvalues is assumed:
λ
≥
≥
1
λ
2
λ
3
. The technique presented here, requires much less user interaction. Now, the
theoretical background is explained leading to a two-fold threshold algorithm where the only
task of the user is to specify two thresholds. Finding the eigenvalues and eigenvectors of the
Hessian matrix is closely related to its decomposition
H
i
= PD P
−1
(2.6)
where P is a matrix and its columns are H’s eigenvectors and
D
i
is a diagonal matrix having
H’s eigenvalues on the Hessian. While computing the gradient magnitude by the Euclidean
norm requires three multiplications, two additions and one square root, the computation of
eigenvalues of the Hessian matrix is more suitable. The explicit formula would require solving
cubic polynomials. In our implementation a numerical technique of fast converging called
Jacobi’s method is used as is recommended in [20] for symmetric matrices. We have proposed
an easy-to-use framework for exploiting eigenvalues of the Hessian matrix to represent volume
data by small subsets.
The relation of eigenvalues to the Laplacian operator is recalled, this shows the suitability of
threshold eigenvalue volumes, and define a two-fold threshold operation to generate sparse data
sets. For data where it can be assumed that objects exhibit higher intensities than background,
we modify the framework taking into account only the smallest eigenvalue. This results in a
further reduction of the representative subsets by selecting just data at the interior side of object
boundaries. For the sake of simplicity, we have omitted the index k for the individual eigenvalue
λ
i that is a function of k. Next, we assume that the negative eigenvalues will not lead to a
physically meaningful solution. They are either caused by errors in
H
k
or are due to the fact
that the iteration has not reached the neighborhood of
θ
*
where the loss function is locally
quadratic. Therefore, we replace them together with the smallest positive eigenvalue with a
descending series of positive eigenvalues:
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