(b) The highest order of polynomial in both non-vanishing component polynomial set F and vanishing component polynomial set V are M order at the most.
Next, the algorithm FindRangeNull used to solve zero space is applied to solve the new non-vanishing component polynomial set F1 and vanishing component polynomial set V1, and later combined with original sets F and V, thus composing the current non-vanishing component polynomial set F and vanishing component polynomial set V. At the same time, a new candidate polynomial set C t is figured out.
The VCA method firstly initializes three sets, i.e., the candidate polynomial set C1 = {f1, …, f n }, in which, f i (x) = x i, non-vanishing component polynomial set ( F=left{fleft(cdot right)=1/sqrt{m}right} ) and vanishing component polynomial set V = φ.
Grouped vanishing component analysis.
After that, the original VCA method is used to obtain vanishing component polynomial respectively, and finally, the vanishing component polynomial in multiple grouped training sets is combined into the vanishing component polynomial in an integral training set.
Afterwards, the order of vanishing component polynomial is set as the integer between [2, 12].
Once acquiring the vanishing component, the natural feature of a data manifold pattern can be captured.
This proves that too high order of vanishing component polynomial exactly greatly affects the classification performance.
Livni et al. put forward a vanishing component analysis method with stable values, i.e., the VCA method, to solve the generator (i.e., vanishing component) for vanishing ideal of fitting data manifold pattern.
In this strategy, the training sets were horizontally or vertically segmented into multiple non-intersecting subsets, which polynomials of vanishing components were later acquired through the VCA method, respectively, and finally combined into an integral set of vanishing component polynomials.
Vanishing component analysis (VCA) method, as an important method integrating commutative algebra with machine learning, utilizes the polynomial of vanishing component to extract the features of manifold, and solves the classification problem in ideal space dual to kernel space.
