Wolfe dual objective of the support vector machine

Formula: L(alpha) = sum_i alpha_i - 0.5 sum_i sum_j alpha_i alpha_j y_i y_j K(x_i, x_j)

Usage

Svmwolfe(alpha, X, y, K = NULL)

Arguments

alpha

Dual variables, length n.

X

One record per row.

y

Class labels coded +1 and -1.

K

Gram matrix; None uses the linear kernel x_i'x_j.

Value

List with dual, linear_term, quadratic_term, constraint_sum, n.

References

Montesinos Lopez, Montesinos Lopez and Crossa (2022), Multivariate Statistical Machine Learning Methods for Genomic Prediction, Springer, doi:10.1007/978-3-030-89010-0. Chapter 9, Eq. (9.32) p. 349. The dual depends on the data only through inner products, which is exactly what lets a kernel replace them; constraint_sum reports sum_i alpha_i y_i, which the dual problem constrains to zero.

Examples

Svmwolfe(alpha = c(1, 2, 3, 4, 5, 6, 7, 8), X = c(1, 2, 3, 4, 5, 6, 7, 8),
  y = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $dual
#> [1] -839772
#> 
#> $linear_term
#> [1] 36
#> 
#> $quadratic_term
#> [1] 839808
#> 
#> $constraint_sum
#> [1] 204
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "SVM Wolfe dual objective, MVSML Eq. (9.32)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"