Table 1 Acronyms

Q

ℝ^{N × N}

a semi-positive definite matrix

C

ℝ^N

a convex set

Ω

ℝ^{N × N}

a combination of multiple semi-positive definite matrices

j

ℕ

the index of kernel matrices

p

ℕ

the number of kernel matrices

θ

[0, 1]

coefficients of kernel matrices

t

[0, + ∞)

dummy variable in optimization problem

ℝ^p

ℝ^p

ℝ^Dor ℝ^Φ

the norm vector of the separating hyperplane

ϕ(·)

ℝ^D→ ℝ^Φ

the feature map

i

ℕ

the index of training samples

ℝ^D

the vector of the i-th training sample

ρ

ℝ

bias term in 1-SVM

ν

ℝ⁺

regularization term of 1-SVM

ξ _i

ℝ

slack variable for the i-th training sample

K

ℝ^{N × N}

kernel matrix

ℝ^D× ℝ^D→ ℝ

kernel function,

ℝ^D

the vector of a test data sample

y _i

-1 or +1

the class label of the i-th training sample

Y

ℝ^{N × N}

the diagonal matrix of class labels Y = diag(y₁, ..., y _N )

C

ℝ⁺

the box constraint on dual variables of SVM

b

ℝ⁺

the bias term in SVM and LSSVM

ℝ^p

k

ℕ

the number of classes

ℝ^p

ℝ^p

variable vector in SIP problem

u

ℝ

dummy variable in SIP problem

q

ℕ

the index of class number in classification problem, q = 1, ..., k

A

ℝ^{N × N}

λ

ℝ⁺

the regularization parameter in LSSVM

e _i

ℝ

the error term of the i-th sample in LSSVM

ℝ^N

the dual variable of LSSVM,

ϵ

ℝ⁺

precision value as the stopping criterion of SIP iteration

τ

ℕ

index parameter of SIP iterations

ℝ^p