Mathematics for Intelligent Systems: Kernels and SVM

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What is the mathematical expression for a hard margin?

y_i(w^T x_i + b) ≥ +1

What needs to be altered if we want to consider a soft-margin instead of a hard-margin?

Eqn.11

What is the geometric representation of the margin (M)?

The perpendicular distance between the lines w^T x + b = −1 and w^T x + b = +1

What is the expression for the margin (M) in terms of the vectors x1 and x2?

M = (x2 − x1)^T w / ||w||^2 Signup and view all the answers

Why is the optimization problem equivalent to minimizing ||w||^2?

Because the margin M is inversely proportional to ||w||^2 Signup and view all the answers

What is the optimization problem that fits a linear model onto the given data?

Minimize 1/2 w^T w subject to yi(w^T xi + b) ≥ +1 for all i = 1, 2, ..., n Signup and view all the answers

What is the key idea of the linear SVM method?

To construct a hyperplane that not only makes the fewest labeling errors for the data, but also optimizes the largest margin between the data. Signup and view all the answers

What is the purpose of the optimization problem associated with SVM?

To optimize a decision line that makes the fewest labeling errors for the data, and to optimize the largest margin between the data. Signup and view all the answers

What do the equations (5) and (6) represent in SVM?

The margin between the two classes. Signup and view all the answers

What is the purpose of normalization in SVM?

To get 1 or -1 on the LHS of the equations. Signup and view all the answers

What do the parameters w and b represent in the hyperplane equation?

The normal vector and bias of the hyperplane, respectively. Signup and view all the answers

What is the significance of the equations (9) and (10) in SVM?

They represent the constraints for the optimization problem in SVM. Signup and view all the answers

What is the condition for a data point to be within the margin space but on the same side of the decision boundary?

ξi < 1 Signup and view all the answers

What happens to the objective function when a data point is within the margin space but on the opposite side of the decision boundary?

The penalty term C ξi is added to the objective function Signup and view all the answers

What is the purpose of the scalar 'C' in the objective function?

It specifies the penalty for considering a data point within the margin space Signup and view all the answers

What are the primal variables in the constrained optimization problem?

w, b, and ξ Signup and view all the answers

What is the purpose of the Lagrangian function in the constrained optimization problem?

To find the optimal solution by imposing the KKT conditions Signup and view all the answers

What is the purpose of imposing the KKT conditions in the constrained optimization problem?

To find the optimal solution by satisfying the necessary conditions for optimality Signup and view all the answers

What is a kernel function in machine learning?

A kernel function is a real-valued function of two arguments that measures the similarity between objects. Signup and view all the answers

What is edit distance in computational linguistics?

Edit distance is a string metric that measures the minimum number of operations required to transform one string into another. Signup and view all the answers

What is a characteristic of a kernel function?

A kernel function is typically symmetric and non-negative. Signup and view all the answers

Why do we need kernel functions in machine learning?

We need kernel functions to compare objects that cannot be represented as fixed-size feature vectors. Signup and view all the answers

What type of problems can kernel functions be applied to?

Kernel functions can be applied to problems involving text documents or protein sequences of variable lengths. Signup and view all the answers

What is the significance of a kernel function being symmetric?

A symmetric kernel function means that κ(x, x′ ) = κ(x′ , x), which simplifies the comparison of objects. Signup and view all the answers

What is the Squared Exponential (SE) Kernel function, also known as the Gaussian Kernel, and how is it formulated?

κ(x, x′) = exp (−(x − x′)T Σ−1 (x − x′) / 2) Signup and view all the answers

What is the Radial Basis Function (RBF) Kernel and how does it relate to the Squared Exponential Kernel?

The RBF Kernel is a special case of the SE Kernel where all diagonal elements of Σ are equal, i.e., σj2 = σ2 Signup and view all the answers

What is the Cosine Similarity kernel function, and how is it used in document classification and retrieval?

κ(xi, x′i) = (xi ⋅ x′i) / (||xi||2 ||x′i||2), which measures the cosine of the angle between xi and x′i Signup and view all the answers

What is the characteristic length scale of the jth dimension in the SE Kernel, and what does it represent?

The characteristic length scale is σj, which represents the importance of the jth dimension in the SE Kernel Signup and view all the answers

What is the parameter σ2 in the RBF Kernel, and what does it control?

σ2 is the bandwidth, which controls the sensitivity of the kernel to differences between inputs Signup and view all the answers

What is the range of values for the Cosine Similarity kernel, and what do they represent?

The Cosine Similarity kernel takes values between 0 and 1, where 0 means the vectors are orthogonal and 1 means they are identical Signup and view all the answers