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Questions and Answers
What are the differences between a soft and a hard margin SVM?
What are the differences between a soft and a hard margin SVM?
What does the perceptron algorithm directly predict?
What does the perceptron algorithm directly predict?
The perceptron algorithm directly predicts 1 or -1.
In the hard margin SVM, we know that the data is perfectly separable and as a result, there will be no data within -1 < wTx + b < 1.
In the hard margin SVM, we know that the data is perfectly separable and as a result, there will be no data within -1 < wTx + b < 1.
True
What is the distance between the two parallel hyperplanes in the hard margin SVM equivalent to?
What is the distance between the two parallel hyperplanes in the hard margin SVM equivalent to?
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The primal problem in SVM is a convex problem.
The primal problem in SVM is a convex problem.
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What are the points called that lie on the hyperplanes in a hard margin SVM, influencing the optimal solution?
What are the points called that lie on the hyperplanes in a hard margin SVM, influencing the optimal solution?
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What is the dual problem of a hard margin SVM used for?
What is the dual problem of a hard margin SVM used for?
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What does the kernel in a SVM do, in terms of data?
What does the kernel in a SVM do, in terms of data?
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The kernel trick is not an essential component of nonlinear SVMs.
The kernel trick is not an essential component of nonlinear SVMs.
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What are some popular SVM kernels? (Select all that apply)
What are some popular SVM kernels? (Select all that apply)
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What does C represent in the objective function of a soft margin SVM?
What does C represent in the objective function of a soft margin SVM?
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In a soft margin SVM, the C term is similar to a regularization term in regression.
In a soft margin SVM, the C term is similar to a regularization term in regression.
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Which of these are considered key hyperparameters in Support Vector Machines? (Select all that apply)
Which of these are considered key hyperparameters in Support Vector Machines? (Select all that apply)
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Study Notes
Support Vector Machines (SVM)
- SVMs are a supervised learning algorithm, originally designed for classification, but can be extended to regression.
- Developed by Vladimir Vapnik and Alexey Chervonenkis in 1963.
- Further developed by Dr. Vapnik at Bell Labs in the 1990s.
SVM Types
- Hard Margin SVM: Used for perfectly separable data. Tries to maximize the margin (distance between the hyperplane and the closest data points of both classes).
- Soft Margin SVM: Extends hard margin to non-perfectly separable data. Includes a penalty term (C) to allow some data points to be on the wrong side of the hyperplane.
Preliminaries
- Space: A set with a defined structure, like the 2-dimensional plane.
- Subspace: A subset of a space, for example, the positive quadrant.
- Hyperplane: A subspace of one dimension less than the containing space. In 2D, it's a line; in 3D, it's a plane.
Logistic Function
- Predicts the probability of an outcome (0 or 1) using the equation: ŷ1 = 1 / (1 + e-(βTx)).
- Assumes a larger value of βTx yields higher confidence in a prediction.
- βTx ≥ 0 predicts y₁ = 1 with high confidence; βTx < 0 predicts y₁ = 0 with high confidence.
Optimal Separating Hyperplane
- The goal of SVM is to find the optimal separating hyperplane.
- The goal is to maximize the margin, which is the distance from the hyperplane to the nearest data point from either class.
Primal Problem
- The primal problem formulates SVM as a quadratic programming problem that seeks to minimize ||w||2 subject to constraints that ensure correct classification and maximize the margin, where ||w|| is the Euclidean norm.
- The constraint ensures that the correct class (y) is on the correct side of a linear transformation w ⋅ x + b ≥ 1.
Kernel Trick
- Kernels are used to map data into higher dimensions where non-linear separation may become possible.
- Used in non-linear SVMs.
- Kernels compute similarities between observations.
- Popular kernels include:
- Linear: K(xᵢ, xⱼ) = xᵢ ⋅ xⱼ + c
- Polynomial: K(xᵢ, xⱼ) = (xᵢ ⋅ xⱼ + c)d.
- Radial Basis Function (RBF): K(xᵢ, xⱼ) = e-||xᵢ - xⱼ||2/γ
- Sigmoid: K(xᵢ, xⱼ) = tanh(γxᵢ ⋅ xⱼ + c)
Dual Problem
- The dual problem is an equivalent reformulation of the primal problem and often is more efficient to solve.
- It simplifies computation by only focusing on support vectors, those observations that lie on the margin (boundary).
Hyperparameters
- Kernel: Crucial for determining the separating hyperplane's shape.
- C (penalty): Controls the penalty for misclassifying data points.
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