Calculating Dual Form of SVM Equation

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In summary, the conversation is discussing the calculation of the dual form of an SVM optimization problem. The speaker mentions having a vector of alphas, a vector of target outputs, and a Kernel matrix. They are unsure of which indices alpha and j should be taking, and if i and j are equal, the result is just squared. The other speaker suspects that the ##\sum_{i,j}## in the equation is shorthand for ##\sum_i\sum_j## and provides an example calculation for clarification.
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Lavace
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I am trying to calculate the dual form of an SVM optimisation problem:

ELWm1.png

Dual Form Optimsation Problem

In my algorithm, I have a vector of alphas, vector of target outputs, and a Kernel matrix computing upfront.

However, I am stuck as to what indices alpha and j should be taking here. If i and j are equal, then everything is just squared. Is this for i =/= j?

If so, could I just duplicate each vector and flip it?
 
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  • #2
I'm going to suspect that the ##\sum_{i,j}## in your equation is being used as a shorthand for ##\sum_i\sum_j##, in which case ##\sum_i\sum_j\alpha_i\alpha_j = \alpha_1\alpha_1 + \alpha_1\alpha_2 + \cdots + \alpha_1\alpha_m + \alpha_2\alpha_1 + \cdots + \alpha_2\alpha_m + \cdots + \alpha_n\alpha_1 + \cdots \alpha_n\alpha_m## if ##i = 1 \ldots n## and ##j = 1 \ldots m##.
 

Related to Calculating Dual Form of SVM Equation

1. What is the dual form of the SVM equation?

The dual form of the SVM equation is an alternative representation of the original SVM equation. It involves solving a constrained optimization problem to find the optimal hyperplane that maximally separates the classes in a given dataset. The dual form is often used for faster and more efficient computation in SVM models.

2. How do you calculate the dual form of the SVM equation?

To calculate the dual form of the SVM equation, you first need to set up and solve the constrained optimization problem. This involves finding the Lagrange multipliers, which are weights assigned to each data point in the training set. Once the Lagrange multipliers are found, the dual form of the SVM equation can be derived by substituting these values into the original equation.

3. What are the benefits of using the dual form of the SVM equation?

The main benefit of using the dual form of the SVM equation is faster and more efficient computation. It also allows for the use of kernel functions, which can help to better separate non-linearly separable data. In addition, the dual form can provide insights into the importance of each data point in the training set, which can aid in feature selection.

4. Can the dual form of the SVM equation handle non-linearly separable data?

Yes, the dual form of the SVM equation can handle non-linearly separable data by using kernel functions. These functions transform the data into a higher-dimensional space where it may be easier to find a hyperplane that separates the classes. This allows for more flexibility in the types of data that can be effectively separated using SVM models.

5. Are there any limitations to using the dual form of the SVM equation?

One limitation of using the dual form of the SVM equation is that it may not be suitable for datasets with a large number of features. This is because the number of Lagrange multipliers to be calculated increases with the number of features, making the computation more complex and time-consuming. In addition, the dual form may not perform well on imbalanced datasets, where one class has significantly more data points than the other.

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