Regarding Lagrange Multiplier in SVM

Hello all, In my work I am using SVM for classification. I had trained the SVM classifier and also obtained the Lagrangian multiplier (α) which is a column vector .

My query is how we can use this Lagrangian multiplier (α) to predict the labels of the data points.

Any help in this regard will be highly appreciated.

[charu shree]

Any hint or suggestion in this regard will be highly appreciated.

[Torsten]

I think there are only very few people here who know what you are talking about:

SVM ?

trained the SVM classifier ?

Lagrangian multiplier (α) which is a column vector ?

how we can use this Lagrangian multiplier (α) to predict the labels of the data points ?

Maybe you could formulate your problem in a more general - maybe mathematical - form.

[charu shree]

Thank you sir for your response....

I am using the paper " Supervised Learning-Based Semi-Blind Detection for Generalized Space Shift Keying MIMO Systems" for my work.

I am using SVM method for classification. The SVM classifier is trained using the training data in vector having dimension , the labels in vector having dimension .

So as per equation (5) of this paper, I had obtained 16 different α of dimension .

My query in mathematical form is as follows:

How can I predict the labels of the test data say in vector having dimension 404 x 4 ?

[charu shree]

Any hint will help me in moving ahead in tackling this problem...

Answer 1

[Parag]

Hi @charu shree,

To predict the labels for your test data (e.g., a 404 × 4 matrix), using SVM trained in a one-vs-all setting with 16 classifiers, you can follow this approach:

• Each SVM classifier corresponds to one class and provides a decision score using the learned parameters (α, support vectors, labels, and bias).
• For each test vector, compute the decision score from all 16 classifiers using the kernel function (e.g., linear).
• Stack all scores in a matrix of size 404 × 16.
• Assign each test vector the class label of the classifier with the highest score (i.e., maximum margin decision function output).

This approach implements the standard prediction step in multi-class SVM, aligned with the formulation in the paper.

% Inputs (assumed precomputed and available): 
% X_test          : 404 x 4 test data matrix 
% supportVectors  : N x 4 training vectors 
% alpha           : 1x16 cell array, each cell contains alpha vector for a class 
% labels          : N x 16 label matrix in one-vs-all format (+1/-1 per class) 
% b               : 16 x 1 bias vector for each classifier 

numTest = size(X_test, 1); 
numClasses = 16; 
F = zeros(numTest, numClasses);  % Stores decision scores 

for k = 1:numClasses 
    % Linear kernel: dot product 
    K = X_test * supportVectors';  % 404 x N 
    F(:, k) = K * (alpha{k} .* labels(:, k)) + b(k); 
end 

% Predict class with highest score 
[~, predicted_labels] = max(F, [], 2);  % Output: 404 x 1 vector of predicted classes 

Hope it helps!