Advanced Regression and SVM Concepts

Which expression correctly represents the gradient of the objective function in ordinary least squares regression?

• Gradient = 2 * X_transpose * (X * beta - y)

• Gradient = X_transpose * X - X_transpose * y

• Gradient = X * (X_transpose * beta)

• Gradient = X_transpose * X * beta - X_transpose * y

In the chain rule for multivariable functions, what is the correct form of the total derivative of z = f(x(t), y(t))?

• dz/dt = df/dx * dx/dt + df/dy * dy/dt

• dz/dt = partial f / partial x + partial f / partial y

• dz/dt = partial f / partial x * dx/dt + partial f / partial y * dy/dt

• dz/dt = gradient f dot dx/dt

Which condition is necessary to ensure that X_transpose * X is invertible in the normal equation for linear regression?

• Matrix X has full row rank

• Matrix X is square

• Matrix X has linearly dependent columns

• Matrix X has full column rank

What is the entropy of a binary variable with values {0,1} occurring with equal probability?

• H(X) = 0

• H(X) = 1

• H(X) = log2(1/2)

• H(X) = -2 * (1/2) * log2(1/2)

In SVM, what condition must support vectors satisfy if 0 < alpha_i < C?

• y_i * (w_transpose * x_i + b) = 0

• y_i * (w_transpose * x_i + b) = 1

• alpha_i = C

• xi_i = 1

The Chebyshev distance is the limit of the Minkowski distance as:

• p approaches 0

• p approaches 1

• p approaches 2

• p approaches infinity

Which of the following is NOT a required property for a distance metric?

• Non-negativity

• Symmetry

• Smoothness

• Triangle inequality

In matrix multiplication, the element C_ij of the product C = AB is given by:

• The sum of the i-th row of A and the j-th column of B

• The product of the i-th row of A and the j-th column of B

• The dot product of the i-th row of A and the j-th column of B

• The maximum value in the i-th row of A and j-th column of B

In k-NN classification, what is a key implication of the curse of dimensionality?

• Distances become meaningless as dimensions increase

• Training time increases exponentially with dimensions

• Accuracy always improves with more features

• All features contribute equally

What is the partial derivative of f(x, y) = x^2 * y + 3 * x * y^2 + 5 with respect to x?

• x^2 + 6 * x * y

• 2 * x * y + 3 * y^2

• 2 * x + 6 * y

• 3 * x^2 + 2 * y