The interview questions are collected from LinkedIn, Twitter, and Quora posts.

1. What is Linear independence?

Linear independence is a concept in linear algebra that refers to the property of a set of vectors, where no vector in the set can be expressed as a linear combination of the other vectors in the set.

In other words, a set of vectors is said to be linearly independent if and only if none of the vectors in the set can be represented as a linear combination of the other vectors in the set. This is an important concept in linear algebra as it has many applications in areas such as solving systems of linear equations, finding the dimension of a vector space, and analyzing the behavior of linear transformations.

For example, consider the set of vectors {(1,0), (0,1)}. These two vectors are linearly independent because they cannot be expressed as a linear combination of each other. However, if we had the vector (2,2) in the set, it could be expressed as a linear combination of (1,0) and (0,1), so the set would no longer be linearly independent.

2. Tell me about Determinant.

The determinant is a scalar value in linear algebra that represents the magnitude of a square matrix. It is used to calculate the inverse of a matrix, to find the eigenvalues and eigenvectors, and to solve linear equations. The determinant of a matrix is calculated by multiplying and adding or subtracting the elements of the matrix in a specific order, which is known as the expansion of the determinant. The value of the determinant is zero if the matrix is singular, meaning that it is not invertible.

3. What are Eigenvalues and Eigenvectors?

Eigenvalues and Eigenvectors are important mathematical concepts used in linear algebra and other areas such as physics, engineering, and computer science.

An eigenvalue is a scalar value associated with a linear transformation or a square matrix. It represents the factor by which the eigenvector (which is also associated with the same linear transformation or matrix) is scaled during the transformation.

An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scalar multiple of the vector itself. In other words, the eigenvector remains on the same line after the transformation, and the eigenvalue determines the factor by which it is scaled.

The relationship between eigenvalues and eigenvectors can be represented by the equation Ax = λx, where A is the matrix, x is the eigenvector, λ is the eigenvalue, and A is multiplied by x to produce the transformed vector.

Eigenvalues and eigenvectors have many applications in various fields, including but not limited to the calculation of eigenvectors of a covariance matrix to determine the principal components in PCA (Principal Component Analysis), the calculation of eigenvalues of a matrix to determine its stability in control theory, and the calculation of eigenvectors of a graph to determine its connectivity.

4. What is SVD?

SVD stands for Singular Value Decomposition, which is a matrix factorization technique used in many different areas, including linear algebra, computer vision, and natural language processing.

In mathematics, the singular value decomposition of a matrix is a factorization of that matrix into three matrices: U, Σ, and V. These matrices are related to the eigenvalues and eigenvectors of the matrix, and they represent the structure of the original matrix in a way that can be useful for various applications.

For example, in image compression, SVD can be used to reduce the number of elements in a matrix that represents an image, while preserving its most important features. In recommendation systems, SVD can be used to factorize a large user-item matrix into smaller matrices that capture the relationships between users and items.

Let A be a m x n matrix. The Singular Value Decomposition of A is given by:Let A be a m x n matrix.

The Singular Value Decomposition of A is given by:

A = U * Σ * V^T

Where:
U is a m x m orthogonal matrix (i.e., its columns are orthonormal)
Σ is a m x n diagonal matrix with the singular values of A on its diagonal
V is a n x n orthogonal matrix (i.e., its columns are orthonormal)
V^T is the transpose of V

The singular values of A are the positive square roots of the non-zero eigenvalues of A^T * A (or equivalently, of A * A^T). They measure the magnitude of the deviation of A from being a rank-deficient matrix, and they give us a way to quantify the "importance" of each singular vector in representing the data in A.

U and V have the property that they are orthogonal matrices, meaning that their columns are orthonormal. This means that their columns form an orthonormal basis for the space of the data represented by A. The columns of U and V can be thought of as representing the "directions" in which the data in A varies the most and the least, respectively.

The matrix Σ is a diagonal matrix whose elements are the singular values of A, arranged in decreasing order. The singular values give us a way to quantify the "importance" of each direction of variation represented by the corresponding columns of U and V.

In summary, the SVD of a matrix provides a way to decompose a matrix into its orthogonal components, each with a different level of importance, which can be useful for various applications.

5. Can you explain Expectation and variance?

Yes, I'd be happy to explain expectation and variance!

Expectation, also known as expected value, is a measure of the central tendency of a random variable. In other words, it represents the average outcome of a random process, such as the roll of a die or the outcome of a coin flip. The expectation of a random variable X is denoted as E(X) and is calculated as the sum of the product of each possible outcome of X and its corresponding probability.

For example, if X represents the outcome of a single roll of a fair six-sided die, then E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5. This means that, on average, a single roll of a fair die will result in a 3.5.

Variance, on the other hand, is a measure of the spread or dispersion of a random variable. It quantifies how far the possible outcomes of a random process deviate from the expected value. The variance of a random variable X is denoted as Var(X) and is calculated as the expected value of the squared deviation of X from its mean, E(X).

In other words, Var(X) = E((X - E(X))^2). A high variance indicates that the possible outcomes of a random process are spread out, while a low variance indicates that the outcomes are clustered around the expected value.

In conclusion, expectation represents the average outcome of a random process, while variance quantifies how spread out the possible outcomes are from the average.

6. Explain the Central limit theorem.

The central limit theorem is a fundamental principle in statistics that states that the distribution of the sum of a large number of independent, identically distributed random variables approaches a normal distribution, regardless of the distribution of the individual variables. In other words, if you have a large number of random variables that have similar characteristics and you sum them up, the sum will have a normal distribution.

The central limit theorem is important because many real-world processes can be modeled as the sum of many independent, small random variables. For example, the height of a stack of cards is the sum of the height of each individual card, and the time it takes for a car to travel a certain distance is the sum of the time it takes for each individual segment of the trip.

The central limit theorem has many practical applications, such as in hypothesis testing, where it is used to calculate the probability of observing a particular value if the underlying distribution is normal. It also plays a crucial role in statistical inference, where it is used to estimate the parameters of a population based on a sample.

Mathematically, the central limit theorem states that if X1, X2, ..., Xn are independent, identically distributed random variables with mean μ and standard deviation σ, then the sum of these variables, Y = X1 + X2 + ... + Xn, will have a normal distribution with mean μ_n and standard deviation σ/√n. As n becomes larger, the distribution of Y will approach a normal distribution with mean μ_n and standard deviation σ/√n, regardless of the shape of the original distribution of X1, X2, ..., Xn.

The central limit theorem can be written as:

lim_n→∞ P(Y) = 1/√2πσ/√n * e^(-(Y-μ_n)²/(2σ/√n)²)

where P(Y) is the probability density function of Y.

The central limit theorem is important because it provides a way to approximate the distribution of the sum of many independent random variables, even if the individual variables do not have a normal distribution. This makes it possible to make predictions and draw conclusions about real-world processes, such as the height of a stack of cards or the time it takes for a car to travel a certain distance, based on the distributions of the individual components.

Overall, the central limit theorem is a fundamental concept in statistics that provides a basis for understanding the behavior of random variables and their distributions, and helps us make predictions and draw conclusions based on our data.

7. KL divergence, other divergences.

KL divergence, also known as Kullback-Leibler divergence or relative entropy, is a measure of the difference between two probability distributions. It is a non-symmetric measure that quantifies the amount of information lost when approximating one distribution with another. KL divergence is commonly used in machine learning and information theory to evaluate the quality of approximations and to determine the distance between distributions.

Some other divergences that are used in probability and statistics include:

1. Jensen-Shannon divergence: This is a symmetric version of KL divergence that measures the similarity between two distributions. It is a popular choice for comparing distributions because it is always positive and provides a meaningful measure of similarity even if the distributions are different.

2. Hellinger distance: This is another symmetric measure of the difference between two distributions, defined as the square root of the sum of squared differences between the distributions.

3. Total variation distance: This measures the difference between two probability distributions by computing the maximum difference between the cumulative distribution functions of the two distributions.

4. Wasserstein distance: Also known as the Earth Mover's distance, this measures the distance between two distributions based on the cost of transforming one distribution into the other.

Each of these divergences has its own strengths and weaknesses, and the choice of which divergence to use depends on the specific problem being analyzed and the goals of the analysis. However, KL divergence is one of the most widely used divergences in machine learning and information theory because it provides a flexible and intuitive way to compare the distance between distributions.

8. Tell me more about the Kolmogorov complexity.

Kolmogorov complexity is a measure of the amount of information or randomness contained in a string of data. It is a theoretical concept that quantifies the minimum amount of information required to generate a particular string. In other words, it is the length of the shortest possible computer program that can generate the string.

The Kolmogorov complexity of a string cannot be computed exactly because it involves finding the shortest possible program to generate the string, which is an undecidable problem. However, it can be approximated using algorithmic information theory, which is a branch of computer science that studies the theory of algorithms and complexity.

Kolmogorov complexity has important applications in many fields, including information theory, cryptography, and data compression. For example, in data compression, it can be used to determine the amount of information that can be removed from a data string without losing any important information. In cryptography, it can be used to measure the randomness of a key, which is an important factor in the security of cryptographic algorithms.

Kolmogorov complexity provides a way to quantify the amount of information in a string and provides a framework for understanding the randomness of data. It is a fundamental concept in the study of information and complexity, and its applications continue to expand as researchers find new ways to use it in various fields.

9. Can you give me an example of that?

Sure! Here's a simple example to illustrate Kolmogorov complexity:

Suppose we have the string "hello world". The shortest computer program that can generate this string is simply a program that outputs the string "hello world". The length of this program would be the number of characters in the string, which is 11. Therefore, the Kolmogorov complexity of the string "hello world" would be 11.

Now let's consider another string, "aaa...aa" (where there are 1000 "a"s). The shortest program that generates this string is a program that outputs "a" 1000 times. This program would have a length proportional to the number of "a"s, which is 1000. Therefore, the Kolmogorov complexity of this string is 1000.

This example illustrates that the Kolmogorov complexity of a string is proportional to the amount of information required to generate it. In the first example, the Kolmogorov complexity is 11 because the shortest program to generate the string is 11 characters long. In the second example, the Kolmogorov complexity is 1000 because the shortest program to generate the string is 1000 characters long.

10. Explain the terms: Jacobian and Hessian

The Jacobian and the Hessian are two important mathematical concepts in multivariable calculus.

The Jacobian is a matrix that contains the partial derivatives of a multivariable function. It represents the local linear approximation of the function at a point. The Jacobian is used in optimization problems and in solving systems of nonlinear equations.

The Hessian is a square matrix that contains the second partial derivatives of a multivariable function. It represents the local curvature of the function at a point. The Hessian is used in optimization problems to determine the nature of the critical points of a function, such as whether the critical point is a maximum, minimum, or saddle point.

In summary, the Jacobian provides information about the linear approximation of a function at a point, while the Hessian provides information about the curvature of a function at a point. Both concepts are important in various areas of mathematics, including optimization and numerical analysis, and play a crucial role in understanding the behavior of functions in multiple dimensions.

11. Explain the concept with a matrix example.

Here's an example to illustrate the Jacobian and the Hessian using matrices:

Consider a multivariable function f(x,y) = x^2 + y^2. The Jacobian of this function is given by the matrix of partial derivatives:

J(x,y) = [df/dx, df/dy] = [2x, 2y]

The Jacobian provides a local linear approximation of the function. For example, at the point (x,y) = (1,1), the Jacobian is [2, 2], which means that the local linear approximation of the function at this point is a straight line with slope 2.

The Hessian of the function is given by the matrix of second partial derivatives:

H(x,y) = [d^2f/dx^2, d^2f/dxdy; d^2f/dydx, d^2f/dy^2] = [2, 0; 0, 2]

The Hessian provides information about the local curvature of the function. For example, in this case, the Hessian is a constant matrix with all entries equal to 2, which means that the local curvature of the function is constant and the function has a constant upward parabolic shape.

In conclusion, the Jacobian and the Hessian are two important mathematical concepts that provide important information about the behavior of functions in multiple dimensions. The Jacobian provides a linear approximation of the function, while the Hessian provides information about the local curvature of the function.

12. What is Gradient descent and SGD?

Gradient descent and Stochastic Gradient Descent (SGD) are optimization algorithms used to find the minimum of a cost or loss function in machine learning.

Gradient Descent is a first-order optimization algorithm that updates the parameters of a model based on the negative gradient of the cost function. The gradient is a vector of the partial derivatives of the cost function with respect to the parameters. The gradient descent algorithm updates the parameters in the direction of the negative gradient, which leads towards the minimum of the cost function.

Stochastic Gradient Descent (SGD) is a variation of gradient descent that updates the parameters of the model based on a randomly selected subset of the data instead of the entire data. This randomization makes the optimization process more efficient and faster, especially when the data is large.

Both Gradient Descent and SGD are widely used optimization algorithms in machine learning, especially in training neural networks. However, SGD is more commonly used in deep learning models because it can handle large datasets and can converge faster to the minimum of the cost function.