(quant)
What do we mean when we say a matrix is ‘sparse’
What do we mean when we say a matrix has high cardinality
What is Singular Value Decomposition and how is it different from Principal Component Analysis (PCA)?
How can SVD be used to reduce the dimensionality of a dataset?
Explain how matrix factorization is utilized in recommendation systems.
What are the challenges in implementing matrix factorization techniques in large datasets?
How do linear transformations relate to machine learning models?
What is a basis in linear algebra and how does the concept apply in the context of machine learning?
Explain the concept of orthogonality and its relevance in regression analysis.
How is the method of least squares used in training machine learning models?
In the context of neural networks, how are linear algebra operations applied during forward and backward propagation?
Discuss how linear algebra is applied in natural language processing, particularly in the context of word embeddings.
Can you explain the concept of tensor decomposition and its application in machine learning?
How do you use linear algebra for optimizing machine learning algorithms, particularly in gradient descent?
Why can’t you multiply vectors?
https://youtu.be/htYh-Tq7ZBI?si=0lBepSQq_sTP4ENk
Can you explain the Cauchy-Schwarz inequality?
Cauchy-Schwarz inequality
- An upper bound on the inner product between two vectors
- Considered one of the most important and widely used inequalities in mathematics.
\[ |\langle \mathbf{u}, \mathbf{v} \rangle|^2 \leq \langle \mathbf{u}, \mathbf{u} \rangle \cdot \langle \mathbf{v}, \mathbf{v} \rangle \]
What is singular value decomposition?
What is the formula for calculating the covariance matrix of a dataset, given observation vectors
The full formula for the covariance matrix of a dataset without assuming that the mean of the data is zero involves subtracting the mean from each data point before calculating the outer product. For a dataset with observations (data points) each having variables (dimensions), the formula for the covariance matrix is as follows:
Here:
- ( x_i ) is a vector representing the ( i )-th observation in the dataset.
- ( \bar{x} ) is the mean vector of the dataset, where each element of ( \bar{x} ) is the mean of the corresponding variable across all observations.
- ( (x_i - \bar{x}) ) is the deviation of ( x_i ) from the mean.
- ( (x_i - \bar{x})^T ) is the transpose of the deviation vector.
- ( \frac{1}{n - 1} ) is a scaling factor used to get an unbiased estimate of the covariance when working with a sample from a larger population. For the entire population, this would be ( \frac{1}{n} ).
Each element ( C_{jk} ) of the covariance matrix ( C ) is calculated by the following formula:
[ C_{jk} = \frac{1}{n - 1} \sum_{i=1}^{n} (x_{ij} - \bar{x}j)(x{ik} - \bar{x}_k) ]
Where:
- ( x_{ij} ) is the value of the ( j )-th variable in the ( i )-th observation.
- ( x_{ik} ) is the value of the ( k )-th variable in the ( i )-th observation.
- ( \bar{x}_j ) is the mean of the ( j )-th variable across all observations.
- ( \bar{x}_k ) is the mean of the ( k )-th variable across all observations.
This formula ensures that the covariance matrix reflects the true variability and relationships between the different variables in the dataset.