Lecture 03

Statistics / linear algebra / calculus review

Today’s Topics:

• Today’s Topics:
• 1. Tensors in Deep Learning
• 2. Tensors and PyTorch
• 3. Vectors, Matrices, and Broadcasting
• 4. Probability Basics
• 5. Estimation Methods
• 6. Linear Regression

1. Tensors in Deep Learning

• A tensor is a multidimensional array
• Dimensionality (order) = number of indices
• It is generalization of scalars(order-0 tensor), vectors(order-1 tensor), matrices(order-2 tensor).
• Images are common examples of tensor used as input in deep learning.
• 3D tensor : a single color image (height, width, channels)
• 4D tensor : a batch of images (batch_size, height, width, channels)

2. Tensors and PyTorch

• NumPy vs PyTorch: They have similar syntax but PyTorch adds:
  • GPU support,
  • automatic differentiation,
  • deep learning convenience functions.
• Data types: mappings exist (e.g., NumPy float64 → PyTorch DoubleTensor).
• Matrix multiplication in Pytorch: b.matmul(b), b.dot(b), b @ b .
• Check GPU availability in PyTorch and move tensors between CPU and GPU using b.to(torch.device('cpu')) or b.to(torch.device('cuda:0'))

3. Vectors, Matrices, and Broadcasting

• Notations:
  • x (lower case): vector
  • X (upper case): matrix
  • w: weights (also a vector)
  • b: bias (a constant)
  • z: pre-activation value
• Vectors:
  • A vector is an n-by-1 matrix, where n is the number of row, and 1 is the number of column.
  • In deep learning, we often represent vectors as column vectors.
  • The linear combination (pre-activation value) is written: $z = \mathbf{w}^\top \mathbf{x} + b$, where
  • \[\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix} \in \mathbb{R}^{m \times 1}, \quad \mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_m \end{bmatrix} \in \mathbb{R}^{m \times 1}, \quad z \in \mathbb{R}\]
• Matrices:
  • A matrix is an n-by-m arrays of numbers, where n is the number of row, and m is the number of column.
  • The linear combination is written: $\mathbf{z} = \mathbf{Xw} + b$, where

\(\mathbf{X} = \begin{bmatrix} x^{[1]}_1 & x^{[1]}_2 & \cdots & x^{[1]}_m \\ x^{[2]}_1 & x^{[2]}_2 & \cdots & x^{[2]}_m \\ \vdots & \vdots & \ddots & \vdots \\ x^{[n]}_1 & x^{[n]}_2 & \cdots & x^{[n]}_m \end{bmatrix} \in \mathbb{R}^{n \times m}, \quad \mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_m \end{bmatrix} \in \mathbb{R}^{m \times 1}, \quad \mathbf{z} = \begin{bmatrix} z^{[1]} \\ z^{[2]} \\ \vdots \\ z^{[n]} \end{bmatrix} \in \mathbb{R}^{n \times 1}\)

• Time Complexity of N-by-N matrices multiplication by naive algorithms: $O(n^3)$.
• Broadcasting:
  • The rigorous math formula of linear combination is: $\mathbf{z} = \mathbf{Xw} + \mathbf{1}_n b$.
  • Using NumPy / PyTorch / TensorFlow, broadcasting helps to automatically expands the scalar $b$ into an $n \times 1$ vector.
  • Thus, in deep learning notation, we usually drop $\mathbf{1}_n$ and simply write $\mathbf{z} = \mathbf{Xw} + b$.
• Broadcasting Usage:
  • Be cautious when debugging, since broadcasting expands tensors automatically.
  • Example:

    • # 1 is broadcast to [1, 1, 1] to match the vector
torch.tensor([1, 2, 3]) + 1
      → tensor([2, 3, 4])

    • # [1, 2, 3] is broadcast to [[ 1, 2, 3], [ 1, 2, 3]] to match the matrix
t = torch.tensor([[4, 5, 6], [7, 8, 9]])
t + torch.tensor([1, 2, 3])
      → tensor([[ 5, 7, 9], [ 8, 10, 12]])

4. Probability Basics

• Random Variables and Distributions:
  • Discrete Random Variables: Values from a countable set (e.g. coin flip).
    • Described by: Probability Mass Function (PMF): $P(X = x)$.
    • Example: Bernoulli Distribution $P(X = x) = \theta^x (1-\theta)^{1-x}, \; x \in {0,1}$.
      • Example: fair coin flip ($\theta = 0.5$).
  • Continuous Random Variables: Values from an interval (e.g. a height).
    • Described by Probability Density Function (PDF): $f(x)$.
    • Example: Gaussian (Normal) Distribution $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
      • Called “Normal” because of the Central Limit Theorem.
      • Standard Normal: when $\mu= 0, \sigma= 1$.
• Central Limit Theorem (CLT):
  • Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed (i.i.d.) random variables with mean $\mu$ and finite variance $\sigma^2$.
  • Define the sample mean: $\bar{X}_n = \frac{1}{n}(X_1 + X_2 + \cdots + X_n)$
  • Then we have: $\frac{\bar{X}_n - \mu}{\frac{\sigma}{\sqrt{n}}} \to N(0,1)$ as $n \to \infty$
• Joint, Marginal, and Conditional Probabilities:
  • Joint: $P(A,B)$, probability of two events occurring together.
  • Marginal: $P(A) = \sum_B P(A, B)$, sum of joint probabilities over one variable.
  • Conditional: $P(A \mid B) = \frac{P(A, B)}{P(B)}$, probability of A given B.
• Expectation:
  • Formula:
    • Discrete: $E[X] = \sum_x x P(X = x)$
    • Continuous: $E[X] = \int_{-\infty}^{\infty} x f(x)\, dx$
  • Linearity:
    • $E[aX + b] = aE[X] + b$
    • $E[X_1 + X_2] = E[X_1] + E[X_2]$
  • Expectation of Functions:
    • Discrete:
      • $E[g(X)] = \sum_x g(x) P(X = x)$
      • Example: $X \sim \text{Bernoulli}(\theta), g(X) = X^2$, then $E[g(X)] = 1^2 \theta + 0^2 (1-\theta) = \theta$
    • Continuous:
      • $E[g(X)] = \int_x g(x) f(x)\, dx$
      • Example: $X \sim \text{Uniform}(0,1), \, g(X) = X^2$, then $E[g(X)] = \int_0^1 x^2 dx = \tfrac{1}{3}$
• Variance:
  • Formula: $Var(X) = E[(X - E[X])^2] \equiv E[X^2] - (E[X])^2$
  • Variance of Functions: $Var(g(X)) = E[(g(X) - E[g(X)])^2] \equiv E[g(X)^2] - E[g(X)]^2$
• Covariance:
  • $Cov(X,Y) = E[(X - E[X])(Y - E[Y])]$
  • If $X, Y$ are independent, then $Cov(X,Y) = 0$
  • $Cov(X,X) = Var(X)$
• Correlation:
  • $\rho(X, Y) = \dfrac{\mathrm{Cov}(X,Y)}{\sqrt{Var(X)Var(Y)}}$
    • $\rho = 1$: Perfect positive linear relationship.
    • $\rho = 0$: No linear relationship.
    • $\rho = -1$: Perfect negative linear relationship.
    • Always in the range $[-1, 1]$.
• Bayes’Rule:

  • $P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}$

  • Example: Medical Test:

    • $P(\text{disease} \mid \text{positive test}) = \frac{P(\text{positive test} \mid \text{disease}) \, P(\text{disease})} {P(\text{positive test})} $

5. Estimation Methods

The goal of estimation is to infer unknown parameters $\theta$ from observed data.

• Types of Estimation

  • Point Estimation: Provides a single best guess of the parameter.
  • Interval Estimation: Provides a range of plausible values (e.g., confidence intervals, Bayesian credible intervals).

• Maximum Likelihood Estimation (MLE)

  • Definition:

\[\hat{\theta}_{\text{MAP}} = \arg\max_{\theta} P(\theta \mid \text{data}) = \arg\max_{\theta} \big[ P(\text{data} \mid \theta)\, P(\theta) \big].\]

• Interpretation:
  MLE chooses $\theta$ that makes the observed data most “likely.”
• Log-likelihood:

\[\ell(\theta) = \log L(\theta) = \sum_i \Big[ x_i \log \theta + (1-x_i)\log(1-\theta) \Big]. </d-math>\]

• Example (Bernoulli):
  Suppose we observe $k$ successes in $n$ Bernoulli trials. Then

\[\hat{\theta}_{\text{MLE}} = \frac{k}{n}.\]

• Notes:
  • MLE does not always exist.
  • MLE may not be unique.
  • MLE may not always be admissible.

• Maximum A Posteriori (MAP)

  • Definition:

\[\hat{\theta}{\text{MAP}} = \arg\max{\theta} P(\theta \mid \text{data}) = \arg\max_{\theta} P(\text{data} \mid \theta),P(\theta).\]

• MAP incorporates a prior distribution $P(\theta)$.

• MLE ignores the prior.

• Regularization as MAP

  • Adding a regularization term to MLE is equivalent to doing MAP estimation with a prior:
    • L2 penalty (ridge regression): corresponds to a Gaussian prior on parameters.
    • L1 penalty (lasso regression): corresponds to a Laplace prior on parameters.

Formally:

\[\hat{\theta}_{\text{reg}} = \arg\max_{\theta} \Big[ \log L(\theta) - \lambda R(\theta) \Big] \quad\Longleftrightarrow\quad \hat{\theta}_{\text{MAP}}\]

6. Linear Regression

Linear regression models the relationship between inputs (features) and outputs (responses).

• Model Definition

  \(y = X\beta + \epsilon\)

  • $y$: response variable (dependent variable).
  • $X$: design matrix (independent variables or features).
  • $\beta$: coefficients (parameters to estimate).

  • $\epsilon$: error term, often assumed $\epsilon \sim N(0, \sigma^2)$.

  • Goal: Estimate $\beta$.

• Evaluation Metrics

  • Coefficient of Determination ($R^2$):

\[R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}\]

• Measures the proportion of variance in $y$ explained by the model.

• Mean Squared Error (MSE):

\[MSE = \frac{1}{n} \sum_i (y_i - \hat{y}_i)^2\]

• Mean Absolute Error (MAE):

\[MAE = \frac{1}{n} \sum_i |y_i - \hat{y}_i|\]

• Ordinary Least Squares (OLS)

  • Objective:

\[\hat{\beta}_{\text{OLS}} = \arg\min_{\beta} \|y - X\beta\|^2\]

• Residuals: $e_i = y_i - \hat{y}_i$.

• Closed-form solution:

\[\hat{\beta}_{\text{OLS}} = (X^TX)^{-1}X^Ty\]

• Regularization in Linear Regression

  • Ridge Regression (L2):

\[\hat{\beta}_{\text{ridge}} = \arg\min_{\beta} \|y - X\beta\|^2 + \lambda \|\beta\|_2^2\]

Equivalent MAP interpretation: Gaussian prior $\beta \sim N(0, \frac{\sigma^2}{\lambda})$.

• Lasso Regression (L1):

\[\hat{\beta}_{\text{lasso}} = \arg\min_{\beta} \|y - X\beta\|^2 + \lambda \|\beta\|_1\]

Equivalent MAP interpretation: Laplace prior $\beta \sim \text{Laplace}(0, \frac{\sigma}{\lambda})$.
Encourages sparsity (many coefficients shrink to 0).