Forward Propagation

For any layer $j$:

Linear Step

Calculate pre activation term

$$z^{(j)} = \Theta^{(j-1)} a^{(j-1)}$$

Activation Step

Apply activation

$$a^{(j)} = g(z^{(j)})$$

This process is repeated until the output layer.

---

From Scalar Equations to Vector Form

graph LR

%% Input Layer
    subgraph Input Layer
        x1(((x1)))
        x2(((x2)))
        x3(((x3)))
    end

%% Hidden Layer 1
    subgraph Hidden Layer 1
        a1{a1}
        a2{a2}
        a3{a3}
    end



%% Output Layer
    subgraph Output Layer
        y(((hθx)))
    end

%% Connections: Input → Hidden 1
    x1 --> a1
    x1 --> a2
    x1 --> a3
    
    x2 --> a1
    x2 --> a2
    x2 --> a3
    
    x3 --> a1
    x3 --> a2
    x3 --> a3

%% Connections: Hidden 2 → Output
    a1 --> y
    a2 --> y
    a3 --> y

Previously, we wrote each neuron separately.

For the hidden layer:

$$a^{(2)}_1 = g\left(\Theta^{(1)}_{10}x_0 + \Theta^{(1)}_{11}x_1 + \Theta^{(1)}_{12}x_2 + \Theta^{(1)}_{13}x_3\right)$$ $$a^{(2)}_2 = g\left(\Theta^{(1)}_{20}x_0 + \Theta^{(1)}_{21}x_1 + \Theta^{(1)}_{22}x_2 + \Theta^{(1)}_{23}x_3\right)$$ $$a^{(2)}_3 = g\left(\Theta^{(1)}_{30}x_0 + \Theta^{(1)}_{31}x_1 + \Theta^{(1)}_{32}x_2 + \Theta^{(1)}_{33}x_3\right)$$

Where

• Superscript $a^{(2)}$ indicates layer 2 (hidden layer)
• $g(\cdot)$ is the sigmoid function

Final hypothesis is:

$$h_\Theta(x) = a^{(3)}_1$$

Where

$$a^{(3)}_1 = g\left(\Theta^{(2)}_{10}a^{(2)}_0 + \Theta^{(2)}_{11}a^{(2)}_1 + \Theta^{(2)}_{12}a^{(2)}_2 + \Theta^{(2)}_{13}a^{(2)}_3\right)$$

It does not scale. So we need to vectorize it for more complex use cases.

---

Pre Activation Term $z_k^{(j)}$

Intermediate Variable contain the weighted sum before activation:

Suppose

$$z_1^{(2)} = \Theta^{(1)}_{10}x_0 + \Theta^{(1)}_{11}x_1 + \Theta^{(1)}_{12}x_2 + \Theta^{(1)}_{13}x_3$$ $$z_2^{(2)} = g\left(\Theta^{(1)}_{20}x_0 + \Theta^{(1)}_{21}x_1 + \Theta^{(1)}_{22}x_2 + \Theta^{(1)}_{23}x_3\right)$$ $$z_3^{(2)} = g\left(\Theta^{(1)}_{20}x_0 + \Theta^{(1)}_{21}x_1 + \Theta^{(1)}_{22}x_2 + \Theta^{(1)}_{23}x_3\right)$$

🧠 Generalized preactivation term

$$z^{(j)}_k = \Theta^{(j-1)}_{k,0} a^{(j-1)}_0 + \Theta^{(j-1)}_{k,1} a^{(j-1)}_1 + \dots + \Theta^{(j-1)}_{k,n} a^{(j-1)}_n$$

Then

$$a^{(2)}_1 = g(z_1^{(2)})$$ $$a^{(2)}_2 = g(z_2^{(2)})$$ $$a^{(2)}_3 = g(z_3^{(2)})$$

🧠 Generalized Activation

$$a^{(j)}_k = g\left(z^{(j)}_k\right)$$

This separates:

• Linear computation
• Nonlinear activation

Vector Representation

Input layer:

$$x = \begin{bmatrix} x_0 \\ x_1 \\ \vdots \\ x_n \end{bmatrix}$$

Where $x_0$ = 1

Let

$$a^{(1)} = x$$

Weighted sum vector:

$$z^{(j)} = \begin{bmatrix} z^{(j)}_1 \\ z^{(j)}_2 \\ \vdots \\ z^{(j)}_{s_j} \end{bmatrix}$$

Where:

$$s_j = \text{number of units in layer } j$$

We can calculate $z$ as

$$z^{(2)} = \Theta^{(1)}x$$

Since x = $a^{(1)}$, so we can rewrite it:

$$z^{(2)} = \Theta^{(1)} a^{(1)}$$

🧠 Generalized Vectorized Preactivation term:

$$z^{(j)} = \Theta^{(j-1)} a^{(j-1)}$$

Where Vector Dimensions:

$$\Theta^{(j-1)} \in \mathbb{R}^{s_j \times (n+1)}$$ $$a^{(j-1)} \in \mathbb{R}^{(n+1) \times 1}$$ $$z^{(j)} \in \mathbb{R}^{s_j \times 1}$$

Activation Function

Since

$$a^{(2)}_2 = g(z_2^{(2)})$$

Generalized Activation Function:

$$a^{(j)} = g\left(z^{(j)}\right)$$

If using sigmoid:

$$g(z) = \frac{1}{1 + e^{-z}}$$

Add Bias Unit

After computing $a^{(2)}$, add:

$$a_0^{(2)} = 1$$

Now:

$$a^{(j)} = \begin{bmatrix} 1 \\ a^{(j)}_1 \\ \vdots \\ a^{(j)}_{s_j} \end{bmatrix}$$

Output Layer

Repeat the same process:

calculate Linear Term z

$$z^{(j+1)} = \Theta^{(j)} a^{(j)}$$

Apply activation sigmoid of z

$$a^{(j+1)} = g\left(z^{(j+1)}\right)$$

Final hypothesis:

$$h_\Theta(x) = a^{(3)} = g(z^{(3)})$$

🧠 Generalized Hypothesis

$$h_\Theta(x) = a^{(j+1)} = g\left(z^{(j+1)}\right)$$

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The Big Picture

Each layer performs:

$$\text{Linear transformation} \quad z = \Theta a$$

followed by

$$\text{Nonlinearity} \quad a = g(z)$$

Stacking these layers allows neural networks to represent complex nonlinear functions.

Intuition

If we remove the hidden layer, the model becomes logistic regression:

$$h(x) = g(\theta^T x)$$

With hidden layers, the network instead uses learned features:

$$a_1, a_2, a_3$$

These are:

• Computed by the hidden layer
• Learned from data
• Controlled by parameters $\Theta^{(1)}$

So a neural network is:

Logistic regression on learned features.