Upload README.md

README.md

@@ -11,4 +11,49 @@ license: mit
 short_description: 'This project implements a Generative Adversarial Network '
 ---
 
-Check out the configuration reference at https://huggingface.co/docs/hub/spaces-config-reference
+## **GenDigit**
+
+This project implements a **Generative Adversarial Network (GAN)** that generates images of handwritten digits using the **MNIST dataset**.
+The network consists of two primary components:
+
+- **Generator (G):** A neural network that generates fake images based on random noise and the desired label (digit).
+- **Discriminator (D):** A neural network that evaluates whether an image is real (from the dataset) or fake (generated by the generator).
+
+
+### **Generator (G) Loss Function**:
+The Generator tries to minimize the **binary cross-entropy loss** by generating images that fool the Discriminator. The loss for the Generator is:
+
+$$
+ L_G = -\mathbb{E}_{z}[ \log D(G(z)) ]
+$$
+
+Where:
+- $$ z $$ is the random noise input to the Generator.
+- $$ G(z) $$ is the generated image.
+- $$ D(G(z)) $$ is the Discriminator's probability that the generated image is real.
+
+The Generator is trained to minimize this loss by improving its ability to generate realistic images.
+
+### **Discriminator (D) Loss Function**:
+The Discriminator tries to distinguish between real and fake images. Its loss consists of two parts:
+1. **Loss for real images**:
+   $$
+   L_{D_{\text{real}}} = -\mathbb{E}_{x_{\text{real}}}[ \log D(x_{\text{real}}) ]
+   $$
+2. **Loss for fake images**:
+   $$
+   L_{D_{\text{fake}}} = -\mathbb{E}_{z}[ \log (1 - D(G(z))) ]
+   $$
+
+Where:
+- $$ x_{\text{real}} $$ is a real image from the dataset.
+- $$ D(x_{\text{real}}) $$ is the Discriminator's prediction for a real image.
+- $$ G(z) $$ is the fake image generated by the Generator.
+
+The total loss for the Discriminator is:
+
+$$
+L_D = L_{D_{\text{real}}} + L_{D_{\text{fake}}}
+$$
+
+The Discriminator is trained to minimize this loss by improving its ability to classify real and fake images correctly.