Following this instruction :
1. Strictly adhere to PEP-8 standards, ensuring code is clean, well-structured, and follows Python's style guide for readability.
2. Use appropriate modules like `typing` for type hints, ensuring code clarity, maintainability, and leveraging Python's standard libraries effectively.
3. Focus on time, space, robustness, scalability; write optimized, efficient code that can handle large datasets and scale seamlessly.
4. Implement comprehensive error handling, catching exceptions gracefully, providing meaningful error messages, and ensuring code robustness under unexpected conditions.
5. Ensure the developer agent writes advanced, maintainable Python code, balancing performance and readability while adhering to best practices.

TASK : objective below all api we have develop from scratch using only torch module :

user_input :
1 Model (nn.module) 
2) loss fumnction
3) optimiser
3) adversarical_sample ( batch_size, channel, height, weight)
4) label(batch_size, true_value)


---

### 1. **CutMix**
   - **Purpose**: Mixes two images by cutting a rectangular patch from one image and placing it onto another, with corresponding adjustments to labels.
   - **Equations**:
     \[
     \text{mixed\_image} = M \odot x_A + (1 - M) \odot x_B
     \]
     \[
     \text{mixed\_label} = \lambda \cdot y_A + (1 - \lambda) \cdot y_B
     \]
     where \( M \) is a binary mask for the patch, \( x_A \) and \( x_B \) are input images, \( y_A \) and \( y_B \) are corresponding labels, and \( \lambda \) is a mixing ratio.

### 2. **Cutout**
   - **Purpose**: Masks out a square region in the image, providing robustness by forcing the model to rely on a complete image context.
   - **Equation**:
     \[
     \text{masked\_image} = M \odot x
     \]
     where \( M \) is a binary mask with a square patch of zeros, and \( x \) is the input image.

### 3. **Feature Squeezing**
   - **Purpose**: Reduces the precision of input features by rounding, making small perturbations less impactful.
   - **Equation**:
     \[
     x_{\text{squeezed}} = \text{round}(x, \text{bits})
     \]
     where \( x \) is the input and \(\text{bits}\) indicates precision.

### 4. **Gaussian Augmentation**
   - **Purpose**: Adds Gaussian noise to inputs, increasing robustness by introducing randomness.
   - **Equation**:
     \[
     x_{\text{aug}} = x + \epsilon, \quad \epsilon \sim \mathcal{N}(0, \sigma^2)
     \]
     where \( \sigma \) controls noise level.

### 5. **DefenseGAN / InverseGAN**
   - **Purpose**: Projects inputs onto a GAN-generated manifold, removing noise or adversarial perturbations.
   - **Equation**:
     \[
     x_{\text{projected}} = G(z^*) \quad \text{where} \quad z^* = \arg \min_z \| x - G(z) \|
     \]
     where \( G \) is the generator and \( z \) is optimized for best reconstruction.

### 6. **JPEG Compression**
   - **Purpose**: Compresses inputs via JPEG, removing high-frequency artifacts.
   - **Process**: JPEG compression transforms the image into the frequency domain and discards components below a certain threshold.

### 7. **Label Smoothing**
   - **Purpose**: Reduces overconfidence by making labels less definitive.
   - **Equation**:
     \[
     y_{\text{smooth}} = (1 - \alpha) \cdot y + \frac{\alpha}{K}
     \]
     where \( K \) is the number of classes and \( \alpha \) is the smoothing parameter.

### 8. **Mixup**
   - **Purpose**: Creates mixed inputs by linear interpolation between pairs of inputs.
   - **Equation**:
     \[
     x_{\text{mixed}} = \lambda x_i + (1 - \lambda) x_j
     \]
     \[
     y_{\text{mixed}} = \lambda y_i + (1 - \lambda) y_j
     \]
     where \( \lambda \) is a random mixing coefficient.

### 9. **MP3 Compression**
   - **Purpose**: Compresses audio to remove irrelevant features.
   - **Process**: Converts audio to frequency domain, applying quantization to reduce precision and remove perturbations.

### 10. **PixelDefend**
   - **Purpose**: Projects input to natural image space by moving pixels closer to probable values.
   - **Equation**:
     \[
     x_{\text{defended}} = \arg \min_{x'} D(x', x) + \beta \cdot P(x')
     \]
     where \( D \) is a distance metric and \( P \) represents natural image probability.

### 11. **Resample**
   - **Purpose**: Adjusts the resolution of images, reducing sensitivity to small perturbations.

### 12. **Spatial Smoothing**
   - **Purpose**: Applies local smoothing to reduce high-frequency noise.
   - **Equation**:
     \[
     x_{\text{smoothed}}[i, j] = \frac{1}{|N(i, j)|} \sum_{(m, n) \in N(i, j)} x[m, n]
     \]
     where \( N(i, j) \) is a neighborhood around \((i, j)\).

### 13. **Thermometer Encoding**
   - **Purpose**: Converts each feature into a binary vector.
   - **Equation**:
     \[
     \text{encoded}(x) = \left[\underbrace{1, \dots, 1}_{x \geq t_1}, \underbrace{0, \dots, 0}_{x < t_1}\right]
     \]
     where \( t_i \) are threshold levels.

### 14. **Total Variation Minimization (TotalVarMin)**
   - **Purpose**: Removes noise by minimizing total variation.
   - **Equation**:
     \[
     x_{\text{TV-min}} = \arg \min_x \sum_{i, j} |x[i+1, j] - x[i, j]| + |x[i, j+1] - x[i, j]|
     \]

### 15. **Video Compression**
   - **Purpose**: Compresses video frames to remove perturbations.
   - **Process**: Similar to JPEG but applied to temporal frames.