Here is a detailed pseudocode and breakdown of the algorithm presented in the Wave Network paper, covering each essential element for implementation. I'll include equations, observations, and insights drawn from the methodology and experiments provided in the paper. Following these provide me the implementation code in python.

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### Pseudocode for Implementing the Wave Network

#### Assumptions and Dependencies
1. **Complex Vector Representation**: Each token in the input sequence is represented by a complex vector composed of:
   - **Magnitude Vector (G)**: Represents global semantics.
   - **Phase Vector (α)**: Captures relationships between tokens and the global semantic context.

2. **Wave Operations**: 
   - **Interference** and **Modulation** are two types of updates applied to the complex vector representations.

3. **Libraries/Dependencies**:
   - Requires a deep learning library such as PyTorch or TensorFlow.
   - Requires linear algebra and complex number support (available in most scientific computing libraries).

4. **Data Requirements**: Input sequences are tokenized and embedded in a 768-dimensional space (similar to BERT embeddings).

5. **Training Parameters**:
   - Learning rate and batch size (as specified in experiments).
   - Use gradient-based optimization (e.g., Adam optimizer).

### Algorithm Structure

#### 1. **Initialization and Token Embedding**

```pseudo
Initialize Wave_Network(model_type="Wave", embed_dim=768, init_method="random")

Procedure Initialize_Tokens(input_text)
    tokens ← Tokenize(input_text)
    embeddings ← Random_Embeddings(tokens, dimensions=embed_dim)
    Return embeddings
End Procedure
```

- **Input**: Text is tokenized, and each token is assigned a 768-dimensional embedding.
- **Embedding Initialization**: Embeddings are initialized randomly using a method such as `torch.nn.Embedding` in PyTorch.

#### 2. **Calculate Global Semantic Vector**

**Equation for Global Semantic Vector \( G \)**:

\[
G_k = \sqrt{\sum_{j=1}^{n} w_{j,k}^2}
\]

Where:
- \( G_k \): Global semantic component along the \( k \)-th dimension.
- \( w_{j,k} \): Token embedding component for the \( j \)-th token along the \( k \)-th dimension.

```pseudo
Procedure Calculate_Global_Semantics(embeddings)
    G ← Vector of zeros with size=embed_dim
    For each dimension k in 1 to embed_dim
        G_k ← √(sum of square of embeddings[:, k])
    End For
    Return G
End Procedure
```

- **Output**: `G` is a vector of global semantic values across all dimensions of the token embeddings.

#### 3. **Calculate Local Semantic Phase Vector for Each Token**

**Equation for Phase Vector \( α \)**:

\[
\alpha_{j,k} = \arctan2\left(\sqrt{1 - \left(\frac{w_{j,k}}{G_k}\right)^2}, \frac{w_{j,k}}{G_k}\right)
\]

Where:
- \( \alpha_{j,k} \): Phase for the \( j \)-th token along the \( k \)-th dimension.
- \( G_k \): Global semantics for dimension \( k \).

```pseudo
Procedure Calculate_Phase_Vector(embeddings, G)
    Phase_Matrix ← Matrix of zeros with size=(num_tokens, embed_dim)
    For each token j and each dimension k
        Phase_Matrix[j, k] ← arctan2(√(1 - (embeddings[j, k] / G[k])^2), embeddings[j, k] / G[k])
    End For
    Return Phase_Matrix
End Procedure
```

- **Output**: Phase matrix storing the phase \( \alpha \) for each token across all dimensions.

#### 4. **Construct Complex Vector Representation**

Construct the complex vector representation \( Z_j \) for each token \( j \):

\[
Z_j = G \cdot e^{i \cdot \alpha_j}
\]

```pseudo
Procedure Construct_Complex_Vector(G, Phase_Matrix)
    Complex_Vectors ← Matrix of zeros with size=(num_tokens, embed_dim) (complex type)
    For each token j
        Complex_Vectors[j] ← G * exp(i * Phase_Matrix[j])
    End For
    Return Complex_Vectors
End Procedure
```

- **Output**: `Complex_Vectors`, the representation of each token in complex vector form.

#### 5. **Wave Interference Operation**

Perform wave interference by adding two complex vector variants \( Z_j \) and \( Z_j' \):

\[
\text{Interference}_j = Z_j + Z_j'
\]

Where:
- **Constructive or Destructive Interference**: Determined by phase difference between \( Z_j \) and \( Z_j' \).

```pseudo
Procedure Wave_Interference(Z, Z_prime)
    Interference_Vector ← Z + Z_prime
    Return Interference_Vector
End Procedure
```

- **Output**: Interference result of combining two complex vectors for each token.

#### 6. **Wave Modulation Operation**

Perform wave modulation by multiplying two complex vector variants \( Z_j \) and \( Z_j' \):

\[
\text{Modulation}_j = Z_j \times Z_j'
\]

Where:
- **Amplitude Modulation**: Affects the overall magnitude.
- **Phase Modulation**: Alters the phase by the sum of phases of \( Z_j \) and \( Z_j' \).

```pseudo
Procedure Wave_Modulation(Z, Z_prime)
    Modulation_Vector ← Z * Z_prime
    Return Modulation_Vector
End Procedure
```

- **Output**: Modulation result, which modulates both the amplitude and phase of each token’s representation.

#### 7. **Training the Model**

This procedure covers the training process, updating embeddings using gradient descent.

```pseudo
Procedure Train_Wave_Network(data_loader, epochs, learning_rate)
    Initialize optimizer
    For each epoch
        For each batch of data in data_loader
            embeddings ← Initialize_Tokens(batch_input)
            G ← Calculate_Global_Semantics(embeddings)
            Phase_Matrix ← Calculate_Phase_Vector(embeddings, G)
            Complex_Vectors ← Construct_Complex_Vector(G, Phase_Matrix)
            
            # Apply wave operations
            Interference_Vector ← Wave_Interference(Complex_Vectors, Complex_Vectors)
            Modulation_Vector ← Wave_Modulation(Complex_Vectors, Complex_Vectors)

            # Compute loss based on task (e.g., classification)
            loss ← Compute_Loss(Interference_Vector, Modulation_Vector, labels)
            loss.backward()
            optimizer.step()
    End For
End Procedure
```

- **Loss Computation**: Customize `Compute_Loss` according to the NLP task, such as classification or language modeling.
- **Output**: Trained model with optimized token representations.

---

### Insights and Observations

1. **Efficiency**: The Wave Network reduces memory and computation by representing both local and global semantics in a single-layer, complex vector format. Experimental results showed memory savings of up to 77.34% and time savings of 85.62% compared to BERT.

2. **Signal Processing Analogy**: By interpreting tokens as signals, the model efficiently encodes relationships in complex vectors, using physical wave properties to process these signals, enabling operations like interference and modulation.

3. **Resource Savings for NLP Tasks**: The compact architecture means it could enable NLP capabilities in hardware-limited devices, such as mobile or IoT devices, where traditional models are infeasible due to resource constraints.

4. **Limitations**: The model’s accuracy may not match that of large models in all tasks, especially those requiring extensive context across long sequences. Additionally, the approach may be sensitive to initialization due to reliance on random embeddings, suggesting potential benefits from pre-trained embeddings in some settings.

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This pseudocode should provide a comprehensive guide for implementing the Wave Network, covering all critical stages from initialization to the training process. Let me know if you need further clarification on any part.