claude paper

>>>>>>>>>>>>>> Read this paper. I am trying to do something similar, but the methodological steps are a bit abstract. Can you create a step-by-step extraction of the method of analysis the researchers take? be concrete, eg. how did they inspect the embedding layer, etc.

>>>>>>>>>>>>>> Certainly! I'll provide a step-by-step breakdown of the methodological approach taken by the researchers in this paper:

1. Dataset Preparation:
   - Selected a subset of Wikipedia articles in multiple languages (English, Spanish, Arabic, Turkish, Sundanese, and Swahili).
   - Used this dataset for analysis experiments in Sections 3.1 to 3.3.

2. Model Selection:
   - Used multilingual BERT (mBERT) model with 12-layer transformer-based architecture.
   - Extracted representations from the last layer of mBERT.

3. Isotropy Analysis:
   a. Calculated isotropy using two metrics:
      - Cosine Similarity (I_Cos(W)): Randomly sampled 1000 token pairs and computed average cosine similarity.
      - Principal Components (I_PC(W)): Used eigenvectors of the embedding matrix to approximate isotropy.
   b. Visualized embedding distributions using PCA (Principal Component Analysis).

4. Dimension Sensitivity Analysis:
   - Computed the contribution of individual dimensions to cosine similarity between random embeddings.
   - Reported average contribution of top-three dimensions to I_Cos(W).

5. Outlier Analysis:
   - Averaged 10,000 randomly selected representations.
   - Calculated mean and standard deviation (σ) of dimensions' distribution.
   - Considered a dimension as an outlier if its value was at least 3σ larger/smaller than the mean.
   - Visualized average representations, highlighting the 3σ range.

6. Word Frequency Bias Analysis:
   - Sampled 500 sentences from Wikipedia datasets for English, Spanish, and Arabic.
   - Obtained word representations by averaging over all sub-token representations.
   - Used the wordfreq library to determine word frequencies.
   - Visualized word representations colored by frequency using PCA.

7. Isotropy Enhancement:
   - Applied a cluster-based approach to improve isotropy:
     a. Clustered embeddings using k-means algorithm (7 clusters).
     b. Calculated dominant directions for each cluster using PCA.
     c. Nullified 12 dominant directions in each cluster.
   - Evaluated performance on Semantic Textual Similarity (STS) task:
     a. Used multi- and cross-lingual STS datasets.
     b. Represented sentences by averaging all token embeddings.
     c. Computed cosine similarity between sentence representations as the semantic similarity score.
   - Conducted experiments in two settings:
     a. Individual: Applied method to each language separately.
     b. Zero-shot: Used English clusters and directions for other languages.

8. Comparative Analysis:
   - Repeated analyses for XLM-R model and compared results with mBERT.
   - Compared multilingual models with their monolingual counterparts (BERT and RoBERTa).

By following these steps, the researchers were able to analyze the geometry of multilingual embedding spaces, focusing on isotropy, dimension sensitivity, outliers, and frequency bias. They also demonstrated the effects of improving isotropy on downstream task performance.

>>>>>>>>>>>>>> Lay out a step by step of the experiments in Sections 3.1 to 3.3. e.g., be specific on how did they use the dataset as input or whatever.

>>>>>>>>>>>>>> Certainly! I'll provide a detailed step-by-step breakdown of the experiments in Sections 3.1 to 3.3:

Section 3.1: Probing isotropy

1. Dataset preparation:
   - Used a subset of Wikipedia articles in English, Spanish, Arabic, Turkish, Sundanese, and Swahili.

2. Model setup:
   - Used multilingual BERT (mBERT) model.
   - Extracted representations from the last layer of mBERT.

3. Isotropy calculation:
   a. Cosine Similarity metric (I_Cos(W)):
      - Randomly sampled 1000 token pairs from the dataset.
      - Computed cosine similarity between each pair of token embeddings.
      - Calculated the average of these 1000 cosine similarities.

   b. Principal Components metric (I_PC(W)):
      - Constructed the embedding matrix W using all token embeddings from the dataset.
      - Computed eigenvectors U of the embedding matrix.
      - Calculated F(u) for each eigenvector u using the formula in Equation 2.
      - Computed I_PC(W) as the ratio of min F(u) to max F(u).

4. Visualization:
   - Applied PCA to the token embeddings.
   - Plotted the first two principal components to visualize the embedding distribution.

Section 3.2: Sensitivity to Rogue Dimensions

1. Dataset and model setup:
   - Used the same Wikipedia subset and mBERT model as in Section 3.1.

2. Dimension contribution analysis:
   - Randomly sampled 1000 token pairs from the dataset.
   - For each pair (x, y):
     - Computed the contribution of each dimension i as CC_i = x_i * y_i / (||x|| * ||y||).
     - Summed these contributions to get the total cosine similarity.
   - Averaged the contributions across all 1000 pairs.

3. Reporting:
   - Identified the top three dimensions with the highest average contributions.
   - Reported these contributions along with the overall I_Cos(W) value.

Section 3.3: Outlier Dimensions

1. Dataset and model setup:
   - Used the same Wikipedia subset and mBERT model as before.

2. Representation averaging:
   - Randomly selected 10,000 tokens from the dataset.
   - Extracted their embeddings using mBERT.
   - Computed the average embedding across these 10,000 tokens.

3. Outlier detection:
   - Calculated the mean and standard deviation (σ) across all dimensions of the average embedding.
   - Identified dimensions as outliers if their value was at least 3σ larger or smaller than the mean.

4. Visualization:
   - Plotted the average embedding vector.
   - Highlighted the ±3σ range around the mean as a shaded area.
   - Marked any dimensions exceeding this range as outliers.

5. Comparative analysis:
   - Repeated the process for both mBERT and monolingual BERT.
   - Compared the presence or absence of outliers between the two models.

In each of these experiments, the researchers used the Wikipedia dataset as a source of text input for the mBERT model. They then analyzed the resulting embeddings to investigate different aspects of the multilingual embedding space geometry, focusing on isotropy, dimension sensitivity, and outliers.

>>>>>>>>>>>>>> Can you write an example code that can reproduce section 3.1, using the model and code from huggingface?

>>>>>>>>>>>>>> Certainly! Here's an example Python code that reproduces the experiments in Section 3.1 using the Hugging Face transformers library. This code calculates the isotropy metrics and visualizes the embedding distribution for mBERT.

python
import torch
from transformers import BertTokenizer, BertModel
import numpy as np
from sklearn.decomposition import PCA
import matplotlib.pyplot as plt
from tqdm import tqdm

# Load mBERT model and tokenizer
model_name = 'bert-base-multilingual-cased'
tokenizer = BertTokenizer.from_pretrained(model_name)
model = BertModel.from_pretrained(model_name)
model.eval()

# Function to get embeddings
def get_embeddings(text, layer=-1):
    inputs = tokenizer(text, return_tensors="pt", padding=True, truncation=True)
    with torch.no_grad():
        outputs = model(**inputs)
    return outputs.last_hidden_state[0, 1:-1, :].numpy()  # Exclude [CLS] and [SEP]

# Load your Wikipedia subset (replace with your actual data loading)
wiki_texts = [
    "This is an English sentence.",
    "Esta es una oración en español.",
    "هذه جملة باللغة العربية.",
    "Bu bir Türkçe cümledir.",
    "Ieu kalimat dina basa Sunda.",
    "Hii ni sentensi ya Kiswahili."
]

# Collect embeddings
all_embeddings = []
for text in wiki_texts:
    all_embeddings.extend(get_embeddings(text))
all_embeddings = np.array(all_embeddings)

# Calculate I_Cos(W)
def cosine_similarity(a, b):
    return np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b))

n_samples = 1000
i_cos_w = 0
for _ in range(n_samples):
    idx1, idx2 = np.random.choice(len(all_embeddings), 2, replace=False)
    i_cos_w += cosine_similarity(all_embeddings[idx1], all_embeddings[idx2])
i_cos_w /= n_samples

print(f"I_Cos(W): {i_cos_w}")

# Calculate I_PC(W)
def calculate_i_pc_w(embeddings):
    cov = np.cov(embeddings.T)
    eigenvalues, eigenvectors = np.linalg.eig(cov)

    def f(u):
        return np.sum(np.exp(np.dot(embeddings, u)))

    f_values = [f(eigenvectors[:, i]) for i in range(len(eigenvalues))]
    return np.min(f_values) / np.max(f_values)

i_pc_w = calculate_i_pc_w(all_embeddings)
print(f"I_PC(W): {i_pc_w}")

# Visualize embedding distribution
pca = PCA(n_components=2)
embeddings_2d = pca.fit_transform(all_embeddings)

plt.figure(figsize=(10, 8))
plt.scatter(embeddings_2d[:, 0], embeddings_2d[:, 1], alpha=0.5)
plt.title("mBERT Embedding Distribution")
plt.xlabel("First Principal Component")
plt.ylabel("Second Principal Component")
plt.show()


This code does the following:

1. Loads the mBERT model and tokenizer from Hugging Face.
2. Defines a function to get embeddings for input text.
3. Uses a small sample of multilingual sentences (replace with your actual Wikipedia subset).
4. Collects embeddings for all tokens in the dataset.
5. Calculates I_Cos(W) by sampling random pairs of embeddings and computing their average cosine similarity.
6. Calculates I_PC(W) using the method described in the paper.
7. Visualizes the embedding distribution using PCA.

To run this code, you'll need to install the required libraries:


pip install torch transformers numpy scikit-learn matplotlib tqdm


Note that this is a simplified version of the experiment. For a more comprehensive analysis:
- Use a larger subset of Wikipedia data in multiple languages.
- Increase the number of samples for I_Cos(W) calculation.
- Implement error handling and logging.
- Optimize the code for larger datasets (e.g., batch processing, using GPU if available).

Also, remember that the results might slightly differ from those in the paper due to potential differences in the exact dataset used and random sampling.