**Practical breakdown** of embeddings: how they work, how they’re trained, and concrete examples you can actually implement.

---

# 🧠 What embeddings actually are (intuitively + formally)

![Image](images/a.jpg)

![Image](images/b.jpg)

![Image](images/c.jpg)

![Image](images/d.jpg)

![Image](images/e.jpg)

![Image](images/f.jpg)

At the core:

* An embedding is a **function**
  [
  f(x) \rightarrow \mathbb{R}^d
  ]

* It maps raw data (text, image, etc.) into a **dense vector**.

Key property:

* **Semantic structure is encoded as geometry**

  * Similar things → vectors close together
  * Different things → far apart

This is the core idea behind vector search, RAG, clustering, etc. ([ibm.com][1])

---

# ⚙️ Why embeddings work (the underlying theory)

### 1. Distributional hypothesis

> “You shall know a word by the company it keeps”

* Words appearing in similar contexts → similar vectors
* This is the **foundation of Word2Vec, GloVe, BERT, etc.** ([ibm.com][2])

---

### 2. Geometry encodes meaning

Classic example:

```
king - man + woman ≈ queen
```

This works because relationships become **linear directions in vector space**.

---

### 3. Dense vs sparse representations

| Method     | Problem            |
| ---------- | ------------------ |
| One-hot    | huge, no meaning   |
| TF-IDF     | frequency only     |
| Embeddings | compact + semantic |

Embeddings are **low-dimensional but information-rich** representations. ([GeeksforGeeks][3])

---

# 🏗️ How embeddings are trained (core methods)

## 1. Prediction-based models (most important)

### Word2Vec (classic foundation)

![Image](images/a1.jpg)

![Image](images/b1.jpg)

![Image](images/c1.jpg)

![Image](images/d1.jpg)

![Image](images/e1.jpg)

![Image](images/f1.jpg)

Two main training strategies:

### (a) Skip-gram

Predict context from a word:

[
P(context \mid word)
]

### (b) CBOW

Predict word from context:

[
P(word \mid context)
]

Mechanism:

* Input = one-hot vector
* Hidden layer = embedding
* Train via gradient descent

👉 The embedding is literally the **weights of the hidden layer** ([Medium][4])

---

### Example (Skip-gram training loop)

```python
# pseudo-code
for word in corpus:
    context = get_context_window(word)
    loss = -log P(context | word)
    update_weights()
```

---

## 2. Matrix factorization (GloVe)

Instead of prediction:

* Build co-occurrence matrix
* Factorize it into lower dimensions

Captures **global statistics**, not just local context. ([Medium][5])

---

## 3. Neural embedding layers (modern approach)

Used in:

* Transformers (BERT, GPT)
* Recommender systems

Mechanism:

* Embedding = **lookup table**
* Trained jointly with model

```python
embedding = torch.nn.Embedding(vocab_size, dim)
vector = embedding(token_id)
```

---

## 4. Contrastive learning (modern SOTA)

Used in:

* sentence embeddings
* CLIP (image-text)
* OpenAI embeddings

Core idea:

[
\text{similar pairs} \rightarrow \text{closer}
]
[
\text{different pairs} \rightarrow \text{farther}
]

Loss function:

[
\mathcal{L} = -\log \frac{e^{sim(x_i, x_j)}}{\sum_k e^{sim(x_i, x_k)}}
]

---

# 🔬 How modern embeddings (LLMs) differ

Older:

* static embeddings (Word2Vec)

Modern:

* **contextual embeddings**

Example:

* “bank” (river vs finance) → different vectors

This is why models like BERT/GPT outperform Word2Vec.

---

# 🧪 Practical training examples

## Example 1 — Train Word2Vec (Gensim)

```python
from gensim.models import Word2Vec

sentences = [["cat", "sat", "mat"], ["dog", "sat", "floor"]]

model = Word2Vec(sentences, vector_size=100, window=5, min_count=1)

vector = model.wv["cat"]
```

---

## Example 2 — Train embeddings in PyTorch

```python
import torch
import torch.nn as nn

embedding = nn.Embedding(10000, 128)  # vocab, dim

input_ids = torch.tensor([1, 5, 23])
vectors = embedding(input_ids)
```

---

## Example 3 — Train contrastive embeddings

```python
# pseudo
anchor = model(text1)
positive = model(text2)
negative = model(text3)

loss = contrastive_loss(anchor, positive, negative)
```

---

## Example 4 — PCA reduction (your earlier question)

```python
from sklearn.decomposition import PCA

pca = PCA(n_components=256)
X_reduced = pca.fit_transform(X)
```

---

# 📊 Types of embeddings

| Type       | Example         |
| ---------- | --------------- |
| Word       | Word2Vec, GloVe |
| Sentence   | SBERT           |
| Document   | Doc2Vec         |
| Image      | CLIP            |
| Graph      | Node2Vec        |
| Multimodal | CLIP, Gemini    |

---

# 🧩 Key properties you should care about (engineering perspective)

### 1. Dimensionality

* Typical: 128–1536
* Tradeoff: memory vs accuracy

---

### 2. Distance metric

* cosine similarity (most common)
* dot product
* Euclidean

---

### 3. Normalization

Critical for:

* search quality
* clustering

---

### 4. Training data distribution

Embeddings are only as good as:

* corpus size
* domain relevance

---

# ⚠️ Common pitfalls (important)

### ❌ Mixing embedding spaces

* embeddings from different models are **not compatible**

---

### ❌ Assuming linear compression is harmless

* PCA can distort semantic relationships

---

### ❌ Ignoring normalization

* cosine similarity breaks without it

---

### ❌ Using embeddings without evaluation

Always test:

* retrieval accuracy
* clustering quality

---

# 🧠 Mental model (most useful takeaway)

Think of embeddings as:

> A learned coordinate system where **meaning = position**

Training = learning that coordinate system so that:

* similar things cluster
* relationships become directions

---


[1]: https://www.ibm.com/think/topics/vector-embedding?utm_source=chatgpt.com "What is Vector Embedding? | IBM"
[2]: https://www.ibm.com/think/topics/word-embeddings?utm_source=chatgpt.com "What Are Word Embeddings? | IBM"
[3]: https://www.geeksforgeeks.org/nlp/word-embeddings-in-nlp/?utm_source=chatgpt.com "Word Embeddings in NLP"
[4]: https://medium.com/%40manansuri/a-dummys-guide-to-word2vec-456444f3c673?utm_source=chatgpt.com "A Dummy's Guide to Word2Vec - Medium"
[5]: https://medium.com/%40neri.vvo/word-embedding-a-powerful-tool-word2vec-glove-fasttext-dd6e2171d5?utm_source=chatgpt.com "Word Embedding Explained — Word2Vec GloVe, FastText"

