# Label and one-hot encoding: three complete examples

Subject: Natural Language Processing

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/nlp/#tee/formulas/f-vectors/u2-label-onehot

Canonical: https://deep-learning-study-room.vercel.app/read/nlp/tee/formulas/f-vectors--u2-label-onehot/

Formulas explained through full worked examples

**Label encoding** assigns each category or word a unique integer ID. **One-hot encoding** assigns a length-

$$
V
$$

 vector with one 1 at the word's vocabulary position and zeros elsewhere.



$$
\operatorname{onehot}(w)_j=\begin{cases}1,&j=\operatorname{index}(w)\\0,&\text{otherwise.}\end{cases}
$$



The index is an ID, not a meaning score. One-hot represents one token/category; a multiword document generally needs multiple token vectors or an aggregation scheme.

**Question. Encode “dog cat dog” using vocabulary [cat,dog,fish], IDs starting at 0. Give label IDs and a one-hot vector for every token.**

#### Given

cat→0, dog→1, fish→2. Vector columns have the same order.

#### To find

The ID sequence and three one-hot rows.

#### Formula and why it applies

Replace each word by its ID for labels; set its vocabulary position to 1 for one-hot.

#### Step 1. Label IDs

dog→1, cat→0, dog→1 → [1,0,1].

#### Step 2. First dog

[0,1,0].

#### Step 3. Cat

[1,0,0].

#### Step 4. Second dog

[0,1,0]. Repeated tokens keep the same representation.

**Answer.** IDs [1,0,1]; one-hot rows [0,1,0], [1,0,0], [0,1,0].

**Question. Encode the city sequence Mumbai, Delhi, Mumbai, Pune using vocabulary [Delhi,Mumbai,Pune] and label IDs starting at 1.**

#### Given

Delhi→1, Mumbai→2, Pune→3.

#### To find

IDs and one-hot rows.

#### Formula and why it applies

One-hot uses three coordinates regardless of whether label IDs begin at 0 or 1.

#### Step 1. IDs

[2,1,2,3].

#### Step 2. Mumbai rows

Each Mumbai → [0,1,0].

#### Step 3. Delhi and Pune

Delhi→[1,0,0]; Pune→[0,0,1].

**Answer.** Rows [0,1,0], [1,0,0], [0,1,0], [0,0,1].

**Question. One-hot encode categories [red,green,blue] for a linear regression with an intercept. Then give dummy coding after dropping blue.**

#### Given

Full column order [red,green,blue]. Blue is the baseline.

#### To find

Full and reduced codes.

#### Formula and why it applies

Full one-hot rows sum to 1. With an intercept this creates a linear dependency; dropping one category column gives k−1 independent dummy indicators.

#### Step 1. Full encoding

| Category | red | green | blue |
| --- | --- | --- | --- |
| red | 1 | 0 | 0 |
| green | 0 | 1 | 0 |
| blue | 0 | 0 | 1 |

#### Step 2. Drop blue column

| Category | red | green |
| --- | --- | --- |
| red | 1 | 0 |
| green | 0 | 1 |
| blue | 0 | 0 |

#### Step 3. Interpret zero row

[0,0] now identifies the baseline blue, not missing data.

**Answer.** Full width 3; reduced dummy width 2. Dropping a column is model-dependent, not a requirement for every neural embedding input.

**Limitations:** one-hot is large and sparse and does not express semantic similarity. Unseen words need a defined unknown-token or vocabulary-update policy. Label IDs are compact but should not be treated as an ordered scalar feature for nominal words.

Label = ID number. One-hot = us ID ke box mein 1, baaki 0.