---
title: "Dimension of data; curse of dimensionality"
subject: "Data Mining"
chapter: 2
key: "semester_7/data_mining:dimension-curse"
source: https://www.bctengineer.com/q/semester_7/data_mining/dimension-curse
site: BCT Engineering
---

# Dimension of data; curse of dimensionality

Data Mining · Chapter 2: Data Pre-processing

- Asked in: year not given
- Video: [Curse Of Dimensionality Explained with Examples in Hindi ll Machine Learning Course](https://www.youtube.com/watch?v=SMaa3pnQmbg) — 5 Minutes Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/data_mining/dimension-curse
- Whole chapter: https://www.bctengineer.com/md/semester_7/data_mining/chapter-2

## Question

What do you mean by dimension of data? Briefly explain about curse of Dimensionality.

## Short answer

Dimension is the number of attributes per object. The curse of dimensionality means that as dimensions increase, data becomes sparse, distances lose meaning, models overfit, and computation grows, making analysis unreliable.

## Answer

The **dimension** (dimensionality) of a data set is the number of attributes (features) used to describe each object. A table with 4 attributes per record has dimension 4, and each record is a point in 4-dimensional space.

#### Curse of dimensionality
As the number of dimensions increases, data analysis becomes much harder. The term was coined by Richard Bellman.

- **Sparsity**: the volume of the space grows exponentially, so a fixed number of points becomes very sparse. To keep the same density the number of samples must grow exponentially with the dimension.
- **Distance loses meaning**: in high dimensions the distances to the nearest and farthest points become almost equal, so (max − min)/min → 0. Nearest neighbour, clustering and outlier methods based on distance or density become unreliable.
- **Statistical problems**: with few samples and many features, models overfit and give poor accuracy on new data.
- **Computation**: time and memory needed rise sharply with more attributes.
- **Visualisation and interpretation** are difficult beyond 3 dimensions.

*Example*: 10 intervals per attribute give 10 cells in 1-D, 100 cells in 2-D and 10¹⁰ cells in 10-D. 1000 records fill the first densely but leave almost all cells of the last empty.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the dimension of a data set?
   A. Number of records
   B. Number of attributes describing each object
   C. Size in GB
   D. Number of clusters
   Answer: B. A table with 4 attributes per record has dimension 4. (Key point: Dimension)

2. Why does data become sparse in high dimensions?
   A. The data are deleted
   B. Attributes become identical
   C. The volume of the space grows exponentially, so a fixed number of points covers little of it
   D. Distances grow linearly
   Answer: C. To keep density, samples must grow exponentially with dimension. (Key point: Sparsity)

3. What happens to distances in high dimensions?
   A. Nearest points get much closer than others
   B. All distances become 0 exactly
   C. Distances become negative
   D. Nearest and farthest distances become almost equal, so (max - min)/min tends to 0
   Answer: D. Distance-based methods like nearest neighbour become unreliable. (Key point: Distance concentration)

4. With 10 intervals per attribute, how many cells are there in 10-D?
   A. 10^10
   B. 100
   C. 10^5
   D. 1000
   Answer: A. 10^d cells in d dimensions, so 1000 records leave almost all cells empty. (Key point: Cell counting example)

## Related in Data Mining

- [Avoiding curse of dimensionality](https://www.bctengineer.com/question.md?key=semester_7%2Fdata_mining%3Aavoid-curse-dimensionality) — Data Mining, chapter 2
- [Dimensionality reduction techniques](https://www.bctengineer.com/question.md?key=semester_7%2Fdata_mining%3Adimensionality-reduction) — Data Mining, chapter 2
- [OLAP vs OLTP; OLAP operations, fact, dimension](https://www.bctengineer.com/question.md?key=semester_7%2Fdata_mining%3Aolap-oltp-operations-fact-dimension) — Data Mining, chapter 2
- [Define dimension table](https://www.bctengineer.com/question.md?key=semester_7%2Fdata_mining%3Adefine-dimension-table) — Data Mining, chapter 1
- [OLAP vs OLTP; schema, fact, dimension](https://www.bctengineer.com/question.md?key=semester_7%2Fdata_mining%3Aolap-oltp-schema-fact-dimension) — Data Mining, chapter 2

## Related in other subjects

- [Forebay dimension for 3 minutes storage](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aforebay-dimension-3min-storage-2-24m) — Hydropower Engineering, chapter 6