What is high-dimensional data, and why is it hard to work with?

In this article

1. Before you start
2. Why interviewers ask this
3. High-dimensional data from first principles
4. Worked example
5. Trade-offs and edge cases
6. How to answer it in an interview
7. Common mistakes
8. Follow-up questions
9. Sources

Before you start

This page is one step of the deep learning learning path. It uses a little probability and statistics, mainly variance.

Why interviewers ask this

Candidates who say "more features always give a better model" lose the point at once, because the next question exposes it. The curse of dimensionality is a standard topic in machine learning rounds, and candidates report it as a warm-up before nearest-neighbor, embedding or regularization questions.

A weak answer says "high dimensions are slow." A strong answer says what actually breaks: data becomes sparse, distances lose contrast, and a nearest neighbor stops being meaningfully nearer than anything else. A Staff-level answer adds the correction: real data often lies near a low-dimensional surface, so the damage is smaller than the mathematics for random points predicts, and the remedies work for that reason.

High-dimensional data from first principles

High-dimensional data

High-dimensional data is a set of records that each have a large number of features, commonly so many that features rival or outnumber records, so that each record is a point in a space with one axis per feature and ordinary geometric intuition from two or three dimensions fails.

Intrinsic dimension

The intrinsic dimension of a dataset is the number of independent directions the data actually varies along, which can be far smaller than the number of features when the points lie near a lower-dimensional surface.

In plain words

Picture a spreadsheet where every row is a person and every column is a measurement. Two columns let you draw each person on a flat map. Three columns need a room. Beyond that you cannot draw it. The idea still holds: each row is a location, and "similar people" means "nearby locations." The analogy breaks in one place: in a room, being nearby is meaningful, while in thousands of columns almost everyone is about equally far from everyone else.

In one precise sentence, high-dimensional data lives in a space whose volume grows so fast with each added axis that a fixed number of records covers a vanishing share of it.

Step 1: Each feature adds an axis, so a record becomes a high dimensional vector

A high dimensional space is a space with many axes, and a record with p features is one point in it. A table with 20 columns is a point cloud in 20 dimensions. A text embedding is a list of numbers that stands for a piece of text; one of 768 numbers (illustrative) is a point in 768 dimensions. These high dimensional vectors are the normal currency of modern machine learning, and an image is another case, with one number per pixel per color channel.

In practice, "high-dimensional" usually means the number of features p is large compared with the number of records, as in genomics (thousands of genes per patient), text and images.

Input: p measurements per record. Output: one point in a p-dimensional space. Distance between two records is then the Euclidean distance over all p coordinates. You now know what "high-dimensional" means: many axes, one point per record. The question for the next step is how much room all those axes create.

Step 2: Space empties, so data becomes sparse

Space empties because volume grows exponentially with the number of axes, so a fixed number of records covers a smaller and smaller share of it. Take data spread uniformly inside a unit cube and ask how long a smaller cube's edge must be to hold 10% of the points. The edge is $e={0.1}^{1/p}$. For one dimension that is 0.1; for two, 0.3162; for ten, 0.7943; for 100, 0.9772.¹ A "local" neighborhood holding a tenth of the data therefore spans almost the whole range of every axis, and it is no longer local.

The same growth shows in counts. Placing just 10 points along each axis takes 10 points in one dimension, 100 in two, 10 billion in ten and a 1 followed by 100 zeros in 100 dimensions.¹ Real datasets never reach such sizes, so the data stays sparse. You now know why local methods starve; the next step shows what happens to distances.

Step 3: Distances bunch together, so the nearest neighbor loses meaning

Distance concentration is the effect that, as dimensions grow, the distances from a query point to the other points become nearly equal. A simple measure is the relative contrast, which is the farthest distance minus the nearest distance, divided by the nearest distance. A large value means the nearest point is clearly nearer than the farthest one.

In a simulation of 500 uniformly random points, the average relative contrast is 82.4 in two dimensions, 2.5 in ten, 0.40 in 100 and 0.11 in 1,000.¹ At 1,000 dimensions the farthest point is only about 11% farther than the nearest. Beyer et al. (1999) define a nearest-neighbor query as unstable when most points lie within a factor of (1 + ε) of the nearest one, as Ting and Zhu (2021) summarize. Beyer et al. prove that if the variance of the distance divided by the square of its mean shrinks to zero as dimensions grow, the nearest and farthest distances become equal in the limit. The paper reports that the effect can already appear at 10–15 dimensions.

Step 4: Real data often has low intrinsic dimension, which is why the remedies work

Real data is milder than random points because it often lies near a low-dimensional surface. To test this, the same simulation was repeated with points on a flat two-dimensional sheet placed inside 2, 10, 100 and 1,000 dimensions. At 10, 100 and 1,000 dimensions the relative contrast stays between about 65 and 71. The comparison starts at 10 dimensions, where the sheet is a thin slice of the space.¹ Only the intrinsic dimension, 2, matters. Ting and Zhu (2021) report, citing earlier work, that k-nearest-neighbors (kNN) classifiers can still reach reasonable accuracy in high dimensions, and suggest low intrinsic dimension as one possible explanation.

So the practical recipe is to find the few directions that carry the variation and work there. You now have the full picture: many axes, sparse space, bunched distances, and a way out when the data has structure.

1 · Axes2 · Sparse3 · Bunched4 · Low intrinsic dimensionReplay

Step-by-step description

1. Each feature adds one axis to the space.
2. Volume grows fast, so a fixed set of records covers little.
3. Distances from a query become nearly equal.
4. Data near a surface keeps its contrast.

• Each feature adds one axis.
• At 100 dimensions a box holding 10% of uniform data spans 98% of each axis.
• Distances between uniform random points bunch together as dimensions grow.
• Distance contrast of points on a plane stays high from 10 to 1,000 dimensions.

Worked example

The numbers above come from one short script. It places points at random, measures distances and applies two formulas. Everything in the table can be redone with a calculator and a random number generator.

The ball adds one more surprise. A ball inscribed in a cube fills 78.5% of the cube's volume in two dimensions and 52.4% in three, but only 0.249% in ten and about 0.0000025% in 20.¹ In high dimensions almost all the volume sits in the corners, far from the center, which is another way of saying that points spread out and "near" becomes rare.

Text description of the diagram

• Relative contrast of uniform points falls from 82.4 at 2 dimensions to 0.11 at 1,000.
• At 10, 100 and 1,000 dimensions the relative contrast of points on a plane stays between about 65 and 71.

Trade-offs and edge cases

Every remedy for high-dimensional data buys something and has a limit. Principal component analysis (PCA) is a method that finds the directions along which the data varies most and keeps only the top few, which works well when the structure is roughly linear. Random projection is a method that multiplies the data by a random matrix to reach fewer dimensions. Its theory is the Johnson-Lindenstrauss lemma. The Johnson-Lindenstrauss lemma is a result that a small set of points can be embedded in a lower-dimensional space with distances nearly preserved, according to the scikit-learn guide.

The catch is the size of the target. For 1,000,000 points, the minimum dimension is 663 at a tolerance of 0.5, 11,841 at 0.1 and 1,112,658 at 0.01, and it depends on the number of points and the tolerance, not on the original dimension.¹ Ting and Zhu (2021) also report, citing earlier work, that random projection preserves structure yet can still suffer concentration, so projection alone does not remove the curse.

Practice in 2025–26 shows the other side. Vector search runs approximate nearest-neighbor indexes, such as the graph index of Malkov and Yashunin (2016), on embeddings with hundreds of dimensions. It works because learned embeddings tend to have low intrinsic dimension and because the task needs good neighbors, not exact ones.

Text description of the diagram

• Principal component analysis keeps the directions of largest variance.
• Random projection can still suffer distance concentration.
• Finding the right cluster stays meaningful only when clusters are well separated.

Edge cases: when the data has well-separated clusters, the distance from a query to its own cluster stays clearly smaller than to other clusters, so finding the right cluster stays meaningful even when ranking points inside it does not; this holds only when clusters are well separated. Having more features than records lets a flexible model fit the training data perfectly while risking learning little that transfers. Image models such as convolutional networks and diffusion models work on data with huge raw dimension, and their designs exploit locality and shared weights rather than treating every pixel as independent.

How to answer it in an interview

The spoken structure: define it, name the two failures, add the low-intrinsic-dimension correction, then say what you would do.

Common mistakes

• "More features always means more information." (wrong) Each added feature also thins the data, so with a fixed number of records the model can learn less per feature. (right) "More features help only if they carry signal and I have enough records to cover them."
• "Dimensionality reduction removes the curse." (wrong) Random projection can still concentrate distances. (right) "Reduction helps when the data has low intrinsic dimension; it is not a universal fix."
• "Nearest-neighbor methods are hopeless in high dimensions." (wrong) kNN often still works on real data. (right) "Concentration is milder when intrinsic dimension is low, and matching a query to a well-separated cluster stays meaningful."
• "High-dimensional means thousands of columns." (wrong) There is no fixed threshold. (right) "It is relative to the number of records and the structure of the data."

Follow-up questions

What is a high dimensional space?

A high dimensional space is a space with many axes, one per feature, in which each record is a point. Intuition from two or three dimensions, such as "nearby points are similar", becomes unreliable there.

Why does high-dimensional data make nearest neighbor search harder?

Distances from a query to the other points become nearly equal, so the nearest point is hardly nearer than the rest. In the simulation here the farthest point was only about 11% farther than the nearest at 1,000 dimensions.¹

How do you handle high dimensional data?

Reduce it with principal component analysis or learned embeddings, drop uninformative features, regularize the model, and collect more records. Check the intrinsic dimension first, since low intrinsic dimension makes every one of these work better. The deep learning interview questions page covers related prompts.

Does dimensionality reduction remove the curse of dimensionality?

No. It helps when the data lies near a low-dimensional surface, but random projection can still suffer distance concentration, as Ting and Zhu (2021) report, citing earlier work.

Why does vector search work on 768-dimensional embeddings?

Learned embeddings tend to have low intrinsic dimension, so distances keep useful contrast, and approximate indexes only need good neighbors. The 768 is illustrative; the raw dimension matters less than the intrinsic one.

Is having more features always better?

No. Extra features that carry no signal thin the data and add noise to distances. A neural network learns which directions matter by training with backpropagation, but it still needs enough records to find them.

Sources

1. Breaking the curse of dimensionality with Isolation Kernel (arXiv 2109.14198), Section 2.1 — arXiv (Ting and Zhu), 29 Sep 2021
2. When Is Nearest Neighbor Meaningful? — Springer (Beyer et al., ICDT 1999), 15 Jan 1999
3. Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs (arXiv 1603.09320) — arXiv (Malkov and Yashunin), 30 Mar 2016
4. scikit-learn user guide: Random Projection — scikit-learn documentation, 26 Oct 2026