Distance As Dissimilarity

Definition

Distance can be used as a measure of dissimilarity.

If two points are close, they may be similar.

If two points are far apart, they may be different.

Example:

A = [2, 3]
B = [3, 4]
C = [20, 30]

A is closer to B than to C.

Distance as dissimilarity means farther points are treated as more different.

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When This Works

Distance is useful when the vector components have consistent meaning.

Example:

[height, weight]

Then:

[70, 180]
[71, 182]

are close in the chosen coordinates.

They describe similar measurements.

Distance works when the geometry reflects the data structure.

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When This Can Mislead

Distance can mislead if components are not comparable.

Example:

[rooms, price]

Price may have much larger raw values than rooms.

[3, 300000]
[4, 320000]

The price component dominates the distance.

That may or may not match the comparison we want.

Raw distance can be controlled by scale.

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Distance vs Alignment

Distance and alignment measure different things.

Example:

A = [3, 4]
B = [30, 40]

These vectors are far apart by distance.

But they point in the same direction.

So they are:

different in size
similar in direction

Distance compares separation. Alignment compares direction.

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Dissimilarity Depends On The Question

There is no single universal meaning of dissimilarity.

For some tasks, size matters.

For others, direction matters more.

Example:

[3, 4]
[30, 40]

If we care about total amount, they are different.

If we care about pattern, they are similar.

Dissimilarity depends on what the comparison is meant to preserve.

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Normalization Can Change Distance

Normalization can change what distance measures.

Example:

A = [3, 4]
B = [30, 40]

Before normalization, the distance is large.

After length normalization:

A -> [0.6, 0.8]
B -> [0.6, 0.8]

The distance becomes 0.

Normalization changes dissimilarity by changing what information is kept.

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Why Distance Matters For Data

Distance gives a practical way to compare vector data.

It can help answer:

Which data points are close?
Which examples are unusual?
Which groups form clusters?
Which point changed the most?

But distance is only useful if the representation makes the distance meaningful.

Distance is a comparison tool, not automatic truth.

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Final Summary

• Distance can represent dissimilarity.
• Close points may be treated as similar.
• Far points may be treated as different.
• Distance depends on component meaning and scale.
• Distance differs from directional alignment.
• Normalization can change distance.
• Dissimilarity is meaningful only relative to a representation and question.