How Lossy Works | Projection — When a High-Dimensional World Becomes a Shadow

Hold your hand between a lamp and a wall.

Your hand is three-dimensional.

The shadow is two-dimensional.

The shadow can tell you something true about the hand.

It cannot tell you everything.

That is projection.

Quick Read

Projection maps information from a richer space into a lower-dimensional one. A 3D object can be projected onto a 2D image. A dataset with hundreds of variables can be projected onto two axes. A globe can be projected onto a flat map. The operation is useful because lower-dimensional representations are easier to see, compare and compute with. It is lossy because not every relationship in the original space can be preserved simultaneously.

One-sentence answer: projection works by preserving selected structures while collapsing dimensions that the new representation cannot carry, so every useful shadow is also an omission.

A Shadow Is Accurate and Incomplete at the Same Time

Rotate your hand and the shadow changes.

The hand did not.

This reveals the first law of projection: the representation depends not only on the object, but also on the direction and rule of projection.

A finger can hide behind another finger. Two points far apart in depth can land on the same place on the wall. A shape can look shorter because it points partly toward the light.

Flattening creates ambiguity.

Maps Have Always Lived With This Problem

The Earth is curved. Paper is flat.

A map projection must therefore choose which properties to preserve well and which to distort. Area, shape, distance and direction cannot all be globally preserved perfectly on a flat map.

This is why different map projections exist.

A navigation map and an equal-area map answer different receiver needs. Neither is simply “the true map”. Each is a disciplined distortion designed for a job.

High-Dimensional Data Has the Same Geometry Problem

Suppose every student is described by fifty variables: vocabulary, grammar, reading speed, inference, writing structure, arithmetic accuracy, working speed, revision frequency, sleep, attendance, confidence, error types and many more.

That student does not live naturally on one line.

Plotting all fifty dimensions at once is impossible for ordinary human vision.

So analysts project the data into fewer dimensions.

The price of visibility is that some relationships must be compressed.

PCA: Keep Directions of Large Variance

Principal Component Analysis constructs new orthogonal directions that capture as much variance as possible in sequence.

If a cloud of data lies mostly along a diagonal, PCA rotates the coordinate system so the first component follows the direction with the greatest variance.

Keeping the first few components can provide a compact approximation of the original data.

But variance is not the same as importance.

A low-variance direction can still carry a rare but decisive distinction.

What PCA Preserves—and What It Does Not Promise

PCA is linear. It is good at representing data that can be approximated by lower-dimensional linear structure.

It does not promise to preserve every nonlinear neighbourhood, cluster boundary or manifold shape.

A clean two-dimensional PCA plot can therefore be informative without being a complete portrait of the data geometry.

t-SNE: Protect Local Neighbours, Sacrifice Global Geography

t-distributed Stochastic Neighbor Embedding is designed primarily to preserve local neighbourhood relationships when placing high-dimensional points into a low-dimensional visualisation.

This can make clusters visually striking.

But the distances between clusters in a t-SNE plot should not automatically be interpreted as faithful high-dimensional distances. Cluster sizes and large-scale arrangement can also be misleading.

The picture can be excellent for one question and unreliable for another.

UMAP: Another Useful Projection With Its Own Trade-Offs

UMAP is widely used for nonlinear dimensionality reduction and visualisation. It often preserves local structure well and can be computationally efficient on large datasets.

Recent research continues to examine the balance between local and global structure preservation because a low-dimensional embedding cannot guarantee that every inter-cluster distance or broad geometry remains faithful.

The lesson is not “do not trust UMAP”.

The lesson is “know which structure the projection was designed to protect”.

Local Truth and Global Truth Can Conflict

Imagine a map of a city that preserves walking relationships inside each neighbourhood but moves the neighbourhoods closer together so the whole city fits on one page.

Within-neighbourhood navigation may remain useful.

Cross-city distance may become nonsense.

Dimensionality-reduction methods face the same choice: preserve local neighbours, global distances, variance, topology, interpretability or some compromise among them.

A Beautiful Plot Can Be More Persuasive Than It Is Faithful

Humans trust pictures quickly.

Two clusters separated by white space feel objectively different. A gradient feels continuous. An island feels isolated.

But a projection may have created part of that geometry.

This is why responsible visualisation treats the low-dimensional picture as evidence to investigate, not as the final ontology of the data.

Rotation Changes the Story

Project a cube from one angle and it looks like a square.

From another angle, it looks hexagonal.

Likewise, different projection methods, parameters or initialisations can produce different visual arrangements from the same high-dimensional source.

If the scientific claim changes dramatically when the projection changes modestly, the claim may belong to the projection rather than the source.

Projection and Feature Selection Are Not the Same

Feature selection keeps some original dimensions and discards others.

Projection can combine many original dimensions into new axes.

A principal component may contain information from every original variable. That can preserve distributed structure while making the new axis harder to interpret in ordinary language.

Selection preserves original meaning more directly. Projection can preserve more distributed information. Different losses, different gains.

Projection in Cameras

A photograph projects a three-dimensional scene onto a two-dimensional sensor.

Depth is inferred from perspective, occlusion, shading and prior knowledge rather than stored as ordinary visible depth in a single flat image.

Two objects of different real sizes can occupy the same number of pixels if they are at different distances.

The image is faithful to one optical projection, not a complete inventory of spatial reality.

Projection in Language

A description of a person is also a projection.

“Quiet, careful, excellent at mathematics” selects some axes and omits thousands of others.

This is an analogy, not a claim that human description performs PCA.

The structural insight is the same: a multidimensional person becomes a lower-dimensional representation designed for a receiver.

Education: Report Cards Are Projections of Learners

A learner is far richer than English 78, Mathematics 84, Science 81.

Those scores are useful projections onto a few institutional axes.

They can guide decisions. They cannot reveal every misconception, motivation, strength, friendship, curiosity, transfer ability or future trajectory.

A responsible teacher uses the projection without confusing it for the child.

When Projection Is Exactly the Right Tool

Projection is indispensable when the source is too complex to inspect directly.

A two-dimensional visualisation can reveal clusters worth investigating. A map can make navigation possible. A photograph can preserve a scene. A low-dimensional model can reduce noise and computation.

Loss is not failure when the preserved relationships are the relationships the receiver needs.

When Projection Becomes Dangerous

Danger begins when the receiver forgets the projection rule.

• A cluster map becomes proof of natural categories.
• A report-card score becomes a complete judgement of a learner.
• A flat map becomes intuition about true area.
• A photograph becomes proof of distance without scale cues.
• A two-axis dashboard becomes the organisation itself.

The projection becomes most misleading when it is so useful that nobody remembers what disappeared.

A Practical Projection Audit

• Source space: how many dimensions or properties existed upstream?
• Projection rule: what transformation produced the lower-dimensional view?
• Preserved structure: does the method prioritise variance, neighbourhoods, topology, distance or another property?
• Lost structure: which relationships are not guaranteed to survive?
• Stability: does the picture change substantially under reasonable parameters or methods?
• Receiver: what question is this projection actually fit to answer?
• Return path: can important claims be checked against richer upstream data?

Sources and Further Reading

• Comprehensive review of dimensionality-reduction algorithms, including PCA, t-SNE, UMAP and trade-offs in interpretability and structure preservation.
• Recent work on global versus local structure preservation in nonlinear dimensionality reduction.
• Foundational literature on PCA, t-SNE, UMAP and cartographic projection.

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Continue with How Lossy Works | Feature Selection, How Lossy Works | Resolution, and How Lossy Works | Aggregation.

Final Thought: A Shadow Can Be Enough—If You Remember the Hand

Projection is one of intelligence’s most useful tricks.

It makes the invisible visible.

But every projection is a bargain.

The wise reader enjoys the shadow and keeps a route back to the object that cast it.