Two different hands can cast almost the same shadow.
Two different three-dimensional scenes can produce the same photograph.
Two different student profiles can land on the same point in a two-axis chart.
The projection did not lie.
It simply no longer contains enough dimensions to tell those worlds apart.
A projection collision occurs when distinct source states map to the same—or effectively indistinguishable—lower-dimensional representation.
This is the second pillar beneath How Lossy Works | Projection. The master owns loss through dimensional reduction. This article owns non-uniqueness: why the shadow may be accurate yet insufficient to reconstruct the object that produced it.
Quick Read
Projection collisions are a direct consequence of many-to-one mappings. If a richer source is compressed into fewer dimensions, different source states can differ only along dimensions that are discarded and therefore land on the same projected point. This makes the projection non-invertible without additional assumptions or side information. Collisions appear as occlusion in images, overlap in scatterplots, equal scores for different profiles, and ambiguous solutions in inverse problems. The safe response is to preserve uncertainty and, where the decision matters, return to richer measurements.
different sources → same projection → inverse ambiguity → priors / side information / additional views → candidate reconstruction, not guaranteed original
Many-to-One Is the Core Geometry
Project a point from three dimensions to two.
If the discarded dimension is depth, then every point lying on the same viewing ray can land at the same image coordinate.
One pixel can therefore correspond to many possible real depths.
This is not an implementation bug.
It is what lower dimensionality means.
Occlusion Is a Physical Projection Collision
A person stands behind a wall.
The camera records the wall.
Many hidden scenes behind that wall could generate the same visible pixels.
The photograph cannot tell you which hidden scene is present because the projection never captured it.
One Image Does Not Contain Absolute Depth by Itself
Human vision infers depth from:
- perspective;
- known object size;
- occlusion;
- shading;
- texture gradients;
- stereo disparity;
- motion;
- prior knowledge.
The flat projection supplies cues.
The receiver supplies a model.
Reconstruction from projection is often projection plus prior.
A Shadow Can Have Several Sources
A narrow object close to the lamp and a wider object farther away may cast similar silhouettes.
Silhouette preserves boundary from one direction.
It does not preserve all internal shape or depth.
This is the simplest mental model for inverse ambiguity.
Linear Projection Has a Null Space
In a linear projection, some source directions may map to zero in the retained representation.
Move the source along one of those discarded directions and the projected point does not change.
All such differences are invisible to the projection.
The set of invisible directions forms a mathematical description of what the representation cannot distinguish.
PCA Collisions Live in Discarded Components
Keep two principal components from a hundred-dimensional dataset.
Two observations can share almost identical values on those retained components while differing substantially across discarded components.
They look like neighbours in the PCA plot.
They are not necessarily near in every original dimension.
Explained Variance Is Not Collision Immunity
Suppose two retained components explain 85% of total variance.
The remaining 15% can still contain the distinction relevant to a rare class, safety event or subgroup.
High explained variance says much variation is represented.
It does not guarantee every decision-relevant distinction survives.
t-SNE Can Put Distinct Global States Near Each Other
t-SNE is designed around neighbourhood similarities rather than preserving every global distance. Current scikit-learn documentation warns that results can differ with initialization and that parameter choices can significantly alter the embedding.
Points that appear nearby in the two-dimensional view should therefore be interpreted through the method’s objective, not as guaranteed globally faithful Euclidean neighbours in the source space.
Cluster Overlap Can Be a Collision or a Real Continuum
Two labelled groups overlap in the projection.
Possibility one: they genuinely overlap in the source.
Possibility two: they differ along discarded dimensions.
Possibility three: the method compressed global relationships to preserve something else.
The plot alone cannot decide among these explanations.
Separated Clusters Can Hide Collisions Too
A beautiful low-dimensional plot displays three islands.
Inside each island, many distinct high-dimensional states may occupy nearly the same local coordinates.
Visual separation between islands does not restore information lost inside each island.
A Score Is a Projection Collision Machine
Student A: excellent algebra, weak geometry.
Student B: weak algebra, excellent geometry.
Both score 75.
The one-dimensional mark is useful for ranking a certain performance aggregate.
It cannot uniquely reconstruct the learning profile.
Aggregation owns the broader many-to-one averaging problem. Projection Collisions owns the geometry of indistinguishability after dimension reduction.
Ranking Creates Even More Collisions
Rank 5 tells you relative order.
It does not tell you absolute distance from rank 4 or 6.
Many different score configurations can produce the same ranking.
Ranking owns that specific representation.
Thresholds Collapse Whole Regions Into One Label
69.9 and 0 both become “fail” under one threshold.
70.0 and 100 both become “pass.”
Distinct source states collide because the representation intentionally keeps only which side of a boundary they occupy.
Thresholding owns this binary special case.
Quantisation Makes Nearby Values Indistinguishable
Infinite possible intensities become 256 digital levels.
Every source value within one quantisation interval maps to the same output code.
This is another many-to-one collision, now along precision rather than geometric dimension.
Quantisation owns it in depth.
Anonymisation Intentionally Creates Identity Collisions
Remove direct identifiers and coarsen quasi-identifiers.
Now several people may share the same released demographic pattern.
That collision can be a privacy feature.
The goal is precisely to make individual reconstruction harder.
Anonymisation owns the privacy-specific trade-off.
Loss Can Be the Job
A road map deliberately makes many buildings equivalent because precise building interiors are irrelevant to route planning.
A privacy release deliberately makes several individuals indistinguishable.
A diagnostic summary may intentionally collapse dozens of variables into a small risk band.
Collision is not automatically error.
It becomes error when the receiver later needs a distinction the representation was designed to erase.
Inverse Problems Begin Where Projection Ends
We observe the shadow.
We want the source.
That is an inverse problem.
If many sources could generate the observation, the problem is ill-posed unless additional constraints narrow the possibilities.
Priors Resolve Ambiguity by Adding Assumptions
Among all three-dimensional scenes compatible with an image, prefer scenes that resemble ordinary physical environments.
Among all signals compatible with sparse observations, prefer smooth or structured signals.
The prior does not recover information magically.
It supplies extra structure that the projection itself did not contain.
Regularisation Is a Controlled Preference Among Colliding Sources
When many source candidates explain the same projection, regularisation chooses according to a rule such as simplicity, smoothness or sparsity.
This can make an inverse problem solvable in practice.
It should not be described as proving that the chosen reconstruction was the unique original source.
Additional Views Can Break Collisions
One photograph leaves depth ambiguous.
Two separated cameras add stereo information.
One medical projection overlaps anatomy.
Multiple angles can support tomographic reconstruction.
The first pillar, Projection Direction, explains why new views reveal dimensions hidden in the first.
Time Can Break Spatial Collisions
Two objects overlap in one frame.
They move differently in the next.
Temporal information reveals that the original overlap contained two objects.
Additional dimensions can arrive through time, not only through another camera.
Metadata Can Break Semantic Collisions
Two plotted points overlap.
One belongs to a child and one to an adult.
A label, timestamp or provenance record restores a distinction absent from the coordinates.
Side information can recover usability without recovering every lost dimension.
Receiver Knowledge Can Hide the Collision Problem
An expert looks at a compressed chart and silently supplies domain knowledge.
A novice sees only the chart.
The representation can therefore appear more self-sufficient than it is because expert priors are invisible in the finished interpretation.
AI Reconstruction Can Produce One Plausible Source Among Many
A generative model can fill missing detail.
It may produce a coherent image, sentence or structure consistent with the projection.
Coherence is not provenance.
If several originals are compatible with the observed representation, the generated completion should be treated as a hypothesis unless independent evidence identifies the source.
The Collision Set Is the Right Object of Uncertainty
Instead of asking only:
What source produced this projection?
ask:
What family of source states remains compatible with what we observed?
This shift prevents false certainty after lossy representation.
Collision Risk Depends on the Decision
If two source states collide but would trigger the same safe action, the loss may be acceptable.
If they require opposite actions, the projection is too coarse for that decision.
The fourth pillar, Projection Objectives, owns this task-fit test.
A Better Projection-Collision Model
source states → projection → equivalence classes of indistinguishable outputs → decision risk → add view / metadata / prior / measurement → shrink candidate set → preserve remaining uncertainty
A 30-Lens Projection Collision Audit
- Source dimension: how rich is the original state?
- Output dimension: how much smaller is the representation?
- Mapping: what projection rule is used?
- Discarded dimensions: what can vary invisibly?
- Null directions: which changes leave output unchanged?
- Occlusion: what can hide behind what?
- Quantisation: which nearby values share a code?
- Aggregation: which profiles share the same summary?
- Threshold: which source range becomes one label?
- Ranking: which absolute differences disappear?
- Anonymisation: which identities are intentionally merged?
- PCA loss: what lies in discarded components?
- Nonlinear embedding: what relationships are not preserved by objective?
- Collision pair: can two concrete examples share one output?
- Decision: would those examples require different actions?
- Inverse problem: is unique reconstruction possible?
- Prior: what assumptions choose among sources?
- Regularisation: what source properties are preferred?
- Second view: what ambiguity would another projection remove?
- Time: can dynamics separate overlapping states?
- Metadata: can labels or provenance restore distinctions?
- Receiver knowledge: what hidden prior is being supplied?
- AI completion: is a generated reconstruction being mistaken for source evidence?
- Uncertainty set: how many source possibilities remain?
- Source validation: can richer data be inspected?
- Irreversibility: was the lost information ever stored elsewhere?
- Acceptability: is collision harmless for this job?
- Escalation: when must the system request richer evidence?
- Auditability: is the loss mechanism documented?
- Return: did additional evidence identify or narrow the source state?
Laboratory 1: Same Shadow, Different Object
Use paper cut-outs or objects to produce similar silhouettes from one direction. Then rotate them. The second view reveals what the first projection collapsed.
Laboratory 2: Same Score, Different Learner
Create four skill profiles that all average to 75. Ask which tutoring decision would fail if only the total score were retained.
Laboratory 3: Candidate Reconstruction Set
Show a simple two-dimensional projection of a three-dimensional point. List several possible source depths consistent with the same observed coordinates, then identify what extra measurement would distinguish them.
For Primary Readers
Make two different objects cast the same shadow. Ask why the wall cannot tell which object was used.
For Secondary Readers
Explain many-to-one mapping with a score or shadow and identify what extra information would reverse the ambiguity.
For Advanced Readers
Model projection collisions as equivalence classes under a non-injective map. Inverse inference requires priors, constraints or additional measurements because the observation identifies a fibre of possible source states rather than a unique point.
Common Misconceptions
- “If the projection is accurate, reconstruction should be possible.” Accuracy about retained structure does not restore discarded dimensions.
- “High explained variance means collisions do not matter.” decision-critical distinctions can live in low-variance directions.
- “AI can recover the missing original.” It can infer a plausible completion, not guarantee provenance when several sources fit.
- “Overlap in a 2D plot proves the groups are identical.” they may differ in omitted dimensions.
- “Projection collisions are always bad.” intentional indistinguishability can be useful for privacy, simplification or task-focused representation.
Research Corridor
- scikit-learn — PCA — retained components and explained variance.
- scikit-learn — t-SNE — neighbourhood-based embedding and non-convex optimisation.
- eduKateSG — Quantisation.
- eduKateSG — Projection Master.
Frequently Asked Questions
What is a projection collision?
It is a case where two or more distinct source states produce the same or effectively indistinguishable projected representation.
Why are projections often non-invertible?
Because reducing dimensions removes degrees of freedom. Several source states can differ only along discarded directions while sharing the same retained coordinates.
How can projection ambiguity be reduced?
Add independent views, richer measurements, metadata, temporal information or justified prior constraints, while preserving uncertainty about what remains unresolved.
Final Thought: A Shadow Does Not Owe You a Unique Object
The projection can be perfectly faithful to its rule.
The rule can still erase the difference you later wish you had.
The discipline of lossy reasoning is knowing when one visible shadow still corresponds to several possible worlds.
PROJECTION · FOUR PILLAR LEGS
Return to Projection, or continue through Projection Direction, Projection Stability and Projection Objectives. Return to the Information & Representation Hub.