📊 statistics

Rethinking Intrinsic Dimension Estimation in Neural Representations

This paper challenges the validity of common intrinsic dimension estimators in neural representation analysis by demonstrating a critical discrepancy between theory and practice, revealing that these methods fail to track true underlying dimensions and instead reflect other confounding factors.

Original authors: Rickmer Schulte, David Rügamer

Published 2026-04-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rickmer Schulte, David Rügamer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: The "Fake" Compass

Imagine you are exploring a vast, foggy forest (the world of Artificial Intelligence). Scientists have been using a special compass called the Intrinsic Dimension (ID) to figure out how "complex" the terrain is.

The theory goes like this: Even though the forest looks huge and 3D, maybe the actual path you need to walk is just a thin, winding trail (a low-dimensional "manifold"). If you can find that trail, you understand the data better.

For years, researchers have used this compass to map neural networks (AI brains). They noticed a strange pattern: As the AI gets deeper into its thinking process, the compass says the terrain gets "bigger" and more complex. They thought this meant the AI was creating more abstract, complex ideas as it went deeper.

This paper says: "Stop. Your compass is broken."

The authors, Rickmer Schulte and David Rügamer, prove that the compass isn't actually measuring the complexity of the terrain. It's just measuring how much the AI is stretching the map.

The Core Problem: The Stretchy Rubber Sheet

To understand why the compass is broken, imagine the neural network is a giant, stretchy rubber sheet.

The Input: You put a crumpled piece of paper (the data) on the sheet.
The Layers: As the data moves through the AI, it passes through layers of "stretchers."
The Result: By the time the data reaches the end, the rubber sheet has been pulled so tight that every single point on the paper is now very far away from its neighbors.

The "Broken Compass" (The Estimators):
The tools scientists use to measure complexity (called TwoNN and MLE) work by looking at how far apart neighbors are.

The Trap: Because the rubber sheet has been stretched so much, the neighbors look incredibly far apart.
The False Conclusion: The compass thinks, "Wow, these points are so far apart! The space must be huge and complex!"
The Reality: The space isn't complex; it's just stretched. The "Intrinsic Dimension" (the true complexity) is actually shrinking or staying the same, but the distance between points is growing.

The Mathematical Proof:
The authors proved mathematically that neural networks act like "Lipschitz mappings." In plain English, this means they can squish or stretch things, but they cannot create new complexity out of thin air. If you start with a simple shape, you can't turn it into a more complex shape just by stretching it. Therefore, the true complexity cannot go up as the AI gets deeper.

But the compass says it goes up. Why? Because the compass is confused by the stretching.

The "Token" Surprise: The Library of Discrete Books

The paper also looked at Large Language Models (LLMs) like the ones that write this text.

The Analogy: Imagine a library where every book is a unique word.
The Discovery: In an LLM, the "data" is just a finite list of words (tokens). Even though the AI creates a "space" for them, it's like a library with a finite number of books.
The Twist: Mathematically, a finite collection of points has zero complexity. It's like a single dot.
The Conclusion: If you measure the "Intrinsic Dimension" of an LLM's internal thoughts, the true answer should be zero. But the broken compass still says it's high because it's just measuring how far apart the "books" are on the shelf.

What Is Actually Happening? (The Real Story)

If the compass is lying, what is the AI actually doing?

The authors found that the "stretching" is driven by Entropy (a measure of how spread out the data is).

The Metaphor: Imagine a group of friends standing in a circle (the input). As they walk through the AI, they start walking further and further apart from each other, spreading out across a massive field.
The Cause: The AI is trying to make sure every different sentence or image gets its own unique spot in this field so it doesn't get confused.
The Result: The "Intrinsic Dimension" tools are just measuring how far the friends have spread out, not how complex their conversation is.

The paper shows that the "Intrinsic Dimension" curve (which goes up and then down) looks almost exactly like the Entropy curve. They are measuring the same thing: The expansion of space.

The Takeaway: A New Map for the Future

1. The Old Map is Wrong:
We can no longer trust the "Intrinsic Dimension" numbers we've seen in papers for the last few years. When a paper says, "The AI's complexity increases in the middle layers," they are actually saying, "The AI is stretching the data apart."

2. The New Perspective:
Instead of asking "How complex is the shape?", we should ask, "How much is the AI spreading the data out?"

Early Layers: The AI stretches the data out to separate different ideas (High expansion).
Late Layers: The AI starts to group similar ideas back together to make a final decision (Lower expansion).

3. The Future:
Scientists need to stop using these old compasses. Instead, they should use Entropy (spread) or other metrics that actually tell us what the AI is doing. The "Intrinsic Dimension" isn't a measure of the AI's intelligence; it's just a measure of how much the AI is stretching the rubber sheet.

Summary in One Sentence

The paper reveals that the popular tools used to measure the "complexity" of AI brains are actually just measuring how much the AI is stretching its data apart, meaning all the previous conclusions about AI "growing more complex" as it thinks were likely an illusion caused by a broken measuring tape.

1. Problem Statement

The paper addresses a critical discrepancy between the theoretical understanding of Intrinsic Dimensions (IDs) in neural networks and the empirical results reported in recent literature.

The Manifold Hypothesis: It is widely believed that high-dimensional data (e.g., images, text) lies on or near a low-dimensional manifold. The dimension of this manifold is the "intrinsic dimension" ( $d_M$ ).
The Empirical Observation: Numerous studies using common ID estimators (such as TwoNN and Maximum Likelihood Estimators) have observed a consistent pattern: estimated IDs increase in the early layers of neural networks and decrease in later layers. This is often interpreted as the network learning abstractions or undergoing phase transitions.
The Core Conflict: The authors argue that this observed "increase" contradicts fundamental mathematical properties of neural networks. If neural networks are composed of Lipschitz continuous mappings, the true intrinsic dimension cannot increase as data passes through layers. Therefore, the widely accepted empirical patterns must be artifacts of the estimation methods rather than reflections of the true underlying geometry.

2. Methodology

The authors employ a combination of theoretical proofs and empirical validation across various architectures (CNNs, ResNets, Vision Transformers, and Large Language Models).

A. Theoretical Framework

Lipschitz Continuity: The authors establish that standard neural network components (linear/conv layers, ReLU, Softmax, pooling, normalization) are Lipschitz continuous mappings. Consequently, the composition of these layers (the entire network) is also Lipschitz.
Dimensionality under Mappings: They utilize the mathematical property that Lipschitz mappings cannot increase the Hausdorff dimension or the pointwise dimension of a set.
- Lemma 1: For a Lipschitz map $f$ , the pointwise dimension of the image measure $\nu$ is less than or equal to the pointwise dimension of the source measure $\mu$ ( $d_\nu \leq d_\mu$ ).
- Theorem 1: In any Lipschitz neural network, the intrinsic dimension of layer-wise representations is non-increasing.
Extension to Union of Manifolds: The authors extend this proof to the "Union of Manifolds" hypothesis (where data lies on multiple disconnected manifolds, e.g., different classes). They prove that even in this case, the dimension cannot increase under Lipschitz mappings.
LLM Specifics: They prove that for Large Language Models (LLMs) with finite vocabularies, the set of possible token embeddings is finite (and thus countable). Since the Hausdorff and pointwise dimensions of countable sets are zero, the true ID of LLM embeddings is theoretically zero, regardless of the layer.

B. Empirical Analysis

Bias Analysis: They demonstrate that common estimators (TwoNN, MLE) suffer from severe negative bias in high dimensions, underestimating the true ID. Crucially, they show this bias is inconsistent, meaning the estimators do not track the true ID even relatively.
Driver Investigation: To explain why estimators show an "increase" despite the theoretical "decrease," the authors analyze factors driving the estimators:
- Ambient Space Dimension: Ruled out as the primary driver (LLM hidden sizes are constant; CNN ID patterns persist despite varying ambient dimensions).
- Cosine Similarity: Ruled out (similarity patterns do not match ID patterns).
- Nearest Neighbor (NN) Distances: Found that NN distances grow significantly in early layers.
- Representation Size (L2 Norm): Found that the L2 norm of representations (distance from the origin) expands in early layers.
- Entropy: They compute the von Neumann entropy of the eigenvalue distribution of layer representations.

3. Key Contributions

Theoretical Refutation: Proved that for Lipschitz neural networks, the true intrinsic dimension is non-increasing. This invalidates the interpretation of "increasing ID" in early layers as a real geometric phenomenon.
Estimator Critique: Demonstrated that popular ID estimators (TwoNN, MLE, Gride) are not only biased but fail to track the true underlying ID, even in a relative sense. The observed patterns are artifacts of the estimator's sensitivity to distance ratios.
Identification of Driving Forces: Identified that the "increasing ID" pattern is actually driven by the expansion of representations in the latent space (increasing L2 norms) and the resulting growth in nearest-neighbor distances.
Entropy Correlation: Discovered a striking correlation between estimated ID patterns and von Neumann entropy. Both metrics rise in early layers and fall in later layers.
LLM Specific Insight: Proved that for token-based LLMs, the true intrinsic dimension is zero due to the finite nature of the vocabulary, rendering standard manifold hypotheses inapplicable in the strict topological sense.

4. Key Results

Contradiction Confirmed: Theoretical proofs confirm that IDs cannot increase in Lipschitz networks. Empirical data shows estimators do increase, confirming the estimators are misleading.
Bias Visualization: Experiments on synthetic manifolds show that TwoNN and MLE estimators drastically underestimate true IDs as dimensionality increases, and the magnitude of this error varies non-linearly.
The "Expansion" Mechanism:
- In LLMs, the L2 norm of hidden states grows in early layers, pushing representations further apart in the latent space.
- This expansion causes nearest-neighbor distances to increase.
- ID estimators rely on the ratios of these distances. When all distances grow proportionally, the ratios shrink toward 1, which mathematically forces the ID estimate to increase.
Entropy vs. ID: The paper shows that von Neumann entropy (measuring the spread of variance across eigendirections) follows the exact same layer-wise trajectory as the flawed ID estimates.
- Interpretation: Early layers spread variance across many directions (high entropy), while later layers concentrate variance (low entropy). The ID estimators are inadvertently measuring this variance spread, not the manifold dimension.

5. Significance and Future Outlook

Re-evaluation of Literature: The paper suggests that a vast body of literature interpreting "increasing IDs" as the emergence of abstraction or phase transitions is likely misinterpreting the data. These patterns reflect the geometric expansion of representations, not a change in manifold dimension.
New Metric Proposal: The authors propose that entropy-based metrics (like von Neumann entropy) are a more theoretically sound and interpretable way to analyze the geometry of neural representations than current ID estimators.
Benchmarking Standard: They propose a new necessary condition for any future ID estimator: Layer-wise ID estimates must be non-increasing in Lipschitz networks. Any estimator that shows an increase is fundamentally flawed for this application.
Impact on LLMs: The finding that LLM embeddings have a true ID of zero challenges the application of manifold learning techniques to token-based models, suggesting that the "manifold hypothesis" may need a different formulation for discrete token spaces.

In summary, the paper fundamentally shifts the perspective on neural representation geometry: what was thought to be a dynamic change in intrinsic dimension is actually a static geometric expansion of the latent space, best captured by entropy rather than biased distance-based estimators.