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Unambiguous Representations in Neural Networks: An Information-Theoretic Approach to Intentionality

This paper employs information theory to demonstrate that neural networks can encode unambiguous, intrinsic representations decodable from their relational connectivity structures, thereby providing a quantitative method to measure the low-ambiguity states posited by theories of consciousness like IIT and narrow representationalism.

Original authors: Francesco Lässig

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Francesco Lässig

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

The Big Idea: How Does a Brain "Know" What It's Seeing?

Imagine you have a box of digital files. One file is a string of numbers: 010101.

If you use a JPEG decoder, that string becomes a picture of a cat.
If you use an MP3 decoder, that same string becomes a song of a dog barking.

The string itself doesn't know what it is. It is ambiguous. Its meaning depends entirely on the "decoder" you choose to apply from the outside. This is how computers usually work.

Consciousness is different.
When you look at a red square, your brain doesn't need an external manual to tell you, "Okay, this neural pattern means 'red square'." The experience of seeing red is intrinsic to the brain state itself. You can't accidentally interpret that same brain state as "seeing a green triangle."

The paper asks a big question: How does a brain state lock in its meaning so it can't be misinterpreted? The authors call this "unambiguous representation."

The Experiment: The "Scrambled Puzzle" Test

To test this, the researchers trained simple computer brains (neural networks) to recognize handwritten numbers (like the digits 0–9). They then did something tricky: they scrambled the neurons.

Imagine you have a team of 10 workers (output neurons) who are supposed to shout out the number they see.

In a normal setup, Worker #1 always shouts "1", Worker #2 always shouts "2," etc.
In the experiment, the researchers shuffled the workers so that Worker #1 might now be the one shouting "7," and Worker #5 might be shouting "2."

The Challenge: If you only look at how the workers talk to each other (their connections and relationships), can you figure out which number each worker is actually shouting?

If the workers' relationships are just random noise, you can't guess. But if their relationships form a unique, rigid structure that only fits one specific arrangement, then the meaning is "unambiguous."

The Key Discovery: The "Dropout" Magic

The researchers tried two different ways to train these computer brains:

Standard Training: The brain learns normally.
Dropout Training: During learning, the brain is forced to "forget" random parts of itself (like a student who has to study with their eyes closed half the time). This forces the brain to build robust, distributed connections.

The Results:

Standard Training: When they scrambled the neurons, they could only guess the correct number about 38% of the time (which is barely better than random guessing). The relationships between neurons were fuzzy and ambiguous.
Dropout Training: When they scrambled the neurons, they could guess the correct number 100% of the time.

The Analogy:
Think of the Standard Training brain like a group of people holding a rope. If you pull one person, the whole group moves, but the shape is loose and wobbly. You can't tell exactly who is who just by looking at the rope's shape.

Think of the Dropout Training brain like a crystal. The atoms (neurons) are locked into a very specific, rigid geometric shape. If you scramble the atoms, the crystal falls apart or looks completely wrong. But if you look at the pattern of the connections, it's so unique that you can instantly tell, "Ah, this specific atom must be the corner piece, and this one must be the center." The structure itself dictates the meaning.

What About "Where" Things Are?

The researchers also tested if the brain could figure out where on a screen a pixel was located just by looking at the connections.

They found that the brain could decode the location of a pixel with high accuracy (up to 84% precision) just by looking at the web of connections between input neurons.
This suggests that the "map" of the visual field is built into the geometry of the connections, not just the activity.

Why Does This Matter?

The paper doesn't claim these computer brains are "conscious" or that we should use this to cure diseases. Instead, it offers a quantitative tool.

Ambiguity is a real thing: You can measure how "confused" a neural network's internal language is.
Training matters: How you train a system changes how clearly it represents information, even if the system performs the same task (like recognizing digits) equally well.
A test for consciousness theories: Some theories of consciousness (like "Narrow Representationalism" or "IIT") argue that for a system to be conscious, its internal states must be unambiguous. This paper proves that neural networks can achieve this unambiguous state through specific training methods.

Summary in One Sentence

The paper shows that by training computer brains in a specific way (using "dropout"), the internal connections between neurons become so rigid and unique that the meaning of every neuron is locked in by its position in the network, making the representation "unambiguous" just like our conscious experience.

Technical Summary: Unambiguous Representations in Neural Networks

Problem Statement

Conventional representations (e.g., bit strings, letters) are inherently ambiguous; their meaning depends entirely on an external decoder or convention. A single bit string can encode a JPEG or an MP3 depending on the decoding scheme. In contrast, conscious experience is posited to be fundamentally different: a neural state corresponding to the perception of a red square cannot simultaneously encode the perception of a green triangle. This intrinsic nature of consciousness suggests that conscious representations must be unambiguous, where the content is determined solely by the neural state itself, not by external interpretation.

The paper addresses the question of whether neural networks can encode representational content unambiguously through their internal relational structure. Specifically, it investigates if the content of a neuron (e.g., the digit class it represents or the spatial position of an input it encodes) can be uniquely determined by its connectivity patterns relative to other neurons, without knowledge of the training procedure or external labeling.

Methodology

Theoretical Framework

The authors formalize representational ambiguity using information theory. Ambiguity is defined as the conditional entropy $H(I|R)$ , where $I$ represents possible interpretations and $R$ is the representation.

High Ambiguity: Uniform distribution over interpretations (minimal information about content).
Low Ambiguity: Concentrated probability (high confidence in content).
The hypothesis is that the relational structure (the web of internal relations) within a network can constrain possible interpretations, thereby reducing $H(I|R)$ .

Experimental Setup

Two primary experiments were conducted using fully-connected neural networks trained on the MNIST digit classification task (784-50-50-10 architecture). Three training paradigms were compared:

Untrained: Random initialization (control).
Standard Backpropagation: Standard training.
Dropout Training: Backpropagation with 20% dropout on hidden layers.

Experiment 1: Decoding Output Neuron Class Identity

Task: Determine which digit class (0–9) an output neuron represents based solely on its connectivity.
Procedure: Neurons in the output layer were randomly permuted to remove positional information. Relational structure was extracted via the Gram matrix of cosine similarities between neuron weight vectors.
Decoding Methods:
1. Geometric Matching: A reference Gram matrix (averaged from networks with known ordering) was compared to test networks. The permutation minimizing the Frobenius distance was selected.
2. Learned Decoder: A transformer-based decoder (without positional encodings to ensure permutation invariance) was trained to predict class identity from the relational structure.
Ablation: Tested whether local neighborhoods (similarity of a target neuron to all others) suffice or if the full global geometry is required.
Scaling: Varied the number of output neurons (2 to 10) to test if increasing relational complexity reduces ambiguity.
Cross-Architecture: Tested transferability across different hidden layer architectures (50-50, 25-25, 100).

Experiment 2: Decoding Input Neuron Spatial Position

Task: Decode the Euclidean distance of an input neuron's corresponding pixel from the image center.
Procedure: Similar to Experiment 1 but applied to the input layer weights ( $784 \times 50$ ). Due to the large number of permutations ( $784!$ ), only the learned decoder (regression task) was used.
Metric: $R^2$ score for predicting spatial distance.

Quantifying Ambiguity

The authors computed Ambiguity Reduction Scores (ARS) to quantify the reduction in conditional entropy relative to maximum entropy.

For classification: Derived from accuracy using Fano's inequality.
For regression: Derived from $R^2$ assuming Gaussian residuals.
An ARS of 1.0 indicates complete disambiguation ( $H(I|R, C) = 0$ ).

Key Results

1. Decoding Output Neuron Class Identity

Untrained Networks: Decoding accuracy was at chance level (10%), validating that random weights contain no recoverable structure.
Standard Backpropagation: Achieved 25% accuracy with the learned decoder and 38% with geometric matching. While above chance, the relational geometry showed significant ambiguity (multiple permutations yielded similar distances to the reference).
Dropout Training: Achieved 100% accuracy with geometric matching (zero variance) and 75% with the learned decoder. The true permutation was clearly separated from all incorrect alternatives in the geometric space.
Orthogonality to Performance: Despite these drastic differences in representational ambiguity, the underlying MNIST classification accuracy was nearly identical across all training paradigms.

2. Structural Necessity and Complexity

Global vs. Local: Removing the global geometric context (providing only local similarity vectors) caused a substantial drop in decoding accuracy, indicating that a neuron's content is determined by its position within the entire relational geometry, not just local neighbors.
Relational Complexity: Accuracy scaled monotonically with the number of neurons used to compute the relational structure. With 2 neurons, performance was at chance (50%); with 10 neurons, it reached 100%. This demonstrates that richer relational geometries provide more constraints on interpretations.

3. Invariance and Generalization

Architecture Invariance: Geometric matching achieved strong decoding accuracy across different hidden layer architectures (e.g., decoding a 25-25 network using a 50-50 reference), suggesting the relational encoding of content is largely architecture-invariant.
Dataset Discrimination: A binary classifier could distinguish between networks trained on MNIST vs. Fashion-MNIST with near-perfect accuracy (99.8% for dropout) based solely on relational structure.

4. Input Neuron Spatial Position

Contrary to the output layer results, standard backpropagation yielded higher spatial decoding accuracy ( $R^2 = 0.844$ ) than dropout training ( $R^2 = 0.695$ ).
However, in both cases, the full relational structure significantly outperformed local neighborhood information.
Ambiguity Reduction Scores (ARS) for input positions ranged from 0.419 to 0.654, indicating substantial but not perfect disambiguation.

Significance and Claims

The paper claims to provide a quantitative method for measuring representational ambiguity in neural systems. Its primary contributions are:

Operationalizing Unambiguity: It demonstrates that neural networks can encode information in a way that is unambiguous with respect to relational structure. For dropout-trained networks, the relational geometry alone completely specifies the representational content (class identity), achieving an ARS of 1.0.
Orthogonality of Ambiguity and Performance: The study shows that representational ambiguity is a dimension of neural representation that can vary independently of behavioral task performance. A network can perform a task perfectly while encoding information either ambiguously (standard backprop) or unambiguously (dropout).
Theoretical Alignment: The results support theoretical accounts of consciousness (such as Narrow Representationalism and Integrated Information Theory) which posit that conscious states must be unambiguous. The paper argues that if consciousness supervenes on brain states, those states must unambiguously determine their content. The experimental finding that relational structure can fix content without external decoding aligns with the "intrinsic" nature of phenomenal experience.
Structural Determinacy: The work suggests that content is not merely conventionally assigned but can be structurally determined by the internal relations of the system. The decoder used in the experiments does not impose meaning but discovers consistent relational patterns inherent to the network instances.

The authors remain modest in their claims, explicitly stating that they do not assert MNIST networks are conscious. Instead, they propose that their framework offers a necessary (though not sufficient) condition for phenomenal experience: the ability of a neural substrate to encode content unambiguously through its internal structure.