🧬 biology

Signal-to-Noise Ratio and Sample Size Govern Representational Alignment in Neural Networks

This paper demonstrates that the representational alignment across neural networks is governed by signal-to-noise ratio and sample size in a non-monotonic manner, particularly showing minimized alignment near the interpolation threshold, a phenomenon that is decoupled from generalization performance.

Original authors: Ali Hussaini Umar, Alessandro Laio

Published 2026-05-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ali Hussaini Umar, Alessandro Laio

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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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: Do Different Students Learn the Same Way?

Imagine you have a classroom of students (neural networks) all trying to learn the same subject (like math or recognizing cats in photos). You give each student a slightly different textbook (training data) and let them study on their own.

Usually, we assume that if two students get the same test score, they must have learned the material in the same way. But this paper asks a deeper question: Even if they get the same score, do they actually "see" the world in the same way inside their heads?

The researchers found that the answer is no. Two students can ace the test but have completely different mental maps of the subject. Furthermore, the quality of their textbooks (how clear the information is) and the size of their textbooks (how many examples they study) change how they align their thinking, but not always in the way you'd expect.

The Tools: Measuring "Mental Similarity"

To figure out if two students are thinking alike, the researchers needed a special ruler. Standard rulers (like checking if two maps look similar) didn't work well because the students' internal maps were curved and twisted like a rollercoaster, not flat like a piece of paper.

Instead, they used a tool called Conditional Copula Entropy (CCE).

The Analogy: Imagine Student A has a map of a city. If you tell Student A, "I am standing right next to the library," can Student B guess where you are?
- If Student B can easily guess, their maps are aligned (they see the neighborhood structure the same way).
- If Student B is totally lost, their maps are misaligned.
The researchers measured how well one student's "neighborhood" predictions matched the other's.

The Two Main Factors

The paper tested two things that change how these students learn:

1. Signal-to-Noise Ratio (SNR): "The Clarity of the Textbook"

The Analogy: Imagine studying for a test.
- High SNR: The textbook is clear, the teacher speaks loudly, and there is no background noise.
- Low SNR: The textbook is blurry, the teacher is mumbling, and there is construction noise outside.
The Finding: When the "textbook" is clearer (High SNR), the students' internal maps become more similar to each other. They all agree on what the neighborhood looks like. When the noise is high, their internal maps become chaotic and different from one another.

2. Sample Size: "The Size of the Textbook"

The Analogy: How many pages of examples does the student study?
- Too few pages: They haven't seen enough examples to form a solid opinion.
- Just enough pages: They have exactly enough to memorize the answers perfectly.
- Too many pages: They have seen so many examples they start finding patterns that aren't even there.
The Finding: This is where it gets weird. The alignment of their thinking does not go up and down in a straight line.
- It drops to its lowest point right when the student has just enough data to memorize the training set perfectly (the "interpolation threshold").
- At this exact moment, even though the students might be getting perfect scores on the practice test, their internal maps are completely different from each other. They are all memorizing the answers in their own unique, chaotic ways.
- Once they study more than that (overparameterized), their maps start to align again.

The Surprising Twist: Good Grades $\neq$ Good Thinking

The most important discovery is that getting a good grade does not mean you have a good understanding.

The Scenario: The researchers compared a "Linear Network" (a simple student) and a "Non-Linear Network" (a complex, creative student).
The Result: The simple student got better test scores (lower error). However, the complex student had better, more structured internal maps.
The Takeaway: The complex student learned a richer, more informative way of seeing the data, even though they made more mistakes on the final test.
The Warning: If you only look at test scores (generalization error), you might think the simple student is "smarter" or has a better model. But in reality, the complex student has a more useful internal representation of the world.

Summary in Plain English

Alignment: Neural networks trained on different data often end up with similar internal "maps" of the world.
Clarity Matters: The clearer the data (less noise), the more similar these maps become.
Quantity is Tricky: The size of the dataset changes alignment in a "U-shape." The maps are most different (least aligned) right when the network is just big enough to memorize the training data perfectly.
Performance is a Bad Proxy: A network can get a higher test score but still have a "dumber" or less informative internal map than a network with a lower score. You cannot judge the quality of a network's internal thinking just by looking at its test results.

The paper concludes that to truly understand how AI learns, we need to look inside the "brain" (the representations), not just at the final test score.

Technical Summary: Signal-to-Noise Ratio and Sample Size Govern Representational Alignment in Neural Networks

Problem Statement

Deep learning models trained independently often converge to structurally similar internal representations, a phenomenon known as representational alignment. While empirical evidence for this "Platonic Representation Hypothesis" is robust, the theoretical mechanisms governing it remain elusive. Specifically, fundamental questions persist regarding:

How alignment depends on the quality (noise levels) and quantity (sample size) of training data.
Whether the mechanisms driving alignment differ between underparameterized and overparameterized regimes.
Whether representational alignment is a reliable predictor of model quality (generalization error).

Existing studies primarily quantify alignment using tools like Representational Similarity Analysis (RSA), Singular Vector Canonical Correlation Analysis (SVCCA), and Centered Kernel Alignment (CKA). However, these metrics often assume linear correspondences and may be ill-suited for representations lying on highly curved, nonlinear manifolds. Furthermore, it is unclear if better generalization performance necessarily implies more aligned internal representations.

Methodology

Theoretical Framework and Metric

The authors propose a novel statistical measure for alignment called Conditional Copula Entropy (CCE). Unlike symmetric measures (e.g., CKA), CCE is asymmetric and quantifies how well the neighborhood structure of one representation predicts the neighborhood structure of another.

Definition: CCE is derived from the Information Imbalance (II) and defined as the negative conditional entropy of copula variables: $CCE(A \to B) = -\lim_{\eta \to 0} H(c_B | c_A = \eta)$ .
Rationale: It captures local representational alignment by measuring the reduction in uncertainty about distances in space $B$ given that points are infinitesimally close in space $A$ .
Analytic Tractability: Crucially, for one-dimensional projections of Gaussian vectors, CCE admits a closed-form expression dependent on the correlation coefficient $\rho$ : $CCE = -\frac{1}{2}\log(1-\rho^2) - \frac{1}{2}\rho^2$ .

Experimental Setup

The study employs a multi-tiered approach to validate the theory:

Linear Teacher-Student Setting (Analytic):
- Model: A two-layer deep linear network trained via gradient descent on a noisy linear regression task ( $y = w^* \cdot x + \epsilon$ ).
- Protocol: Two identical student networks are trained independently on independent datasets drawn from the same teacher distribution.
- Analysis: The authors derive the asymptotic correlation ( $\rho^*$ ) between the hidden representations of the two networks at the global minimum. This allows for the analytic calculation of CCE as a function of Signal-to-Noise Ratio (SNR) and the ratio of training samples to input dimension ( $\alpha = n/d$ ).
Nonlinear Simulations (Numerical):
- Model: Two-layer networks with ReLU activation functions.
- Task: The same synthetic regression task as the linear setting.
- Comparison: Direct comparison of alignment and generalization error between linear and ReLU networks trained on the same data.
Real-World Classification:
- Datasets: CIFAR-10 (using a CNN) and MNIST (using a Fully Connected Neural Network).
- Protocol: Networks are trained on independent subsets of data with varying levels of label noise (acting as inverse SNR).
- Metric: CCE is computed between the penultimate layer representations.

Key Contributions

Theoretical Derivation: The paper establishes a theoretical framework for analyzing representational alignment in deep linear networks, deriving closed-form expressions for CCE based on the correlation of learned weights.
Governance Factors: The authors demonstrate that alignment is governed by two primary factors:
- Signal-to-Noise Ratio (SNR): Alignment increases monotonically with SNR across all regimes (linear and nonlinear, regression and classification).
- Sample Size: Alignment varies non-monotonically with training sample size. It attains a minimum near the interpolation threshold ( $n \approx d$ or $n \approx p$ ), mirroring the "double descent" behavior observed in generalization error.
Decoupling of Alignment and Generalization: The study provides evidence that high generalization performance does not necessarily imply high representational alignment.
- In the underparameterized regime, networks with different SNRs can achieve identical generalization errors but exhibit different degrees of alignment.
- In the overparameterized regime, networks with lower SNR (better generalization) may have less aligned representations than those with higher SNR.
Informativeness vs. Performance: The paper shows that a network can achieve higher task performance while learning less informative internal representations than a network with lower performance. This highlights that task performance is an insufficient proxy for representation quality.

Results

Linear Networks

Generalization Error: Exhibits the characteristic double descent curve, peaking at the interpolation threshold ( $\alpha = 1$ ).
Alignment (CCE): Also exhibits a non-monotonic dependence on sample size, reaching a minimum (near zero alignment) at the interpolation threshold.
SNR Dependence: Higher SNR consistently leads to stronger alignment. Notably, in the underparameterized regime ( $\alpha > 1$ ), different SNRs yield the same generalization error but distinct alignment levels.

Nonlinear Networks (ReLU)

Qualitative Similarity: The behavior of CCE and generalization error in ReLU networks qualitatively mirrors the linear case: non-monotonic dependence on sample size with a minimum near the interpolation threshold, and monotonic improvement with SNR.
Quantitative Differences: Unlike the linear case, the generalization error does not diverge at the interpolation threshold, and CCE does not collapse to zero. Instead, it converges to an SNR-dependent value, suggesting that even at the threshold, nonlinear networks retain some shared information.
Linear vs. ReLU Comparison: When trained on the same dataset, the linear network achieves lower generalization error (closer to the Bayes-optimal learner for this linear problem) but learns less informative representations (lower CCE) compared to the ReLU network. The ReLU network, despite higher error, captures richer structural information.

Real-World Data (CIFAR-10, MNIST)

Label Noise: Introducing label noise (inverse SNR) increases generalization error and reduces representational alignment.
Sample Size: The non-monotonic relationship between sample size and alignment holds, with a minimum near the interpolation threshold.
Consistency: The findings extend from synthetic linear settings to complex, real-world classification tasks using standard architectures.

Significance and Claims

The paper claims to provide a non-trivial dependence of alignment on data quality and quantity, which is decoupled from generalization performance. By introducing CCE, the authors offer a metric capable of capturing alignment on highly curved manifolds where traditional linear measures (CKA, SVCCA) may fail or report spurious similarities.

The core significance lies in challenging the assumption that better task performance equates to better or more aligned internal representations. The authors argue that:

Task performance is an insufficient proxy for representation quality. A network can outperform another in a specific task yet encode less structured or informative internal representations.
Alignment is not a universal indicator of "good" learning. The mechanisms driving alignment (SNR, sample size) operate independently of the mechanisms driving generalization error in certain regimes.

These findings contribute to building a principled theory of representation learning by identifying specific conditions (SNR and sample size relative to dimensionality) under which independently trained networks converge to similarly structured internal representations. The work suggests that the "Platonic" convergence of representations is not guaranteed but is strictly governed by the statistical properties of the data and the training regime.