📄 molecular biology

The projection basis determines the information ceiling for perturbation prediction

The paper demonstrates that the predictive performance of deep-learning models for perturbation responses is fundamentally limited by the information captured in the projection basis, revealing that while PCA effectively models chemical perturbations by capturing high-variance modes, network-based bases are superior for genetic perturbations, and graph wavelets can unify both by accessing the full graph spectrum.

Original authors: BIANCO, S.

Published 2026-07-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: BIANCO, S.

Original paper licensed under CC BY 4.0 (https://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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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

Imagine you are trying to predict how a city will change after a specific event, like a new law or a natural disaster. You have a massive amount of data about the city: traffic patterns, power usage, weather, and social media posts. To make a prediction, you first need to summarize this huge amount of information into a smaller, manageable "map."

This paper argues that the quality of your map matters more than the complexity of the weatherman (the AI model) trying to read it.

Here is the breakdown of the paper's findings using simple analogies:

1. The "Information Ceiling" (The Map Limit)

The authors discovered a hard rule: You cannot predict what you cannot see.

If you use a map that only shows the major highways (a "low-resolution" map), you will miss the side streets where the actual changes are happening. No matter how smart your AI is, if you feed it a map that has thrown away 90% of the details, the AI can only guess at best.

The Analogy: Imagine trying to guess the flavor of a complex soup, but you are only allowed to taste the water it was boiled in. No matter how sophisticated your taste buds (the AI model) are, you will never get the right answer because the flavor (the signal) was filtered out before it reached you.
The Math: The paper proves that the accuracy of a prediction is capped by how much of the original "variance" (the interesting changes) your map preserves. If your map keeps only 10% of the information, your prediction accuracy can never exceed the ceiling set by that 10%.

2. The Drug Problem: "Static Noise" vs. "Smooth Waves"

The researchers tested this on chemical drugs (like medicine).

The Old Way (PCA): Most scientists use a method called PCA to make their map. Think of PCA as a "wide-angle lens" that captures the biggest, most obvious trends in the data. For chemical drugs, this works great because drugs often cause widespread, scattered changes across the cell, like static noise on a radio. The wide-angle lens catches most of this noise.
The "Biological" Way (Network Basis): Some scientists tried to use a map based on a "gene network" (a map of how genes talk to each other). They thought this would be smarter. However, this map acts like a low-pass filter. It only lets through "smooth, slow waves" of information and blocks out the "sharp, fast spikes."
The Result: Chemical drugs create "sharp spikes" (scattered, high-frequency changes) in the cell. The biological map filtered these out, leaving the AI with a blank slate. The AI performed no better than random guessing. The "smart" biological map actually threw away the most important clues.

3. The Solution: The "Graph Wavelet" (The Multi-Scale Map)

The authors asked: Can we keep the biological map but fix the filter?

They tried a new tool called Graph Wavelets. Imagine this as a map that doesn't just show smooth waves; it also zooms in to show the sharp spikes and high-frequency details.
The Result: When they used this new map, the AI suddenly became very good at predicting drug effects. It recovered the "sharp spikes" that the old biological map had thrown away. This proved that the biological structure did contain the answer, but the old map was too "blurry" to see it.

4. The Twist: Genetic Perturbations (The "Domino Effect")

The paper then tested a different type of change: CRISPR gene editing (turning specific genes on or off).

The Difference: Unlike drugs, which hit scattered targets, turning on a gene is like pushing the first domino in a long line. The effect ripples through the network in a smooth, connected chain.
The Inversion: In this case, the "smooth wave" map (the biological network basis) was actually the best map. It captured the "ripple" perfectly. The "wide-angle lens" (PCA) was too broad and missed the specific direction of the ripple.
The Lesson: There is no single "best" map.
- For drugs (scattered noise), you need a map that catches high-frequency details (like PCA or Wavelets).
- For gene editing (smooth ripples), you need a map that understands the network connections (the biological basis).

Summary

The paper concludes that the choice of "map" (projection basis) is the most important decision, not the complexity of the AI model.

If you use a map that filters out the signal, even the most powerful supercomputer will fail.
If you use a map that matches the type of change (scattered noise vs. smooth ripples), even a simple model can succeed.
The "biological" way isn't wrong; it just needs the right "lens" (like Graph Wavelets) to see the whole picture without throwing away the high-frequency details.

In short: You can't build a better prediction by building a smarter brain if you are feeding it a blurry picture. You need to fix the picture first.

Technical Summary: The Projection Basis Determines the Information Ceiling for Perturbation Prediction

Problem Statement
Recent benchmarks indicate that deep-learning models for single-cell perturbation prediction (e.g., GEARS, CPA, scGen) often fail to outperform simple linear baselines operating in Principal Component Analysis (PCA) space. While previous work has attributed this to insufficient model capacity or a lack of appropriate inductive bias, this paper posits that the primary limitation lies not in the model architecture, but in the projection basis itself. The authors argue that any method reducing gene expression to a fixed low-dimensional coordinate system before prediction passes the signal through a linear bottleneck, creating an information-theoretic ceiling that no downstream model complexity can overcome.

Methodology and Framework
The study employs a Waddington Potential Network (WPN) framework, inspired by the Waddington epigenetic landscape, to isolate the effects of the projection basis. In this framework, cellular velocity is parameterized as the gradient of a learned scalar potential ( $v = -\nabla U$ ). The model projects gene expression onto a $k$ -dimensional basis $\Phi$ , learns a potential over these coordinates, and derives velocity via automatic differentiation. Crucially, the authors hold all hyperparameters and architectures constant, varying only the projection basis to ensure observed performance differences are attributable solely to the basis choice.

The core theoretical contribution is a projection bound derived from information theory:
$r^2 \leq VE(\Phi) := \frac{\text{Var}(\Phi\Phi^\top \delta)}{\text{Var}(\delta)}$
where $r^2$ is the squared correlation between prediction and truth, and $VE(\Phi)$ is the fraction of task-relevant variance (drug-response delta $\delta$ ) explained by the basis. This bound establishes that if a basis discards signal (low $VE$), no model can recover it. The authors distinguish between Information Availability (the variance preserved by the basis, $VE$) and Information Exploitation (how well a model utilizes the available signal, $r^2/VE$ ).

Experimental Design
The authors validate this framework through three complementary tests across multiple datasets (sciPlex3, LINCS L1000, and Norman et al. 2019 CRISPRa) and cell lines (K562, A549, MCF7):

Test 1 (Chemical Perturbations): Evaluates whether a low-variance basis (gene-association network eigenbasis) can support above-chance prediction compared to PCA.
Test 2 (Spectral Recovery): Tests if expanding the basis to include high-frequency modes (via graph wavelets) on the same network topology recovers the missing signal.
Test 3 (Genetic Perturbations): Investigates if the ranking inverts for CRISPRa genetic perturbations, where responses may align with network geometry rather than being spectrally diffuse.

Mechanistic controls include degree-preserving network permutations, synthetic ground-truth networks, and PCA fitted solely on untreated control cells to rule out circularity and degree-distribution artifacts.

Key Results

Chemical Perturbations (SciPlex3 & LINCS L1000):
- The eigenbasis of a composite gene-association network (GRN) explains only 10–12% of drug-response variance, whereas PCA explains 90–99%.
- Models using the GRN eigenbasis perform at or below chance levels ( $r \approx 0$ ), while PCA-based models achieve significant accuracy.
- This failure is attributed to the spectral diffuseness of drug responses: chemical perturbations act as "impulse noise" affecting sparse, scattered targets, generating high-frequency signals on the graph that the low-pass network eigenbasis discards.
- Wavelet Rescue: When the same network topology is used with graph wavelets (band-pass or diffusion) that access the full spectrum, the variance explained rises to ~88%, and prediction performance recovers to match PCA. This confirms the failure was a spectral-range limitation, not a topology failure.
Genetic Perturbations (CRISPRa):
- The ranking inverts for single-gene CRISPRa activations. Here, the GRN eigenbasis (explaining ~19% of variance) outperforms PCA in parameterized models (WPN and MLP), despite PCA explaining 100% of the variance.
- This suggests that genetic perturbations propagate along regulatory cascades, aligning with the low-frequency, pathway-coherent modes captured by the network basis. The network geometry acts as an informative inductive bias that parameterized models can exploit, even with lower variance coverage.
Architecture Independence:
- The performance gap is consistent across architectures, including the WPN, simple residual MLPs, and variational autoencoders. Even a zero-parameter mean predictor follows the $r^2 \leq VE$ bound, confirming that the ceiling is set by the projection, not the model.

Significance and Claims
The paper claims to resolve the paradox of why biologically motivated network bases often underperform simple PCA baselines in drug perturbation tasks. The authors assert that:

Basis Choice is the Primary Bottleneck: For methods with fixed linear bottlenecks, the projection basis sets a hard information ceiling ($VE$) that dictates the maximum achievable accuracy.
Modality Dependence: The optimal basis depends on the perturbation modality. PCA captures the variance driving chemical responses (spectrally diffuse), while network bases capture the cascade structure driving genetic responses (spectrally coherent).
Spectral Coverage is Critical: Standard network eigenbases act as low-pass filters that discard high-frequency drug signals. Methods like graph wavelets, which span the full graph spectrum, can recover biological structure without sacrificing variance coverage.
Reframing Benchmarks: The authors propose that future benchmarks must report the Variance Explained (VE) of a method's internal representation on task-relevant variation, separating "availability" (basis quality) from "exploitation" (model capability). They suggest that the lack of improvement in full-gene models (like GEARS) over PCA baselines may be due to exploitation limits at current sample sizes, rather than availability limits.

The work concludes that basis selection is a modality-dependent design decision with a provable ceiling, and that graph wavelets offer a concrete path to incorporating biological structure without the variance loss inherent in standard low-pass eigenbases.