---
title: "USC's Zeta Law Predicts Exactly How Much Data Biomedical AI Needs"
subtitle: "A new theoretical framework ties biomedical AI scaling curves to the Riemann zeta function, predicting when more data, better encoders, or extra modalities will pay off."
url: "https://alphasignal.ai/news/usc-s-zeta-law-predicts-exactly-how-much-data-biomedical-ai-needs"
author: "AlphaSignal Newsroom"
author_url: "https://alphasignal.ai/editorial-team"
company: "AlphaSignal"
company_url: "https://alphasignal.ai/company/alphasignal"
published: "2026-04-19T19:08:53.000Z"
updated: "2026-08-21T19:52:29.913Z"
type: "Paper"
topics:
  - "Data"
  - "Benchmarks"
subtopics:
  - "Datasets"
  - "Synthetic Data"
upvotes: 22
source_url: "https://arxiv.org/abs/2604.17581"
comments: 0
truncated: true
---

# USC's Zeta Law Predicts Exactly How Much Data Biomedical AI Needs

> A new theoretical framework ties biomedical AI scaling curves to the Riemann zeta function, predicting when more data, better encoders, or extra modalities will pay off.

How much data does it actually take to make a scientific discovery? A new theoretical paper from Paul Thompson at the University of Southern California's Stevens Institute for Neuroimaging & Informatics proposes a scaling-law framework called the [zeta law of discoverability](https://arxiv.org/abs/2604.17581), linking the accuracy of biomedical classifiers to a truncated form of the Riemann zeta function, the object at the center of one of the oldest unsolved problems in mathematics.

## Why trial-and-error benchmarking keeps failing

Biomedical AI has a scaling problem. Consortia are pooling millions of brain scans and genomes, models keep growing, and yet advances often rely on trial-and-error benchmarking across architectures, modalities, and sample sizes, with little theoretical guidance on when performance should jump and when it will flatten. A team running a diagnostic study on Alzheimer's, schizophrenia, or a rare psychiatric condition rarely knows whether to collect more subjects, swap the encoder, or add a new modality such as clinical text.

Scaling laws borrowed from language modeling do not translate cleanly. Biomedical signals are high-dimensional, heterogeneous across modalities, and often spread across many weak patterns rather than a few strong ones.

## Signal that piles up like a zeta sum

The core idea: many performance metrics, including AUC, can be written as a sum of signal-to-noise contributions from the eigenmodes (the coherent patterns of variation) of the data covariance. Each mode contributes signal proportional to how well it aligns with the disease contrast, divided by its noise variance. With finite data you can only trust the top

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## Sources

- [How Much Data is Enough? The Zeta Law of Discoverability in Biomedical Data](https://arxiv.org/abs/2604.17581)
- [Zeta Law paper (HTML)](https://arxiv.org/html/2604.17581)
- [ENIGMA Consortium](https://enigma.ini.usc.edu)
- [Learning Transferable Visual Models From Natural Language Supervision (CLIP)](https://arxiv.org/abs/2103.00020)
- [Situating the default-mode network along a principal gradient of macroscale cortical organization](https://doi.org/10.1073/pnas.1608282113)