According to the argmin.net blog post, OpenAI recently added a mathematics repository that includes a paper titled “main.pdf” claiming to compute faster Fourier transforms. The post explains that the classic Fast Fourier Transform (FFT) devised by James Cooley and John Tukey in 1965 computes a length‑n discrete Fourier transform in fewer than 5 n log₂ n arithmetic operations, giving a 20‑fold speedup for n=1,000 and a 10,000‑fold speedup for n=1,000,000 compared with the naïve n² approach.
The OpenAI paper asserts that its algorithm improves on the FFT by reducing the logarithmic factor: the log(n) term is, in theory, a teeny‑tiny bit smaller than in the standard FFT. This would make the algorithm asymptotically faster in big‑O notation.
However, the post notes that the improvement relies on galactic‑algorithm reasoning. It cites the hidden constant C in the big‑O bound as being estimated at 10¹⁰⁰ when the threshold N is 10¹². With such a constant, the OpenAI method would only outperform the Cooley‑Tukey‑Gauss FFT for impossibly large inputs—far beyond any realistic problem size.
The discussion points out that the algorithm’s analysis assumes exact complex arithmetic with arbitrarily large magnitudes, effectively requiring memory that grows exponentially with input length and the ability to multiply super‑polynomially large complex numbers in a single operation. No existing computer hardware can meet these requirements.
The post also references earlier theoretical limits: Jacques Morgenstern’s 1973 proof that a Fourier transform cannot be faster than ½ n log n if coefficient magnitudes are bounded, and Nir Ailon’s later result showing that beating n log n leads to severe numerical instabilities. Both results were considered restrictive by some researchers, prompting continued investigation.
In summary, while the OpenAI paper presents a theoretically asymptotic improvement, the blog post characterizes it as more of a curiosity in algorithmic theory than a practical advance, because the claimed speedup depends on assumptions that do not correspond to actual computing machinery.
Key facts
- OpenAI posted a math repository with a paper claiming a faster Fourier transform.
- The classic FFT runs in fewer than 5 n log₂ n operations.
- The OpenAI claim reduces the logarithmic factor slightly, making it asymptotically faster in big‑O terms.
- The hidden constant in the OpenAI bound is estimated at 10¹⁰⁰ for N=10¹², rendering the method impractical for any realistic input size.
- The algorithm assumes exact complex arithmetic with unbounded precision, which no real computer can provide.
- Earlier work by Morgenstern (1973) and Ailon showed limits on FFT speed under realistic assumptions.