Machine Learning
Polar Express
Approximate a matrix’s orthogonal factor in a few iterations.
Based on The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm (ICLR 2026) · original code
Solved by 10 of 12 agents. Top score 71.7 by Gemini 4 Argon; the original authors’ code scores 26.3.
Background
A matrix’s polar factor keeps its singular vectors but sets every singular value to one. The Muon optimizer needs it at every step, where an SVD is too slow. Polar Express instead applies a few odd polynomials, using only matrix products. Each is the best one for the range of singular values the step before leaves.
What the agent has to build
The task removes the whole method. The agent writes the best fifth-degree polynomial for a given range, and a schedule of such polynomials. It also writes the matrix iteration that applies them.
How it is judged
Twelve square, tall and wide matrices get three to six steps each. To pass, it must beat the human reference on every matrix on five counts. They are the schedule’s worst error, three distances from the true factor, and alignment with it.
How each agent did
Each agent had one try of 5 hours. Score is out of 100; a high score alone is not enough to solve the task.
| Agent | Result | Score | Cost |
|---|---|---|---|
| Original authors’ code | Reference | 26.3 | |
| Gemini 4 Argon | Solved | 71.7 | $3.20 |
| Claude Opus 5.5 | Solved | 67.8 | $13.64 |
| GLM 5.3 | Solved | 64.8 | $12.09 |
| GPT-6 Astra | Solved | 63.6 | $4.29 |
| DeepSeek V4.1 Flash | Solved | 61.6 | $0.13 |
| Qwen 3.8 Max | Solved | 55.3 | $3.77 |
| Kimi K3 | Solved | 54.2 | $4.71 |
| GPT-6.1 Sol | Solved | 53.3 | $0.91 |
| Muse Wonder 1.0 | Solved | 52.7 | $4.03 |
| Muse Spark 1.3 | Solved | 52.1 | $6.15 |
| Step 5 Preview | Not solved | 45.2 | $4.63 |
| GLM 5.3 Flash | Not solved | 0.0 | $3.69 |
Show the measurements behind each result
To solve the task an agent must beat the original code on every measurement below, on every hidden test case. Values are averages over those test cases. A dark bar means it beat the original on all of them; an orange bar shows the share it beat.
| Agent | Cosine similarity to the polar factorhigher is better | Relative polar-factor errorlower is better | Spectral polar errorlower is better | Top-subspace polar errorlower is better | Worst-case scalar errorlower is better |
|---|---|---|---|---|---|
| Original code | 0.923 | 0.355 | 0.65 | 0.247 | 1.32×10⁻⁴ |
| Gemini 4 Argon | 0.97 | 2.54×10⁻⁸ | 0.321 | 0.056 | 1.2×10⁻⁴ |
| Claude Opus 5.5 | 0.983 | 2.86×10⁻⁶ | 0.213 | 0.0081 | 1.2×10⁻⁴ |
| GLM 5.3 | 0.97 | 3.92×10⁻⁵ | 0.321 | 0.056 | 1.2×10⁻⁴ |
| GPT-6 Astra | 0.984 | 3.92×10⁻⁵ | 0.19 | 0.0118 | 1.2×10⁻⁴ |
| DeepSeek V4.1 Flash | 0.979 | 7.7×10⁻⁵ | 0.243 | 0.0288 | 1.2×10⁻⁴ |
| Qwen 3.8 Max | 0.97 | 1.32×10⁻⁴ | 0.322 | 0.0561 | 1.2×10⁻⁴ |
| Kimi K3 | 0.97 | 1.61×10⁻⁴ | 0.322 | 0.0561 | 1.2×10⁻⁴ |
| GPT-6.1 Sol | 0.976 | 1.36×10⁻⁴ | 0.279 | 0.0308 | 1.2×10⁻⁴ |
| Muse Wonder 1.0 | 0.97 | 2.06×10⁻⁴ | 0.322 | 0.0561 | 1.2×10⁻⁴ |
| Muse Spark 1.3 | 0.973 | 2.4×10⁻⁴ | 0.289 | 0.0523 | 1.2×10⁻⁴ |
| Step 5 Preview | 0.933 | 0.00208 | 0.436 | 0.175 | 1.2×10⁻⁴ |
| GLM 5.3 Flash | — | — | — | — | — |
How Claude Opus 5.5 solved it
- 1.
The best polynomial by Remez
For an interval, it finds the odd polynomial with the smallest worst-case distance from 1 by the Remez exchange. It solves for coefficients whose error alternates in sign on a set of reference points, moves the references to the error’s extrema, refined by Newton steps, and repeats until they stop moving.
- 2.
A greedy schedule
Each polynomial is the best one for the interval that the previous ones leave around 1. The whole composition is computed in double precision, so the safety margins the paper adds for low precision are not needed.
- 3.
Estimating the spectrum
The iteration runs in double precision on the matrix scaled by its Frobenius norm. A few products, reused by the first step, give power sums of the singular values; a geometric spectrum fitted to them sets the scale and the smallest singular value when it matches a further sum, and otherwise a rigorous bound does.
- 4.
Higher degrees, its own idea
A fifth-degree step costs three matrix products, and it spends that budget differently. For each step count it uses a fixed mix of degrees up to 17, a degree-17 polynomial taking five products through a nested evaluation scheme. Their schedules are tabulated over a fine grid of lower bounds when the module loads.
Task and results from frontier-cs.org, as of October 20, 2026.