The 97%-rank update that sits 4.5× below noise

A fine-tuning update can use almost every available direction and still put most of its energy into a small head. Those statements sound contradictory only if “rank” is treated as one number. It is not: rank measures ask different questions of the same singular values.

For an update matrix ΔW, write its singular values as σ₁ ≥ … ≥ σᵣ. Singular values tell us how strongly the update acts along orthogonal directions. The threshold rank counts every σᵢ at least 1% as large as σ₁. Stable rank instead divides the squared Frobenius norm—the total spectral energy—by σ₁². The paper reports both as fractions of the maximum possible rank r.

def rank_summary(singular_values):
    values = sorted(singular_values, reverse=True)
    energy = [value * value for value in values]
    total = sum(energy)
    shares = [value / total for value in energy]
    threshold = sum(value >= 0.01 * values[0] for value in values) / len(values)
    stable = (total / (values[0] * values[0])) / len(values)
    return threshold, stable, shares[0]

def gap_below_noise(update, noise):
    return 1.0 - update / noise

The TypeScript implementation above drives this page; Python and C++ are faithful translations. Our eight-direction toy has one singular value of 10 and seven of 1. Every direction clears the 1% threshold, just as in the flat spectrum, but the largest direction holds most of the energy.

The paper evaluates five summaries on each attention and MLP projection. Moving from threshold count to entropy effective rank, 90%-energy rank, participation ratio, and stable rank progressively gives the strongest directions more influence. The raw values must not be compared as if they shared a scale; their distance from the same-shape noise baseline is the informative quantity.

Threshold counting places the SFT update only 2.3% below noise. Stable rank puts it 77.8% below noise—about 4.5 times smaller in the paper's unrounded calculation. The update's first direction holds 4.2% of its energy versus 0.6% for noise, while many weak directions remain to be counted. That is the entire apparent contradiction.

A statistic can separate fine-tuning from Gaussian noise and still fail to identify what the model learned. On Qwen, supervised instruction tuning, Wikipedia, and Python code have stable ranks 0.097, 0.093, and 0.094. More sharply, DPO—Direct Preference Optimization—looks almost identical when half its preference labels are swapped, removing the net preference signal.

The strangest control is shuffled Wikipedia: destroying word order makes the update more concentrated, with stable rank 0.042 on Qwen and 0.009 on SmolLM2. The paper does not explain why. That unexplained result is exactly why “concentrated update” cannot be promoted to “alignment lives in a few directions.”

There is a second trap below the metric choice. Released checkpoints are often stored in bfloat16, which keeps only eight significant bits. If an update is smaller than half a storage unit, subtraction returns zero. In the author's first low-precision runs, 77–98% of weights appeared unchanged; keeping master weights in fp32 reduced that fraction below 1.01 × 10⁻⁵.

The study repeats most fine-tunes only twice, its shuffled-word result is unexplained, and its editing audit is narrow: one model, non-thinking mode, and 198 GPQA questions with intervals around ±3 points. A stronger follow-up would vary optimizers, training duration, and precision while connecting individual spectral directions to behavior rather than only their count.

The practical rule is modest: treat spectral rank as a description, not a semantic label. If you want to see the architecture that produces these attention and MLP matrices, the site's interactive Transformer walkthrough exposes those components directly.