Alpha-Stable Normalize

The AlphaStableNormalize transform applies a per-row robust affine normalization (v - δ) / γ to 2D tensors, backed by an exact XOR residual to guarantee lossless reconstruction.

Theory

This transform isolates heavy-tailed, row-wise scales and offsets. It calculates:

• δ: The median of the row.
• γ: The median absolute deviation (MAD) from δ.

By using the median and MAD instead of the mean and variance, the fit remains robust against extreme outliers. The transform splits the input into three planes:

1. Scale: The (γ, δ) pair for each row, stored as f32.
2. Normalized: The normalized elements (v - δ) / γ, quantized to bf16.
3. Residual: The bit-exact XOR difference between the original float bytes and the bf16-reconstructed bytes.

Because the γ and δ parameters are stored per row and the residual repairs all round-off errors, reconstruction is bit-exact regardless of the underlying distribution.

Usage

The transform requires the input to have a Rows layout and supports f32, bf16, and fp16 dtypes.

References

• This robust affine normalization approach (using median and MAD with exact XOR residual) is an original contribution to PTWM, inspired by robust statistics for heavy-tailed distributions.