Stage-wise Sparse Coding with the Out-of-Place Fast Walsh–Hadamard Transform (FWHT)

Slide 1 — Key idea

  • Run the out-of-place FWHT and inspect all intermediate stage outputs (not just the final transform).

  • Treat each large-magnitude entry observed at any intermediate stage as a coefficient for a corresponding basis vector (the column of the stage operator).

  • Keep the top kk candidates (or greedily select + remove, iteratively) and use them for sparse reconstruction.

Why do this?

  • Intermediate stages expose coarse-to-fine structure and may reveal transient, locally large components missed by looking only at the final basis ordering.

  • The out-of-place schedule with contiguous sums/diffs gives convenient, streaming-friendly access to large groups of coefficients at each stage.

Consequence:

  • If you select coefficients only from the final transform (normalized Hn/nH_n/\sqrt{n}), the squared-error from zeroing the rest equals the sum of omitted squared coefficients (Parseval).

  • If you select coefficients coming from arbitrary intermediate rows, the simple sum-of-squares error bound no longer applies because these selected rows are not an orthonormal set — you are effectively selecting from a (redundant) frame. That makes reconstruction and error analysis different.


Additional Note — Extended Intermediate Space for Compression

  • The out-of-place FWHT is self-inverse: applying the same transform matrix S repeatedly will cycle the data back to the original input after 2log⁡2N2 \log_2 N stages.

  • This means we can treat the transform process not just as a single pass, but as a loop through multiple intermediate states:

    1. Original input data.

    2. All intermediate arrays from the forward FWHT stages.

    3. Final FWHT output (full transform domain).

    4. All intermediate arrays from the inverse FWHT stages.

  • Each of these intermediate states can contain high-magnitude coefficients that do not appear prominently in the final transform.

  • By scanning the entire cycle of intermediate states for large-magnitude values, we enlarge the candidate set for sparse selection or iterative max-removal compression.

  • This effectively blends forward and inverse domain sparsity into a single framework, potentially increasing compression efficiency for signals with structure that is not strictly concentrated in the standard transform output.


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