Page 2 — The Pipeline

Run the DCT once per dimension. The output is a T x D matrix of coefficients ordered from DC (index 0) to the highest representable frequency (index T-1). Plot the variance of each coefficient across the training distribution and the curve falls off exponentially: the first few coefficients carry nearly all the signal energy, the tail is near zero. This is the same property JPEG exploits on image patches — and the paper invokes the analogy literally, not rhetorically:

"since pixels often vary smoothly, DCT can often represent most of the information of an input signal in only a few coefficients." (FAST §3)

For smooth motor trajectories the prior holds with little modification. The mechanical consequence is that information has now been sorted by importance. Any fixed-precision quantization applied next will land most bits where they matter and zero out the rest, without handcrafting a Q-table.

Sources: FAST §3 (arXiv 2501.09747).

After DCT, apply uniform scalar quantization per coefficient: a fixed scale, round to integer bins. Because the DCT has already sorted energy by frequency, a uniform step on coefficients becomes an implicitly resolution-aware operation on the original signal. Low-frequency coefficients retain real precision; high-frequency coefficients whose magnitude sits below the step size round to exactly zero. After quantization the matrix is dominated by zeros, with a small minority of nonzero low-frequency entries and a handful of mid-range ones. The paper puts the sparsity plainly:

"The DCT coefficient matrix is typically sparse, with most entries being zero and only a few significant coefficients remaining per action dimension." (FAST §3)

This sparsity is the precondition for the next stage. BPE's compression ratio on dense random integers would be mediocre; on a mostly-zero stream with repeating low-frequency patterns, it is substantial.

Sources: FAST §3 (arXiv 2501.09747).

Flatten the quantized integer matrix into a 1D sequence. Run BPE — byte-pair encoding trained once on a large action corpus. BPE is algorithmically agnostic to symbol meaning; it sees integers and counts adjacent-pair frequencies, greedily merging the most common pairs into new symbols. Runs of zeros collapse to single tokens; recurring low-frequency patterns that appear across the training distribution become motor-primitive tokens. The paper is direct about why BPE and not a better-compressing code:

"We choose BPE to compress the DCT matrix, since many efficient implementations exist and it can produce a fixed-size output vocabulary that can be easily integrated into the existing vocabulary of vision-language models." (FAST §3)

That last clause is the actual engineering win. Huffman would compress comparably, but BPE emits a fixed-size vocabulary (FAST uses 1024–2048 tokens) whose entries drop cleanly into a pretrained VLM's embedding table. The tokenizer is designed as an interface layer, not just a codec.

Sources: FAST §3 (arXiv 2501.09747); FAST §VI-D ablation notes.

End to end: a chunk of 50 timesteps by 7 dimensions enters as 350 floats. DCT reshapes it into a sparse 50×7 coefficient matrix. Quantization drops most of it to zero. BPE flattens and merges the result into 30 to 60 tokens drawn from a fixed 1024–2048 vocabulary (FAST paper numbers; FAST blog for chunk-level token count). The headline gains, as reported by PI: roughly 10× compression versus naive per-dim per-timestep binning, and roughly 5× training speedup on comparable tasks (Pertsch X thread, Jan 2025). Two framing notes, kept brief here and expanded on Page 3: the 5× is a training speedup, not inference — inference is in fact slower, at roughly 750 ms per chunk per the FAST paper itself — and the 10× compression comes with codebook utilization around 48%, per FASTer benchmarks. The pipeline is coherent; its deployment profile is where the honest caveats live.

Sources: FAST §3, §VI-D (arXiv 2501.09747); Pertsch X thread, Jan 2025; FASTer (arXiv 2512.04952) for 48% codebook utilization.