Aperture: Merge-Consistent Rotary States for Compressed Tokens
Organizations: The Chinese University of Hong Kong, Shenzhen · Shenzhen Loop Area Institute
Abstract
Token compression combines content from several positions, yet rotary position embeddings usually assign the merged token one coordinate. We ask what positional information must survive later merges. Aperture stores Fourier moments of the token's weighted support at the model's rotary frequencies. We prove that these moments have minimal real dimension among continuous states sufficient for the selected expected rotary interactions. Represented mass makes updates additive; attention normalisation remains a separate readout choice. Uniform intervals give a centre rotation times a sinc gain. We characterise when centres determine interval widths and construct matched examples where they do not. Numerical checks verify the weighted-support implementation. In trained temporal readers, compression transfer varies with gain calibration and feature placement. In a prespecified native video question-answering comparison, stored support reaches accuracy versus for the deployed merging rule. These results separate exact positional preservation under compression from downstream benefit.
Figures & tables
| Readout | Tree discrepancy | Affine defect |
|---|---|---|
| Centre RoPE | ||
| exact Aperture | ||
| Aperture RMS | ||
| MN-sinc | ||
| Hard cutoff |
| Observation | ||
|---|---|---|
| Relative offsets only | 1 | 1 |
| Absolute centres | 1 | 1 |
| Absolute centres, left boundary | 0 | 0 |
| Absolute centres, total span | 1 | 0 |
| Paired accuracy contrast | 3B | 7B |
|---|---|---|
| exact Aperture centre only | ||
| exact Aperture width feature | ||
| exact Aperture MN-sinc | ||
| Matched-RMS MN-sinc |
| Comparison | Difference | 95% seed interval | 95% crossed bootstrap |
|---|---|---|---|
| Qwen2.5-Omni-3B: unseen policy | |||
| Qwen2.5-Omni-3B: unseen rate | |||
| Qwen2.5-Omni-3B: width feature | |||
| Qwen2.5-Omni-7B: unseen policy | |||
| Qwen2.5-VL-7B: spatial / PPE |
Appendix figures & tables32 assets
Supplementary material from the paper’s appendix.
Appendix
| Code | A1: affinity | A2: translation | Violated |
|---|---|---|---|
| Aperture | — | ||
| Centre only | A1 | ||
| Hard cutoff | A1 | ||
| Aperture RMS | A1 | ||
| MN-sinc | A1 | ||
| Width-capped phase | A1+A2 |
| Support domain | Tree / quadrature error | Interval-summary error |
|---|---|---|
| 1D mixtures | ||
| 2D mixtures |
| Crossing / payload | Cell accuracies | Paired effects |
|---|---|---|
| Overlap / metadata Centre | TR: 0.526 [0.502, 0.550] TM: 0.646 [0.628, 0.665] GR: 0.526 [0.502, 0.550] GM: 0.646 [0.628, 0.665] | G: 0.000 [0.000, 0.000] M: 0.121 [0.113, 0.128] I: 0.000 [0.000, 0.000] |
| Overlap / metadata Stored support | TR: 0.521 [0.507, 0.536] TM: 0.636 [0.622, 0.650] GR: 0.523 [0.508, 0.537] GM: 0.640 [0.623, 0.656] | G: 0.002 [0.000, 0.005] M: 0.116 [0.110, 0.122] I: 0.002 [-0.002, 0.005] |
| Gap / metadata Centre | TR: 0.527 [0.506, 0.548] TM: 0.651 [0.631, 0.670] GR: 0.527 [0.506, 0.548] GM: 0.651 [0.631, 0.670] | G: 0.000 [0.000, 0.000] M: 0.124 [0.117, 0.131] I: 0.000 [0.000, 0.000] |
| Gap / metadata Stored support | TR: 0.524 [0.507, 0.542] TM: 0.641 [0.626, 0.657] GR: 0.523 [0.505, 0.541] GM: 0.642 [0.625, 0.659] | G: 0.000 [-0.002, 0.001] M: 0.118 [0.112, 0.125] I: 0.002 [-0.001, 0.005] |
| Overlap / physical Centre | TR: 0.521 [0.506, 0.537] TM: 0.651 [0.635, 0.666] GR: 0.524 [0.510, 0.539] GM: 0.656 [0.642, 0.670] | G: 0.004 [0.001, 0.006] M: 0.130 [0.124, 0.136] I: 0.002 [-0.002, 0.006] |
| Overlap / physical Stored support | TR: 0.524 [0.514, 0.534] TM: 0.646 [0.630, 0.662] GR: 0.529 [0.519, 0.540] GM: 0.652 [0.636, 0.668] | G: 0.006 [0.003, 0.009] M: 0.122 [0.113, 0.131] I: 0.001 [-0.006, 0.007] |
| Arm | Overall | 80-cell range | Policy cell | Rate cell |
|---|---|---|---|---|
| G2: Qwen2.5-Omni-3B (13 arms) | ||||
| centre | 0.084 | 0.083–0.084 | 0.084 [0.081, 0.087] | 0.083 [0.080, 0.087] |
| stored_support | 0.083 | 0.083–0.084 | 0.084 [0.079, 0.090] | 0.084 [0.078, 0.090] |
| exact_interval † | 0.088 | 0.087–0.088 | 0.088 [0.080, 0.096] | 0.088 [0.080, 0.096] |
| learned_interval | 0.084 | 0.084–0.085 | 0.084 [0.082, 0.087] | 0.085 [0.083, 0.087] |
| start_end | 0.083 | 0.083–0.084 | 0.083 [0.079, 0.088] | 0.083 [0.078, 0.088] |
| arm | final train loss | test mean | chance |
|---|---|---|---|
| centre-only ( centre ) | |||
| stored support ( stored_support ) | |||
| exact Aperture ( exact_interval ) | |||
| learned interval ( learned_interval ) | |||
| PPE-ranked ( ppe_ranked ) |
| Model / retention / rate | Readout | Median / p90 | Peak | State |
|---|---|---|---|---|
| 3B / 0.5 / 16 fps | Centre-only | 6.61 / 6.87 | 24.08 | 48 |
| 3B / 0.5 / 16 fps | Stored support | 6.60 / 6.88 | 24.08 | 17 |
| 3B / 0.25 / 8 fps | Centre-only | 5.19 / 5.46 | 22.48 | 48 |
| 3B / 0.25 / 8 fps | Stored support | 5.20 / 5.44 | 22.48 | 4.5 |
| 7B / 0.25 / 8 fps | Centre-only | 5.19 / 5.42 | 34.71 | 48 |
| 7B / 0.25 / 8 fps | Stored support | 5.17 / 5.42 | 34.71 | 4.5 |
| Method | Accuracy | Med./p90 | Req./s | GPU | KV/state |
|---|---|---|---|---|---|
| Centre | 0.676 | 930.8/4144.2 | 0.46 | 25.51 | 320.41/0.527 |
| Stored support | 0.692 | 1333.8/8622.2 | 0.20 | 25.51 | 320.41/0.527 |
| PPE | 0.777 | 984.1/3194.9 | 0.30 | 25.51 | 320.41/4.237 |
| Readout | Mean tokens | Median tokens | Requests at cap |
|---|---|---|---|
| Centre-only | 4.36 | 2 | 8 |
| Stored support | 12.86 | 3 | 79 |
| PPE | 8.95 | 3 | 56 |
| Code | Contrast | Estimate | 95% interval |
|---|---|---|---|
| Aperture | |||
| Aperture | |||
| Aperture | |||
| Aperture RMS | |||
| Aperture RMS | |||
| Aperture RMS |
| Code | 8 fps | 32 fps | Minimum at 32 fps | |
|---|---|---|---|---|
| Token index | 5 | 0.174 | ||
| Interleaved index | 5 | 0.195 | ||
| Centre only | 15 | 0.677 | ||
| Width feature | 15 | 0.333 | ||
| Aperture | 15 | 0.662 | ||
| Aperture RMS | 15 | 0.981 |
| Trained on tilings | ||||
|---|---|---|---|---|
| Code | Retain 1.00 | Retain 0.50 | Retain 0.25 | |
| Token index | 5 | |||
| Interleaved index | 5 | |||
| Centre only | 5 | |||
| Width feature | 5 | |||
| Aperture | 5 | |||
| Training | Code | 32 fps | 64 fps | 128 fps | |
|---|---|---|---|---|---|
| Single | Aperture | 15 | |||
| Single | Learned gain | 15 | |||
| Single | Width feature | 15 | |||
| Covered | Aperture | 15 | |||
| Covered | Learned gain | 15 | |||
| Covered | Width feature | 15 |
| Code | |||||
|---|---|---|---|---|---|
| Centre only | 24 | ||||
| Width feature | 24 | ||||
| Aperture | 48 | ||||
| MN-sinc | 24 | ||||
| Width-capped phase | 24 | ||||
| Aperture RMS | 24 |
| Code | |||||
|---|---|---|---|---|---|
| Centre only | 24 | ||||
| Width feature | 24 | ||||
| Aperture | 24 | ||||
| MN-sinc | 24 | ||||
| Aperture RMS | 24 | ||||
| Aperture, matched RMS | 24 |
| Budget | ||||
|---|---|---|---|---|
| Aperture clock, B | ||||
| Aperture clock, B | ||||
| Aperture RMS clock, B |
| (a) Width reconstruction error / | ||||
| Policy | Streams | Recursion | Linear, | Linear, |
| Contiguous tiling | 2048 | |||
| Unanchored window | 2048 | |||
| Overlap, | 2048 | |||
| Overlap, | 2048 | |||
| Retain | 2048 | |||
| kernel of the width offset map | minimax floor / | ||||||
|---|---|---|---|---|---|---|---|
| Identifiable | Full span | ||||||
| 4 | 1 | 3.9 | 0.000 | 0.880 | — | — | 0.480 |
| 8 | 1 | 8.4 | 0.000 | 0.813 | 0.318 | — | 0.246 |
| 16 | 1 | 16.5 | 0.000 | 0.781 | 0.269 | — | 0.121 |
| 32 | 1 | 32.6 | 0.000 | 0.767 | 0.249 | 0.085 | 0.062 |
| 64 | 1 | 64.4 | 0.000 | 0.759 | 0.237 | 0.073 | 0.030 |
| Regime | ||||||||
|---|---|---|---|---|---|---|---|---|
| Tiling | 4 | 0.518 | 0.554 | — | — | — | 0.554 | 2 |
| 8 | 0.580 | 0.660 | 0.676 | — | — | 0.676 | 4 | |
| 16 | 0.618 | 0.726 | 0.758 | 0.759 | — | 0.759 | 4 | |
| 32 | 0.648 | 0.764 | 0.812 | 0.815 | 0.813 | 0.815 | 4 | |
| 64 | 0.656 | 0.783 | 0.849 | 0.860 | 0.860 | 0.860 | 8 | |
| Eviction | 4 | 0.120 | 0.131 | — | — | — | 0.131 | 2 |
| Policy | Support | Content | Aperture | Aperture RMS |
|---|---|---|---|---|
| Coarsen | Tiling | Retained | ||
| Dilate | Overlap | Retained | ||
| Mask | Tiling | Masked | ||
| Evict | Gaps | Removed |
| Code | Accuracy | ||
|---|---|---|---|
| Coarsen | |||
| Token index | 15 | 0.4434 (0.0210) | +0.0000 (0.0000) |
| Centre only | 15 | 0.4643 (0.0166) | +0.0000 (0.0000) |
| Width feature | 15 | 0.5214 (0.0242) | +0.2626 (0.0298) |
| Hard cutoff RMS | 5 | 0.5213 (0.0309) | +0.1678 (0.0305) |
| Aperture | 15 | 0.5174 (0.0210) | +0.1696 (0.0220) |
| Retained | Masked tokens | Centre only | Aperture RMS | |
|---|---|---|---|---|
| 1 | 0.00 | 0.4643 | 0.5238 | +0.2049 (0.0233) |
| 0.875 | 1.34 | 0.4168 | 0.4748 | +0.1316 (0.0394) |
| 0.75 | 3.33 | 0.4137 | 0.4250 | +0.0990 (0.0245) |
| 0.5 | 7.14 | 0.4627 | 0.4932 | +0.1265 (0.0378) |
| 0.25 | 10.58 | 0.3750 | 0.3969 | +0.1033 (0.0282) |
| Code | Coarsen | Dilate | Mask | Evict | Raw |
|---|---|---|---|---|---|
| (a) Width sensitivity | |||||
| Token index | † | — | |||
| Centre only | † | — | |||
| Width feature | † | — | |||
| Hard cutoff RMS | † | — | |||
| Aperture | † | — | |||
| Code | Coarsen | Dilate | Mask | Evict | Raw |
|---|---|---|---|---|---|
| (a) Width sensitivity | |||||
| Centre only | † | — | |||
| Width feature | † | — | |||
| Aperture | † | — | |||
| Aperture RMS | † | — | |||
| (b) Accuracy minus centre only, mean (seed s.d.) | |||||
| Code | Coarsen | Dilate | Mask | Evict | Raw |
|---|---|---|---|---|---|
| (a) Width sensitivity | |||||
| Token index | † | — | |||
| Centre only | † | — | |||
| Width feature | † | — | |||
| Hard cutoff RMS | † | — | |||
| Aperture | † | — | |||
| Code | Coarsen | Dilate | Mask | Evict | Raw |
|---|---|---|---|---|---|
| (a) Width sensitivity | |||||
| Centre only | † | — | |||
| Width feature | † | — | |||
| Aperture | † | — | |||
| Aperture RMS | † | — | |||
| (b) Accuracy minus centre only, mean (seed s.d.) | |||||
| Permutation | Moved | Coarsen | Dilate | Mask | Evict | ||
|---|---|---|---|---|---|---|---|
| (a) Padding and live-slot controls | |||||||
| All slots | 0.855 | 1.036 | 0.325 | +0.2033 | +0.2168 | +0.1234 | +0.1092 |
| Live only | 0.809 | 0.890 | 0.266 | +0.1746 | +0.1975 | +0.0891 | +0.0670 |
| (b) Fraction of live slots permuted | |||||||
| One swap | 0.146 | 0.368 | 0.047 | +0.0330 | +0.0451 | +0.0105 | +0.0156 |
| 0.149 | 0.381 | 0.049 | +0.0424 | +0.0480 | +0.0244 | +0.0178 | |
| (a) Per-condition means | ||||||
| Cell | Sham gap | Real gap | Sham acc. | Real acc. | acc. | Positive |
| Coarsen | 0.4398 | 0.5199 | 5/5 | |||
| Dilate | 0.4197 | 0.5080 | 5/5 | |||
| Mask | 0.4318 | 0.4766 | 3/5 | |||
| Evict | 0.4004 | 0.4369 | 5/5 | |||
| (b) Four-policy contrasts and raw -values | ||||||
| (a) Matched-policy evaluation | |||||||
|---|---|---|---|---|---|---|---|
| Policy | Avail. | S.d. | Accuracy | Clock | Error | No query | |
| Coarsen | 0.0000 | +0.2096 | 0.0093 | 0.5379 | 0.4463 | 0.00 s | 0.036 |
| Similarity merge | 0.0000 | +0.3113 | 0.0134 | 0.4900 | 0.4141 | 0.00 s | 0.046 |
| Evict | 0.2363 | +0.1084 | 0.0053 | 0.4646 | 0.4209 | 1.75 s | 0.054 |
| Salience evict | 0.2227 | +0.2473 | 0.0077 | 0.6480 | 0.5703 | 0.00 s | 0.214 |
| (b) Test accuracy under policy transfer | |||||||
| Policy | Quantity | 1 layer | 2 layers | 4 layers | 8 layers |
|---|---|---|---|---|---|
| Coarsen | Clock accuracy | ||||
| Coarsen | Width sensitivity | ||||
| Evict | Clock accuracy | ||||
| Evict | Width sensitivity |
| Policy | Code | ||||
|---|---|---|---|---|---|
| Coarsen | Aperture RMS | [5] | [5] | [5] | [5] |
| Coarsen | Centre only | [5] | [5] | [5] | [5] |
| Evict | Aperture RMS | [5] | [5] | [5] | [5] |
| Evict | Centre only | [5] | [5] | [5] | [5] |