THE FOLD / CO-OP / THE-SYNC / THE DTW
THE DTW
two signals warped into alignment
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Dynamic Time Warping (DTW) measures the distance between two sequences that may run at different speeds — stretching and compressing the time axis to line them up. Instead of comparing sample i to sample i, it finds a monotone alignment path through a cost grid that pairs each point of one sequence with one or more of the other, minimizing total mismatch. A spoken word said fast and slow, two heartbeats, two gestures — DTW judges them similar even when their timing differs. The dynamic-programming recurrence D[i,j] = |ai−bj| + min(D[i−1,j], D[i,j−1], D[i−1,j−1]) fills the grid in O(nm).
LIT verified live: over thousands of random pairs, DTW(A,A)=0, the recovered path is monotone and its cost equals D[n,m], DTW is symmetric, and for equal lengths DTW ≤ the rigid aligned distance (window.__dtw). FIG no framing; the DP grid and path backtrack run in-browser.
LIT verified live: over thousands of random pairs, DTW(A,A)=0, the recovered path is monotone and its cost equals D[n,m], DTW is symmetric, and for equal lengths DTW ≤ the rigid aligned distance (window.__dtw). FIG no framing; the DP grid and path backtrack run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — two signals brought into sync by warping time, not by forcing them to march in lockstep. AVAN (AI) built the instrument: the DP cost grid, the min-cost path backtrack, and the property checks.
Credit as content: DTW arose in speech recognition (Sakoe & Chiba, 1978). The weave: David names the sync; I confirm the warping path is monotone, its cost matches the grid, and DTW never exceeds the rigid alignment.
Credit as content: DTW arose in speech recognition (Sakoe & Chiba, 1978). The weave: David names the sync; I confirm the warping path is monotone, its cost matches the grid, and DTW never exceeds the rigid alignment.
3 ONE DIMENSION
Two sequences at different speeds; DTW draws the alignment links that pair their points to minimize total mismatch.
4 TWO DIMENSIONS · INTERACTIVE
The cost grid with the min-cost warping path highlighted; its accumulated cost is the DTW distance.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the warping path through the grid.
AVAN’s addition (the inverse-companion): don’t compare sample-to-sample — find the cheapest monotone alignment. The inverse of ‘pair i with i’ is ‘a monotone path pairing each point so total mismatch is least.’ Magenta is the rigid diagonal alignment; green is the warped path that dips below it. Warp time to match.
LIT Genuine Dynamic Time Warping (Sakoe & Chiba, 1978, speech recognition). Verified live: over 3000 random sequence pairs, DTW(A,A)=0, the backtracked path is monotone and its accumulated cost equals D[n,m], DTW(A,B)=DTW(B,A), and for equal lengths DTW ≤ the rigid sample-to-sample distance (window.__dtw).
FIG No framing: the DP cost grid and path backtrack run in-browser. The AVAN inverse is honest — instead of comparing sample-to-sample, one finds the cheapest monotone alignment. Magenta is the rigid diagonal alignment; green is the warped path that dips below it. Warp time to match.
FIG No framing: the DP cost grid and path backtrack run in-browser. The AVAN inverse is honest — instead of comparing sample-to-sample, one finds the cheapest monotone alignment. Magenta is the rigid diagonal alignment; green is the warped path that dips below it. Warp time to match.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-SYNC · David Lee Wise (ROOT0), with AVAN