This experiment asks whether the weighted three-source Shape Budget control object is recoverable from boundary data when the source positions and weights are not given in advance.
The forward weighted family was:
[ w_1|x-p_1| + w_2|x-p_2| + w_3|x-p_3| = S ]
with positive normalized weights.
The recovery target is not the absolute placement or absolute scale.
It is the normalized control object:
- the normalized source triangle relative to budget
- the normalized weight vector in the simplex
Can a boundary-only inverse recover the compact weighted multi-source control object well enough to be useful, and does it outperform an equal-weight baseline that ignores the weight degrees of freedom?
This is a deliberate first inverse test, not the hardest possible one.
The observations are:
- boundary-only
- centered and scale-normalized from the boundary itself
- kept in canonical pose
So this experiment removes translation and scale from the observed boundary, but it does not randomize orientation.
That is intentional.
The question here is whether the weighted control object is operational at all under a simple boundary-only encoding, not whether the fully arbitrary-pose inverse is already solved.
The experiment script is run_weighted_multisource_inverse_experiment.py.
The inverse is deliberately simple:
- build a weighted reference bank of forward models
- encode each boundary as a centroid-centered, mean-radius-normalized radial signature
- recover the nearest weighted reference signature under masked L2 distance
There is no custom optimizer here.
That is part of the point.
If a simple reference-bank inverse works, that is strong evidence that the weighted control object is genuinely recoverable and not just elegant on paper.
The baseline is an equal-weight reference bank:
- same geometric ranges
- weights fixed to
(1/3, 1/3, 1/3)
This asks a very clean question:
- if the true family is weighted, how much do we lose by forcing the inverse to pretend that all sources matter equally?
Reference banks:
- weighted bank size:
300 - equal-weight baseline bank size:
150
Test set:
40weighted test cases per observation regime
Observation regimes:
full_cleanfull_noisypartial_arc_noisysparse_full_noisysparse_partial_high_noise
Signature encoding:
- boundary samples per forward curve:
96 - radial-signature bins:
64
The experiment records three kinds of quantities.
Control-object recovery:
- geometry MAE on the normalized edge-length triple
- weight MAE on the normalized weight vector
Fit quality:
- clean-signature RMSE of the recovered weighted-bank model
- clean-signature RMSE of the equal-weight baseline model
Because the canonical three-source setup has a left-right reflection symmetry, the geometry and weight errors are evaluated with a symmetry-aware source-1/source-2 swap when that gives the smaller error.
The inverse works meaningfully well.
In the canonical-pose boundary-only setting, a simple weighted reference-bank inverse can recover the normalized three-source geometry and normalized weights to a useful degree, and it consistently outperforms an equal-weight baseline.
The summary file is weighted_multisource_inverse_summary.json.
Global summary:
- reference bank size:
300 - equal-weight baseline bank size:
150 - trials per regime:
40 - best mean geometry MAE:
6.2280e-02 - worst mean geometry MAE:
8.5950e-02 - best mean weight MAE:
8.6077e-02 - worst mean weight MAE:
1.2484e-01 - best mean weighted-fit RMSE:
2.9443e-03 - worst mean weighted-fit RMSE:
1.5177e-02 - smallest mean fit-improvement factor over the equal-weight baseline:
1.8771 - largest mean fit-improvement factor:
5.5255
By regime:
-
full_clean- geometry MAE mean:
6.6360e-02 - weight MAE mean:
9.9616e-02 - weighted-fit RMSE mean:
2.9443e-03 - equal-weight baseline RMSE mean:
1.3205e-02 - mean improvement factor:
5.5255
- geometry MAE mean:
-
full_noisy- geometry MAE mean:
7.5033e-02 - weight MAE mean:
9.3243e-02 - weighted-fit RMSE mean:
3.1231e-03 - equal-weight baseline RMSE mean:
1.2232e-02 - mean improvement factor:
4.8779
- geometry MAE mean:
-
partial_arc_noisy- geometry MAE mean:
6.2280e-02 - weight MAE mean:
8.6077e-02 - weighted-fit RMSE mean:
4.2799e-03 - equal-weight baseline RMSE mean:
1.5237e-02 - mean improvement factor:
4.2167
- geometry MAE mean:
-
sparse_full_noisy- geometry MAE mean:
7.7342e-02 - weight MAE mean:
9.8146e-02 - weighted-fit RMSE mean:
6.4296e-03 - equal-weight baseline RMSE mean:
1.4043e-02 - mean improvement factor:
2.9858
- geometry MAE mean:
-
sparse_partial_high_noise- geometry MAE mean:
8.5950e-02 - weight MAE mean:
1.2484e-01 - weighted-fit RMSE mean:
1.5177e-02 - equal-weight baseline RMSE mean:
1.8457e-02 - mean improvement factor:
1.8771
- geometry MAE mean:
These numbers establish:
- the inverse is not perfect
- the control object is still recoverable in a meaningful way
- the weight degrees of freedom are operational, not decorative
This is a real step forward for the project.
The weighted multi-source forward experiment showed that normalized placement plus normalized participation organizes the family.
This inverse experiment shows that the same compact object is not just a forward description.
It is recoverable from boundary data by a simple method.
That matters because it turns the weighted multi-source control object into something you can infer, not just something you can write down after the fact.
The strongest reading is:
normalized placement plus normalized participation is an operational state variable for the weighted three-source family, at least in the canonical-pose boundary-only setting tested here.
Before this experiment, the multi-source program had strong forward structure but only a hypothesis about inverse usefulness.
After this experiment, there is direct evidence that:
- the weighted control object can be recovered from boundary-only data
- the recovery remains useful under noise and partial observation
- forcing equal weights leaves measurable predictive performance on the table
That moves the project closer to a genuine inferential framework.
This experiment does show:
- boundary-only recovery of the normalized weighted control object in canonical pose
- useful geometry and weight recovery across multiple observation regimes
- consistent improvement over an equal-weight inverse baseline
This experiment does not address:
- full arbitrary-pose recovery
- recovery of absolute scale
- recovery when the source count exceeds three
- recovery in anisotropic or otherwise non-Euclidean media
- weighted_multisource_inverse_heatmap.png
- weighted_multisource_inverse_baseline.png
- weighted_multisource_inverse_examples.png
The clearest figures are:
- weighted_multisource_inverse_heatmap.png for the actual recovery error scale
- weighted_multisource_inverse_baseline.png for the evidence that the weighted control object beats the equal-weight shortcut
Data:
- weighted_multisource_inverse_trials.csv
- weighted_multisource_inverse_summary.csv
- weighted_multisource_inverse_summary.json
Code:
The cleanest next step is pose-free weighted inversion.
At this point the project has:
- strong forward structure for weighted multi-source geometry
- a first boundary-only inverse in canonical pose
- a clear signal that the weight simplex is inferentially useful
The next question is whether the same recovery result survives once rotation is also unknown, and then whether the inverse remains stable when the medium itself is anisotropic or otherwise warped.