# Tangent-Cone Score Target

- ID: 406
- Canonical URL: https://synthcore.org/idea/406/tangent-cone-score-target
- API JSON: https://synthcore.org/api/idea/406.json
- API Markdown: https://synthcore.org/api/idea/406.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.04514](https://arxiv.org/abs/2607.04514)
- Category: geometry
- Solves: accuracy, stability, sample-efficiency
- ML areas: diffusion, diffusion-sampling, loss, geometry
- Math tags: geometry, differential-geometry, probability, asymptotic-analysis
- Ratings: usefulness 7/10; difficulty 6/10; novelty 6/10

## Idea description

Use the Gaussian mass of the local inward tangent cone to construct an analytic score target for noisy points lying within O(\sigma) of a support boundary or corner. This prevents a score network from learning an incorrect full-manifold or Euclidean approximation in the region where diffusion sampling is most sensitive to support truncation.

## Mathematical statement

For a support manifold \mathcal{M}\subset\mathbb{R}^{d} of dimension m, let x lie on a codimension-c stratum, let k=d-m, and let \zeta=(\zeta_{\mathcal S},\zeta_{\mathcal C})\in\mathbb{R}^{m-c}    imes[0,\infty)^c. Let \mathbf{S}(x) and \mathbf{C}(x) be orthonormal tangent frames along the stratum and its c inward transverse tangent directions, let \mathbf{N}(x) be an ambient normal frame, and let \mathbf{L}_{\mathcal S}(x),\mathbf{L}_{\mathcal C}(x) be the coordinate Jacobian blocks of the local chart. For a boundary-layer observation y=x+\sigma\mathbf{C}(x)a_{\mathcal C}+\sigma\mathbf{N}(x)a_{\mathcal N}, the paper gives the limiting exponent \Psi(\zeta;a,x)=\frac12[\|\mathbf{L}_{\mathcal S}(x)\zeta\|^2+\|a_{\mathcal C}-\mathbf{L}_{\mathcal C}(x)\zeta\|^2+\|a_{\mathcal N}\|^2]. If \rho(x) is the support density and J(x) the adapted chart Jacobian, then the leading smoothed density is proportional to \sigma^{m-d}\rho(x)J(x)\int_{\mathbb{R}^{m-c}    imes\mathbb{R}_{+}^{c}}e^{-\Psi(\zeta;a,x)}d\zeta. The exact Gaussian posterior identity is 
abla_y\log p_\sigma(y)=(\mathbb{E}[X\mid Y=y]-y)/\sigma^2 for Y=X+\sigma\varepsilon. Replacing the posterior by the cone posterior gives the adapted score approximation \widehat{\boldsymbol{s}}_\sigma(y)=\sigma^{-1}[\bar v-\mathbf{C}a_{\mathcal C}-\mathbf{N}a_{\mathcal N}], where v(\zeta)=\mathbf{S}\mathbf{L}_{\mathcal S}\zeta+\mathbf{C}\mathbf{L}_{\mathcal C}\zeta and \bar v is its expectation under weights proportional to e^{-\Psi}.

## Key formulas

- $$\Psi(\zeta;a,x)=\frac{1}{2}\left[\|\mathbf{L}_{\mathcal{S}}(x)\,\zeta\|^{2}+\|a_{\mathcal{C}}-\mathbf{L}_{\mathcal{C}}(x)\,\zeta\|^{2}+\|a_{\mathcal{N}}\|^{2}\right].$$
- $$p_\sigma\!\left(x+\sigma\mathbf{C}(x)a_{\mathcal C}+\sigma\mathbf{N}(x)a_{\mathcal N}\right)\approx (2\pi\sigma^2)^{-d/2}\sigma^m\rho(x)J(x)\int_{\mathbb{R}^{m-c}\times[0,\infty)^c}e^{-\Psi(\zeta;a,x)}\,d\zeta.$$
- $$\boldsymbol{s}_\sigma(y)=\frac{\mathbb{E}[X\mid Y=y]-y}{\sigma^2},\qquad \widehat{\boldsymbol{s}}_\sigma(y)=\frac{1}{\sigma}\left(\mathbb{E}_{w}[v(\zeta)]-\mathbf{C}(x)a_{\mathcal C}-\mathbf{N}(x)a_{\mathcal N}\right).$$
- $$w(\zeta\mid a,x)=\frac{\exp[-\Psi(\zeta;a,x)]}{\int_{\mathbb{R}^{m-c}\times\mathbb{R}_{+}^c}\exp[-\Psi(\zeta';a,x)]\,d\zeta'},\qquad v(\zeta)=\mathbf{S}(x)\mathbf{L}_{\mathcal S}(x)\zeta+\mathbf{C}(x)\mathbf{L}_{\mathcal C}(x)\zeta.$$

## Implementation notes

Integrate this at the score-target construction stage of a diffusion or denoising-score model, before the neural network sees the noisy sample. First support a known or estimated local boundary representation. For each clean sample x, assign a stratum codimension c, estimate an orthonormal tangent frame by local PCA, identify inward directions C and ambient normals N, and estimate the chart blocks L_S and L_C. Draw y=x+sigma*epsilon in the usual way, then compute tubular coordinates approximately by projection onto the nearest boundary stratum: a_C=C^T(y-x)/sigma and a_N=N^T(y-x)/sigma. If the sample is farther than r*sigma from every boundary, retain the ordinary denoising target (x-y)/sigma^2. Otherwise evaluate the cone posterior using Gauss-Hermite quadrature for unconstrained zeta_S and truncated Gauss-Hermite or importance samples for zeta_C>=0. The pseudocode is: `zeta <- samples_from(R^(m-c) x R_+^c); logw <- -0.5*(norm(L_S@zeta)^2 + norm(a_C-L_C@zeta)^2 + norm(a_N)^2); w <- softmax(logw); vbar <- sum(w*(S@L_S@zeta + C@L_C@zeta)); target <- (vbar-C@a_C-N@a_N)/sigma; loss <- ||score_net(y,sigma)-target||^2`. The paper supplies the exponent and cone domain; local frames, projection, density, and Jacobians are estimated from data or known geometry. Start with a 2D score network on the uniform distribution over a square or quarter-disk, comparing standard denoising score matching against the cone target at sigma values from 0.01 to 0.2. Measure score error within 2sigma of edges, normal-component error, constraint violations during reverse sampling, and sliced Wasserstein distance. Success means lower boundary score error and fewer samples leaving the support at equal network size and sampling steps.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: inconclusive
- Verdict: Built an analytic tangent-cone score target for the 2D square, with exact half-line cone posterior calculations and a matched CUDA toy score-network experiment. Prediction 1 (cone mass = Phi(a)) matched at a=-1,0,1,2 with maximum error 2e-16; prediction 2 (sigma times score is invariant) matched Mills ratios across sigma=.01,.05,.2; prediction 3 (omitted-cone normal error decays with inward distance) matched exactly from 7.979 at a=0 to 0.0444 at a=3 for sigma=.1. The learned cone target reduced boundary normal RMSE from 11.43 to 7.72 at sigma=.03 and 3.24 to 1.88 at .07, but worsened it at .15 (1.93 to 2.86), so the mechanism is real but not a universal training win.

### Mechanism check

- Verdict: Built an analytic tangent-cone score target for the 2D square, with exact half-line cone posterior calculations and a matched CUDA toy score-network experiment. Prediction 1 (cone mass = Phi(a)) matched at a=-1,0,1,2 with maximum error 2e-16; prediction 2 (sigma times score is invariant) matched Mills ratios across sigma=.01,.05,.2; prediction 3 (omitted-cone normal error decays with inward distance) matched exactly from 7.979 at a=0 to 0.0444 at a=3 for sigma=.1. The learned cone target reduced boundary normal RMSE from 11.43 to 7.72 at sigma=.03 and 3.24 to 1.88 at .07, but worsened it at .15 (1.93 to 2.86), so the mechanism is real but not a universal training win.
- Confidence: 8/10
- Limitations: Only a flat 2D square and boundary/corner-local analytic targets were tested; no learned geometry, curved manifolds, quarter-disk, reverse diffusion sampling, constraint-violation rate, sliced Wasserstein distance, ablations over quadrature/sample estimation, or larger models were evaluated. The high-noise regime showed the local target can be inferior when opposite boundaries interact.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.41055
- Paired wins: 5/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_experiment.py](https://synthcore.org/code/206/bench_experiment.py)
- [bench_report.json](https://synthcore.org/code/206/bench_report.json)
- [cone_bench_track.py](https://synthcore.org/code/206/cone_bench_track.py)
- [experiment.py](https://synthcore.org/code/206/experiment.py)
- [report.md](https://synthcore.org/code/206/report.md)
- [report_bench_2026-08-30T200851.md](https://synthcore.org/code/206/report_bench_2026-08-30T200851.md)
- [results.json](https://synthcore.org/code/206/results.json)
- [Download all files as ZIP](https://synthcore.org/download/206)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.