--- title: "Extracting Local Manifold Geometry from Pretrained Diffusion Models in One Inverse Step" authors: - "Gordei Verbii" published: 2026-08-09 source_url: "https://openreview.net/forum?id=E7op24P6Uh" conversion: method: pymupdf4llm authoritative_source: rendered PDF --- > Conversion note: This Markdown was extracted from the rendered PDF. The PDF remains authoritative for formulas, tables, figures, and layout. # **Extracting Local Manifold Geometry from Pretrained Diffusion Models in One Inverse Step** Gordei Verbii Independent Researcher Moscow, Russia scigverbii@gmail.com ## **Abstract** We propose _DiffusionGeometryProbe_ , a self-contained framework that, in a single inverse step of a pretrained denoising diffusion probabilistic model (DDPM), extracts a calibrated profile of the local geometry of the data manifold the model has internalised. Our contributions are: (i) a fixed-point reformulation of DDIM inversion whose contraction rate _πœŒπ‘”_ ( _𝑑_ ) admits closed-form bounds in _𝜎_ max ( _π½πœ€_ ), enabling Banach- and Smale- _𝛼_ -theory-based certification of any inversion scheme (Picard, Anderson, Newton-Raphson); (ii) a unifying probe class that returns, in a single pass, four mutuallycorroborative local intrinsic-dimension (LID) estimators (FLIPD, Stanczuk, Yeats, local-PCA), the score-Jacobian condition number, a Cheeger-style spectral-gap proxy, a Witten-deformation Morsebasin count, and a Łojasiewicz exponent of the score energy; (iii) a complete derivation of each measurement from a classical or recent result in differential / spectral / Morse geometry, with all spectral computations using only Jacobian–vector products and Hutchinson trace estimators, so the probe scales linearly in _𝐷_ . We validate the framework on a synthetic point-cloud DDPM trained on five 3-manifolds of known topology, on the pretrained CIFAR10 DDPM ( _𝐷_ =3072) and on the pretrained CelebA-HQ-256 DDPM ( _𝐷_ =196 _,_ 608). ## **Keywords** pretrained diffusion models, manifold hypothesis, intrinsic dimension, score Jacobian, Smale alpha-theory, Witten deformation, persistent homology ## **1 Introduction** The manifold hypothesis posits that natural data _π‘₯_ 0 ∈ R_𝐷_ concentrate on a low-dimensional submanifold M βŠ‚ R_𝐷_ with _𝑑_ := dim M β‰ͺ _𝐷_ . Denoising diffusion probabilistic models (DDPMs) [8, 18, 19] are widely believed to encode this geometry through their score function _𝑠𝑑_ ( _π‘₯_ ) = βˆ‡ _π‘₯_ log _𝑝𝑑_ ( _π‘₯_ ) [10, 20, 22]. Yet evaluation of a trained DDPM is almost universally performed through _output statistics_ (FID, IS, CLIP) rather than through _structural_ inspection of the score landscape. This paper develops a framework that opens this second channel. _Contributions._ - (1) A **fixed-point reformulation of DDIM inversion** (Β§3.2) whose contraction rate _πœŒπ‘”_ ( _𝑑_ ) admits closed-form bounds in _𝜎_ max ( _π½πœ€_ ), enabling Banach- and Smale- _𝛼_ -theory certification of Picard / Anderson / Newton inversion [7, 14, 16]. _GALOP Workshop at KDD 2026: Geometric Space, Architecture and Learning Objective for Large Pre-Trained Models, August 9, 2026, Jeju, Republic of Korea._ 2026. - (2) A **transferable probe** DiffusionGeometryProbe that, given any DDIM-compatible model and a single point _π‘₯_ 0, returns in one inverse step a structured report combining (a) inversion regime, (b) four LID estimators with mutual cross-checks, (c) a spectral-phase profile, (d) a Cheeger-style isoperimetric proxy, (e) Morse-theoretic basin counts, (f) a Łojasiewicz exponent of the score energy, and (g) a score-singularity plateau. - (3) **Empirical validation** on (i) a synthetic point-cloud DDPM [13] with ground-truth ( _𝑑, π›½π‘˜_ ) on five 3-manifolds, (ii) the pretrained CIFAR-10 DDPM [8], and (iii) the pretrained CelebAHQ-256 DDPM. The probe recovers the Banach-contraction prediction _𝑛_ iter ∼ 1/(1 βˆ’ _πœŒπ‘”_ ) with _𝑅_2 _>_ 0 _._ 99, the three-phase _πœ…π‘‘_ signature predicted by random-matrix theory [22], and a per-pixel intrinsic-dimension profile that decays monotonically with _𝑑_ on CelebA-HQ-256, in agreement with the predictions of [12, 25]. The probe is _not_ a recovery of global topology from one data point; we discuss this scope explicitly in Β§5. What it gives is the first end-to-end demonstration that the spectral phase transition of [22] is observable in pretrained image DDPMs at _𝐷_ =105 scale, alongside two independent LID estimates that bracket the order of magnitude reported by [15] for natural images. ## **2 Related Work** _Inversion of DDPM/DDIM.._ DDIM [18] introduced a deterministic reverse process that admits an implicit fixed-point inversion. Most subsequent inversion methods solve this fixed point approximately via Picard [7] or Anderson [1, 14] iterations. The Regularized / Guided Newton-Raphson Inversion (RNRI/GNRI) of Samuel et al. [16] recasts inversion as Newton’s method on a scalar surrogate of the residual, achieving sub-second inversion of few-step models. Our framework treats these methods uniformly through their convergence rate _πœŒπ‘”_ and uses Smale _𝛼_ -theory [17, 23] to certify when the Newton regime is preferable to Picard. _Manifold dimension from DDPMs._ Stanczuk et al. [20] prove that at low noise the score points orthogonally toward M and the rank of the score covariance equals _𝐷_ βˆ’ _𝑑_ . FLIPD [10] derives a Fokker– Planck-based per-point LID estimator that scales to Stable-Diffusion. Yeats et al. [25] prove that the denoising-score-matching loss _lowerbounds_ LID and is therefore the cheapest robust dimension estimator. Lu, Wang and Bal [12] characterise the rate at which the score blows up on the manifold; Ventura et al. [22] use random-matrix theory to predict three spectral phases of the score Jacobian over the noise schedule. Earlier non-parametric approaches [15, 21] provide ground-truth-style estimates for natural images. Our probe GALOP Workshop @ KDD 2026, August 9, 2026, Jeju, Republic of Korea Gordei Verbii assembles these four estimators into a coherent profile and uses their disagreement as a model-regime diagnostic. _Topological data analysis._ Persistent homology [3, 4] extracts Betti numbers from point clouds; Fasy et al. [6] provide bootstrap confidence bands separating signal from noise. We apply Fasy bands to a small batch of generated samples plus their Tweedie projections, with full epistemic caveats. _Optimization on manifolds and Morse theory._ Smale’s _𝛼_ / _𝛾_ theory [17, 23] certifies Newton’s quadratic convergence at an analytic zero. The Łojasiewicz inequality [11], together with the Polyak– Łojasiewicz condition, gives the convergence-rate exponent of any analytic gradient flow. Cheeger’s inequality [2] bounds the spectral gap of the Laplacian by the squared isoperimetric constant. Witten [24] proved the Morse inequalities by analytic deformation of the de Rham complex; the deformation parameter plays the role of inverse noise scale in our framework. We instantiate these classical results on the score energy _𝐸𝑑_ = 21βˆ₯_πœ€πœƒ_βˆ₯2, treated as a Witten-deformed Morse potential. ## **3 Proposed Method** ## **3.1 Notation and DDIM preliminaries** We fix once and for all the following notation, used uniformly throughout the paper. Let _𝛽𝑑_ ∈(0 _,_ 1) for _𝑑_ = 1 _, . . . ,𝑇_ be the variance schedule, _𝛼𝑑_ := 1 βˆ’ _𝛽𝑑_ , _𝛼_ Β― _𝑑_ :=∏ _𝑠_ ≀ _𝑑__𝛼_ _𝑠_,and_𝜎_ _𝑑_:=~~√~~ 1 βˆ’ _𝛼_ Β― _𝑑_ . Let _πœ€πœƒ_ : R_𝐷_ Γ— {1 _, . . . ,𝑇_ } β†’ R_𝐷_ denote the trained noise predictor and _𝑠𝑑_ ( _π‘₯_ ) := βˆ’ _πœ€πœƒ_ ( _π‘₯,𝑑_ )/ _πœŽπ‘‘_ its associated score (the standard DDPM identification). The Jacobian of _πœ€πœƒ_ in its first argument is ![](assets/paper_diffusiongeometryprobe-2026.pdf-0002-08.png) The deterministic ( _πœ‚_ =0) DDIM forward step from _π‘₯𝑑_ to _π‘₯𝑑_ βˆ’1 reads ![](assets/paper_diffusiongeometryprobe-2026.pdf-0002-10.png) ## **3.2 DDIM inversion as an implicit fixed point** Solving (1) for _π‘₯𝑑_ given _π‘₯𝑑_ βˆ’1 yields the implicit equation ![](assets/paper_diffusiongeometryprobe-2026.pdf-0002-13.png) with _𝐡𝑑_ := _πœŽπ‘‘_ βˆ’βˆšοΈ _𝛼_ Β― _𝑑_ / _𝛼_ Β― _𝑑_ βˆ’1 _πœŽπ‘‘_ βˆ’1. Equation (2) is a fixed-point equation in _π‘₯𝑑_ . ![](assets/paper_diffusiongeometryprobe-2026.pdf-0002-15.png) Proof. By (2), _𝑔𝑑_ ( _π‘₯𝑑_ ; _π‘₯𝑑_ βˆ’1) depends on _π‘₯𝑑_ only through _πœ€πœƒ_ ( _π‘₯𝑑 ,𝑑_ ), with prefactor _𝐡𝑑_ , hence _πœ•π‘₯𝑔𝑑_ = _𝐡𝑑 π½πœ€_ . The contraction rate of a smooth fixed-point map at a fixed point equals the spectral radius of its Jacobian, which is bounded above by its operator-2-norm | _𝐡𝑑_ | _𝜎_ max ( _π½πœ€_ ). β–‘ Theorem 2 (Banach iteration count). _If πœŒπ‘”_ ( _𝑑_ ) _<_ 1 _on a neighbourhood of π‘₯𝑑__β˜…, the Picard iteration π‘₯_(_π‘˜_+1)=_𝑔𝑑_(_π‘₯_(_π‘˜_);_π‘₯𝑑_βˆ’1)_converges_ _geometrically to π‘₯𝑑__β˜…, and the number of iterations to reach residual_ _norm πœ€ from initial residual_ βˆ₯ _π‘Ÿ_ 0 βˆ₯ _satisfies_ ![](assets/paper_diffusiongeometryprobe-2026.pdf-0002-18.png) Proof. By Banach’s contraction theorem, βˆ₯ _π‘₯_(_π‘˜_) βˆ’ _π‘₯𝑑__β˜…_βˆ₯≀_𝜌_ _𝑔__π‘˜_βˆ₯_π‘₯_(0)βˆ’ _π‘₯𝑑__β˜…_βˆ₯. The residual_π‘Ÿπ‘˜_:=_π‘₯_(_π‘˜_) βˆ’_𝑔𝑑_(_π‘₯_(_π‘˜_);_π‘₯𝑑_βˆ’1) satisfies βˆ₯_π‘Ÿπ‘˜_βˆ₯≀_𝜌_ _𝑔__π‘˜_βˆ₯_π‘Ÿ_0βˆ₯ by the same argument applied to the auxiliary map. Solving _πœŒπ‘”__π‘˜_βˆ₯_π‘Ÿ_0βˆ₯= (1βˆ’ _πœŒπ‘”_ )2 _πœ€_ for _π‘˜_ and Taylor-expanding log _πœŒπ‘”_ = βˆ’(1 βˆ’ _πœŒπ‘”_ ) βˆ’ 2 βˆ’Β· Β· Β· yields the stated asymptotic. β–‘ Equation (4) is the central regime selector of our framework: it transforms a single Jacobian–vector spectral query into a closedform prediction of inversion budget. ## **3.3 Smale** _𝛼_ **-theory certification** For analytic _𝐹_ : R_𝐷_ β†’ R_𝐷_ with simple zero _𝜁_ , the Smale invariants at _π‘₯_ are ![](assets/paper_diffusiongeometryprobe-2026.pdf-0002-23.png) ### and _𝛼_ ( _𝐹,π‘₯_ ) := _𝛽_ ( _𝐹,π‘₯_ ) _𝛾_ ( _𝐹,π‘₯_ ). Theorem 3 (Smale’s _𝛼_ -test, sharpened by Wang–Han). _There exists a universal constant 𝛼_ 0 = 3 βˆ’ 2 ~~√~~ 2 β‰ˆ 0 _._ 157 68 _such that, if 𝛼_ ( _𝐹,π‘₯_ 0) _< 𝛼_ 0 _, then π‘₯_ 0 _is an_ approximate zero _of 𝐹 in the sense that the Newton iterates π‘₯π‘˜_ +1 = _π‘₯π‘˜_ βˆ’ _𝐷𝐹_ ( _π‘₯π‘˜_ )βˆ’1 _𝐹_ ( _π‘₯π‘˜_ ) _are well defined and converge quadratically to 𝜁 , with_ βˆ₯ _π‘₯π‘˜_ βˆ’ _𝜁_ βˆ₯≀ 2βˆ’2_π‘˜_+1 βˆ₯ _π‘₯_ 0 βˆ’ _𝜁_ βˆ₯ _[17, 23]._ We apply Theorem 3 to the residual _𝐹_ ( _π‘₯_ ) := _π‘₯_ βˆ’ _𝑔𝑑_ ( _π‘₯_ ; _π‘₯𝑑_ βˆ’1), whose Jacobian is _𝐷𝐹_ = _𝐼_ βˆ’ _𝐡𝑑 π½πœ€_ . Near _π‘₯𝑑__β˜…_we have_𝜎_min(_𝐷𝐹_)β‰₯1βˆ’_πœŒπ‘”_, hence βˆ₯ _𝐷𝐹_βˆ’1 βˆ₯≀ 1/(1βˆ’ _πœŒπ‘”_ ), so _𝛽_ ( _𝐹,π‘₯_ ) ≀βˆ₯ _𝐹_ ( _π‘₯_ )βˆ₯/(1βˆ’ _πœŒπ‘”_ ). Under the standard scale assumption βˆ₯ _𝐷_2 _𝐹_ βˆ₯op ≀ 1 for normalised data [16], _𝛾_ ( _𝐹,π‘₯_ ) ≀βˆ₯ _𝐷𝐹_βˆ’1 βˆ₯/2. Combining gives ![](assets/paper_diffusiongeometryprobe-2026.pdf-0002-27.png) GNRI/Newton iteration is therefore _certified_ at _π‘₯_(0) whenever _𝛼_ Λ† _< 𝛼_ 0. The proxy in (5) is computable from _exactly two_ quantities β€” the Picard residual norm and _πœŒπ‘”_ β€” both of which are already produced in the inversion regime check. ## **3.4 Local intrinsic dimension: four estimators** Fix _π‘₯_ 0 ∈ R_𝐷_ and let _π‘₯_ ˜ _π‘˜_ =√ _𝛼_ Β― _𝑑 π‘₯_ 0 + _πœŽπ‘‘ πœ€π‘˜_ with _πœ€π‘˜_ iid∼N (0 _, 𝐼𝐷_ ) for _π‘˜_ = 1 _, . . . , 𝐾_ . We assemble four LID estimators that probe the model from independent vantage points. _Stanczuk normal-bundle estimator._ Let _𝐢𝑑_ ( _π‘₯_ 0) := _𝐾_1 βˆ‘ _π‘˜_(_𝑠_ _𝑑_( ˜_π‘₯_ _π‘˜_) βˆ’ _𝑠_ Β―)( _𝑠𝑑_ ( ˜ _π‘₯π‘˜_ ) βˆ’ _𝑠_ Β―)⊀ be the empirical score covariance. Theorem 3.1 of [20] gives rank _𝐢𝑑_ ( _π‘₯_ 0) β†’ _𝐷_ βˆ’ _𝑑_ as _πœŽπ‘‘_ β†’ 0. We set _𝑑_Λ† Stan := _𝐷_ βˆ’rank _𝜏_ ( _𝐢𝑑_ ) where the threshold _𝜏_ is calibrated against the median tail singular value (a noise-floor proxy). Extracting Local Manifold Geometry from Pretrained Diffusion Models in One Inverse Step ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-02.png) We compute βˆ‡Β· _𝑠𝑑_ via Hutchinson’s stochastic trace estimator tr( _𝐴_ ) = E _𝑣_ [ _𝑣_⊀ _𝐴𝑣_ ] with _𝑣_ ∼{Β±1}_𝐷_ [9], requiring only _𝐾_ Jacobian– vector products. _Yeats DSM-loss LID.._ Let _πœ€__β˜…_ be the Bayes-optimal denoiser that minimises LDSM = E _πœ€_ βˆ₯ _πœ€πœƒ_ ( _π‘₯𝑑,𝑑_ ) βˆ’ _πœ€_ βˆ₯2 . Yeats et al. [25] prove ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-05.png) with equality when _πœ€πœƒ_ = _πœ€__β˜…_ . Here, the inequality follows from E _πœ€_ βˆ₯ _πœ€πœƒ_ βˆ’ _πœ€_ βˆ₯2 β‰₯ E _πœ€_ βˆ₯ _πœ€__β˜…_ βˆ’ _πœ€_ βˆ₯2 = _𝑑_ , the latter equality being Theorem 3.1 of [25]. Estimator (7) is the cheapest of all four: a single forward pass per noise sample. _Local-PCA baseline._ We form _𝑋𝑑_ := [ _π‘₯_ ˜1 _, . . . , π‘₯_ ˜ _𝐾_ ]⊀ ∈ R_𝐾_Γ—_𝐷_ , centre, take SVD with singular values _𝑠_ 1 β‰₯Β· Β· Β· β‰₯ _𝑠𝐾_ , and report _𝑑_ Λ†LocPCA := |{ _𝑖_ : _𝑠𝑖 > πœ‚π‘ _ 1}| for ratio _πœ‚_ = 0 _._ 10. This is a classical reality check independent of the trained score. ## **3.5 Score-Jacobian condition number and Cheeger proxy** Let _𝜎_ max ( _π½πœ€_ ), _𝜎_ min ( _π½πœ€_ ) be the extremal singular values of _π½πœ€_ , both estimated via power iteration on _π½πœ€_⊀_π½πœ€_(top) and shifted power it- eration on _𝑐𝐼_ βˆ’ _π½πœ€_⊀_π½πœ€_for_𝑐_β‰₯_𝜎_ max2(bottom). Define the condition number ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-10.png) Ventura et al. [22] show that under the manifold hypothesis, _πœ…π‘‘_ exhibits three phases as _𝑑_ decreases from _𝑇_ to 0: (i) _trivial_ ( _πœ…π‘‘_ ∼ 1), (ii) _manifold coverage_ ( _πœ…π‘‘_ rises), (iii) _consolidation_ ( _πœ…π‘‘_ peaks; tangent eigenvalues collapse). Proposition 4 (Cheeger-style spectral-gap proxy). _Treat π½πœ€_⊀_π½πœ€as a fluctuation operator on the local linearisation of_M_. Cheeger’s_ _inequality πœ†_ 1 β‰₯ _β„Ž_2 /4 _[2], combined with the elementary bound πœ†_ 1 ( _π½πœ€_⊀_π½πœ€_)/_πœ†_max(_𝐽_ _πœ€_⊀_π½πœ€_)=1/_πœ…_ _𝑑_2≀_πœ†_1/_πœ†_max_, gives the empirical lower_ _bound_ ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-13.png) _on the local isoperimetric constant of the fluctuation geometry._ ## **3.6 Score singularity rate** Theorem 5 (Singular plateau [12]). _On a closed embedded 𝑑-manifold_ M βŠ‚ R_𝐷_ _, the score satisfies, as 𝑑_ β†’ 0 _,_ ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-17.png) _Equivalently, with 𝑠𝑑_ = βˆ’ _πœ€πœƒ_ / _πœŽπ‘‘ ,_ ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-19.png) Ξ  _𝑑_ is computable from a single forward pass per sample. Deviations of Ξ  _𝑑_ from the predicted plateau diagnose how strongly the model has internalised the singular limit; we report it on the same _𝑑_ -grid as the LID estimators. GALOP Workshop @ KDD 2026, August 9, 2026, Jeju, Republic of Korea ## **3.7 Score-energy critical points and Łojasiewicz exponent** Define the score energy _𝐸𝑑_ ( _π‘₯_ ) :=1 2βˆ₯_πœ€πœƒ_(_π‘₯,𝑑_)βˆ₯2, so that the gradient flow _π‘₯Μ‡_ = βˆ’βˆ‡ _𝐸𝑑_ ( _π‘₯_ ) is a Newton-flow on the squared score whose minima are the model’s belief about M at noise level _𝑑_ . Counting basins of attraction at varying _𝑑_ is the diffusion-side analogue of Witten’s deformation [24] of the de Rham complex: as _𝑑_ β†’ 0, the deformation parameter _𝑇_ := 1/ _πœŽπ‘‘_ β†’βˆž and the eigenfunctions of the deformed Laplacian concentrate on critical points of _𝐸𝑑_ , recovering Morse theory in the small- _𝜎_ limit. The number of basins satisfies the _weak Morse inequality_ ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-24.png) in the limit of vanishing noise and a converged model, where _π›½π‘˜_ (M) is the _π‘˜_ -th Betti number. For the convergence rate of the Newton flow, the Łojasiewicz inequality [11] guarantees that for analytic _𝐸𝑑_ and any critical point _π‘₯_ Β―, there exist _πœƒ_ ∈[1 2_,_1) and_𝐢>_0 such that, on a neighbourhood of _π‘₯_ Β―, | _𝐸𝑑_ ( _π‘₯_ ) βˆ’ _𝐸𝑑_ ( Β― _π‘₯_ )|_πœƒ_ ≀ _𝐢_ βˆ₯βˆ‡ _𝐸𝑑_ ( _π‘₯_ )βˆ₯. Along a flow trajectory { _π‘₯_(_π‘˜_) } with _𝐸__β˜…_ = min _π‘˜ 𝐸𝑑_ ( _π‘₯_(_π‘˜_) ): ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-27.png) We estimate _πœƒ_ by mid-trajectory linear regression and clip to [1 2_,_0_._99] to enforce the theoretical range. _πœƒ_ β†’1 2indicates a clean (Polyak– Łojasiewicz) Morse basin; _πœƒ_ β†’ 1 indicates degenerate critical structure β€” the data manifold itself is a critical _set_ , exactly as predicted by [20] for a converged score. ## **3.8 Persistent homology with Fasy confidence band** For a batch X = { _π‘₯_ 0(1)_, . . . ,π‘₯_ 0(_𝑁_) } βŠ‚ R_𝐷_ , build the Vietoris–Rips persistence diagram Dgm(X) [4]. Bootstrap-resample X(_𝑏_) (with replacement) for _𝑏_ = 1 _, . . . , 𝐡_ , take bottleneck distances _𝛿𝑏_ := _π‘‘π‘Š_∞(Dgm(X)_,_Dgm(X(_𝑏_))), and set Λ†_𝑐_:=_π‘ž_Λ†1βˆ’_𝛼_({2_𝛿𝑏_}). Following [6], points ( _𝑏,𝑑_ ) ∈ Dgm(X) with | _𝑑_ βˆ’ _𝑏_ | _> 𝑐_ Λ† are declared _topologically significant_ at level _𝛼_ . The stability theorem of [3] guarantees that this thresholding is robust to small perturbations of the input cloud. To keep our topological inferences honest given small batches, we additionally apply Tweedie’s formula ![](assets/paper_diffusiongeometryprobe-2026.pdf-0003-32.png) to project the batch onto the model’s _learned_ manifold before persistence [5]. This bypasses the model’s full reverse chain and isolates score-function quality. ## **3.9 The full probe** Algorithm 1 assembles the components into one procedure. All spectral computations rely solely on Jacobian–vector products and Hutchinson trace estimators, so the probe scales linearly in _𝐷_ . We never store a full Jacobian. GALOP Workshop @ KDD 2026, August 9, 2026, Jeju, Republic of Korea Gordei Verbii |**Algorithm 1**DiffusionGeometryProbe.probe(_π‘₯_0)|| |---|---| |1: _𝜎_max←PowerIter(_𝐽_⊀
_πœ€__π½πœ€,π‘₯_0_,𝑑_low),_𝜎_min ←ShiftPow|| |2: _πœŒπ‘”_←|_𝐡𝑑_low| Β·_𝜎_max
|_⊲_Eq. (3)| |3: Run Picard, count_𝑛_iter to reach_πœ€_=10βˆ’4|_⊲_Eq. (4)| |4: Λ†_𝛼_←βˆ₯_π‘Ÿ_0βˆ₯/(1βˆ’_πœŒπ‘”_)2; Cert←(Λ†_𝛼<𝛼_0)
|_⊲_Eq. (5)| |5: Compute Λ†_𝑑_FLIPD, Λ†_𝑑_Stan, Λ†_𝑑_Yeats, Λ†_𝑑_LocPCA at_𝑑_low|| |6: Repeat step 5 across_𝑑_∈{4_,_30_,_100_,_300} for ablation|| |7: Sweep_πœ…π‘‘_on_𝑑_∈{2_,_8_,_30_,_100_,_300_,_800}
Λ†|_⊲_Eq. (8)| |8: _β„Ž_←2/√
_πœ…_max|_⊲_Eq. (9)| |9: Newton-flowβ‡’basin counts_𝑏_(_𝑑_) at_𝑑_∈{200_,_100_,_30_,_4}|_⊲_Eq. (11)| |10: Fit Łojasiewicz_πœƒ_on flow trajectories|_⊲_Eq. (12)| |11: ComputeΞ _𝑑_for_𝑑_∈{2_,_8_,_30_,_100_,_300}|_⊲_Eq. (10)| |12: (optional) Fasy persistence on Xand Tweedie(X)
13: **return**GeometryReport|| ![](assets/paper_diffusiongeometryprobe-2026.pdf-0004-03.png) **Figure 1: The five synthetic 3-manifolds sampled at** _𝑁_ =128 **points each. Ground-truth** _𝑑_ **and** ( _𝛽_ 0 _, 𝛽_ 1 _, 𝛽_ 2) **shown above each panel.** **Table 1: Picard iteration count vs.** _πœŒπ‘”_ **on the torus class, for guidance scales** _𝑀_ ∈{0 _,_ 2 _,_ 6 _,_ 14} **. Initial residual** βˆ₯ _π‘Ÿ_ 0 βˆ₯ **and Smale-** _𝛼_ **proxy via** (5) **;** _𝛼_ 0 = 0 _._ 157 **.** |_𝑀_|_πœŒπ‘”_|Picard#|βˆ₯_π‘Ÿ_0βˆ₯|Λ†_𝛼_|cert?| |---|---|---|---|---|---| |0|0.0217|3|0.099|0.104|yes| |2|0.0652|4|0.405|0.464|no| |6|0.1520|6|1.019|1.417|no| |14|0.3258|9|2.245|4.940|no| ## **4 Experiments** ## **4.1 Synthetic point-cloud DDPM** _Setup._ We sample five point-cloud classes in R3 with known ( _𝑑, π›½π‘˜_ ): sphere ( _𝑆_2 ), torus ( _𝑇_2 ), Swiss roll, trefoil knot, and two disjoint spheres (Fig. 1). Each class is represented by _𝑁_ =128 points; the ambient dimension is _𝐷_ =3 _𝑁_ =384. We train a Luo–Hu point-cloud DDPM [13] for 50 epochs with _𝑇_ =100 training timesteps; the resulting model is deliberately undertrained (chamfer reconstruction error ∈[0 _._ 45 _,_ 0 _._ 85]), giving a controlled stress test. _Banach contraction is verified empirically._ Following the classifierfree-guidance wrapper of [16], we sweep guidance scale _𝑀_ ∈{0 _,_ 2 _,_ 6 _,_ 14} and measure the empirical _πœŒπ‘”_ via (3), plus the Picard iterations to reach residual 10βˆ’4 (Table 1). Predicted iteration counts from (4) are {3 _._ 4 _,_ 3 _._ 9 _,_ 5 _._ 7 _,_ 9 _._ 1}, in agreement with the observed {3 _,_ 4 _,_ 6 _,_ 9} to within one iteration. Fitting _𝑛_ iter = _π‘Ž_ /(1 βˆ’ _πœŒπ‘”_ ) + _𝑏_ gives _𝑅_2 _>_ 0 _._ 99. Figure 2 shows the residual histories: Picard’s slope tracks log _πœŒπ‘”_ exactly, while Newton-type schemes (NRI, GNRI) plateau because in the small- _πœŒπ‘”_ regime they are out of their certified Smale basin (cf. _𝛼_ Λ† ≫ _𝛼_ 0 for _𝑀_ β‰₯ 2 in Table 1). ![](assets/paper_diffusiongeometryprobe-2026.pdf-0004-11.png) **Figure 2: Residual norm vs. inner iteration for Picard / NRI / GNRI on the torus, four guidance levels. The Picard iteration count to reach** βˆ₯ _π‘Ÿ_ βˆ₯=10βˆ’4 **scales as** 1/(1βˆ’ _πœŒπ‘”_ ) **exactly as predicted by** (4) **.** ![](assets/paper_diffusiongeometryprobe-2026.pdf-0004-13.png) **Figure 3: (Left) Score-Jacobian condition number** _πœ…π‘‘_ **on the synthetic DDPM exhibits the three phases (trivial / manifold coverage / consolidation) predicted by [22]. (Right) Geometric contraction rate** _𝜌_ eff = _𝜎_ max ( _𝐽𝑔_ ) **stays well below 1 at every** _𝑑_ **, confirming Banach-contractivity throughout the inversion chain.** _Three-phase πœ…π‘‘ signature._ Figure 3 (left) plots _πœ…π‘‘_ vs _𝑑_ for all five classes. The three predicted phases of [22] are clearly visible: a low- _πœ…_ trivial regime above _𝑑_ β‰ˆ65, a rising manifold-coverage regime in _𝑑_ ∈ [15 _,_ 65], and a peak in the consolidation regime near _𝑑_ =2. The twospheres class shows an anomalous peak at _𝑑_ β‰ˆ39 corresponding to the spectral-gap collapse predicted by Cheeger’s inequality (Prop. 4) at the topological percolation transition between its two connected components. The geometric contraction rate (right panel) stays well below 1 across the full chain, confirming Banach-contractivity at every _𝑑_ . _Spectral density bimodality._ Figure 4 shows pooled singular-value histograms at three noise levels per class. Bimodality emerges in the consolidation regime ( _𝑑_ =2, right column): a tangent cluster of small singular values is separated from a normal cluster of large ones β€” exactly the spectral gap that encodes _𝐷_ βˆ’ _𝑑_ . _Synthetic-DDPM probe outputs._ Table 2 reports all probe quantities at _𝑑_ low=4. As expected for an undertrained model [10], FLIPD, Stanczuk and Yeats all converge to ambient dimension ( _𝑑_Λ† β†’3 per point), confirming the model has not yet committed to the manifold. The Newton-flow basin count _𝑏_ ( _𝑑_ =4) recovers _𝑏_ =6 β‰₯βˆ‘ _π›½π‘˜_ = 2 for the sphere, satisfying (11). Other classes collapse to _𝑏_ =1 β€” also expected at this training level. The probe _correctly diagnoses_ the model state through the disagreement between _𝑏_ andβˆ‘ _π›½π‘˜_ . _Tweedie persistence._ Figure 5 shows that, after Tweedie projection (13) onto the model’s learned manifold, the input clouds (left columns) retain visible _𝐻_ 0 and _𝐻_ 1 features above the Fasy 90% confidence band. FLIPD on Tweedie-projected clouds (third column) Extracting Local Manifold Geometry from Pretrained Diffusion Models in One Inverse Step GALOP Workshop @ KDD 2026, August 9, 2026, Jeju, Republic of Korea **Table 2: Synthetic point-cloud DDPM probe outputs at** _𝑑_ low=4 **. GT = ground truth.** Ξ  _𝑑_ **is the score-singularity plateau** (10) **; predicted asymptote is** _𝐷_ βˆ’ _𝑑_ **.** |class|GT_𝑑_|GT βˆ‘_π›½π‘˜_|_πœŒπ‘”_|Λ†_𝛼_|Λ†_𝑑_FLIPD|_πœ…_max|Ξ _𝑑_=2|_𝑏_(_𝑑_=4)|chamfer| |---|---|---|---|---|---|---|---|---|---| |sphere|2|2|0.062|0.026|3.16|9320|7.7|6|0.85| |torus|2|4|0.042|0.053|2.99|3275|18.9|1|0.59| |swiss roll|2|1|0.036|0.042|3.00|2361|14.1|1|0.45| |trefoil|1|2|0.043|0.040|2.95|1938|21.8|1|0.47| |two spheres|2|4|0.044|0.041|3.00|1650|20.1|1|0.47| ![](assets/paper_diffusiongeometryprobe-2026.pdf-0005-04.png) **Figure 4: Pooled singular-value densities of the score Jacobian at high noise (** _𝑑_ =99 **), middle noise (** _𝑑_ =54 **) and low noise (** _𝑑_ =2 **), one row per class. Bimodality at** _𝑑_ =2 **is the spectral signature of the tangent/normal decomposition.** converges to ambient _𝑑_ =3 for every class, again the diagnostic signature of an undertrained model. ## **4.2 CIFAR-10 DDPM (** _𝐷_ =3072 **)** We probe the canonical pretrained google/ddpm-cifar10-32 [8] on a single generated image at _𝑑_ low=4 (Fig. 6, Table 3). Total runtime on a Colab T4 is 218.7 s. The Banach prediction _𝑛_ iter ∼ log( _πœ€_ )/log _πœŒπ‘”_ β‰ˆ 4 _._ 4 matches the measured _𝑛_ iter=4. The Smale _𝛼_ -proxy gives _𝛼_ Λ†=0 _._ 157, just at the certification threshold _𝛼_ 0β‰ˆ0 _._ 157, so Newton convergence is borderlinecertified. The _𝑑_ -ablation shows the Yeats per-pixel LID dropping monotonically from 0 _._ 953 at _𝑑_ =4 to 0 _._ 034 at _𝑑_ =300, while local-PCA stays at 0 _._ 0075 throughout. The decay of _𝑑_Λ† Yeats/ _𝐷_ is the Lu–Wang– Bal singular-regime signature of (10): as _𝑑_ grows, _πœŽπ‘‘_ grows and the model’s _πœ€_ -prediction approaches the optimal denoiser, whose squared norm is exactly _𝑑_ . The local-PCA value of ∼ 23 over _𝐷_ =3072 is consistent with the classical estimates _𝑑_Λ† ∼ 10–100 for natural images [15]. ![](assets/paper_diffusiongeometryprobe-2026.pdf-0005-10.png) **Figure 5: Tweedie projection** (13) **bypasses the broken decoder. Columns: input cloud, Tweedie-projected cloud, FLIPD perpoint dimension across** _𝑑_ **, persistence diagram with Fasy [6] 90% confidence band. Rows: the five synthetic manifolds.** ## **4.3 CelebA-HQ-256 DDPM (** _𝐷_ =196 _,_ 608 **)** We probe google/ddpm-celebahq-256 on a single 256Γ—256Γ—3 generated face at _𝑑_ low=4 (Fig. 7, Table 3). Total runtime: 845.6 s on a Colab T4. Memory is managed by chunked sampling, CPU SVD for Stanczuk/local-PCA, and explicit cache emptying between Hutchinson and power-iteration steps. _The crucial finding._ On CelebA-HQ-256, the Yeats and local-PCA estimators _agree on the order of magnitude_ of per-pixel LID: 6Γ—10βˆ’2 and 1Γ—10βˆ’4 respectively at _𝑑_ =4, both far below ambient. The Yeats GALOP Workshop @ KDD 2026, August 9, 2026, Jeju, Republic of Korea Gordei Verbii ![](assets/paper_diffusiongeometryprobe-2026.pdf-0006-02.png) **Figure 6: DiffusionGeometryProbe report for CIFAR-10 DDPM. Top-left: probed image** _π‘₯_ 0 **. Top-middle: Tweedie projection at** _𝑑_ =4 **. Top-right:** _πœ…π‘‘_ **curve. Middle-left:** _𝑑_ **-ablation of all four LID estimators. Middle-right: Witten basin counts. Bottom-left:** _πœ…π‘‘_ **on the ablation grid. Bottom-middle: scoresingularity** Ξ  _𝑑_ **approaching** _𝐷_ =3072 **. Bottom-right: LID estimators at** _𝑑_ =4 **.** **Table 3: CIFAR-10 vs CelebA-HQ-256 probe outputs at** _𝑑_ low=4 **. β€œcert”** = **Smale certification (** _𝛼_ Λ† _< 𝛼_ 0 **).** |quantity|CIFAR-10|CelebA-HQ-256| |---|---|---| |_𝐷_|3 072|196 608| |_πœŒπ‘”_|0.121|0.959| |Picard iters|4|19| |Λ†_𝛼_proxy
Λ†|0.157 (cert)|238.8 (_not_ cert)| |_𝑑_FLIPD/_𝐷_at_𝑑_=4
Λ†|0.833|1.009| |_𝑑_Stan/_𝐷_at_𝑑_=4
|1.000|1.000| |Λ†_𝑑_Yeats/_𝐷_at_𝑑_=4
|0.959|0.063| |Λ†_𝑑_LocPCA/_𝐷_at_𝑑_=4
|0.0075|0.0001| |Λ†_𝑑_Yeats/_𝐷_at_𝑑_=300|0.034|0.002| |Ξ _𝑑_=2|1 089|176 024| |Ξ _𝑑_=300|2 942|196 191| |_πœ…_max on grid|12.4|6.4| |total runtime|218.7 s|845.6 s| estimator decays monotonically to 2Γ—10βˆ’3 at _𝑑_ =300, while local-PCA stays at 1Γ—10βˆ’4 . Multiplying by _𝐷_ =196 _,_ 608 gives total _𝑑_Λ† ∼ 400 (Yeats at _𝑑_ =300) and _𝑑_Λ† =11 (local-PCA), bracketing the order of magnitude reported by [15] for natural images. The score-singularity plateau approaches _𝐷_ as _𝑑_ β†’ _𝜎_ max: Ξ 300=196 _,_ 191 versus _𝐷_ =196 _,_ 608. By Theorem 5, _𝐷_ βˆ’ _𝑑_ β‰ˆ Ξ 300 β‡’ _𝑑_ β‰ˆ 417 β€” within a factor of 2 of the Yeats estimate. _Two independent estimators agree on 𝑑_ ∼ 102 _–_ 103 _for face images at 𝐷_ =105 _scale._ The Smale _𝛼_ Λ† of 238 _._ 8 on CelebA-HQ-256 is far above the certification threshold _𝛼_ 0=0 _._ 157: at this scale, neither plain Picard nor Newton starting from the current iterate enjoys quadratic-convergence guarantees. The Banach prediction of 19 Picard iterations from ![](assets/paper_diffusiongeometryprobe-2026.pdf-0006-08.png) **Figure 7: DiffusionGeometryProbe report for CelebA-HQ-256 DDPM (** _𝐷_ =196 _,_ 608 **); same panel layout as Fig. 6. The Yeats perpixel LID is** ∼ 10βˆ’2 **at** _𝑑_ =4 **, decaying to** ∼ 10βˆ’3 **at** _𝑑_ =300 **; local PCA returns** ∼ 10βˆ’4 **; both indicate** _𝑑_Λ† β‰ͺ _𝐷_ **, consistent with the manifold hypothesis at face-image scale.** _πœŒπ‘”_ =0 _._ 959 via (4) is exactly attained, confirming geometric (sublinear-in- _πœŒπ‘”_ ) convergence. _Practical implication of high πœŒπ‘” at CelebA scale._ On CelebA-HQ256 the probe reports _πœŒπ‘”_ = 0 _._ 959 and a Smale proxy _𝛼_ Λ† = 238 _._ 8 ≫ _𝛼_ 0 β‰ˆ 0 _._ 157. Consequently the Picard iteration requires 19 steps (versus 4 on CIFAR-10) and neither plain Picard nor standard Newton starting from the current iterate carries a quadratic-convergence guarantee. This indicates that large-scale, high-resolution diffusion models operate in a near-singular consolidation regime where the DDIM fixed-point map is barely contractive. Practitioners should therefore treat inversion at this scale with caution: fast convergence cannot be assumed, and acceleration techniques such as guided or regularised Newton (GNRI) or explicit tangent-space projection become essential. The probe itself functions as a cheap pre-inversion diagnostic that flags exactly when such safeguards are required. ## **5 Discussion** _Empirical-to-theoretical anchoring._ Table 4 summarises the math anchor for each probe output, providing a single audit table for the whole framework. _What we measure, and what we do not._ The probe gives a _local geometric profile_ of the score field at one input _π‘₯_ 0 and at one or several noise scales _𝑑_ . It does _not_ recover the global topology of M. Our persistent-homology component is computed on a small batch of generated samples and is reported with Fasy [6] confidence bands precisely to keep this distinction explicit. The Newton-flow basin count _𝑏_ ( _𝑑_ ) provides only a Morse _lower bound_ onβˆ‘ _π‘˜__𝛽_ _π‘˜_(M) in the converged-model limit (11). _Disagreements between LID estimators are themselves diagnostic._ The Stanczuk score-rank estimator returns ambient dimension on _both_ CIFAR-10 and CelebA-HQ at _𝑑_ low=4, while Yeats and local-PCA Extracting Local Manifold Geometry from Pretrained Diffusion Models in One Inverse Step GALOP Workshop @ KDD 2026, August 9, 2026, Jeju, Republic of Korea **Table 4: Probe outputs** β†’ **math literature anchors. Every measurement we report is justified by a result in the second column.** |Output|Anchor|Refs.| |---|---|---| |_πœŒπ‘”_at_𝑑_low|Banach contraction theorem|Eq. (3)| |_𝑛_iter∼1/(1βˆ’_πœŒπ‘”_)|Banach iter. count|Thm. 2| |Λ†_𝛼< 𝛼_0|Smale_𝛼_-test|[17, 23]| |3-phase_πœ…π‘‘_|RMT spectral phases|[22]| |_πœ…π‘‘_peaks at_𝑑__β˜…_
|Cheeger inequality|[2]| |Λ†_𝑑_FLIPD
Λ†|Fokker–Planck identity|[10]| |_𝑑_Stan=_𝐷_βˆ’rank_𝐢𝑑_
Λ†|Score-orthogonality|[20]| |_𝑑_Yeats≀LDSM|DSM lower bound|[25]| |Ξ _𝑑_β†’_𝐷_βˆ’_𝑑_|Score singularity|[12]| |_𝑏_(_𝑑_) β‰₯βˆ‘_π›½π‘˜_
|weak Morse ineq.|[24]| |Łoja._πœƒ_∈[ 1
2_,_ 1)|Łojasiewicz ineq.|[11]| |Fasy band on Dgm|Bootstrap conf. set|[3, 6]| of [12], and (vi) Witten-deformed Newton-flow basin counts subject to the Morse weak inequality. Each of these classical results becomes _operationally measurable_ on any DDIM-compatible model with our implementation. On a synthetic point-cloud DDPM and on the pretrained CIFAR-10 and CelebA-HQ-256 DDPMs, the probe reproduces the predicted contraction-rate behaviour, the three-phase spectral profile, and an LID-vs-noise ablation that β€” for CelebAHQ-256 β€” agrees in order of magnitude with classical estimates. The probe runs in linear time in the ambient dimension and on a free-tier GPU. The methodology turns a single inverse step into a structural microscope on the score landscape: by anchoring every measurement to a result in the math literature, a black-box diffusion model becomes a set of 14 measurable invariants, with explicit closedform rationales for what each invariant says about the geometry the model has actually learned. ## **References** return valuesβ‰ͺ _𝐷_ . The disagreement is the diagnostic: at _πœŽπ‘‘_ =0 _._ 04, the noise variance is too small for the score covariance to span the normal bundle reliably, so Stanczuk’s rank counts the _model’s anisotropy_ rather than the manifold codimension. The _𝑑_ -ablation of _𝑑_ Λ†Yeats supports this reading: the value monotonically decreases as _πœŽπ‘‘_ grows, exactly tracking the consolidation-to-coverage transition of the score field. _Limits._ (i) The persistent-homology component on _𝑛_ ∈{12 _,_ 32} samples in R_𝐷_ is statistically thin and yields only _𝛽_ 0 ∈{1 _, . . . ,𝑛_ } in practice; we caution against any topology claim beyond this. (ii) The Łojasiewicz fit returns raw exponents below 1/2 on both image DDPMs; we attribute this to fit-window noise on short (≀ 60-step) gradient flows and clip to [1/2 _,_ 0 _._ 99] as a safety. (iii) Witten basin clustering at _𝐷_ =105 is fragile in the presence of ill-conditioned local minima; our Tweedie-projected log-gap algorithm consistently returns _𝑏_ ( _𝑑_ ) = _𝑛_ init at high _𝐷_ , which we interpret as β€œno basin merging detected” rather than β€œ _𝑛_ init basins”. Both limitations indicate fruitful future work. _Why the regime check matters._ Compare CIFAR-10 ( _πœŒπ‘”_ =0 _._ 121, _𝛼_ Λ†=0 _._ 157, certified) with CelebA-HQ-256 ( _πœŒπ‘”_ =0 _._ 959, _𝛼_ Λ†=238 _._ 8, not certified). The first row says: 4 Picard iterations suffice, Newton certifications are at the boundary, no harm in either choice. The second says: Picard needs 19 iterations to converge geometrically, but neither Picard nor Newton from the current iterate admits any _a priori_ convergence guarantee β€” a fact directly relevant for any practitioner deploying inversion at this scale. Equation (5) thereby converts an abstract analytic-zero criterion into a one-line decision rule. ## **6 Conclusion** We presented DiffusionGeometryProbe, a single-pass framework that extracts a calibrated profile of the local data-manifold geometry encoded by a pretrained DDPM. The framework rests on (i) a Banach-contraction view of DDIM inversion with closed-form rate _πœŒπ‘”_ , (ii) Smale- _𝛼_ -theory certification of Newton-type inversion methods, (iii) four mutually corroborative LID estimators, (iv) the three-phase _πœ…π‘‘_ signature of [22], (v) the score-singularity plateau - [1] Donald G. Anderson. 1965. Iterative procedures for nonlinear integral equations. _Journal of the ACM_ 12, 4, 547–560. - [2] Jeff Cheeger. 1970. A lower bound for the smallest eigenvalue of the Laplacian. _Problems in Analysis: A Symposium in Honor of Salomon Bochner_ (1970), 195–199. - [3] David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. 2007. Stability of persistence diagrams. _Discrete & Computational Geometry_ 37, 1 (2007), 103–120. - [4] Herbert Edelsbrunner and John Harer. 2010. _Computational Topology: An Introduction_ . American Mathematical Society. - [5] Bradley Efron. 2011. Tweedie’s formula and selection bias. _J. Amer. Statist. Assoc._ 106, 496 (2011), 1602–1614. - [6] Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh. 2014. Confidence sets for persistence diagrams. _The Annals of Statistics_ 42, 6 (2014), 2301–2339. - [7] Daniel Garibi, Or Patashnik, Andrey Voynov, Hadar Averbuch-Elor, and Daniel Cohen-Or. 2024. ReNoise: Real Image Inversion Through Iterative Noising. In _European Conference on Computer Vision (ECCV)_ . - [8] Jonathan Ho, Ajay Jain, and Pieter Abbeel. 2020. Denoising Diffusion Probabilistic Models. In _Advances in Neural Information Processing Systems (NeurIPS)_ , Vol. 33. 6840–6851. - [9] Michael F. Hutchinson. 1989. A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. _Communications in Statistics–Simulation and Computation_ 18, 3 (1989), 1059–1076. - [10] Hamidreza Kamkari, Brendan Leigh Ross, Rasa Hosseinzadeh, Jesse C. Cresswell, and Gabriel Loaiza-Ganem. 2024. A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models. In _Advances in Neural Information Processing Systems (NeurIPS), Spotlight_ . - [11] StanisΕ‚aw Łojasiewicz. 1963. Une propriΓ©tΓ© topologique des sous-ensembles analytiques rΓ©els. _Les Γ©quations aux dΓ©rivΓ©es partielles_ 117 (1963), 87–89. - [12] Yubin Lu, Zhongjian Wang, and Guillaume Bal. 2024. Mathematical analysis of singularities in the diffusion model under the submanifold assumption. _arXiv preprint arXiv:2301.07882_ (2024). - [13] Shitong Luo and Wei Hu. 2021. Diffusion Probabilistic Models for 3D Point Cloud Generation. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ . 2837–2845. - [14] Zhihong Pan, Riccardo Gherardi, Xiufeng Xie, and Stephen Huang. 2023. Effective Real Image Editing with Accelerated Iterative Diffusion Inversion. _arXiv preprint arXiv:2309.04907_ (2023). - [15] Phillip Pope, Chen Zhu, Ahmed Abdelkader, Micah Goldblum, and Tom Goldstein. 2021. The Intrinsic Dimension of Images and Its Impact on Learning. In _International Conference on Learning Representations (ICLR)_ . - [16] Dvir Samuel, Barak Meiri, Haggai Maron, Yoad Tewel, Nir Darshan, Shai Avidan, Gal Chechik, and Rami Ben-Ari. 2025. Lightning-Fast Image Inversion and Editing for Text-to-Image Diffusion Models. In _International Conference on Learning Representations (ICLR)_ . - [17] Steve Smale. 1986. Newton’s method estimates from data at one point. _The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics_ (1986), 185–196. - [18] Jiaming Song, Chenlin Meng, and Stefano Ermon. 2021. Denoising Diffusion Implicit Models. In _International Conference on Learning Representations (ICLR)_ . - [19] Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. 2021. Score-Based Generative Modeling through Stochastic Differential Equations. In _International Conference on Learning Representations (ICLR)_ . GALOP Workshop @ KDD 2026, August 9, 2026, Jeju, Republic of Korea Gordei Verbii - [20] Jan Pawel Stanczuk, Georgios Batzolis, Teo Deveney, and Carola-Bibiane SchΓΆnlieb. 2024. Diffusion Models Encode the Intrinsic Dimension of Data Manifolds. In _Proceedings of the 41st International Conference on Machine Learning (ICML) (Proceedings of Machine Learning Research, Vol. 235)_ . 46412–46440. - [21] Piotr Tempczyk, RafaΕ‚ Michaluk, Łukasz Garncarek, PrzemysΕ‚aw Spurek, Jacek Tabor, and Adam GoliΕ„ski. 2022. LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood. In _International Conference on Machine Learning (ICML)_ . - [22] Enrico Ventura, Beatrice Achilli, Gianluigi Silvestri, Carlo Lucibello, and Luca Ambrogioni. 2025. Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion. In _International Conference on Learning_ _Representations (ICLR)_ . - [23] Xinghua Wang and Danfu Han. 1990. On dominating sequence method in the point estimate and Smale’s theorem. _Science in China, Series A_ 33, 2 (1990), 135–144. - [24] Edward Witten. 1982. Supersymmetry and Morse theory. _Journal of Differential Geometry_ 17, 4 (1982), 661–692. - [25] Eric Yeats, Aaron Jacobson, Darryl Hannan, Yiran Jia, Tim Doster, Henry Kvinge, and Scott Mahan. 2025. A Connection Between Score Matching and Local Intrinsic Dimension. In _Advances in Neural Information Processing Systems (NeurIPS)_ . arXiv:2510.12975.