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Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Subspace Constrained Mean Shift algorithm converges to a stable ridge defined by the Jacobian of the projected gradient, not to the classical static ridge.

desk verdict A rigorous paper that correctly identifies the stable ridge as the SCMS limit under a clearly stated regularity class; the main gap is that the convergence condition is stronger than the defining eigenvalue condition, which is worth flagging but is an explicit assumption, not a hidden flaw. read the letter →

arxiv 2608.05112 v1 pith:EA72WUY2 submitted 2026-08-05 stat.ML cs.LG

Wanli Qiao This is my paper
classification stat.MLcs.LG MSC 62G0762G2037C10
keywords SubspaceConstrainedMeanShiftstabledensityridgestaticestimationnonparametricalgorithmdynamicalsystemsR-linearconvergence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper re-derives the theoretical target of the Subspace Constrained Mean Shift (SCMS) algorithm. It argues that SCMS trajectories do not, in general, converge to the classical 'static' density ridge defined by the Hessian's trailing eigenvalue, because the algorithm's vector field carries the trailing eigenspace along with it as it rotates. The true limit is the 'stable ridge', the set of points where the projected gradient vanishes and the Jacobian of that projected gradient has its (k+1)-th eigenvalue with negative real part. If correct, all existing SCMS-based ridge estimation should be re-interpreted as estimating this dynamically defined object, and a constant-step-size version of SCMS is both statistically consistent and far cheaper to run than the original.

What carries the argument

The carrying object is the projected gradient vector field $\xi(x)=\Pi(x)\nabla f(x)$, where $\Pi$ is the orthogonal projection onto the trailing eigenspace of the Hessian, and its Jacobian $J_\xi(x)=\nabla\xi(x)$. The eigenvalues $\mu_1,\ldots,\mu_d$ of $J_\xi$, sorted by real part, encode not only the local curvature of $f$ but also the rotation of the eigenspaces along the flow, which is the mechanism the static ridge misses. The stable ridge is the set of equilibria $\xi=0$ whose normal Jacobian eigenvalues have negative real parts, exactly the attractors of the dynamical system $\dot{x}=\xi(x)$. The ridge-regular class $\mathcal{F}(\eta,\gamma,\kappa,\mathcal{D})$ supplies the three properties that make the argument run: a spectral gap for a smooth projection, uniform normal contraction (the symmetric part of $J_\xi$ is $\le -\gamma$ on the trailing eigenspace), and a uniform positive lower bound on $\|\xi\|$ away from the ridge.

What would settle it

Run the generalized SCMS on the Appendix A.1.2 density, where the vertical axis has $\lambda_2>0$ (a static valley) but $\mathrm{Re}(\mu_2)<0$; the claim predicts convergence to that axis, whereas the static-ridge theory predicts no ridge there. Alternatively, in the A.1.1 pitchfork density, initialize on the lower vertical axis $v<1/\sqrt{2}$; the claim predicts trajectories escape to the symmetric wings, and any trajectory sticking to the axis would refute it.

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Extended reading notes

Core claim

The central claim is that SCMS, both in its original log-density form and in a new constant-step-size generalization, converges to the stable ridge $R_{\mathrm{stable}}(f) = \{x : \xi(x)=0,\ \mathrm{Re}(\mu_{k+1}(x))<0\}$, where $\xi(x)=\Pi(x)\nabla f(x)$ is the projection of the density gradient onto the trailing eigenspace of the Hessian and $\mu_{k+1}(x)$ is the $(k+1)$-th eigenvalue of the Jacobian $J_\xi(x)=\nabla\xi(x)$. The static ridge condition $\lambda_{k+1}(x)<0$ is replaced by the dynamic stability condition $\mathrm{Re}(\mu_{k+1}(x))<0$; the difference is exactly the rotation term of the trailing eigenspace along the flow. Under ridge-regular assumptions, the stable ridge is a $C^2$, $k$-dimensional submanifold, the continuous flow and the discrete generalized SCMS converge to it R-linearly, and the limit map from the boundary of an $\epsilon$-neighborhood is surjective onto the ridge. Plug-in estimation attains Hausdorff error $O\!\left(h^2 + \sqrt{\log n/(n h^{d+4})}\right)$, and the original SCMS is shown to need $O(h^{-2}\log(1/\epsilon))$ iterations because its step size is implicitly $h^2$, while the generalized version needs only $O(\log n)$ iterations.

Load-bearing premise

The argument hinges on Assumption (A2), Property 2 of the ridge-regular class: the symmetric part of the Jacobian of the projected gradient must be uniformly negative definite (with margin $\gamma$) on the trailing eigenspace across the whole domain; without this uniform normal contraction, the exponential decay of $\|\xi\|$ along trajectories and the convergence theorems fail.

Editorial extensions

If this is right

  • If the paper is right, any SCMS trajectory that appears to converge to a static ridge is actually converging to the stable ridge; the two sets coincide only when the trailing eigenspace does not rotate (for example, radially symmetric densities).
  • The generalized SCMS with constant step size $\alpha$ converges R-linearly with rate $1-\alpha\gamma/4$ and needs $O(\log n)$ iterations to match the statistical error, instead of the original algorithm's $O(h^{-2}) = O(n^{2/(d+8)})$ iterations.
  • The estimated stable ridge is consistent in Hausdorff distance, with rate $O(h^2 + \sqrt{\log n/(n h^{d+4})})$; balancing bias and variance under the required bandwidth smoothness gives $O(((\log n)^{1+\delta}/n)^{2/(d+8)})$.
  • The stable ridge is a $C^2$ $k$-dimensional submanifold under the ridge-regular assumptions, and it can be recovered in full by initializing the flow from the boundary of a thin tubular neighborhood.
  • For $k=0$ the stable ridge reduces to the local maxima of the density, so the dynamical definition generalizes mode-seeking in a way the static definition does not.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The static/stable distinction suggests a general principle: any ridge- or filament-extraction algorithm built on repeated projection should be analyzed by the Jacobian of its own vector field, not by the Hessian of the density; otherwise the reported 'ridge' may be a different object.
  • The $O(h^2)$ step-size coupling identified in the original SCMS is likely shared by other mean-shift-type algorithms that derive their step from the kernel bandwidth; a constant-step reformulation may yield comparable speedups there.
  • One can test the theory directly on the Appendix A.1.2 example: the vertical axis is a static valley ($\lambda_2>0$) yet is predicted to be a stable ridge ($\mathrm{Re}(\mu_2)<0$), so SCMS should converge there rather than slide off.
  • The stable ridge, being defined by the flow of the projected gradient, may be better suited than the static ridge for tracking filamentary structures in time-varying or noisy data where eigenspace rotation is significant.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a new notion of density ridge, the 'stable ridge', defined as the zero set of the projected density gradient ξ with the (k+1)-th eigenvalue of its Jacobian Jξ having negative real part. It argues that the SCMS algorithm converges to this stable ridge rather than the classical static ridge defined via the Hessian. Under a ridge-regular class F(η,γ,κ,D), the paper proves: (i) the stable ridge is a C^2 k-dimensional submanifold; (ii) continuous and discrete SCMS flows converge R-linearly to the ridge and the limit map is surjective from the boundary of a tubular neighborhood; (iii) a plug-in KDE-based estimator yields Hausdorff rates O(h^2 + sqrt(log n/(n h^{d+4}))); and (iv) the original SCMS with bandwidth-coupled step size requires polynomial O(n^{2/(d+8)}) iterations, while a constant-step-size generalization needs O(log n) iterations. The proofs are collected in Appendix B and numerical experiments on a circle model illustrate the complexity and consistency claims.

Significance. If the results hold, the paper resolves a long-standing open question about the theoretical target of SCMS and provides a new dynamical-systems perspective on ridge estimation, together with concrete convergence rates and a computational-complexity comparison. The paper includes complete proofs in Appendix B, clear assumptions, and numerical validation. However, the universality of the paradigm-shift claim is currently restricted by the ridge-regular class assumption, and one proof step in the statistical transfer is flawed as written, so the significance is conditional on a successful repair of those points.

major comments (3)
  1. [§5.2, proof of Corollary 2, part (d)] The assertion 'For j≤k, Re(μ_j(x))=0' is stated for an arbitrary x∈D with ∥bξ(x)∥<κ/2, but the zero-eigenvalue structure of Jξ is only guaranteed on the zero set ξ=0 (or on the ridge manifold), not throughout the neighborhood R_κ(f). This makes the Elsner-based eigenvalue perturbation argument invalid as written, and since Corollary 2 is the step that transfers ridge-regularity to the estimated density bf, the error propagates into Theorems 4–7. The conclusion is nevertheless recoverable: because Re(μ_{k+1}(x))≤−γ/2 on {∥ξ(x)∥≤κ}, at most k eigenvalues of Jξ(x) can have real part larger than −γ/2; choosing the eigenvalue perturbation ε_J(n)<γ/8 then forces at most k eigenvalues of Jbξ(x) to have real part larger than −γ/4, yielding Re(bμ_{k+1}(x))<−γ/4 without the spurious zero-real-part claim.
  2. [Definition 1, Property 2, and Eq. (3)] The ridge-regular class requires sup_{u∈E_⊥,∥u∥=1} u^T Jξ(x)u ≤ −γ, which is strictly stronger than the defining stable-ridge condition Re(μ_{k+1}(x))<0; the Jacobian Jξ need not be normal, so a nonnormal matrix can have all eigenvalues in the left half-plane while its symmetric part is positive on some unit directions. Since Lemmas 2 and 4 and Theorems 1–3 and 5–7 all rely on this uniform numerical-range condition for the Lyapunov decay of ∥ξ∥, the paper establishes convergence to the stable ridge only for densities in the restricted class F(η,γ,κ,D), not for all densities that satisfy the eigenvalue definition in (3). The manuscript should either explicitly restrict the definition of the target to the regular class, prove the convergence under the weaker eigenvalue condition, or add a counterexample showing that SCMS can fail for eigenvalue-stable but non-regular densities; as written, this gap sits exactly at the interface between the new geometric definition and the convergence theorems.
  3. [Section 6 / Theorem 8] The claimed R-linear contraction with rate ρ_n = 1 − C_K γ_log h^2/16 for the original SCMS is asserted with a proof that says only 'similar to Theorem 5 and Corollary 3'; however, the original algorithm uses the state-dependent step size α_n(x) (Eq. (38)), so the Lyapunov argument of Lemma 4, which assumes a constant α, does not apply directly. The proof requires a uniform perturbation argument controlling the extra term bξ_log(x)∇α_n(x)^T in Eq. (109) and the variation of α_n along the trajectory; Lemma 7 states a uniform bound on ∥∇α_n∥ and a bound on α_n, but the detailed contraction proof is omitted. Since the polynomial iteration complexity m*=O(n^{2/(d+8)}) in Remark 5 is one of the paper's headline claims, this step needs a full proof or a more detailed derivation.
minor comments (4)
  1. [Section 3, before Lemma 3] There is a duplicated word in 'the the continuous differentiability of Φ'; please fix the typo.
  2. [Section 6, Eq. (35) and surrounding text] The profile function h(u) and the bandwidth h share the same symbol; consider denoting the profile by something like ψ(u) or h_profile to avoid confusion in the derivation of the Mean Shift vector.
  3. [Appendix A.1.1] The constant C in f(u,v)=C+3/8(1-u^2)v is said to make f a valid probability density, but the normalization is never specified; since ridges are invariant to affine transformations of f this is harmless, but a sentence clarifying that the example is considered up to normalization would be helpful.
  4. [Section 7.1] The step-size selection for the generalized SCMS is described as trial and error; a short remark linking the practical range α∈{0.1,0.2,0.3,0.4} to the theoretical bounds in (15) and (20) would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stable ridge is defined geometrically and the convergence theorems are proved from explicit assumptions, not from the algorithm's own target by construction.

full rationale

The paper's central claim is that SCMS converges to the stable ridge R_stable(f) = {x : ξ(x)=0, Re(μ_{k+1}(x))<0}, introduced as a geometric/dynamical object in Eq. (3), independently of the algorithm's iterates. The convergence results (Lemma 2, Lemma 4, Theorems 1–3, 5–8) are derived under Assumption (A2) / (A2'), which impose a uniform negative-definite numerical range on Jξ. These are substantive assumptions, not a restatement of the target definition. The observation that Property 2 is stronger than the eigenvalue condition Re(μ_{k+1})<0 is a possible gap in generality or a correctness risk for non-normal Jacobians, but it is not a circular reduction: the paper does not define the stable ridge by Property 2, nor does it fit any parameter to force the convergence result. The citation of [31] (Qiao & Polonik 2025, sharing the present author) supplies a motivating counterexample but does not carry the burden of the main proofs, which are self-contained. No fitted quantity is renamed as a prediction, and no load-bearing claim reduces to a self-citation. Hence no circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims are derived from explicit smoothness, regularity, and bandwidth assumptions; no numbers are fitted to data. The stable ridge is defined, not postulated, so it is not listed as an invented entity.

assumptions (7)
  • domain assumption Assumption (A1): f ∈ C^4 with uniformly bounded derivatives and f ≥ c0 on D.
    Required for the definition of the stable ridge and the perturbation analysis.
  • domain assumption Assumption (A2): f ∈ F(η,γ,κ,D), the ridge-regular class.
    Properties 1-3 provide the spectral gap, uniform contraction of the projected Jacobian, and separation of the ridge; all convergence theorems rest on this.
  • domain assumption Assumption (A2'): log-density p ∈ F(η_log,γ_log,κ_log,D) for analysis of the original SCMS.
    Used in Section 6 for the log-density stable ridge.
  • domain assumption Assumption (B1)/(B1'): kernel smoothness, VC subgraph classes, radial symmetry for mean shift.
    Standard KDE conditions for uniform convergence of derivatives.
  • domain assumption Assumption (B2): nh^{d+8}/log n → ∞.
    Ensures uniform consistency of fourth derivatives.
  • standard math Center Manifold Theorem and discrete analog.
    Invoked in Theorem 1 and Theorem 3 to prove surjectivity of the limit maps.
  • standard math Inverse Function Theorem, Weyl's inequality, Davis-Kahan sin Θ theorem, Elsner's spectral variation bound.
    Used in proofs of Lemma 6 and Corollary 2.

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Cite this review

Pith. "Pith review of Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift." pith.science (2026). https://pith.science/paper/EA72WUY2

@misc{pith2026260805112,
  author       = {Pith},
  title        = {Pith review of: Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EA72WUY2}},
  note         = {Machine review of arXiv:2608.05112}
}
read the original abstract

The Subspace Constrained Mean Shift (SCMS) algorithm is a popular nonparametric method for extracting density ridges, which serve as a low-dimensional representation of high-dimensional data. It is a widely held belief in the literature that SCMS trajectories converge to the classical density ridge, which we call the "static ridge", defined via the density gradient and the eigenvalues and eigenvectors of the density's Hessian. In this paper, we demonstrate that this assumption does not hold in general, as the static definition fails to account for the rotation of the trailing eigenspace along the continuous flow of the algorithm's underlying vector field. To resolve this, we propose a paradigm shift by introducing the "stable ridge", a novel geometric structure defined through the lens of dynamical systems and the Jacobian of the projected density gradient. We prove that this stable ridge is the true theoretical target of the SCMS algorithm. Building upon this foundation, we develop a generalized SCMS framework utilizing a constant step size, establishing its uniform R-linear convergence and topological surjectivity onto the stable ridge. We further derive the rates of convergence for estimating the stable ridge in terms of the Hausdorff distance. Finally, we expose that the original SCMS algorithm suffers from polynomial-time computational complexity, which is caused by implicitly coupling the step size to the smoothing bandwidth via the Mean Shift operator, and demonstrate how our generalized framework provides a statistically consistent and more efficient solution.

Figures

Figures reproduced from arXiv: 2608.05112 by the authors.

Figure 1
Figure 1. Visualization of the static ridge and SCMS. Left panel: A 3D perspective plot of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Number of iterations to convergence across varying sample sizes [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Trajectories of the original and generalized SCMS algorithms. The black dot denotes the [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Hausdorff distance between the estimated ridge [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: Three examples illustrating the difference and coincidence between static and stable ridges. [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

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Works this paper leans on

40 extracted references · 39 canonical work pages

  1. [1]

    Mason, and B

    Arias-Castro, E., D. Mason, and B. Pelletier (2016). Errata: On the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm.Journal of Machine Learning Research 17(206), 1–4

    work page 2016

  2. [2]

    Arias-Castro, E. and W. Qiao (2023a). Moving up the cluster tree with the gradient flow.SIAM Journal on Mathematics of Data Science 5(2), 400–421

    work page 2023

  3. [3]

    Arias-Castro, E. and W. Qiao (2023b). A unifying view of modal clustering.Information and Inference: A Journal of the IMA 12(2), 897–920

    work page 2023

  4. [4]

    Arias-Castro, E. and W. Qiao (2025). Clustering by hill-climbing: Consistency results.The Annals of Statistics 53(6), 2536–2562

    work page 2025

  5. [5]

    Chac´ on, J. E. (2015). A population background for nonparametric density-based clustering. Statistical Science 30(4), 518–532

    work page 2015

  6. [6]

    Genovese, and L

    Chen, Y., C. Genovese, and L. Wasserman (2015). Asymptotic theory for density ridges.Annals of Statistics 43(5), 1896–1928

    work page 2015

  7. [7]

    Chen, Y.-C., C. R. Genovese, and L. Wasserman (2017). Statistical inference using the morse- smale complex.Electronic Journal of Statistics 11, 1390–1433

    work page 2017

  8. [8]

    Chen, Y.-C., S. Ho, P. E. Freeman, C. R. Genovese, and L. Wasserman (2015). Cosmic web reconstruction through density ridges: method and algorithm.Monthly Notices of the Royal Astronomical Society 454(1), 1140–1156

    work page 2015

Show all 40 references
  1. [9]

    Cheng, Y. (1995). Mean shift, mode seeking, and clustering.IEEE Transactions on Pattern Analysis and Machine Intelligence 17(8), 790–799

    1995

  2. [10]

    Comaniciu, D. and P. Meer (2002). Mean shift: A robust approach toward feature space analysis.IEEE Transactions on Pattern Analysis and Machine Intelligence 24(5), 603–619

    2002

  3. [11]

    (1996).Ridges in Image and Data Analysis

    Eberly, D. (1996).Ridges in Image and Data Analysis. Kluwer Academic Publishers

    1996

  4. [12]

    Einmahl, U. and D. M. Mason (2005). Uniform in bandwidth consistency of kernel-type function estimators.The Annals of Statistics 33(3), 1380–1403

    2005

  5. [13]

    Elsner, L. (1985). An optimal bound for the spectral variation of two matrices.Linear algebra and its applications 71, 77–80

    1985

  6. [14]

    Ivanov, Y

    Fefferman, C., S. Ivanov, Y. Kurylev, M. Lassas, and H. Narayanan (2018). Fitting a putative manifold to noisy data. InConference On Learning Theory, pp. 688–720. PMLR. 21

    2018

  7. [15]

    Fukunaga, K. and L. Hostetler (1975). The estimation of the gradient of a density function, with applications in pattern recognition.IEEE Transactions on Information Theory 21(1), 32–40

    1975

  8. [16]

    Genovese, C. R., M. Perone Pacifico, I. Verdinelli, L. Wasserman, et al. (2014). Nonparametric ridge estimation.The Annals of Statistics 42(4), 1511–1545

    2014

  9. [17]

    Ghassabeh, Y. A., T. Linder, and G. Takahara (2013). On some convergence properties of the subspace constrained mean shift.Pattern Recognition 46(11), 3140–3147

    2013

  10. [18]

    Gin´ e, E. and A. Guillou (2002). Rates of strong uniform consistency for multivariate kernel density estimators.Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques 38(6), 907– 921

    2002

  11. [19]

    Hastie, T. and W. Stuetzle (1989). Principal curves.Journal of the American Statistical Association 84(406), 502–516

    1989

  12. [20]

    Krzyzak, T

    K´ egl, B., A. Krzyzak, T. Linder, and K. Zeger (2000). Learning and design of principal curves. IEEE transactions on pattern analysis and machine intelligence 22(3), 281–297

    2000

  13. [21]

    A.Elements of applied bifurcation theory

    Kuznetsov, Y. A.Elements of applied bifurcation theory. Springer

  14. [22]

    (2012).Introduction to Smooth Manifolds

    Lee, J. (2012).Introduction to Smooth Manifolds. Springer Science & Business Media

    2012

  15. [23]

    Li, W. and S. Ghosal (2020). Posterior contraction and credible sets for filaments of regression functions.Electronic Journal of Statistics 14, 1707–1743

    2020

  16. [24]

    Milnor, J. W. (1963).Morse theory. Number 51. Princeton university press

    1963

  17. [25]

    Moews, B., J. R. Argueta Jr, and A. Gieschen (2021). Filaments of crime: Informing policing via thresholded ridge estimation.Decision Support Systems 144, 113518

    2021

  18. [26]

    Ozertem, U. and D. Erdogmus (2011). Locally defined principal curves and surfaces.Journal of Machine Learning Research 12(34), 1249–1286

    2011

  19. [27]

    (2001).Differential Equations and Dynamical Systems(3rd ed.)

    Perko, L. (2001).Differential Equations and Dynamical Systems(3rd ed.). Texts in Applied Mathematics. Springer-Verlag New York

    2001

  20. [28]

    Qiao, W. (2021). Asymptotic confidence regions for density ridges.Bernoulli 27(2), 946–975

    2021

  21. [29]

    Qiao, W. (2025). Confidence regions for filamentary structures.Information and Inference: A Journal of the IMA 14(4), iaaf030

    2025

  22. [30]

    Qiao, W. and W. Polonik (2016). Theoretical analysis of nonparametric filament estimation. The Annals of Statistics, 1269–1297

    2016

  23. [31]

    Qiao, W. and W. Polonik (2025). Algorithms for ridge estimation with convergence guarantees. Journal of Machine Learning Research 26

    2025

  24. [32]

    Qiao, W. and A. Shehu (2022). Space partitioning and regression maxima seeking via a mean- shift-inspired algorithm.Electronic Journal of Statistics 16(2), 5623–5658

    2022

  25. [33]

    Roweis, S. T. and L. K. Saul (2000). Nonlinear dimensionality reduction by locally linear embedding.science 290(5500), 2323–2326

    2000

  26. [34]

    Sandilya, S. and S. R. Kulkarni (2002). Principal curves with bounded turn.IEEE Transactions on Information Theory 48(10), 2789–2793

    2002

  27. [35]

    Tenenbaum, J. B., V. d. Silva, and J. C. Langford (2000). A global geometric framework for nonlinear dimensionality reduction.science 290(5500), 2319–2323

    2000

  28. [36]

    Yao, Z., J. Su, B. Li, and S.-T. Yau (2023). Manifold fitting.arXiv preprint arXiv:2304.07680

    2023 arXiv

  29. [37]

    Yao, Z. and Y. Xia (2025). Manifold fitting under unbounded noise.Journal of Machine Learning Research 26(45), 1–55

    2025

  30. [38]

    and Y.-C

    Zhang, Y. and Y.-C. Chen (2023). Linear convergence of the subspace constrained mean shift algorithm: from euclidean to directional data.Information and Inference: A Journal of the IMA 12(1), 210–311

    2023

  31. [39]

    and Y.-C

    Zhang, Y. and Y.-C. Chen (2026). Mode and ridge estimation in euclidean and directional product spaces: A mean shift approach.Journal of Computational and Graphical Statistics 35(1), 101–110. 22 A Static vs. Stable Ridges As discussed in the Introduction section, the local con...

    2026

  32. [40]

    A.1.2 Example 2: Static V alley as Stable Ridge ConsiderD= [−0.5,0.5]×[−0.1,0.4] and a density defined as f(u, v) =C+v+ 5v 2 + 1 2 (1 + 18v)u2

    For this example, the stable ridge is a strict subset of the static ridge. A.1.2 Example 2: Static V alley as Stable Ridge ConsiderD= [−0.5,0.5]×[−0.1,0.4] and a density defined as f(u, v) =C+v+ 5v 2 + 1 2 (1 + 18v)u2. The derivatives are∇f(u, v) = u(1 + 18v) 1 + 10v+ 9u 2 and...

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Reviewed August 6, 2026 · model on record in the stance chip above.