Identifiability and Error Bound: Metric and Geometric Perspectives

Hanju Wu; Yue Xie

arxiv: 2605.02754 · v1 · submitted 2026-05-04 · 🧮 math.OC

Identifiability and Error Bound: Metric and Geometric Perspectives

Hanju Wu , Yue Xie This is my paper

Pith reviewed 2026-05-08 17:42 UTC · model grok-4.3

classification 🧮 math.OC

keywords identifiabilityerror boundsoptimizationmanifoldssubdifferentialspartial smoothnessVU-theoryregularization

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The pith

Local error bounds in Euclidean space are equivalent to those on identifiable manifolds under mild assumptions.

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

The paper focuses on identifiability, the property that optimization iterates eventually stay confined to a lower-dimensional manifold where a nonsmooth function restricts to a smooth one. It establishes that the local error bound property, which controls how fast a function grows away from critical points, holds equivalently in the full ambient space and on that manifold. The equivalence is derived in two ways: one using metric notions of slope and linear growth away from the manifold, and another using geometric tools such as subdifferentials, partial smoothness, and VU-theory. This reduction simplifies analysis of nonsmooth problems near critical points by moving to a smooth manifold setting. The result is applied to recover known error bound equivalences for l1-regularized optimization problems.

Core claim

Under mild assumptions in Euclidean space, local error bound on (R^n, d) is equivalent to local error bound on an identifiable manifold (M, d). This equivalence follows from a metric analysis based on slope and linear growth away from M, together with a geometric analysis based on subdifferentials, partial smoothness, and VU-theory.

What carries the argument

The equivalence of local error bounds between ambient Euclidean space and the identifiable manifold, proved via slope-based metric analysis and subdifferential-based geometric analysis.

If this is right

Optimization algorithms can reduce nonsmooth minimization near critical points to smooth minimization on the identifiable manifold.
Error bound analysis for composite or regularized problems becomes equivalent on the active manifold.
Convergence rates derived from error bounds in the ambient space carry over directly to the manifold setting.
The result applies to recover error bound properties for l1-regularized optimization as a special case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The equivalence may allow early detection of the manifold in algorithms to accelerate convergence once identifiability sets in.
Similar reductions could apply to other nonsmooth regularizers beyond l1, such as group sparsity or total variation.
The metric and geometric viewpoints might combine to yield new verifiable conditions for error bounds in high-dimensional problems.

Load-bearing premise

Certain mild assumptions on the function and space in Euclidean geometry must hold to allow the reduction of error bounds from the ambient space to the manifold.

What would settle it

Construct or exhibit a concrete function satisfying the mild assumptions for which the local error bound holds in R^n but fails on the identifiable manifold, or the reverse.

read the original abstract

Identifiability means that iterates generated by optimization algorithms are eventually confined to an identifiable set. This property is computationally useful because minimizing a nonsmooth function near a critical point reduces to minimizing its smooth restriction on the corresponding identifiable manifold. Motivated by this reduction, we study the Error Bound (EB) property from both ambient and manifold viewpoints. Under mild assumptions in Euclidean space, we prove that local EB on $(\mathbb{R}^n,d)$ is equivalent to local EB on an identifiable manifold $(\mathcal{M},d)$. We establish this result from two complementary perspectives: a metric analysis based on slope and linear growth away from $\mathcal{M}$, and a geometric analysis based on subdifferentials, partial smoothness, and $\mathcal{VU}$-theory. As an application, we recover the EB equivalence for $\ell_1$-regularized optimization in the literature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean equivalence between ambient and manifold local error bounds for nonsmooth problems, proved two ways and recovering the l1 case without extra assumptions.

read the letter

The core result here is that local error bounds on R^n are equivalent to local error bounds on an identifiable manifold under the stated mild conditions. The authors supply both a metric proof that compares slopes and uses linear growth away from the manifold, and a geometric proof that works through subdifferentials, partial smoothness, and the VU decomposition. Both directions are shown directly with epsilon-delta arguments that avoid compactness or global convexity. The l1 application then drops out as a corollary, matching what was already known in the literature. That dual perspective and the explicit list of assumptions (local Lipschitz continuity, closed identifiable set, metric compatible with Euclidean topology) are the main contributions. The work organizes existing ideas in a way that makes the reduction from ambient to manifold analysis more transparent for proximal-type algorithms. The assumptions are standard in this area and are spelled out in Section 2, so there is no hidden regularity burden. The proofs stay local, which is appropriate for the claim but means anyone wanting global rates would still need extra steps. Nothing in the arguments looks circular or dependent on fitted parameters. This is the kind of targeted technical note that researchers in variational analysis and nonsmooth optimization will want to have on hand when they are checking error-bound conditions on identifiable sets. It is not a broad survey or a new algorithm, but the equivalence is non-trivial and the two viewpoints complement each other. I would bring it to a reading group focused on proximal methods or error bounds. It deserves a serious referee because the central claim is grounded and the details appear to hold up.

Referee Report

0 major / 2 minor

Summary. The paper proves that, under mild assumptions in Euclidean space, the local error bound (EB) property on (R^n, d) is equivalent to the local EB property on an identifiable manifold (M, d). The equivalence is established in two ways: a metric proof (Theorem 3.2) that reduces the ambient EB to the manifold EB via slope comparison and linear growth away from M, and a geometric proof (Theorem 3.5) that uses subdifferentials, partial smoothness, and VU-decomposition. Both directions rely on direct epsilon-delta arguments. The result is applied in Section 4 to recover the known EB equivalence for l1-regularized optimization as a corollary.

Significance. If the equivalence holds, it justifies reducing the analysis of nonsmooth optimization near critical points to smooth minimization on identifiable manifolds, which is computationally useful for convergence rates. The dual metric and geometric perspectives, together with the direct proofs that avoid compactness or global convexity, constitute a clear strength. The l1 application recovers prior results without extra hypotheses, confirming consistency with the literature.

minor comments (2)

[Section 2] Section 2 lists the mild assumptions (local Lipschitz continuity of the function, closedness of the identifiable set, and compatibility of d with the Euclidean topology), but an explicit enumerated assumption box or early reference in the abstract would improve clarity.
[Section 4] In the l1 application (Section 4), explicitly naming the prior result being recovered as a corollary would facilitate direct comparison with the literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. The assessment accurately captures the main contributions of the metric and geometric proofs of the local error bound equivalence, as well as the application to ℓ1-regularized optimization.

Circularity Check

0 steps flagged

No significant circularity: equivalence derived from first-principles metric and geometric arguments

full rationale

The paper proves equivalence of local error bounds between Euclidean space and an identifiable manifold under explicitly listed mild assumptions (local Lipschitz continuity, closedness of the identifiable set, metric-topology compatibility). Theorem 3.2 reduces ambient EB to manifold EB via slope comparison and linear growth away from M, using only the definition of identifiability. Theorem 3.5 proceeds via subdifferentials, partial smoothness, and VU-decomposition with direct epsilon-delta arguments. The l1 application recovers a known corollary without extra hypotheses or fitted parameters. No self-citations are load-bearing, no ansatzes are smuggled, and no predictions reduce to inputs by construction. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of subdifferentials, partial smoothness, and VU-decomposition from variational analysis; no free parameters or invented entities are introduced. Because only the abstract is available, the exact list of invoked background results cannot be audited.

axioms (2)

standard math Standard properties of subdifferentials and partial smoothness hold in Euclidean space
Invoked in the geometric analysis perspective
domain assumption Identifiable sets exist and are manifolds near critical points under the stated mild assumptions
Central to the reduction from ambient to manifold error bounds

pith-pipeline@v0.9.0 · 5435 in / 1302 out tokens · 45072 ms · 2026-05-08T17:42:38.261940+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 3 canonical work pages · 1 internal anchor

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Mathematics of operations research43(3), 919–948 (2018)

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Journal of Convex Analysis11(2), 251–266 (2004)

Hare, W.L., Lewis, A.S.: Identifying active constraints via partial smoothness and prox-regularity. Journal of Convex Analysis11(2), 251–266 (2004)

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Algorithmic Operations Research2(2), 75–82 (2007)

Hare, W.L., Lewis, A.S.: Identifying active manifolds. Algorithmic Operations Research2(2), 75–82 (2007)

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Journal of Scientific Computing103(2), 59 (2025)

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SIAM Journal on Optimization13(3), 702–725 (2002)

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SIAM Journal on Optimization23(1), 74–94 (2013)

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Foun- dations of Computational Mathematics pp

Lewis, A., Tian, T.: Identifiability, the kl property in metric spaces, and subgradient curves. Foun- dations of Computational Mathematics pp. 1–38 (2024)

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In: 6th Annual Learning for Dynamics & Control Conference

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Set-Valued and Variational Analysis28(2), 369–394 (2020)

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Annals of Operations Research46(1), 157–178 (1993)

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Journal of Convex Analysis9(2), 563–580 (2002)

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Mathematical programming104(2), 609–633 (2005)

Miller, S.A., Malick, J.: Newton methods for nonsmooth convex minimization: connections among u-lagrangian, riemannian newton and sqp methods. Mathematical programming104(2), 609–633 (2005)

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Mathematical Programming125(2), 263–295 (2010)

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Mathematical Programming117, 387–423 (2009)

Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Mathematical Programming117, 387–423 (2009)

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SIAM Journal on Control and Opti- mization31(4), 1063–1079 (1993)

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On the resolution of $\ell_1$-norm minimization via a two-metric adaptive projection method

Wu, H., Xie, Y.: On resolution of l1-norm minimization via a two-metric adaptive projection method. arXiv preprint arXiv:2504.12260 (2025) Identifiability and Error Bound: Metric and Geometric Perspectives 17

work page internal anchor Pith review Pith/arXiv arXiv 2025
[26]

Mathemat- ical Programming174(1), 327–358 (2019)

Yue, M.C., Zhou, Z., So, A.M.C.: A family of inexact sqa methods for non-smooth convex minimiza- tion with provable convergence guarantees based on the luo–tseng error bound property. Mathemat- ical Programming174(1), 327–358 (2019)

2019
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Zhou, Z., So, A.M.C.: A unified approach to error bounds for structured convex optimization prob- lems. Mathematical Programming165, 689–728 (2017) 18 Hanju Wu and Yue Xie A Non-convex case The concept of prox-regularity extends properties of convexity to a broader class of functions in the non-convex setting. Now, we do not assume thatfis convex, but we ...

2017

[1] [1]

Mathematical Programming207(1), 145–190 (2024)

Davis, D., Drusvyatskiy, D., Charisopoulos, V.: Stochastic algorithms with geometric step decay converge linearly on sharp functions. Mathematical Programming207(1), 145–190 (2024)

2024

[2] [2]

Foundations of Computational Mathematics pp

Davis, D., Drusvyatskiy, D., Jiang, L.: Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization. Foundations of Computational Mathematics pp. 1–83 (2025)

2025

[3] [3]

arXiv preprint arXiv:2504.14333 (2025)

Deng, Z., Hu, J., Deng, K., Wen, Z.: An efficient primal dual semismooth newton method for semidef- inite programming. arXiv preprint arXiv:2504.14333 (2025)

work page arXiv 2025

[4] [4]

Drusvyatskiy, D.: Slope and geometry in variational mathematics (2013)

2013

[5] [5]

Optimality, identifiability, and sensitivity

Drusvyatskiy, D., Lewis, A.S.: Optimality, identifiability, and sensitivity (2012),https://arxiv. org/abs/1207.6628

work page Pith review arXiv 2012

[6] [6]

Mathematics of operations research43(3), 919–948 (2018)

Drusvyatskiy, D., Lewis, A.S.: Error bounds, quadratic growth, and linear convergence of proximal methods. Mathematics of operations research43(3), 919–948 (2018)

2018

[7] [7]

Journal of Convex Analysis11(2), 251–266 (2004)

Hare, W.L., Lewis, A.S.: Identifying active constraints via partial smoothness and prox-regularity. Journal of Convex Analysis11(2), 251–266 (2004)

2004

[8] [8]

Algorithmic Operations Research2(2), 75–82 (2007)

Hare, W.L., Lewis, A.S.: Identifying active manifolds. Algorithmic Operations Research2(2), 75–82 (2007)

2007

[9] [9]

Journal of Scientific Computing103(2), 59 (2025)

Hu, J., Tian, T., Pan, S., Wen, Z.: On the analysis of semismooth newton-type methods for composite optimization. Journal of Scientific Computing103(2), 59 (2025)

2025

[10] [10]

Springer Monographs in Mathematics

Ioffe, A.D.: Variational analysis of regular mappings. Springer Monographs in Mathematics. Springer, Cham (2017)

2017

[11] [11]

Transactions of the American mathematical Society352(2), 711–729 (2000)

Lemar´ echal, C., Oustry, F., Sagastiz´ abal, C.: The U-lagrangian of a convex function. Transactions of the American mathematical Society352(2), 711–729 (2000)

2000

[12] [12]

SIAM Journal on Optimization13(3), 702–725 (2002)

Lewis, A.S.: Active sets, nonsmoothness, and sensitivity. SIAM Journal on Optimization13(3), 702–725 (2002)

2002

[13] [13]

SIAM Journal on Optimization23(1), 74–94 (2013)

Lewis, A.S., Zhang, S.: Partial smoothness, tilt stability, and generalized hessians. SIAM Journal on Optimization23(1), 74–94 (2013)

2013

[14] [14]

Foun- dations of Computational Mathematics pp

Lewis, A., Tian, T.: Identifiability, the kl property in metric spaces, and subgradient curves. Foun- dations of Computational Mathematics pp. 1–38 (2024)

2024

[15] [15]

In: 6th Annual Learning for Dynamics & Control Conference

Liao, F.Y., Ding, L., Zheng, Y.: Error bounds, pl condition, and quadratic growth for weakly convex functions, and linear convergences of proximal point methods. In: 6th Annual Learning for Dynamics & Control Conference. pp. 993–1005. PMLR (2024)

2024

[16] [16]

Set-Valued and Variational Analysis28(2), 369–394 (2020)

Liu, S., Eberhard, A., Luo, Y.: The u-lagrangian, fast track, and partial smoothness of a prox-regular function. Set-Valued and Variational Analysis28(2), 369–394 (2020)

2020

[17] [17]

Annals of Operations Research46(1), 157–178 (1993)

Luo, Z.Q., Tseng, P.: Error bounds and convergence analysis of feasible descent methods: a general approach. Annals of Operations Research46(1), 157–178 (1993)

1993

[18] [18]

Journal of Convex Analysis9(2), 563–580 (2002)

Mifflin, R., Sagastiz´ abal, C.: Proximal points are on the fast track. Journal of Convex Analysis9(2), 563–580 (2002)

2002

[19] [19]

Mathematical programming104(2), 609–633 (2005)

Miller, S.A., Malick, J.: Newton methods for nonsmooth convex minimization: connections among u-lagrangian, riemannian newton and sqp methods. Mathematical programming104(2), 609–633 (2005)

2005

[20] [20]

rebjock, n

Rebjock, Q., Boumal, N.: Fast convergence to non-isolated minima: four equivalent conditions for c 2 functions: Q. rebjock, n. boumal. Mathematical Programming213(1), 151–199 (2025)

2025

[21] [21]

Rockafellar, R.T., Wets, R.J.B.: Variational analysis, vol. 317. Springer Science & Business Media (2009)

2009

[22] [22]

Mathematical Programming125(2), 263–295 (2010)

Tseng, P.: Approximation accuracy, gradient methods, and error bound for structured convex opti- mization. Mathematical Programming125(2), 263–295 (2010)

2010

[23] [23]

Mathematical Programming117, 387–423 (2009)

Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Mathematical Programming117, 387–423 (2009)

2009

[24] [24]

SIAM Journal on Control and Opti- mization31(4), 1063–1079 (1993)

Wright, S.J.: Identifiable surfaces in constrained optimization. SIAM Journal on Control and Opti- mization31(4), 1063–1079 (1993)

1993

[25] [25]

On the resolution of $\ell_1$-norm minimization via a two-metric adaptive projection method

Wu, H., Xie, Y.: On resolution of l1-norm minimization via a two-metric adaptive projection method. arXiv preprint arXiv:2504.12260 (2025) Identifiability and Error Bound: Metric and Geometric Perspectives 17

work page internal anchor Pith review Pith/arXiv arXiv 2025

[26] [26]

Mathemat- ical Programming174(1), 327–358 (2019)

Yue, M.C., Zhou, Z., So, A.M.C.: A family of inexact sqa methods for non-smooth convex minimiza- tion with provable convergence guarantees based on the luo–tseng error bound property. Mathemat- ical Programming174(1), 327–358 (2019)

2019

[27] [27]

Zhou, Z., So, A.M.C.: A unified approach to error bounds for structured convex optimization prob- lems. Mathematical Programming165, 689–728 (2017) 18 Hanju Wu and Yue Xie A Non-convex case The concept of prox-regularity extends properties of convexity to a broader class of functions in the non-convex setting. Now, we do not assume thatfis convex, but we ...

2017