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REVIEW 2 major objections 6 minor 33 references

Kurdyka-\L ojasiewicz exponent via square transformation

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Squaring a constrained composite objective yields an explicit Kurdyka–Łojasiewicz exponent for the unconstrained lift, deduced from the original problem's data.

desk verdict A genuinely strong strict-complementarity KL transfer with a broken non-strict-complementarity section that needs a rewrite before this is citable in full. read the letter →

arxiv 2506.10110 v2 pith:CCUMJSCR submitted 2025-06-11 math.OC

Wenqing Ouyang This is my paper · ORCID
classification math.OC MSC 90C2590C2668Q25
keywords Kurdyka–ŁojasiewiczexponentsquaretransformationHadamardparameterizationpolyhedralfunctionsecondsubderivativesubdifferentialdistanceerrorboundstrictcomplementarity
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

This paper studies the oldest trick in constrained optimization: replace $x \ge 0$ by $x = y^2$, turning a constrained composite problem into an unconstrained one. The central claim is that the Kurdyka--\L ojasiewicz (KL) exponent of the squared problem---the number that controls how quickly first-order methods approach a stationary point---can be deduced from the original problem instead of being re-established from scratch. Under a strict-complementarity condition the transfer is exact: the new exponent is $\max\{\alpha, 1/2\}$ when the old one is $\alpha$. In a convex degenerate case satisfying an error-bound condition, the paper derives the exponent $(1+\beta)/2$ with $\beta = 1 - \gamma(1-\alpha)$, where $\gamma$ is the error-bound exponent. The proof matters because the square map has zero or low-rank Jacobian at boundary points, so the usual chain rules for subdifferentials and second subderivatives are not available.

What carries the argument

The machine that carries the argument is the exact distance formula of Proposition 3.2, $\mathrm{dist}(0, \partial\Phi(y)) = 2\,\mathrm{dist}(-y \circ \nabla f(y^2),\, y \circ \partial g(y^2))$, derived by exploiting the evenness of $H(y) = g(y^2)$ to avoid the failed chain rule for $T: y \mapsto y^2$. Alongside it sits the second-subderivative formula of Proposition 4.2: on the subspace $S_I = \{w : w_I = 0\}$, $d^2(g \circ T)(\bar{y} \mid 2\bar{y} \circ v)(w)$ equals $2 \sup_{p \in S(I,v) \cap \partial g(\bar{x})} \langle w_{I^c}^2,\, p_{I^c} \rangle$, where $S(I,v)$ is the set of vectors agreeing with $v$ on $I$. This formula is what allows the paper to single out the stationary points of $\Phi$ that correspond to genuine stationary points of $\phi$. The KL transfer in Section 5 then combines these formulas with finite face decompositions of polyhedral subdifferentials, minimal exposed faces, and projection estimates on relative interiors.

What would settle it

Take a small convex instance with polyhedral $g$ and a degenerate stationary point where strict complementarity fails, and check whether $\mathrm{dist}^2(0, \partial\Phi(y)) \ge c(\Phi(y) - \Phi(\bar{y}))^{1+\beta}$ holds with the stated $\beta$ for all nearby $y$; any failing sequence with no positive $c$ refutes the transfer. Separately, inspect the step of Theorem 6.2 that invokes Lemma 6.1: replacing the printed condition (6.2) by the lemma's $I^c$ version and re-running the proof would show whether the index-set mismatch is a typo or a substantive gap.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is two precise transfer theorems. Theorem 5.12: if $\phi = f + g$ with $f \in C^2$ and $g$ proper polyhedral, $0$ lies in the relative interior of $\partial\phi(\bar{x})$, and $\phi$ has the KL property at $\bar{x}$ with exponent $\alpha$, then $\Phi(y) = f(y^2) + g(y^2)$ has the KL property at $\bar{y}$ (with $\bar{x} = \bar{y}^2$) with exponent $\max\{\alpha, 1/2\}$. Theorem 6.2: if $\phi$ is convex, has KL exponent $\alpha$ at a stationary point $\bar{x}$, and satisfies the H\"older error-bound condition (6.2) with exponent $\gamma$, then $\Phi$ has KL exponent $(1+\beta)/2$, where $\beta = 1 - \gamma(1-\alpha)$. Both theorems rest on the identity $\mathrm{dist}(0, \partial\Phi(y)) = 2\,\mathrm{dist}(-y \circ \nabla f(y^2),\, y \circ \partial g(y^2))$, which makes the subdifferential distance in the lifted problem a weighted subdifferential distance in the original variables, and on a second-subderivative formula for $g \circ T$ along the subspace where the nonzero coordinates are frozen. The paper's contribution is that the exponent is deduced rather than fitted: no new KL analysis of the lifted problem is needed once the original data are known.

Load-bearing premise

The load-bearing premise is the H\"older error-bound condition (6.2) in Theorem 6.2: it is assumed for every $\rho$ rather than proved for the general polyhedral problem, and as printed it constrains the nonzero-coordinate set $I$ while the lemma cited in the proof requires the complementary set $I^c$.

Editorial extensions

If this is right

  • If $\phi$ has KL exponent $\alpha \le 1/2$ at a strict-complementarity stationary point, the squared problem inherits exactly $\alpha$; if $\alpha > 1/2$, the squared problem improves to $1/2$.
  • In the convex degenerate case, the formula $(1+\beta)/2$ converts known original-problem data (KL exponent $\alpha$ and error-bound exponent $\gamma$) into a specific convergence exponent for the lifted problem, so no additional KL computation is required.
  • The identity for $\mathrm{dist}(0, \partial\Phi(y))$ gives a computable stationarity measure for the lifted problem, a practical stopping or certificate quantity for algorithms run directly on $y$.
  • The second-order characterization recovers, as special cases, earlier results for simplex-to-sphere Hadamard parameterizations, basis-pursuit reformulations, and polyhedral Hadamard parameterizations.

Reading between the lines

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

  • If Theorem 5.12 holds, local rate analyses for algorithms on the squared variables can be imported from the original problem, which would save re-deriving KL exponents for each reparameterized model.
  • The face-based proof suggests a testable generalization: replace polyhedral $g$ by a definable (e.g., semi-algebraic) function with known KL exponent; the squared lift may still admit an explicit exponent, though the face machinery would need a tame-geometry substitute.
  • A literal reading of Theorem 6.2 defines $S_\rho = \{x : |x_i - \bar{x}_i| \le \rho_i \text{ for } i \in I\}$ while Lemma 6.1, which the proof invokes unchanged, requires the complementary index set $I^c$; if that is a typo, the theorem should be read with $I^c$, and the numerical predictions of Remark 6.4 still stand.
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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

2 major / 6 minor

Summary. The paper studies the square (Hadamard) reparametrization Φ(y)=f(y^2)+g(y^2) of the composite objective ϕ(x)=f(x)+g(x), where f∈C^2 and g is a proper polyhedral function with dom g⊆R_+^n. The main contributions are: a formula for the distance from 0 to the subdifferential of Φ (Prop. 3.2); a computation of the second subderivative of g∘T on a linear subspace and a characterization of those stationary points of Φ that correspond to stationary points of ϕ (Prop. 4.2 and 4.3); and two KL-exponent transfer theorems. Theorem 5.12 states that under the strict complementarity condition 0∈ri(∂ϕ(\bar{x})), if ϕ has KL exponent α at \bar{x}, then Φ has KL exponent max{α,1/2} at \bar{y} with \bar{x}=\bar{y}^2. Theorem 6.2 claims that under convexity and a Hölderian error-bound condition (6.2), the exponent becomes (1+β)/2 with β=1−γ(1−α), without strict complementarity. The paper is written as a sequence of theorems with detailed proofs for the strict-complementarity part, but the non-strict part depends on an unproved lemma and a hypothesis that is stated on the wrong index block.

Significance. If the main claims are correct, the KL-exponent transfer is a valuable parameter-free result: it gives a principled way to compute the KL exponent of a frequently used reparametrization without fitting constants, and it extends the Hadamard-parametrization analysis of [22] to the nonsmooth case. The first-order distance formula and the face-based second-subderivative computation are substantive original tools, and the paper is honest in stating its assumptions. However, the advertised non-strict-complementarity transfer is not established as printed: the error-bound hypothesis in Theorem 6.2 is defined on the wrong coordinate block relative to Lemma 6.1, and Lemma 6.1 itself is asserted by reference rather than proved. The strict-complementarity theorem (Section 5) appears internally consistent and is developed in detail, which supports a major-revision verdict rather than rejection, but the paper's full advertised scope requires repair of Section 6.

major comments (2)
  1. [6 (Eqs. (6.1)-(6.3))] Theorem 6.2's error-bound hypothesis does not match the lemma it invokes. Lemma 6.1 assumes (6.1) with S_ρ = {x : |x_i − \bar{x}_i| ≤ ρ_i for i ∈ I^c}, while Theorem 6.2 states (6.2) with S_ρ = {x : |x_i − \bar{x}_i| ≤ ρ_i for i ∈ I}. For I = {i : \bar{x}_i ≠ 0}, these are different sets and neither contains the other. The proof of Theorem 6.2 then says “By Lemma 6.1” to obtain (6.3); on a literal reading, (6.2) does not imply the hypothesis of Lemma 6.1. Since (6.3) is exactly the inequality that produces the claimed exponent (1+β)/2, the non-strict-complementarity theorem lacks a valid hypothesis as printed. The fix is not a one-character typo: the geometry of Lemma 6.1 is tailored to the complementary (zero) block, so either (6.2) must be restated with I^c and the proof re-checked, or a genuinely different argument must be supplied.
  2. [6 (Lemma 6.1)] Lemma 6.1 is load-bearing but is not proved in the manuscript. The text says it follows by essentially the same argument as [22, Lemma 3.10], with two listed differences: g is not assumed differentiable, and the separable structure encoded by J_1 in [22] is absent. These are substantive differences, not cosmetic ones: the proof in [22] relies on separable smooth calculus, while the present setting is nonsmooth and nonseparable. The manuscript does not reproduce the modified argument, state which steps change, or indicate how the constants are obtained. Because Theorem 6.2's only route to (6.3) is Lemma 6.1, the advertised non-strict exponent transfer is not verifiable from the manuscript as written. A full proof of Lemma 6.1, or a precise step-by-step reduction to [22, Lemma 3.10] with all adaptations, is required.
minor comments (6)
  1. [3 (Prop. 3.2)] In the proof of Proposition 3.2, the text invokes “Proposition 3.2” to obtain w ∈ ∂g(y^2) from v ∈ ∂(g∘T)(y); this should refer to Proposition 3.1(ii), since Proposition 3.2 is the statement being proved.
  2. [4 (Prop. 4.2)] In the proof of Proposition 4.2, several occurrences of ∂f(\bar{x}) in Equations (4.10)(d), (4.11), and the surrounding display should be ∂g(\bar{x}); f is not polyhedral in the statement and the result concerns g.
  3. [4 (Prop. 4.3)] In the proof of Proposition 4.3, the phrase “d²Φ(y)(w) ≥ 0 for all w ∈ S_{I^c}” should read w ∈ S_I, to match hypothesis (i) and the subsequent display that quantifies over w ∈ S_I.
  4. [2 (Def. 2.2)] Definition 2.2(b) is circular as printed: it defines v ∈ ∂f(\bar{x}) in terms of sequences v^k ∈ ∂f(x^k). The definition should use regular subgradients, v^k ∈ \hat∂ f(x^k), as in [26, Definition 8.3].
  5. [2 (Def. 2.3)] In Definition 2.3, the phrase “the φ(s) in (2.5)” should refer to Equation (2.1), which is the sharpened KL inequality; there is no Equation (2.5) in the paper.
  6. [5 (Lemma 5.8)] In the proof of Lemma 5.8, the sentence “Let V be a neighborhood of s such that (5.4) holds by using Lemma 5.7” appears to refer to Equation (5.3), not to the inequality (5.4) that the lemma is proving.

Circularity Check

1 steps flagged · score 2.0 of 10

No fitted-parameter circularity: the KL-exponent transfers are deduced, not fitted. The single flagged reduction is Theorem 6.2's proof, which rests on Lemma 6.1 — a load-bearing same-author citation with asserted modifications and a printed S_rho mismatch (I vs I^c) — a rigor/correctness defect rather than a circular derivation.

  1. self citation load bearing [Section 6, Lemma 6.1 and Theorem 6.2 (equations (6.1)-(6.3))]
    "The next lemma can be deduced by using essentially the same argument as in [22, Lemma 3.10]. The main differences here are that: first, we do not require the differentiability of g; second, we set J1 = ∅ in [22, Lemma 3.1] since we do not have an explicit separable structure here as in [22]. ... By Lemma 6.1, we know there exists a neighborhood V of ¯x such that (6.3)"

    The advertised non-strict-complementarity transfer (Theorem 6.2) reduces, at its only nontrivial step, to Lemma 6.1: the proof's engine is "By Lemma 6.1", which converts the assumed error bound (6.2) into the weighted bound (6.3); the exponent (1+β)/2 then follows by algebra. But Lemma 6.1 is stated without proof; its only justification is a citation to the author's own prior work [22, Lemma 3.10] with two asserted modifications. The load-bearing estimate is thus taken on the authority of a same-author preprint, not derived here. Separately, as printed the hypotheses do not match: Lemma 6.1's S_ρ is built on I^c, while Theorem 6.2's (6.2) builds S_ρ on I, so the cited lemma is not literally applicable.

full rationale

The central derivation chain is self-contained for Sections 3-5. Proposition 3.2 (dist(0, ∂Φ(y)) = 2dist(−y∘∇f(y²), y∘∂g(y²))) is proved in-paper via variational arguments; Theorem 5.12 (exponent max{α, 1/2} under strict complementarity) is proved from Lemmas 5.8-5.10, all proved in the paper, plus standard Lipschitz facts. No fitted parameters or empirical constants appear anywhere; the error-bound exponent γ and the KL exponent α are stated hypotheses, and the output exponents (max{α, 1/2}, (1+β)/2) are forced by the derived inequalities, not by fitting. The only place the derivation reduces to an external authority is Section 6: Theorem 6.2's proof invokes "By Lemma 6.1" to obtain (6.3), and Lemma 6.1 is itself justified only by "essentially the same argument as in [22, Lemma 3.10]", a preprint by the same first author. Under rule 4, [22] is parameter-free with stated assumptions that do not include the present target result, so this is real evidence rather than full circularity; however, the adaptation is asserted, not demonstrated, and as printed the hypotheses do not match (Lemma 6.1's S_ρ uses I^c; Theorem 6.2's (6.2) uses I), so the printed chain is not a closed argument. The tightness remark likewise leans on [22, Example 3.14]. These are correctness and rigor risks, weighed here as the reason the score is 2 rather than 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants or invented entities are introduced. The central transfer results rest on domain assumptions (C^2 smooth f, polyhedral g with nonnegative domain), strict complementarity or convexity plus an error bound, and on a self-cited lemma from [22] whose proof is not reproduced.

assumptions (6)
  • domain assumption f is in C^2(R^n)
    Used throughout, e.g. in problem (1.5), equation (5.22), and the Lipschitz properties in Theorem 5.12.
  • domain assumption g is proper polyhedral with dom g subset R^n_+
    Core structural assumption in (1.5) and throughout Sections 3 to 6, including Propositions 2.5 and 3.1.
  • standard math Variational analysis calculus from Rockafellar and Wets
    Provides subderivatives, subdifferentials, chain rules and normal cone facts used in almost every proof.
  • domain assumption Strict complementarity: 0 in ri(delta phi(xbar))
    Assumed in Theorem 5.12 and used through Lemmas 5.8 to 5.10 to obtain the max{alpha,1/2} transfer.
  • ad hoc to paper Convexity of phi plus error bound (6.2)
    Introduced in Theorem 6.2 specifically for the non-strict-complementarity result; not proven to hold generally and stated with an index-set inconsistency.
  • standard math Lemma 6.1 from [22]
    Borrowed from [22, Lemma 3.10]; the paper does not reproduce its proof even though it notes differences in the setting.

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Cite this review

Pith. "Pith review of Kurdyka-\L ojasiewicz exponent via square transformation." pith.science (2026). https://pith.science/paper/CCUMJSCR

@misc{pith2026250610110,
  author       = {Pith},
  title        = {Pith review of: Kurdyka-\L ojasiewicz exponent via square transformation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CCUMJSCR}},
  note         = {Machine review of arXiv:2506.10110}
}
read the original abstract

We consider one of the most common reparameterization techniques, the square transformation. Assuming the original objective function is the sum of a smooth function and a polyhedral function, we study the variational properties of the objective function after reparameterization. In particular, we first study the minimal norm of the subdifferential of the reparameterized objective function. Second, we compute the second subderivative of the reparameterized objective function on a linear subspace, which allows for fully characterizing the subclass of stationary points of the reparameterized objective function that correspond to stationary points of the original objective function. Finally, utilizing the representation of the minimal norm of the subdifferential, we show that the Kurdyka-\L ojasiewicz (KL) exponent of the reparameterized function can be deduced from that of the original function.

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Works this paper leans on

33 extracted references · 31 canonical work pages

  1. [22]

    Ouyang, Y

    W. Ouyang, Y. Liu, T. K. Pong, and H. W ang, Kurdyka- Lojasiewicz exponent via Hadamard parametrization, arXiv preprint arXiv:2402.00377, (2024)

    arXiv 2024

  2. [1]

    H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, Springer, 2017

    work page 2017

  3. [2]

    Benko and P

    M. Benko and P. Mehlitz, Why second-order sufficient conditions are, in a way, easy–or– revisiting the calculus for second subderivatives , Journal of Convex Analysis, 30 (2023), pp. 541–589

    work page 2023

  4. [3]

    Bertsekas, Nonlinear Programming, Athena Scientific, 1999

    D. Bertsekas, Nonlinear Programming, Athena Scientific, 1999

    work page 1999

  5. [4]

    J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems , Springer Science & Business Media, 2013

    work page 2013

  6. [5]

    S. S. Chen, Basis pursuit , Stanford University, 1996

    work page 1996

  7. [6]

    S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic decomposition by basis pursuit , SIAM Review, 43 (2001), pp. 129–159

    work page 2001

  8. [7]

    H.-H. Chou, J. Maly, C. M. Verdun, B. F. P. da Costa, and H. Mirandola, Get rid of your constraints and reparametrize: A study in NNLS and implicit bias , in The 28th International Conference on Artificial Intelligence and Statistics, 2025

    work page 2025

Show all 33 references
  1. [8]

    Davis and D

    D. Davis and D. Drusvyatskiy, Proximal methods avoid active strict saddles of weakly convex functions, Foundations of Computational Mathematics, 22 (2022), pp. 561–606

    2022

  2. [9]

    Ding and S

    L. Ding and S. J. Wright, On squared-variable formulations, arXiv preprint arXiv:2310.01784, (2023)

    2023 arXiv

  3. [10]

    D. L. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), pp. 1289–1306

    2006

  4. [11]

    F acchinei and J.-S

    F. F acchinei and J.-S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer Science & Business Media, 2007

    2007

  5. [12]

    E. H. Fukuda and M. Fukushima, A note on the squared slack variables technique for nonlinear optimization, Journal of the Operations Research Society of Japan, 60 (2017), pp. 262–270

    2017

  6. [13]

    P. E. Gill, W. Murray, and M. H. Wright, Practical optimization, Academic Press, 1981

    1981

  7. [14]

    Levin, J

    E. Levin, J. Kileel, and N. Boumal, The effect of smooth parametrizations on nonconvex optimization landscapes, Mathematical Programming, 209 (2025), pp. 63–111

    2025

  8. [15]

    Li and T

    G. Li and T. K. Pong, Calculus of the exponent of Kurdyka– Lojasiewicz inequality and its applications to linear convergence of first-order methods , Foundations of computational mathematics, 18 (2018), pp. 1199–1232

    2018

  9. [16]

    J. Li, T. Nguyen, C. Hegde, and K. W. Wong, Implicit sparse regularization: The impact of depth and early stopping , Advances in Neural Information Processing Systems, 34 (2021)

    2021

  10. [17]

    Q. Li, D. McKenzie, and W. Yin, From the simplex to the sphere: faster constrained optimization using the Hadamard parametrization , Information and Inference: A Journal of the IMA, 12 (2023), pp. 1898–1937

    2023

  11. [18]

    J. Liu, J. Chen, and K. Wei, On the linear convergence of policy gradient under hadamard parameterization, Information and Inference: A Journal of the IMA, 14 (2025), p. iaaf003

    2025

  12. [19]

    Milzarek, Numerical methods and second order theory for nonsmooth problems , PhD thesis, Technische Universit¨ at M¨ unchen, 2016

    A. Milzarek, Numerical methods and second order theory for nonsmooth problems , PhD thesis, Technische Universit¨ at M¨ unchen, 2016

    2016

  13. [20]

    Mohammadi, B

    A. Mohammadi, B. Mordukhovich, and M. Sarabi, Parabolic regularity in geometric varia- tional analysis , Transactions of the American Mathematical Society, 374 (2021), pp. 1711– 1763

    2021

  14. [21]

    B. S. Mordukhovich and M. E. Sarabi, Generalized differentiation of piecewise linear functions in second-order variational analysis , Nonlinear Analysis, 132 (2016), pp. 240–273

    2016

  15. [23]

    Ouyang and A

    W. Ouyang and A. Milzarek, Variational properties of decomposable functions. part i: Strict epi-calculus and applications , arXiv preprint arXiv:2311.07267, (2023)

    2023 arXiv

  16. [24]

    Ouyang and A

    W. Ouyang and A. Milzarek, Variational properties of decomposable functions part ii: Strong second-order theory, arXiv preprint arXiv:2311.07276, (2023). 26 W. OUYANG

    2023

  17. [25]

    R. T. Rockafellar, Convex analysis , vol. 28, Princeton University Press, 1970

    1970

  18. [26]

    R. T. Rockafellar and R. J.-B. Wets, Variational analysis, vol. 317, Springer Science & Business Media, 2009

    2009

  19. [27]

    Schneider, Convex bodies: the Brunn–Minkowski theory , no

    R. Schneider, Convex bodies: the Brunn–Minkowski theory , no. 151, Cambridge University Press, 2014

    2014

  20. [28]

    Soltan, Lectures on convex sets , World Scientific, 2019

    V. Soltan, Lectures on convex sets , World Scientific, 2019

    2019

  21. [29]

    Tang and K.-C

    T. Tang and K.-C. Toh, Optimization over convex polyhedra via Hadamard parametrizations , Mathematical Programming, (2024), pp. 1–41

    2024

  22. [30]

    R. J. Tibshirani, Equivalences between sparse models and neural networks , Working Notes. URL https://www. stat. cmu. edu/ryantibs/papers/sparsitynn. pdf, (2021)

    2021

  23. [31]

    V askevicius, V

    T. V askevicius, V. Kanade, and P. Rebeschini, Implicit regularization for optimal sparse recovery, Advances in Neural Information Processing Systems, 32 (2019)

    2019

  24. [32]

    Weis, A note on touching cones and faces , arXiv preprint arXiv:1010.2991, (2010)

    S. Weis, A note on touching cones and faces , arXiv preprint arXiv:1010.2991, (2010)

    2010 arXiv

  25. [33]

    P. Zhao, Y. Yang, and Q.-C. He, High-dimensional linear regression via implicit regularization, Biometrika, 109 (2022), pp. 1033–1046

    2022

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