Pith. sign in

REVIEW 3 major objections 6 minor 2 cited by

Corner Gradient Descent

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Corner-shaped contours in the complex plane let SGD accelerate beyond plain gradient rates.

desk verdict A genuinely new contour-based framework for accelerating stationary SGD, with a clean phase diagram—but the headline rate formula is proved only under the tau2=0 simplification, and the advertised scope is wider than the proof. read the letter →

arxiv 2504.12519 v1 pith:MZXJ7NPT submitted 2025-04-16 math.OC cs.LG

Dmitry Yarotsky This is my paper · ORCID
classification math.OCcs.LG MSC 68W4090C2565K10
keywords stochasticgradientdescentoptimizationaccelerationheavyballmomentummethodsill-conditionedquadraticproblemspower-lawspectracomplexcontoursrationalapproximation
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 claims that stationary stochastic gradient algorithms with long memory can be encoded as contours in the complex plane, and that a contour with a corner of exterior angle $\theta\pi$ accelerates the loss decay from $O(t^{-\zeta})$ to $O(t^{-\theta\zeta})$. If true, mini-batch SGD can reach rates approaching $O(t^{-2\zeta})$ in the signal-dominated regime, a regime where no stationary finite-memory algorithm is known to accelerate. The paper derives the optimal acceleration factor $\theta_{\max}=\min(2,\nu,2/(\zeta+1/\nu))$, balances signal decay against sampling-noise amplification, and constructs finite-memory rational approximations that reproduce the acceleration numerically on a synthetic problem and on MNIST. The result matters because it gives a geometric design principle for accelerated stochastic optimization on ill-conditioned quadratic problems.

What carries the argument

The central object is the map $\Psi=P/Q$ built from the characteristic polynomial of a memory-$M$ gradient algorithm, together with the contour $\gamma=\Psi(\{|\mu|=1\})$ it traces in the complex plane. A corner map obeys $\Psi(\mu)=-c_\Psi(\mu-1)^\theta(1+o(1))$ near $\mu=1$, producing a contour with external angle $\theta\pi$. This corner condition determines the leading decay of the propagators through Mittag-Leffler functions: signal propagators decay as $t^{-\theta\zeta}$ and noise propagators as $t^{\theta/\nu-2}$, and Theorem 4 balances them to yield $\theta_{\max}$.

What would settle it

Run memory-M corner SGD on a quadratic least-squares problem with power-law spectrum and non-Gaussian features whose fourth moments violate the $\tau_2=0$ identity; fit the empirical loss exponent at the theoretically optimal $\theta_{\max}$. If the exponent remains at the plain SGD value $\zeta$ instead of approaching $\theta_{\max}\zeta$ even with large batch sizes, the $\tau_2=0$ simplification is doing the work.

Watch

Extended reading notes

Core claim

On infinite-dimensional quadratic problems with power-law spectra, any stationary SGD with finite linear memory is confined to the phase diagram of plain SGD, with loss $O(t^{-\zeta})$. This paper identifies generalized SGD algorithms with rational maps $\Psi=P/Q$ and their contour images, and shows that a contour with a corner of external angle $\theta\pi$ produces signal propagators $O(t^{-\theta\zeta})$ and noise propagators $O(t^{\theta/\nu-2})$. Balancing the two exponents gives the maximal achievable acceleration $\theta_{\max}=\min(2,\nu,2/(\zeta+1/\nu))$ in the signal-dominated phase, with rates arbitrarily close to $O(t^{-2\zeta})$ when $\theta_{\max}=2$. The ideal infinite-memory corner algorithm is approximated by memory-$M$ algorithms via rational approximations of power functions with error $O(e^{-c\sqrt{M}})$, and a memory-5 implementation shows acceleration on a synthetic indicator problem and on MNIST.

Load-bearing premise

The load-bearing premise is that fourth-order moment evolution is exactly described by the Spectrally Expressible approximation with $\tau_2=0$, which is true only for translation-invariant or Gaussian data; for general distributions, the main theorem is conditional on this approximation, and the extension to $\tau_2\neq 0$ is only sketched.

Editorial extensions

If this is right

  • Corner SGD attains loss $O(t^{-\theta\zeta})$ with $\theta$ up to 2 in the signal-dominated phase, improving on the plain SGD rate $O(t^{-\zeta})$.
  • In the fully accelerated region $\nu>2,\ 0<\zeta<1-1/\nu$, rates arbitrarily close to $O(t^{-2\zeta})$ become feasible for mini-batch SGD.
  • In the noise-limited region, the finite-noise condition forces $\theta_{\max}=\nu<2$, so the noise propagator boundary, not the signal exponent, caps the achievable acceleration.
  • Finite-memory approximations converge to the ideal corner algorithm with error $O(e^{-c\sqrt{M}})$, making the acceleration practically accessible with small memory.
  • The phase diagram predicts concrete exponent transitions that can be checked on synthetic least-squares problems with known spectral exponents.

Reading between the lines

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

  • A testable extension: if the sketch in Appendix H holds, the acceleration diagram is independent of the fourth-moment constant $\tau_2$, so corner SGD should accelerate on non-Gaussian data too, not only on translation-invariant or Gaussian data.
  • The contour viewpoint suggests a design route for non-stationary schedules, where approximating an ideal corner with time-changing finite memory might restore asymptotic acceleration even where stationary finite memory is provably incapable of it.
  • Because the deterministic rate is governed only by the source exponent $\zeta$, corner algorithms may transfer directly to kernel-ridge and overparameterized linear-model settings with power-law spectra.
  • The paper establishes rates for the limiting propagators of an ideal infinite-memory algorithm; turning this into a concrete infinite-dimensional dynamical system would be a next step, though finite-memory rational approximations may already supply the practical algorithm.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper proposes "Corner Gradient Descent", a family of stationary SGD algorithms with memory for infinite-dimensional quadratic problems with power-law spectra (capacity exponent ν, source exponent ζ). The author identifies stationary finite-memory (S)GD with rational maps Ψ=P/Q (Section 3) and shows that algorithms whose contour γ=Ψ({|μ|=1}) has a corner of external angle θπ at the origin, i.e. Ψ(μ)∼−cΨ(μ−1)^θ, have propagators Ut=O(t^{θ/ν−2}) and Vt=O(t^{−θζ}) (Theorem 3, proved under the Spectrally Expressible closure with τ2=0). Balancing signal and noise exponents gives Theorem 4: in the signal-dominated phase, the supremum of achievable acceleration factors is θmax=min(2,ν,2/(ζ+1/ν)), summarized in a three-region phase diagram (Figure 1 right). Corner maps are approximated by memory-M rational algorithms via midpoint quadrature of an integral representation (Propositions 1–2) and implemented with M=5 on a synthetic indicator problem (predicted/observed exponent 0.45/0.41) and on MNIST (acceleration factor ≈1.5). The paper ends with an explicit discussion of the limitations of the τ2=0 assumption and of finite-memory approximation (Section 7).

Significance. The conceptual claim is substantial: it offers a path beyond the barrier O(t^{−ζ}) that Yarotsky & Velikanov (2024) proved for stationary finite-memory SGD, and it produces a clean, falsifiable prediction (the θmax formula) with a three-region phase diagram that subsumes the earlier phase picture. The contour/Riemann-map viewpoint is elegant and likely to be reused. The proof of Theorem 3 is detailed and self-contained for the τ2=0 case (Appendix C), including explicit coefficient formulas in terms of Mittag-Leffler functions; the stability and separation arguments for the concrete corner map (Proposition 1) are rigorous; and the experiments ship reproducible code and agree qualitatively with the predicted exponents. The paper is also commendably explicit about what it does not prove (Section 7, Appendix H). The significance is real but conditional: the proven rate claims live in the τ2=0 SE propagator model, the natural exact-SE distributions have τ2≠0, and the bridge in Appendix H is a sketch. If either a rigorous τ2≠0 extension or an honest scoping of the claims is provided in revision, this would be a valuable contribution to the optimization literature.

major comments (3)
  1. [Section 7 / Appendix H / Theorem 4] The scope of the headline claim is narrower than what is proved, and the paper's own text confirms this. Section 2 fixes τ2=0 for the main results (Eq. (8)), and Section 7 states that τ2=0 "does not seem to correspond to any specific natural data distribution", whereas the SE closure is exact for translation-invariant (τ1=τ2=1) and Gaussian (τ1=1,τ2=−1) data. Appendix H is explicitly introduced as a sketch "without much rigor"; its conclusion is an upper bound |Ut−U(0)t|≲C t^{θ/ν−3} under the batch-size condition (Eq. (262)) and a still sketchier treatment for Vt. Since Theorem 4's θmax=min(2,ν,2/(ζ+1/ν)) is derived directly from Theorem 3's propagator exponents, the central claim is proved only for a regime containing no natural exact-SE distribution. Within the τ2=0 model the derivation of Theorem 4 from Theorems 1 and 3 is internally coherent; the problem is that the abstract and Theorem 4 assert the result without the τ2 qualification. The paper should either prove the general-τ2 claim or restate Theorem 4 and the abstract with the τ2=0 hypothesis explicit, presenting τ2-independence as a conjecture supported by Appendix H.
  2. [Sections 3–4; Eqs. (20)–(26)] The object carrying the rates is a limiting propagator system, not a concrete stochastic iteration of the form (1). The propagators Ut,Vt in Eqs. (20)–(22) are defined by contour integrals involving an irrational map Ψ, and the corner maps of Section 4 (e.g. Eq. (29)) are not rational, so there is no matrix Sλ and no memory-M system realizing them; moreover, by Eqs. (24) and (26) the effective learning rate αeff diverges when θ>1. Consequently, Theorem 4's "there exists a corner algorithm and batch size B such that Lt=O(t^{−θζ})" is a statement about the SE propagator expansion (7) and the associated infinite-memory limit, not about a runnable finite-dimensional algorithm. The body of the paper is reasonably transparent about this, but the abstract's phrasing "rates up to O(t^{−2ζ}) can be achieved by a generalized stationary SGD with infinite memory" overstates what is constructed. Please state precisely in what sense the ideal corner algorithm exists and which objects achieve the rates.
  3. [Sections 5–6; Propositions 1–2] The bridge from the ideal corner map to the implemented finite-memory algorithm has two unquantified links. First, the midpoint rule (31) is a one-sided quadrature, and the cited O(e^{−c√M}) rational-approximation bounds of Gopal & Trefethen are not shown to apply to this discretization as presented; no theorem bounds the approximation error of Ψ(M) to Ψ in a relevant norm, nor translates such an error into a propagator or loss error over finite horizons. Second, the stability of the Sλ matrices associated with Ψ(M) is asserted by construction but not verified. Section 7 concedes that the finite-memory algorithm does not accelerate the asymptotic exponent, so the experimental exponents in Section 6 — fitted over finite iteration ranges, on a nonlinear finite-dimensional model in the MNIST case — are evidence for, but not a proof of, the claimed acceleration. This does not affect Theorem 4, but the abstract's claim that ideal corner algorithms "can be efficiently approximated" should be softened or substantiated with approximation-error results.
minor comments (6)
  1. [Section 6, Eq. (39)] The pointwise loss is written ℓ(x,w)=½|ŷ(x,w)−y(x)|; the square appears to be missing, unlike the quadratic loss defined in Section 2.
  2. [Figure 1] The annotations "MNIST+MLP" and "CIFAR10+ResNet" in the left phase diagram are never explained in the text or caption.
  3. [Theorem 3, Eq. (20)] The Lebesgue–Stieltjes notation ∫_0^∞ r²F_U²(r) dr^{−θ/ν} (and similarly for F_V) is nonstandard; one sentence of explanation in the main text would help readers who do not consult Appendix C.
  4. [Appendix H, Eq. (237)] The symbol G0(t,λ)≡U2(t,λ) introduces U2 without prior definition; the definition U2(t,λ)=|(1 0^T)Sλ^{t−1}(−α c)|² should be stated explicitly.
  5. [Section 5] The quadrature constant l is fixed to l=5 in the experiments without comment or sensitivity analysis; a brief justification or a robustness check would be useful.
  6. [Section 6, MNIST] For θ=1.3 the measured acceleration factor (≈1.5) exceeds the infinite-dimensional prediction θmax=ν≈1.3 from Theorem 4; the text explains this via finite dimensionality, but spelling out the finite-dimensional expectation would make the comparison easier to assess.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: corner-map acceleration is derived from stated spectral and contour assumptions; the τ2≠0 extension is a scope limitation, not an input-output equivalence.

full rationale

Walking the derivation chain, the inputs are the SE moment closure (Eq. 6), the propagator expansion (Eq. 7), the power-law spectral conditions (11)-(12), and the corner condition (25) fixing the map's local singularity. Theorem 3 then uses explicit Wiener-Hopf/Mittag-Leffler computations (Lemmas 1, 2, 5, 6) to show U_t = O(t^{θ/ν−2}) and V_t = O(t^{−θζ}); these exponents are outputs of the analysis, not imposed decay laws. Theorem 4's θ_max formula follows algebraically from balancing θζ = 2−θ/ν together with the constraints θ<2 and 2−θ/ν>1, and no fitted constant is relabeled as a prediction. In the experiments, θ is fixed a priori and loss exponents are estimated afterward, so the measured exponents are not used to set the theorem constants. The reliance on Yarotsky & Velikanov (2024) for the convergence criterion (Theorem 1) and for the SE expansion is ordinary citation of prior mathematical work: those results are not equivalent to the new acceleration theorem, and no self-citation is used to forbid alternatives. The paper itself flags the τ2≠0 extension as a sketch 'without much rigor' (Appendix H) and notes τ2=0 'does not seem to correspond to any specific natural data distribution' (Section 7), which is a genuine limitation on the advertised scope of Theorem 4 but is a completeness concern rather than circularity. Overall, no equation in the claimed derivation reduces to its own input by construction, so the circularity score is 0.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central derivation is self-contained given the propagator framework, but it rests on a moment-closure assumption (SE τ2=0) that is not universally valid, and the corner map is a constructed object rather than an independently observed phenomenon. No new physical entities are introduced.

free parameters (5)
  • corner angle θ = 1.8 (synthetic), 1.3 and 1.8 (MNIST)
    Design parameter balancing acceleration and noise; not fitted to the loss exponent in the derivation, but chosen by hand in experiments.
  • corner map scaling A = not specified
    Free scale in the corner map (Eq. 29) controlling contour size; rates are independent but stability requires A < λmax/2.
  • rational approximation spacing l = 5
    Spacing in the trapezoidal approximation of the corner map (Eq. 31); chosen heuristically, no theory links it to finite-time performance.
  • memory size M = 5
    Truncation of the infinite-memory ideal; larger M gives better approximation but the asymptotic O(t^{-ζ}) of finite memory remains.
  • batch size |B| = 100 (synthetic), 1000 and 100 (MNIST)
    Chosen to make total noise UΣ < 1; not fitted, but affects the noise-dominated transition.
assumptions (7)
  • domain assumption Propagator expansion (7) and phase diagram from Yarotsky & Velikanov (2024)
    The paper starts from the exact expansion of the loss in terms of scalar signal and noise propagators; this is a prior result by the same author group, not re-proved here.
  • domain assumption Spectrally Expressible approximation with τ2=0 (Eq. 6)
    Needed for the simple second-moment propagator formulas (8); exact only for translation-invariant and Gaussian data, otherwise an approximation.
  • domain assumption Power-law spectral conditions (11), (12)
    Standard capacity and source conditions in kernel methods; the paper assumes ν>1 and 0<ζ<2-1/ν for the signal-dominated phase.
  • ad hoc to paper Corner map Ψ exists, is C1, and is spectrally separated (Eq. 25)
    The construction in Proposition 1 provides one such map; the theorem's proof assumes this regularity rather than deriving it from a concrete algorithm.
  • standard math Riemann mapping theorem and Carathéodory's theorem
    Used to reconstruct Ψ from an arbitrary contour γ; standard results not proved in the paper.
  • standard math Rational approximation bounds for power functions
    Used to justify replacing the irrational corner map by finite-memory rational approximations; taken from Newman (1964) and Gopal & Trefethen (2019).
  • domain assumption Strict stability of Sλ and D, αeff>0
    Needed for finite propagators and the stated convergence/divergence dichotomy; standard stability conditions for iterative algorithms.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Corner Gradient Descent." pith.science (2026). https://pith.science/paper/MZXJ7NPT

@misc{pith2026250412519,
  author       = {Pith},
  title        = {Pith review of: Corner Gradient Descent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZXJ7NPT}},
  note         = {Machine review of arXiv:2504.12519}
}
abstract

We consider SGD-type optimization on infinite-dimensional quadratic problems with power law spectral conditions. It is well-known that on such problems deterministic GD has loss convergence rates $L_t=O(t^{-\zeta})$, which can be improved to $L_t=O(t^{-2\zeta})$ by using Heavy Ball with a non-stationary Jacobi-based schedule (and the latter rate is optimal among fixed schedules). However, in the mini-batch Stochastic GD setting, the sampling noise causes the Jacobi HB to diverge; accordingly no $O(t^{-2\zeta})$ algorithm is known. In this paper we show that rates up to $O(t^{-2\zeta})$ can be achieved by a generalized stationary SGD with infinite memory. We start by identifying generalized (S)GD algorithms with contours in the complex plane. We then show that contours that have a corner with external angle $\theta\pi$ accelerate the plain GD rate $O(t^{-\zeta})$ to $O(t^{-\theta\zeta})$. For deterministic GD, increasing $\theta$ allows to achieve rates arbitrarily close to $O(t^{-2\zeta})$. However, in Stochastic GD, increasing $\theta$ also amplifies the sampling noise, so in general $\theta$ needs to be optimized by balancing the acceleration and noise effects. We prove that the optimal rate is given by $\theta_{\max}=\min(2,\nu,\tfrac{2}{\zeta+1/\nu})$, where $\nu,\zeta$ are the exponents appearing in the capacity and source spectral conditions. Furthermore, using fast rational approximations of the power functions, we show that ideal corner algorithms can be efficiently approximated by finite-memory algorithms, and demonstrate their practical efficiency on a synthetic problem and MNIST.

Figures

Figures reproduced from arXiv: 2504.12519 by the authors.

Figure 1
Figure 1. Left: The phase diagram of stationary finite-memory SGD from Velikanov et al. (2023); Yarotsky & Velikanov (2024). Right: Maximum acceleration factor θmax = min(2, ν, 2 ζ+1/ν ) for Corner SGD in the signal-dominated regime (see Theorem 4). Stability and asymptotics of the propagators. Let us say that a square matrix A is strictly stable if all its eigenvalues are less than 1 in absolute value. It is natural to requi… view at source ↗
Figure 2
Figure 2. Left: The map Ψ = P Q for Heavy Ball with P(µ) = (µ−1)(µ−0.4) and Q(µ) = −µ. The contour γ = Ψ({µ : |µ| = 1}) encircles spec(H). The map Ψ bijectively maps {µ ∈ C : |µ| > 1} to the exterior open domain Dγ with boundary γ. (See Section B for a general discussion of memory￾1 contours.) Right: Contour γ corresponding to a corner map Ψ with angle θπ. We see, in particular, that the values Ut depend on the algorithm para… view at source ↗
Figure 3
Figure 3. Training loss and final predictions of the linear model (36) trained to fit the target [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Training loss of neural network (39) on MNIST classification with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Contours γ = Ψ({µ : |µ| = 1}) corresponding to different memory-1 maps Ψ (see Section B). Left: plain Gradient Descent (a circle). Center: Heavy Ball (an ellipse; β = 0.5). Right: general memory-1 algorithms (a Zhukovsky airfoil; β = 0.65, q0 = 0.125, q1 = −1). A Deriv…
Figure 6
Figure 6. Figure 6: MNIST trajectories of loss (top row) and error rate (bottom row) on train set (lighter colors) and test set (darker colors). Left column: batch size 1000. Right column: batch size 100. map Ψ that were instrumental in proving Theorem 3. While we are not aware of a simil…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dimension-adapted Momentum Outscales SGD

    stat.ML 2025-05 conditional novelty 7.0 of 10

    DANA, with dimension- and time-dependent momentum, provably outscales SGD on power-law random features when 2α>1, improving loss exponents and compute-optimal curves.

  2. Learning from one graph: transductive learning guarantees via the geometry of small random worlds

    stat.ML 2025-09 conditional novelty 6.0 of 10

    The paper derives transductive generalization bounds for Lipschitz graph learners, including GCNs, on a single graph, with O(N^{-1/2}) rates in the number of labeled nodes.

Reference graph

Works this paper leans on

21 extracted references · 19 canonical work pages · cited by 2 Pith papers

  1. [1]

    At t→∞ , ifVt = O(1) (respectively,Vt = o(1)), then alsoLt =O(1) (respectively,Lt =o(1))

    [Convergence] Suppose thatUΣ < 1. At t→∞ , ifVt = O(1) (respectively,Vt = o(1)), then alsoLt =O(1) (respectively,Lt =o(1))

    work page

  2. [2]

    [Divergence] IfUΣ > 1 andVt > 0 for at least onet, then supt=1,2,...Lt =∞

    work page

  3. [3]

    Suppose also that UΣ < 1 andUt = O(t−ξU ) with some ξU > max(ξV, 1)

    [Signal-dominated regime] Suppose that there exist constantsξV,CV > 0 such thatVt = CVt−ξV (1 +o(1)) ast→∞ . Suppose also that UΣ < 1 andUt = O(t−ξU ) with some ξU > max(ξV, 1). Then Lt = CV 2(1−UΣ)t−ξV (1 +o(1)). (9)

    work page

  4. [4]

    infinite-memory

    [Noise-dominated regime] Suppose that there exist constantsξV >ξ U > 1,CU > 0 such thatUt =CUt−ξU (1 +o(1)) andVt =O(t−ξV ) ast→∞ . Let also thatUΣ < 1. Then Lt = VΣCU 2(1−UΣ)2t−ξU (1 +o(1)). (10) Spectral power laws. The detailed convergence results in items 3, 4 of Theorem 1 require us to know the asymptotics of the propagators Ut,Vt. To this end we int...

    work page 2023

  5. [5]

    (Noise propagators)Ut =CUtθ/ν−2(1 +o(1)), with the coefficient CU = τ1 |B|Λ1/ν Z 0 ∞ r2F 2 U(r)dr−θ/ν <∞, F U(r) = 1 2πi Z iR erzdz cΨzθ + 1

    work page

  6. [6]

    one- sided

    (Signal propagators)Vt =CVt−θζ(1 +o(1)), with the coefficient CV =Q Z ∞ 0 F 2 V (r)drθζ <∞, F V (r) = 1 2πi Z iR cΨzθ−1erzdz cΨzθ + 1 . We see that the leading t→∞ asymptotics of the propagators are completely determined by the λ↘ 0 spectral asymptotics of the problem and the µ→ 1 singularity of the map Ψ. The functions FU,FV can be written in terms of th...

    arXiv 1964

  7. [8]

    In this case condition (65) is equivalent to−1≤ q0 q1 , i.e

    r≤ 0 corresponds to−1≤ q0 q1 ≤−β. In this case condition (65) is equivalent to−1≤ q0 q1 , i.e. it holds. However, the special caseq0 q1 =−1 is the degenerate scenario in which the denominator of Ψ vanishes at µ = 1 and the stability condition q1 <−q0 in Eq. (57) is violated, so we will discard this special case

    work page

  8. [9]

    The option q0 q1 <−1 is inconsistent with condition (65), leaving only the option q0 q1 >−β

    r > 0 corresponds to q0 q1 <−1 or q0 q1 >−β. The option q0 q1 <−1 is inconsistent with condition (65), leaving only the option q0 q1 >−β. (a) If q0 q1 ≤ 0, then condition (65) is equivalent to √r≤ 1 +q0 q1 , (66) which holds true thanks to the assumptionβ <1. (b) If q0 q1 ≥ 0, then condition (65) is equivalent to √r≤ 1− q0 q1 , (67) which holds iff q0 q1 ...

    work page 2011

Show all 21 references
  1. [10]

    FU(r) = (1 +o(1)) 1 cΨΓ(θ)rθ−1 asr↘ 0

  2. [11]

    FU(r) = (1 +o(1)) −cΨ Γ(−θ)r−θ−1 asr→ +∞. Proof. 1. Consider the function f(z) integrated in Eq. (91). For any r ∈ R and θ ∈ (1, 2), the function f is holomorphic in any strip Ta ={0 <ℜz < a},a > 0, and is bounded in Ta as|f(z)| = O(|z|−θ). It follows that the integration line...

  3. [12]

    (94) We can find limr↘0u(r) as follows

    By the change of variablesrz =z′, FU(r) =u(r)rθ−1, (93) 19 where u(r) = 1 2πicΨ Z iR ez′ dz′ z′θ +c−1 Ψ rθ. (94) We can find limr↘0u(r) as follows. Observe that the integration line iR can be deformed to the lineγa,a> 0, encircling the negative semi-axis: γa =γa,1∪γa,2∪γa,3, (...

  4. [13]

    (104) Performing again the change of variablesrz =z′, we have FU(r) =v(r)r−θ−1, (105) where v(r) = cΨθ 2πi Z iR ez′ z′θ−1dz′ (cΨ(z′/r)θ + 1)2

    We start by performing integration by parts inFU : FU(r) = −1 2πir Z iR erzd 1 cΨzθ + 1 = cΨθ 2πir Z iR erzzθ−1dz (cΨzθ + 1)2. (104) Performing again the change of variablesrz =z′, we have FU(r) =v(r)r−θ−1, (105) where v(r) = cΨθ 2πi Z iR ez′ z′θ−1dz′ (cΨ(z′/r)θ + 1)2. (106) T...

  5. [14]

    , uniformly in all λ ∈ (0,λ max]

    FU(r,λ ) = O( 1 r ) for all r of the form r = tλ1/θ,t = 1, 2,... , uniformly in all λ ∈ (0,λ max]. Proof. 1. It suffices to show that, asλ↘ 0, the functions fλ(s) =−(2π)−1(−Ψ(eisλ1/θ )/λ + 1)−11[−π/λ1/θ,π/λ1/θ](s) (137) converge inL1(R) to f0(s) =−(2π)−1(cΨei(signs)θπ/2|s|θ + ...

  6. [15]

    (145) We will show that this integral isO( 1 λ)

    We start by performing integration by parts inU(t,λ ): U(t,λ ) = 1 2πit I |µ|=1 dµt Ψ(µ)−λ = 1 2πit I |µ|=1 Ψ′(µ)µtdµ (Ψ(µ)−λ)2 (144) implying |U(t,λ )|≤ 1 2πt Z π −π |Ψ′(eis)|ds |Ψ(eis)−λ|2. (145) We will show that this integral isO( 1 λ). 23 Note first that we can replace th...

  7. [16]

    FV (r) = (1 +o(1)) cΨ Γ(1−θ)r−θ asr→ +∞. Proof. 1. Like in Lemma 1, this follows by deforming the integration line in Eq. (161) towards +∞

  8. [17]

    By the change of variablesrz =z′, FV (r) = 1 2πi Z iR z′θ−1ez′ dz′ z′θ +c−1 Ψ rθ. (164) 25 As in Lemma 1, the integration lineiR can be deformed to the lineγa,a> 0, encircling the negative semi-axis: γa =γa,1∪γa,2∪γa,3, (165) γa,1 ={z∈ C :ℑz =−a,ℜz≤ 0}, (166) γ2,2 ={z∈ C :|z| ...

  9. [18]

    (170) Taking the limitr→ +∞ and deforming the contour to the negative semi-axis as in Lemma 1, lim r→+∞ v(r) = cΨ 2πi Z γ′ z′θ−1ez′ dz′ = cΨ Γ(1−θ)

    Using the same contourγ′ as in Lemma 1, FV (r) =v(r)r−θ, v (r) = 1 2πi Z γ′ cΨz′θ−1ez′ dz′ cΨ(z′/r)θ + 1. (170) Taking the limitr→ +∞ and deforming the contour to the negative semi-axis as in Lemma 1, lim r→+∞ v(r) = cΨ 2πi Z γ′ z′θ−1ez′ dz′ = cΨ Γ(1−θ). (171) The formal leadi...

  10. [19]

    |FV (r,λ )−FV (r)| =o(1) asλ→ 0, uniformly forr>ϵ , for anyϵ> 0

  11. [20]

    and λ ∈ (0,λ max], with some r,λ - independent constantC

    |FV (r,λ )| ≤C min( 1 r, 1) for all t = 1 , 2,... and λ ∈ (0,λ max], with some r,λ - independent constantC. Proof. 1. The proof of this property is more complicated than the earlier proof for FU because the integrals defining FV are not absolutely convergent. Recall the integr...

  12. [21]

    of (196) vanishes and the second converges thanks toθ> 1

    Note that |FV (r,λ )|≤ C r, C <∞, (198) simply by setting u = 0 in the bound (196), since the first term on the r.h.s. of (196) vanishes and the second converges thanks toθ> 1. It remains to prove thatFV (r,λ ) is bounded uniformly inr,λ . It suffices to prove this forr<ϵ with...

    2024

  13. [1970]

    Zhukovsky airfoil

    doi: 10.1070/im1970v004n05abeh000948. URL https://doi.org/10.1070/ im1970v004n05abeh000948. Blake Bordelon and Cengiz Pehlevan. Learning curves for sgd on structured features.arXiv preprint arXiv:2106.02713, 2021. Blake Bordelon, Alexander Atanasov, and Cengiz Pehlevan. How fe...

    2024 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.