Pith. sign in

REVIEW 2 major objections 4 minor 81 references

On The Geometric Analysis of A Quartic-quadratic Optimization Problem under A Spherical Constraint

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper proves that a quartic-quadratic optimization problem on a sphere has no spurious local minima when the quadratic matrix is diagonal, and that the real-case landscape is benign enough for polynomial-time second-order methods…

desk verdict Real-case landscape results are solid and new, but the abstract's strict-saddle claim is unqualified and false in the complex problem; fix the scope and this paper deserves a serious referee. read the letter →

arxiv 1908.00745 v1 pith:U27YHB24 submitted 2019-08-02 math.OC

Haixiang Zhang , Andre Milzarek , Zaiwen Wen , Wotao Yin This is my paper · ORCID
classification math.OC MSC 15A4547H6058K3058C4090C26
keywords quartic-quadraticoptimizationsphereconstraintstrict-saddlepropertyKurdyka-ŁojasiewiczexponentBose-EinsteincondensationnonconvexlandscapesimplexprojectionWirtingercalculus
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 analyzes the landscape of a nonconvex optimization problem: minimize a quadratic-plus-quartic function on the sphere. The central claim is that even though the problem is NP-hard in general, an important structured version behaves well. When the quadratic matrix is diagonal, every local minimum is global and the global solution is given by a simple projection onto the simplex, computable in O(n log n). In the real case, when the quartic term is either very strong or very weak relative to the quadratic term, the objective satisfies the strict-saddle property—every saddle has a direction of negative curvature—which is what second-order methods need for polynomial-time convergence. The paper also proves that the Kurdyka-Łojasiewicz exponent is 1/4 for broad classes of stationary points, giving explicit sublinear rates for first-order methods.

What carries the argument

The argument runs through four devices. (1) Wirtinger calculus gives the Riemannian gradient and Hessian; the curvature formula Hf(z)[v] = v^*[A + 2β diag(|z|^2) − 2λI]v + 4β Σ_k Re(v_k \bar z_k)^2 is the base of all second-order tests. (2) The substitution u_k = |z_k|^2 changes the diagonal case into a strongly convex simplex problem, whose solution is the Euclidean projection onto the n-simplex. (3) For the strict-saddle property, the sphere is partitioned into three regions—strong convexity, large gradient, and negative curvature—and β is chosen so that the quadratic or quartic term dominates, yielding uniform bounds on the Hessian and gradient. (4) The KL analysis uses the identity f(y) − f(z) = 1/2 y^*H y + β/2 ||τ||^2 with τ_k = |y_k|^2 − |z_k|^2 and a refined decomposition of the Riemannian gradient into terms involving H, τ, and projections.

What would settle it

Scan a fixed small real instance, say n = 3 with the paper's matrix A = [[1,0,1],[0,1,0],[1,0,1]], across β values between the two proven regimes. If any stationary point z has gradf(z) = 0 and Hf(z)[v] ≥ 0 for all tangent v, then the strict-saddle property fails at that β; the paper's Example 1 already constructs such a point for β = C $n^{{3/2−ε}}$, so a numerical search could confirm the boundary and test whether the theorem's thresholds are tight.

Watch

Extended reading notes

Core claim

The paper establishes three landscape facts for problem (1.1). First, if A is diagonal, then all local minimizers are global and are exactly points whose squared moduli solve a strongly convex simplex problem; the global minimizer class is computed by projecting -a/(2β) onto the n-simplex. Second, in the real setting with a spectral gap, f is (ξ, ε, ζ)-strict-saddle when β is at least of order ρ $n^{{3/2}}$ or at most a constant times δ, where ρ is the eigenvalue spread and δ is the gap between the two smallest eigenvalues; for large β there are exactly 2n local minima and for small β exactly two, both global. Third, the Kurdyka-Łojasiewicz exponent is 1/4 for every stationary point when A is diagonal, and for any real stationary point whose certification matrix H = A + 2β diag(|z|^2) − 2λI is positive semidefinite.

Load-bearing premise

The load-bearing premise for the algorithmic consequences is that the problem is studied on the real sphere and, for the small-β regime, that the two smallest eigenvalues of A are strictly separated; in the complex setting the phase direction iz makes the strong-convexity condition fail, so the strict-saddle guarantee does not transfer to the complex Bose-Einstein problem.

Editorial extensions

If this is right

  • For diagonal A, global solutions can be computed in O(n log n) by a simplex projection, and no local search can get stuck at a spurious local minimum.
  • In the real regime with β large or small, a Riemannian trust-region method or any second-order method that escapes saddles will find a global minimum in polynomial time; for small β there are exactly two global minima, for large β exactly 2n.
  • The KL exponent 1/4 means first-order descent methods converge at the sublinear rate O(t^{-1/2}) on the diagonal case and on the real positive-semidefinite-certificate cases.
  • In the rank-one case, global minima are unique up to phase shifts, so the solution set is a single orbit under phase changes and the recovery problem is well-posed up to the usual phase ambiguity.

Reading between the lines

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

  • Although the strict-saddle theorem is stated for the real sphere, the underlying obstacle in the complex case is the free phase: the direction i z is always flat. One could quotient the complex sphere by the phase action and define strict-saddle on the quotient, where the same proof strategy would likely yield analogous guarantees for the physically relevant rotating Bose-Einstein condensate probl
  • The diagonal-case result suggests a practical initialization scheme for general A: solve the diagonalized problem in the eigenbasis, use that solution as a warm start, and then refine; the paper's stability estimate indicates the warm start stays close to a true global minimizer under random perturbations.
  • The strict-saddle thresholds leave a middle regime in β where the landscape may be complicated; the paper's Example 1 shows the large-β threshold cannot be improved deterministically, but random A might allow a much smaller threshold—testing this is a natural next step.
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

2 major / 4 minor

Summary. The paper studies the quartic-quadratic sphere-constrained problem (1.1) over the complex sphere, f(z) = (1/2)z*Az + (β/2)Σ|z_k|^4 subject to ||z||=1. It derives second- and fourth-order optimality conditions, gives a complete landscape description when A is diagonal (no spurious local minima, explicit global minimizers via simplex projection in O(n log n) operations), analyzes the rank-one case, establishes a strict-saddle property in the real case for large and small β, provides an example showing that the O(n^{3/2}) dependence in the large-β threshold cannot be improved in general, and proves a Kurdyka-Łojasiewicz exponent of 1/4 for diagonal A and for a class of real global minimizers with positive semidefinite stationary certification matrix. The proofs are detailed and mostly self-contained, with explicit constants in the strict-saddle region arguments.

Significance. If the scope is stated accurately, the paper makes a substantial contribution. The diagonal-case reduction to a convex simplex projection is elegant and parameter-free; the strict-saddle proofs partition the sphere into explicit regions and give quantitative constants; Example 1 demonstrates tightness of the n^{3/2} scaling; and the KL-exponent results are the first explicit estimates for this quartic-quadratic problem class. The main weakness is that the strict-saddle results, which drive the algorithmic interpretation, are proved only for the real sphere and, in the small-β branch, only under a positive spectral gap. The abstract and conclusions state the strict-saddle property without these qualifications even though the problem is introduced over C^n and the rotating-BEC formulation is complex. This is a scope/presentation defect rather than an internal error in the real-case proofs.

major comments (2)
  1. [Abstract; §5, before Definition 5.1] The abstract states that the strict-saddle property is established when β is at least O(n^{3/2}) or at most O(1), with no domain restriction, but the problem (1.1) is defined over C^n. Section 5 is explicitly titled 'the Real Case,' and the paragraph immediately before Definition 5.1 observes that at any complex critical point z, the vector iz lies in the tangent space and satisfies Hf(z)[iz]=0, so the strong-convexity condition in Definition 5.1 can never hold. Consequently, Theorems 5.1 and 5.3 do not establish any strict-saddle guarantee for the complex problem as advertised in the abstract. Since the strict-saddle property is the basis of the claimed polynomial-time convergence of second-order algorithms, this scope mismatch is load-bearing. Please qualify all strict-saddle statements to the real sphere, state explicitly that the algorithmic implications apply to the real (non-rotating) BEC formulation, and revise the abstract and conclusions accordingly.
  2. [§5.2; Theorem 5.3] The small-β strict-saddle theorem is proved only under the positive spectral gap assumption δ = λ_{n-1} - λ_n > 0, which is stated at the beginning of Section 5.2 and used essentially in Lemmas 5.6 and 5.8 and in the covering argument of Theorem 5.3. The abstract's phrase 'not larger than O(1)' omits this condition, and the actual hypothesis is β ≤ b_γ = [2(7/3+γ)+(2/3+γ)ρ/δ]^{-1}δ, which depends on δ and ρ. Please state the spectral-gap condition wherever the small-β strict-saddle property is claimed, including the abstract, or else provide a separate treatment of the δ=0 case.
minor comments (4)
  1. [§3, after Lemma 3.1] The sentence 'By combining Theorems 2.1 and 3.1' should refer to Lemma 3.1 rather than Theorem 3.1, since Theorem 3.1 is stated immediately afterward.
  2. [Abstract; Theorem 5.1] The abstract summarizes the large-β condition as β being 'at least O(n^{3/2})', but Theorem 5.1 requires β ≥ 8n/(n-1)(1+γ)ρ n^{3/2}, involving the spectral spread ρ. Because scaling (A,β) jointly scales the objective, the threshold is meaningful in the scale-invariant form β ≥ C ρ n^{3/2}; please state this in the abstract and introduction.
  3. [Throughout] There are several typographical and OCR-type spacing issues, for example 'minim a' in the abstract and 'f irst-order' in Section 1; these should be corrected in the final version.
  4. [§5.2, first paragraph] The statement that the spectral gap holds 'with probability 1 when A is a Gaussian random matrix' would benefit from a precise matrix ensemble and a reference to the specific theorem; as written, the reader cannot tell whether real symmetric, complex Hermitian, or Wigner-type ensembles are intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: every claim is derived from stated assumptions by self-contained proofs, with no fitted inputs or load-bearing self-citations.

full rationale

The paper's main results are parameter-free mathematical derivations against explicitly stated definitions. Theorem 3.1 characterizes diagonal-case local minimizers by combining first-order stationarity with second-order necessary conditions, then uses the sufficient global-optimality condition of Theorem 2.1; this is a direct proof, not a definitional identity. The strict-saddle results in Section 5 derive region-wise Hessian and gradient bounds from the eigenvalue assumptions on A and the explicit form of the Riemannian Hessian; constants such as gamma, beta, and delta are hypotheses or thresholds, not fitted quantities. The KL-exponent results in Section 6 are proved from Taylor expansions and norm estimates, with no step that assumes the desired exponent. Cited prior work by the authors is contextual (numerical methods and BEC background) and is not load-bearing. The reader's flagged scope issue is real but is not circularity: Section 5 explicitly states that the strong-convexity condition of Definition 5.1 is never satisfied at stationary points in the complex case, so the abstract's unqualified strict-saddle claim is a presentation/scope defect, not a reduction of the claim to its inputs. No equation in the paper is equivalent to its own conclusion by construction, and no parameter is fitted and then renamed a prediction.

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

The central claims rest on standard optimization theory plus two stated domain assumptions: the real-space restriction for strict-saddle results, and the positive spectral gap for the small-beta regime. No entities are invented, and no numbers are fitted to data. The thresholds in the theorems are sufficient conditions chosen analytically, not free parameters.

assumptions (5)
  • standard math A is Hermitian and, without loss of generality, positive definite (Section 1.3).
    Adding a multiple of the identity shifts the objective by a constant on the unit sphere, so any Hermitian matrix can be shifted to positive definite. This normalization is valid and does not change minimizers.
  • domain assumption The strict-saddle analysis restricts to the real sphere S^{n-1} (Section 5).
    The strong-convexity condition in Definition 5.1 is never satisfied at complex critical points because iz gives zero curvature; the real-space restriction is load-bearing for Theorems 5.1 and 5.3.
  • domain assumption For the small-beta strict-saddle theorem, A has a positive spectral gap delta = lambda_{n-1} - lambda_n > 0 (Section 5.2).
    The covering argument R1 union R2 union R3 = S^{n-1} in Theorem 5.3 depends on this gap; the theorem is not proved when the two smallest eigenvalues coincide.
  • domain assumption For the Kurdyka-Lojasiewicz exponent in Theorem 6.2, the stationary point z satisfies H = A + 2 beta diag(|z|^2) - 2 lambda I >= 0.
    This positive semidefiniteness condition is a global-optimality certificate from Theorem 2.1 and is used throughout the real-case proof to handle non-isolated stationary points.
  • standard math Classical second-order optimality conditions on Riemannian manifolds are used (Lemma 2.1, citing Nocedal-Wright [62]).
    The second-order necessary and sufficient conditions for constrained optimization on manifolds are background results, not derived in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On The Geometric Analysis of A Quartic-quadratic Optimization Problem under A Spherical Constraint." pith.science (2026). https://pith.science/paper/U27YHB24

@misc{pith2026190800745,
  author       = {Pith},
  title        = {Pith review of: On The Geometric Analysis of A Quartic-quadratic Optimization Problem under A Spherical Constraint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U27YHB24}},
  note         = {Machine review of arXiv:1908.00745}
}
abstract

This paper considers the problem of solving a special quartic-quadratic optimization problem with a single sphere constraint, namely, finding a global and local minimizer of $\frac{1}{2}\mathbf{z}^{*}A\mathbf{z}+\frac{\beta}{2}\sum_{k=1}^{n}\lvert z_{k}\rvert^{4}$ such that $\lVert\mathbf{z}\rVert_{2}=1$. This problem spans multiple domains including quantum mechanics and chemistry sciences and we investigate the geometric properties of this optimization problem. Fourth-order optimality conditions are derived for characterizing local and global minima. When the matrix in the quadratic term is diagonal, the problem has no spurious local minima and global solutions can be represented explicitly and calculated in $O(n\log{n})$ operations. When $A$ is a rank one matrix, the global minima of the problem are unique under certain phase shift schemes. The strict-saddle property, which can imply polynomial time convergence of second-order-type algorithms, is established when the coefficient $\beta$ of the quartic term is either at least $O(n^{3/2})$ or not larger than $O(1)$. Finally, the Kurdyka-Lojasiewicz exponent of quartic-quadratic problem is estimated and it is shown that the exponent is ${1}/{4}$ for a broad class of stationary points.

Figures

Figures reproduced from arXiv: 1908.00745 by the authors.

Figure 1
Figure 1. Plot of the landscape of the objective function for fix [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Plot of the sets R1–R3 for the problem (5.1). Figure (a) depicts the regions introduced in section 5.2 for a small interaction coefficient β = 0.2. The overlap of the sets R1–R2 and R2–R3 is shown in green. The set R1 is the union of the yellow and the two surrounding green areas, while R2 is the union of all green and light blue areas. The region R3 coincides with the union of the dark blue sets and the enclosing g… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

81 extracted references · 75 canonical work pages

  1. [1]

    Absil, P .A., Mahony, R., Andrews, B.: Convergence of the i terates of descent methods for analytic cost functions. SIAM J. Optim. 16(2), 531–547 (2005)

    2005

  2. [2]

    Princeton University Press (2009)

    Absil, P .A., Mahony, R., Sepulchre, R.: Optimization alg orithms on matrix manifolds. Princeton University Press (2009)

    work page 2009

  3. [3]

    Physics Letters A 265(1-2), 91–96 (2000)

    Adhikari, S.K.: Numerical solution of the two-dimension al Gross–Pitaevskii equation for trapped interacting atoms. Physics Letters A 265(1-2), 91–96 (2000)

    work page 2000

  4. [4]

    In: 29th Annual Conf

    Anandkumar, A., Ge, R.: Efficient approaches for escaping higher order saddle points in non-convex optimization. In: 29th Annual Conf. on Learn. Theory, Proceedings of Mach. Learn. Res., vol. 49, pp. 81–102 (2016)

    work page 2016

  5. [5]

    Arora, S., Ge, R., Ma, T., Moitra, A.: Simple, efficient, an d neural algorithms for sparse coding. J. Mach. Learn. Res. (2015)

    work page 2015

  6. [6]

    Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involv- ing analytic features. Math. Program. 116(1-2), 5–16 (2009)

    work page 2009

  7. [7]

    Attouch, H., Bolte, J., Redont, P ., Soubeyran, A.: Proxim al alternating minimization and projection methods for nonconvex problems: an approach based on the Kur dyka-Łojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010)

    work page 2010

  8. [8]

    Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of des cent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitti ng, and regularized Gauss-Seidel methods. Math. Program. 137(1-2, Ser. A), 91–129 (2013)

    work page 2013

Show all 81 references
  1. [9]

    Bandeira, A.S., Boumal, N., Singer, A.: Tightness of the m aximum likelihood semidefinite relaxation for angular synchronization. Math. Program. 163(1-2), 145–167 (2017)

    2017

  2. [10]

    In: Co nf

    Bandeira, A.S., Boumal, N., V oroninski, V .: On the low-r ank approach for semidefinite programs arising in synchronization and community detection. In: Co nf. Learn. Theory, pp. 361–382 (2016)

    2016

  3. [11]

    Ki- netic and Related Models 6(1), 1–135 (2012)

    Bao, W., Cai, Y .: Mathematical theory and numerical meth ods for Bose-Einstein condensation. Ki- netic and Related Models 6(1), 1–135 (2012)

    2012

  4. [12]

    Bolte, J., Daniilidis, A., Lewis, A.: The Łojasiewicz in equality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J . Optim. 17(4), 1205–1223 (2006)

    2006

  5. [13]

    Bolte, J., Daniilidis, A., Lewis, A.: A nonsmooth Morse- Sard theorem for subanalytic functions. J. Math. Anal. Appl. 321(2), 729–740 (2006) 40 Haixiang Zhang et al

    2006

  6. [14]

    Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556–572 (2007)

    2007

  7. [15]

    Bolte, J., Sabach, S., Teboulle, M.: Proximal alternati ng linearized minimization or nonconvex and nonsmooth problems. Math. Program. 146(1-2), 459–494 (2014)

    2014

  8. [16]

    Inv erse Problems 33(5) (2017)

    Bonettini, S., Loris, I., Porta, F., Prato, M., Rebegold i, S.: On the convergence of a linesearch based proximal-gradient method for nonconvex optimization. Inv erse Problems 33(5) (2017)

    2017

  9. [17]

    Boumal, N.: Nonconvex phase synchronization. SIAM J. Op tim. 26(4), 2355–2377 (2016)

    2016

  10. [18]

    Cai, Y ., Zhang, L., Bai, Z., Li, R.C.: On an eigenvector-d ependent nonlinear eigenvalue problem. SIAM J. Matrix Anal. Appl. 39(3), 1360–1382 (2018)

    2018

  11. [19]

    Cand` es, E.J., Li, X.: Solving quadratic equations via phaselift when there are about as many equations as unknowns. Found. Comput. Math. 14(5), 1017–1026 (2014)

    2014

  12. [20]

    IEEE Trans

    Cand` es, E.J., Li, X., Soltanolkotabi, M.: Phase retrie val via wirtinger flow: Theory and algorithms. IEEE Trans. Inf. Theory 61(4), 1985–2007 (2015)

    2015

  13. [21]

    Cand` es, E.J., Strohmer, T., V oroninski, V .: Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming. Comm. Pure and Appl. M ath. 66(8), 1241–1274 (2013)

    2013

  14. [22]

    IEEE Trans

    Cand` es, E.J., Tao, T.: The power of convex relaxation: Near-optimal matrix completion. IEEE Trans. Inf. Theory 56(5), 2053–2080 (2010)

    2010

  15. [23]

    Cartis, C., Gould, N.I.M., Toint, P .L.: Second-order op timality and beyond: characterization and eval- uation complexity in convexly constrained nonlinear optim ization. Found. Comput. Math. 18(5), 1073–1107 (2018)

    2018

  16. [24]

    Chen, Y ., Chi, Y ., Fan, J., Ma, C.: Gradient descent with random initialization: fast global convergence for nonconvex phase retrieval. Math. Program. 176(1-2, Ser. B), 5–37 (2019)

    2019

  17. [25]

    IEEE Trans

    Chen, Y ., Chi, Y ., Goldsmith, A.J.: Exact and stable cova riance estimation from quadratic sampling via convex programming. IEEE Trans. Inf. Theory 61(7), 4034–4059 (2015)

    2015

  18. [26]

    Preprint, https://arxiv.org/abs/1809.09573

    Chi, Y ., Lu, Y .M., Chen, Y .: Nonconvex optimization meet s low-rank matrix factorization: An overview (2018). Preprint, https://arxiv.org/abs/1809.09573

    2018 arXiv

  19. [27]

    D’Acunto, D., Kurdyka, K.: Explicit bounds for the Łojas iewicz exponent in the gradient inequality for polynomials. In: Ann. Polonici Mathematici, 87, pp. 51– 61 (2005)

    2005

  20. [28]

    Optimization 33(2), 97– 104 (1995)

    Dedieu, J.P .: Third- and fourth-order optimality condi tions in optimization. Optimization 33(2), 97– 104 (1995)

    1995

  21. [29]

    Physical Review A 51(2), 1382 (1995)

    Edwards, M., Burnett, K.: Numerical solution of the nonl inear Schr¨ odinger equation for small samples of trapped neutral atoms. Physical Review A 51(2), 1382 (1995)

    1995

  22. [30]

    IEEE Trans

    Forti, M., Nistri, P ., Quincampoix, M.: Convergence of n eural networks for programming problems via a nonsmooth Łojasiewicz inequality. IEEE Trans. Neural Networks 17(6), 1471–1486 (2006)

    2006

  23. [31]

    Preprint, https://arxiv.or g/abs/1611.08781

    Gao, B., Liu, X., Chen, X., Y uan, Y .: On the Łojasiewicz ex ponent of the quadratic sphere constrained optimization problem (2016). Preprint, https://arxiv.or g/abs/1611.08781

    2016 arXiv

  24. [32]

    Gao, B., Liu, X., Chen, X., Y uan, Y .: A new first-order algo rithmic framework for optimization prob- lems with orthogonality constraints. SIAM J. Optim. 28(1), 302–332 (2018)

    2018

  25. [33]

    Garc´ ıa-Ripoll, J.J., P´ erez-Garc´ ıa, V .M.: Optimizing Schr¨ odinger functionals using sobolev gradients: Applications to quantum mechanics and nonlinear optics. SI AM J. Sci. Comput. 23(4), 1316–1334 (2001)

    2001

  26. [34]

    In: Conf

    Ge, R., Huang, F., Jin, C., Y uan, Y .: Escaping from saddle pointsonline stochastic gradient for tensor decomposition. In: Conf. Learn. Theory, pp. 797–842 (2015)

    2015

  27. [35]

    In: Proceed

    Ge, R., Jin, C., Zheng, Y .: No spurious local minima in non convex low rank problems: A unified geometric analysis. In: Proceed. of the 34th Int. Conf. Mach . Learn., vol. 70, pp. 1233–1242 (2017)

    2017

  28. [36]

    Ge, R., Lee, J.D., Ma, T.: Matrix completion has no spurio us local minimum. In: Adv. Neural Inf. Process. Syst., pp. 2973–2981 (2016)

    2016

  29. [37]

    Ge, R., Ma, T.: On the optimization landscape of tensor de compositions. In: Adv. Neural Inf. Process. Systems, pp. 3653–3663 (2017)

    2017

  30. [38]

    Cambridge University Press (1996)

    Griffin, A., Snoke, D.W., Stringari, S.: Bose-Einstein c ondensation. Cambridge University Press (1996)

    1996

  31. [39]

    Gwo´ zdziewicz, J.: The łojasiewicz exponent of an analy tic function at an isolated zero. Comment. Math. Helv. 74(3), 364–375 (1999)

    1999

  32. [40]

    Hu, J., Jiang, B., Liu, X., Wen, Z.: A note on semidefinite p rogramming relaxations for polynomial optimization over a single sphere. Sci. China Math. 59(8), 1543–1560 (2016)

    2016

  33. [41]

    Hu, J., Milzarek, A., Wen, Z., Y uan, Y .: Adaptive quadrat ically regularized Newton method for Rie- mannian optimization. SIAM J. Matrix Anal. Appl. 39(3), 1181–1207 (2018)

    2018

  34. [42]

    Preprint, https://arxiv.org/abs/1510.07713 Geometric analysis of a quartic-quadratic optimization pr oblem 41

    Jaganathan, K., Eldar, Y .C., Hassibi, B.: Phase retriev al: an overview of recent developments (2015). Preprint, https://arxiv.org/abs/1510.07713 Geometric analysis of a quartic-quadratic optimization pr oblem 41

    2015 arXiv

  35. [43]

    Jain, P ., Kar, P .: Non-convex optimization for machine learning. Found. and Trends R© in Mach. Learn. 10(3-4), 142–336 (2017)

    2017

  36. [44]

    IEEE Trans

    Keshavan, R.H., Montanari, A., Oh, S.: Matrix completio n from a few entries. IEEE Trans. Inf. Theory 56(6), 2980–2998 (2010)

    2010

  37. [45]

    Keshavan, R.H., Montanari, A., Oh, S.: Matrix completio n from noisy entries. J. Mach. Learn. Res. 11(Jul), 2057–2078 (2010)

    2010

  38. [46]

    Preprint, https://arxiv.org/abs/0906.4835

    Kreutz-Delgado, K.: The complex gradient operator and t he CR-calculus (2009). Preprint, https://arxiv.org/abs/0906.4835

    2009 arXiv

  39. [47]

    Kurdyka, K.: On gradients of functions definable in o-min imal structures. Ann. Inst. Fourier (Greno- ble) 48(3), 769–783 (1998)

    1998

  40. [48]

    Lee, J.D., Panageas, I., Piliouras, G., Simchowitz, M., Jordan, M.I., Recht, B.: First-order methods almost always avoid strict saddle points. Math. Program. 176(1-2, Ser. B), 311–337 (2019)

    2019

  41. [49]

    In: Conf

    Lee, J.D., Simchowitz, M., Jordan, M.I., Recht, B.: Grad ient descent converges to minimizers. In: Conf. Learn. Theory, pp. 1246–1257 (2016)

    2016

  42. [50]

    Li, G., Pong, T.K.: Calculus of the exponent of Kurdyka–Ł ojasiewicz inequality and its applications to linear convergence of first-order methods. Found. Comput . Math. 18(5), 1199–1232 (2018)

    2018

  43. [51]

    Preprint, https://arxiv.org/abs/1809.09237

    Li, X., Zhu, Z., So, A.M.C., Vidal, R.: Nonconvex robust l ow-rank matrix recovery (2018). Preprint, https://arxiv.org/abs/1809.09237

    2018 arXiv

  44. [52]

    Liang, S., Sun, R., Li, Y ., Srikant, R.: Understanding th e loss surface of neural networks for binary classification. In: Int. Conf. Mach. Learn. pp. 2835–2843 (2 018)

  45. [53]

    Liu, H., So, A.M.C., Wu, W.: Quadratic optimization with orthogonality constraint: explicit Łojasiewicz exponent and linear convergence of retraction -based line-search and stochastic variance- reduced gradient methods. Math. Program. pp. 1–48 (2017)

    2017

  46. [54]

    Liu, H., Y ue, M.C., Man-Cho So, A.: On the estimation perf ormance and convergence rate of the generalized power method for phase synchronization. SIAM J . Optim. 27(4), 2426–2446 (2017)

    2017

  47. [55]

    Les ´ equations aux d´ eriv´ ees partielles117, 87–89 (1963)

    Łojasiewicz, S.: Une propri´ et´ e topologique des sous- ensembles analytiques r´ eels. Les ´ equations aux d´ eriv´ ees partielles117, 87–89 (1963)

    1963

  48. [56]

    Luo, Z.Q., Pang, J.S.: Error bounds for analytic systems and their applications. Math. Program. 67(1- 3), 1–28 (1994)

    1994

  49. [57]

    In: High Perform

    Luo, Z.Q., Sturm, J.F.: Error bounds for quadratic syste ms. In: High Perform. Optim., pp. 383–404. Springer (2000)

    2000

  50. [58]

    Marˇ cenko, V .A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Math. of the USSR-Sbornik 1(4), 457 (1967)

    1967

  51. [59]

    Merlet, B., Nguyen, T.N., et al.: Convergence to equilib rium for discretizations of gradient-like flows on riemannian manifolds. Diff. and Integral Equations 26(5/6), 571–602 (2013)

    2013

  52. [60]

    More, J.J.: Generalizations of the trust region problem . Optim. Methods and Software 2(3-4), 189– 209 (1993)

    1993

  53. [61]

    Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117–129 (1987)

    1987

  54. [62]

    Springer, New Y ork, NY , USA (2006)

    Nocedal, J., Wright, S.J.: Numerical Optimization, sec ond edn. Springer, New Y ork, NY , USA (2006)

    2006

  55. [63]

    Ochs, P ., Chen, Y ., Brox, T., Pock, T.: iPiano: inertial proximal algorithm for nonconvex optimization. SIAM J. Imaging Sci. 7(2), 1388–1419 (2014)

    2014

  56. [64]

    Preprint, https://arxiv.or g/abs/1605.00405

    Panageas, I., Piliouras, G.: Gradient descent only conv erges to minimizers: Non-isolated critical points and invariant regions (2016). Preprint, https://arxiv.or g/abs/1605.00405

    2016 arXiv

  57. [65]

    Penot, J.P .: Higher-order optimality conditions and hi gher-order tangent sets. SIAM J. Optim. 27(4), 2508–2527 (2017)

    2017

  58. [66]

    Cambridge University Press (2002)

    Pethick, C.J., Smith, H.: Bose-Einstein condensation i n dilute gases. Cambridge University Press (2002)

    2002

  59. [67]

    Cambridge Univer- sity Press (1986)

    Press, W.H., Flannery, B.P ., Teukolsky, S.A., V etterling, W.T.: Numerical recipes. Cambridge Univer- sity Press (1986). The art of scientific computing

    1986

  60. [68]

    Recht, B.: A simpler approach to matrix completion. J. Ma ch. Learn. Res. 12(Dec), 3413–3430 (2011)

    2011

  61. [69]

    Schneider, R., Uschmajew, A.: Convergence results for p rojected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality. SIAM J. Opti m. 25(1), 622–646 (2015)

    2015

  62. [70]

    I EEE Signal Process

    Shechtman, Y ., Eldar, Y .C., Cohen, O., Chapman, H.N., Mi ao, J., Segev, M.: Phase retrieval with application to optical imaging: a contemporary overview. I EEE Signal Process. Mag. 32(3), 87–109 (2015)

    2015

  63. [71]

    Sorber, L., Barel, M.V ., Lathauwer, L.D.: Unconstraine d optimization of real functions in complex variables. SIAM J. Optim. 22(3), 879–898 (2012) 42 Haixiang Zhang et al

    2012

  64. [72]

    Sun, J.: When Are Nonconvex Optimization Problems Not Sc ary? Columbia University (2016)

    2016

  65. [73]

    In: IEEE Int

    Sun, J., Qu, Q., Wright, J.: A geometric analysis of phase retrieval. In: IEEE Int. Symp. Inf. Theory (ISIT), 2016, pp. 2379–2383. IEEE (2016)

    2016

  66. [74]

    IEEE Trans

    Sun, J., Qu, Q., Wright, J.: Complete dictionary recover y over the sphere i: Overview and the geomet- ric picture. IEEE Trans. Inf. Theory 63(2), 853–884 (2017)

    2017

  67. [75]

    IEEE Trans

    Sun, J., Qu, Q., Wright, J.: Complete dictionary recover y over the sphere ii: Recovery by riemannian trust-region method. IEEE Trans. Inf. Theory 63(2), 885–914 (2017)

    2017

  68. [76]

    Sun, J., Qu, Q., Wright, J.: A geometric analysis of phase retrieval. Found. Comput. Math. 18(5), 1131–1198 (2018)

    2018

  69. [77]

    IEEE Trans

    Sun, R., Luo, Z.Q.: Guaranteed matrix completion via non -convex factorization. IEEE Trans. Inf. Theory 62(11), 6535–6579 (2016)

    2016

  70. [78]

    In: Essays and Surv

    Thoai, N.V .: General quadratic programming. In: Essays and Surv. in Glob. Optim., GERAD 25th Anniv. Ser ., vol. 7, pp. 107–129. Springer, New Y ork (2005)

    2005

  71. [79]

    Wen, Z., Yin, W.: A feasible method for optimization with orthogonality constraints. Math. Program. 142(1-2, Ser. A), 397–434 (2013)

    2013

  72. [80]

    Wu, X., Wen, Z., Bao, W.: A regularized Newton method for c omputing ground states of Bose– Einstein condensates. SIAM J. of Sci. Comput. 73(1), 303–329 (2017)

    2017

  73. [81]

    Y ang, W.H.: Error bounds for convex polynomials. SIAM J. Optim. 19(4), 1633–1647 (2008)

    2008

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.