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Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithms

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A new kind of convexity lets curved-space optimization match Euclidean rates, using horoballs and Busemann functions in place of half-spaces and affine functions.

desk verdict A serious, mostly sound contribution to Riemannian optimization: the Euclidean-matching rates are proven for an exact-oracle model, but the unanalyzed weighted-Fréchet-mean subproblem keeps the headline claim one step short of real complexity. read the letter →

arxiv 2505.16970 v1 pith:X5DQEDGG submitted 2025-05-22 math.OC cs.NAmath.DGmath.NA

Christopher Criscitiello , Jungbin Kim This is my paper
classification math.OCcs.NAmath.DGmath.NA MSC 90C4890C2552A4153C20
keywords horosphericalconvexityHadamardmanifoldsBusemannfunctionsfirst-orderoptimizationcurvature-independentratesNesterovaccelerationFrechetmeanoraclehyperbolicspace
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a new class of functions on Hadamard manifolds, called horospherically convex (h-convex) functions, built from horoballs and Busemann functions as the curved-space analogues of half-spaces and affine functions. It proves that first-order methods for minimizing sums of h-convex functions converge at exactly the same rates as in Euclidean convex optimization: gradient descent at $L/(2N)\ d(x_0,x_*)^2$, subgradient descent at $DL/\sqrt{N+1}$, and Nesterov acceleration at $2L/N^2$, with linear rates in the strongly convex case. These rates hold without any bound on the sectional curvature, in contrast to known lower bounds for geodesically convex optimization. The paper also shows that in hyperbolic space h-convex subgradient descent can localize minimizers in $O(\log r)$ steps, which is faster than the Euclidean analogue. The main caveat, stated by the authors, is that the analysis assumes an exact weighted-Fréchet-mean oracle whose internal cost is not analyzed.

What carries the argument

The central object is the scaled Busemann function $B_{y,v}(x)$, the curved-space generalization of the affine function $\langle v,\exp_y^{-1}(x)\rangle$; its sublevel sets are horoballs, the analogues of half-spaces. h-convexity requires that at every point such a function support $f$ from below, and Theorem 1 shows that h-convex functions are exactly the pointwise suprema of scaled Busemann functions. The algorithms replace each Riemannian gradient step by minimizing a sum of quadratic models $Q^{1/s_k}_{x_k,g_i^k}$, which is exactly a weighted Fréchet mean, and the convergence proofs all rest on the 'horospherical law of cosines' (Lemma 2), which plays the role of the Euclidean law of cosines in the Lyapunov arguments.

What would settle it

For instance, take the two-term sum of Busemann functions on the hyperbolic plane used in the paper as the counterexample to additivity of h-convexity, minimize it with Algorithm (16) using an oracle that computes the exact weighted Fréchet mean to machine precision, and compare the function-value gaps against the claimed bound $L/(2N)\,d(x_0,x_*)^2$. If the gap sequence exceeds that bound by a factor that grows with the curvature or with $N$, the theorem's premise fails; if replacing the exact oracle by a cheap approximate one degrades the rate, that would show the oracle assumption is genuinely load-bearing.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that geodesic convexity was the wrong generalization of convexity for optimization complexity, and the right one is horospherical convexity. A function $f$ is h-convex if at every point $y$ there is a tangent vector $v$ with $f(x)\ge f(y)+B_{y,v}(x)$ for all $x$, where $B_{y,v}$ is a scaled Busemann function; strong h-convexity replaces $B_{y,v}$ by the squared-distance model $Q^\mu_{y,v}(x)=-\frac{1}{2\mu}\|v\|^2+\frac{\mu}{2}d(\exp_y(-v/\mu),x)^2$, which converges to $B_{y,v}$ as $\mu\to0$. Because h-convex functions are exactly suprema of scaled Busemann functions, they inherit an outer, global structure like Euclidean convexity while remaining geodesically convex. The paper proves that the natural Fréchet-mean update for a sum of h-convex functions yields the Euclidean iteration counts (Theorems 4 through 9), and that in hyperbolic space the same structure gives much faster localization of minimizers (Propositions 7 through 9).

Load-bearing premise

The convergence theorems presuppose an exact weighted-Fréchet-mean oracle, plus exact h-subgradients, at every iteration, and the cost of running this oracle is not analyzed.

Editorial extensions

If this is right

  • Tyler's M-estimator, geometric median, minimal enclosing ball, and Horn's problem become h-convex optimization problems, so the new curvature-independent rates apply to them directly.
  • Smooth strongly h-convex objectives can be minimized at a linear rate without a lower sectional-curvature bound or a feasibility ball, a guarantee that is unavailable for geodesically convex smooth functions.
  • In hyperbolic space, h-convex Lipschitz minimization needs only $O(\log r)$ subgradient steps to shrink a radius-$r$ search region to constant size, compared with $\Theta(r^2/\delta^2)$ for Euclidean convex and $\Theta(r^3/\delta^2)$ for geodesically convex problems.
  • The Moreau envelope of an h-convex function is h-convex and $(1/\lambda)$-h-smooth, so proximal regularization and smoothing work with the same constants as in Euclidean space.
  • The lower bounds that make geodesic convex optimization depend on curvature do not apply to the h-convex subclass, making h-convexity a concrete candidate escape route for scaling problems, though the paper does not resolve those applications.

Reading between the lines

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

  • Editorial inference: The exact weighted-Fréchet-mean oracle is the true bottleneck; if that subproblem is expensive on a particular manifold, the curvature-independent iteration counts proven here may not translate into curvature-independent running times, and approximating the oracle could reintroduce geometry-dependent constants.
  • Editorial inference: The proof technique suggests a general lifting rule—replace affine supports with Busemann functions and convex combinations with Fréchet means—so other Euclidean convex algorithms whose proofs rely only on a law of cosines may admit h-convex analogues with matching rates.
  • Editorial inference: For acceleration on sums with $m>1$, the paper assumes the whole sum satisfies the descent condition rather than each summand being $L$-h-smooth; testing whether a sum of two Busemann functions satisfies this condition would delimit how far accelerated h-convex optimization extends.
  • Editorial inference: The hyperbolic-space localization result suggests that h-convexity captures objectives that 'point toward the boundary at infinity,' which may make problems like Horn's problem easier in practice than their geodesic-convexity diameter bounds suggest, but converting this into polynomial-time algorithms still requires controlling bit complexity and oracle cost.
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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

4 major / 4 minor

Summary. The paper introduces horospherical convexity (h-convexity) for functions on Hadamard manifolds, building on horoballs and scaled Busemann functions as analogues of half-spaces and affine functions. It develops a calculus for h-convex functions (subgradients, strong convexity, L-h-smoothness, Moreau envelopes, epigraph characterizations) and analyzes first-order methods for minimizing sums of h-convex functions. The central results are curvature-independent iteration-count bounds: gradient descent achieves L/(2N) d(x0,x*)^2 (Theorem 4), subgradient descent achieves DL/sqrt(N+1) (Theorem 6), Nesterov acceleration achieves 2L/N^2 (Theorem 8), and linear rates hold in the strongly convex case (Theorems 5, 7, 9). These guarantees are conditional on an exact weighted-Frechet-mean oracle for the update subproblems. The paper also claims faster rates in hyperbolic space using horoball localization (Propositions 7--9).

Significance. If the results hold, the paper identifies a new structural class of non-Euclidean optimization problems whose oracle complexity matches Euclidean convex optimization, bypassing known curvature-dependent lower bounds for general geodesically convex optimization. The framework is coherent and the main convergence proofs in the appendices are self-contained; the horospherical law of cosines (Lemma 2) is a clean and useful tool. The paper is honest about its main limitation: the convergence theorems assume an exact weighted-Frechet-mean oracle, and Section 6 explicitly warns that this oracle 'may be too strong.' However, this caveat is not reflected in the abstract's 'exactly matching Euclidean convex optimization' phrasing, and the internal cost of the oracle is never analyzed. The hyperbolic-space faster-rate results are interesting but rest on a proof step currently justified by a figure rather than a derivation. Overall, the contribution is substantive and publishable in principle, but the oracle gap and the missing rigorous derivation in Proposition 8 need to be addressed.

major comments (4)
  1. [Section 4.2, Eq. (16); Section 6] The claimed 'exactly matching Euclidean convex optimization' is a statement about iteration counts conditional on an exact weighted-Frechet-mean oracle, not about total complexity. Every iteration of (16) requires minimizing a sum of squared distances to the points exp_{x_k}(-s_k g_{ik}), which is itself a geodesically convex optimization problem. For the paper's own flagship application, the Karcher mean f(x)=sum_i d(x,p_i)^2, the subproblem is another Frechet-mean problem of the same class with no closed form for m>1. The internal cost of this oracle, including its dependence on curvature and diameter, is not analyzed. Since Section 3.3 itself quotes the geodesic smoothness constant of squared distances as 2r sqrt(-K)/tanh(r sqrt(-K)), which is exactly the curvature-and-diameter dependence the outer rates are designed to eliminate, the paper should either provide an oracle-cost analysis, or explicitly and prominently reframe the contribution as iteration-complexity results for an idealized oracle model.
  2. [Remark 5] The claimed extension to approximately solving the subproblems is not quantitative: the remark never states the required additive accuracy delta as a function of N, L, D, and the target epsilon, nor does it bound the inner cost needed to reach that accuracy. As written, it is unclear whether the curvature-independent rates survive for any implementable inner solver. Please provide a concrete delta and a matching inner-complexity bound, or delete the implicit claim that the approximate version preserves the rates.
  3. [Appendix D.2, Proposition 8] The proof of Proposition 8 relies on the containment H_k ∩ H'_k ⊆ B(x_{k+1}, R_k) and the formula R_k = arccosh(e^{r_k/2}), which are asserted from 'Figure 3' and 'a simple calculation' rather than derived. Because Proposition 8 is the basis for the faster hyperbolic rates and for the ellipsoid-method complexity bound in Proposition 9, this step is load-bearing. Please replace the figure-based argument with an explicit coordinate computation (for example, in the Poincare ball model), including a proof of the claimed radius formula.
  4. [Appendix B.9] The counterexample to local-to-global h-convexity depends on the numerical verification of the connected component C (Figure 2). Since this is a theoretical counterexample, the existence of C and the claimed local h-convexity of the constructed function should be proven analytically, not justified by a plot or by referencing code.
minor comments (4)
  1. [Section 4.2, before Eq. (16)] The phrase 'If each f_i is h- 1/s -smooth' is garbled; it should read 'If each f_i is h-convex and (1/s)-h-smooth.'
  2. [Appendix D.1, Proposition 7] The proof says 'It is easy to see' for the containment B_k ∩ H_k ⊆ B(x_{k+1}, r_{k+1}); a one-sentence coordinate justification would make the argument self-contained.
  3. [Table 3] The table compares the manifold results with Euclidean guarantees but does not indicate that the manifold results are conditional on the weighted-Frechet-mean oracle; a footnote would help prevent over-interpretation.
  4. [Abstract and Section 6] The abstract's 'exactly matching those from Euclidean space' is stronger than what is proven, since Euclidean algorithms do not require a Frechet-mean oracle; the Section 6 caveat should be reflected in the abstract or in the first contribution bullet.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence rates are proved self-contained from the definitions; the exact-oracle assumption is an acknowledged complexity gap, not a circular input.

full rationale

After walking the derivation chain, I find no circular step. The h-convexity and L-h-smoothness definitions are new assumptions, and Theorems 4-9 prove convergence from these definitions plus Lemma 2 (the horospherical law of cosines) via self-contained energy arguments. Theorem 1's outer characterization is essentially an immediate rewriting of Definition 1 (the 'if' direction is the definition, and the 'only if' direction uses the supporting Busemann function at each point), but this is a characterization, not a prediction or a fitted input. The self-citations [32,33,35] supply curvature lower bounds and an ellipsoid reduction used only as motivation or comparison; none of the convergence theorems depends on them. The paper explicitly flags in Section 6 that the exact weighted-Frechet-mean oracle 'may be too strong'; this is an acknowledged oracle-complexity gap (the subproblem for sums of squared distances is itself a Frechet-mean problem), not a circular reuse of the target result. Remark 5 asserts an approximate-solution extension without giving the accuracy analysis; that is an omitted proof, not circularity. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No free parameters are fitted to data: the theory is analytic and the step sizes are chosen by proof needs, not by fitting. The main assumptions are the Hadamard setting, standard Busemann-function and comparison geometry, plus exact oracles for h-subgradients and Frechet means. These oracles are the main practical weakness and are acknowledged in Section 6.

assumptions (5)
  • domain assumption M is a Hadamard manifold: complete, simply connected, non-positive sectional curvature.
    All definitions and theorems are stated for this class (Section 2.1); the results rely on Cartan-Hadamard and triangle comparison.
  • standard math Busemann functions are geodesically convex, 1-Lipschitz, C^1 with unit gradient, and the map (x, omega, y) -> B_{x,omega}(y) is continuous on M x CM(infinity) x M.
    These properties are cited from standard references [13,14,23] and used in Lemma 2, Theorem 1, and Appendix B.4.
  • standard math Triangle comparison inequality: d(x,y)^2 >= ||exp_p^{-1}(x) - exp_p^{-1}(y)||^2 (Lemma 1).
    Proved in Section 2.1 via [87, Section 6.2.2]; it underpins the horospherical law of cosines (Lemma 2) used in every convergence proof.
  • domain assumption Weighted Frechet means and the minimization of sums of Q-functions in updates (16), (20), (23) have unique solutions.
    The algorithms require exact solutions of these subproblems. Existence and uniqueness follow from strong geodesic convexity of the objective, but the computational cost of finding them is not bounded in the paper.
  • standard math Geometric containment facts in hyperbolic space: an open horoball is a union of geodesic balls (equation 31), and the intersection of two horoballs used in Proposition 8 is contained in an explicit ball.
    Equation (31) is cited from [14, p.23]; the two-horoball containment is asserted from a figure in Appendix D.2 rather than fully derived.
invented entities (1)
  • Horospherically convex function class and h-subdifferential independent evidence
    purpose: A new function class defined by Busemann-function supports, used to obtain curvature-independent convergence guarantees for optimization on Hadamard manifolds.
    The class is precisely defined in Definition 1 and can be checked for individual functions; the paper provides explicit examples (distance, squared distance, Busemann functions, Tyler M-estimator terms), and concurrent work [50] independently defined the same nonsmooth notion, providing external grounding.

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Cite this review

Pith. "Pith review of Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithms." pith.science (2026). https://pith.science/paper/X5DQEDGG

@misc{pith2026250516970,
  author       = {Pith},
  title        = {Pith review of: Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X5DQEDGG}},
  note         = {Machine review of arXiv:2505.16970}
}
read the original abstract

Geodesic convexity (g-convexity) is a natural generalization of convexity to Riemannian manifolds. However, g-convexity lacks many desirable properties satisfied by Euclidean convexity. For instance, the natural notions of half-spaces and affine functions are themselves not g-convex. Moreover, recent studies have shown that the oracle complexity of geodesically convex optimization necessarily depends on the curvature of the manifold (Criscitiello and Boumal, 2022; Criscitiello and Boumal, 2023; Hamilton and Moitra, 2021), a computational bottleneck for several problems, e.g., tensor scaling. Recently, Lewis et al. (2024) addressed this challenge by proving curvature-independent convergence of subgradient descent, assuming horospherical convexity of the objective's sublevel sets. Using a similar idea, we introduce a generalization of convex functions to Hadamard manifolds, utilizing horoballs and Busemann functions as building blocks (as proxies for half-spaces and affine functions). We refer to this new notion as horospherical convexity (h-convexity). We provide algorithms for both nonsmooth and smooth h-convex optimization, which have curvature-independent guarantees exactly matching those from Euclidean space; this includes generalizations of subgradient descent and Nesterov's accelerated method. Motivated by applications, we extend these algorithms and their convergence rates to minimizing a sum of horospherically convex functions, assuming access to a weighted-Fr\'echet-mean oracle.

Figures

Figures reproduced from arXiv: 2505.16970 by the authors.

Figure 1
Figure 1. A horoball (purple) and half-space (blue) in the hyperbolic plane (Poincar´e disk model). A horoball can be defined as the limit of a sequence of geodesic balls (portrayed with dashed boundaries). In Euclidean spaces, horoballs are simply half-spaces. On general Hadamard, horoballs differ from half￾spaces. Recent progress in bridging the gap between convexity and g-convexity was made by Lewis et al. [74]. Instead of… view at source ↗
Figure 2
Figure 2. The set C is shaded gray. Q. Is it reasonable to assume that the algorithm can obtain information about f through queries to the subgradient oracle that outputs an element of ∂f(x) for given x ∈ M? One might expect a negative answer since the set ∂f(x) is not fully determined by the local behavior of f (see Appendix B.10). However, we provide a positive answer here. A. When f is h-convex, although it may not be poss… view at source ↗
Figure 3
Figure 3. Bounding the intersection of two horoballs with a ball, in the Poincar´e disk model. 59 [PITH_FULL_IMAGE:figures/full_fig_p059_3.png] view at source ↗

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