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arxiv: 2606.07782 · v1 · pith:5TPBP3BUnew · submitted 2026-06-05 · 🧮 math.OC · cs.LG· math.MG

Non-Archimedean Polydisc Spaces and Applications to Optimisation

Paul Lezeau , Yiannis Fam , Anthea Monod , Yue Ren This is my paper

Pith reviewed 2026-06-27 20:52 UTC · model grok-4.3

classification 🧮 math.OC cs.LGmath.MG
keywords non-Archimedean geometrypolydisc spacesoptimizationBerkovich geometrymetric treesgeodesicsuniversal approximationminimisers
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The pith

Polydisc spaces over non-Archimedean fields support optimization with existence of minimizers for certain functions.

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

The paper proposes polydisc spaces, formed as products of closed balls over non-Archimedean fields, to enable optimization while keeping the hierarchical structure of the field and adding useful geometric properties. It defines functions as linear combinations of absolute values of polynomials, which have a piecewise polynomial form along geodesics and can approximate other functions universally. The authors establish that minimizers for these functions exist in the space and provide algorithms to compute them, along with showing that metric trees embed naturally into these spaces.

Core claim

We introduce polydisc spaces as products of closed balls over a non-Archimedean field. Metric trees embed naturally into these spaces. The spaces have unique geodesics. Functions given by linear combinations of absolute values of polynomials admit piecewise polynomial descriptions along geodesics and a universal approximation property. We prove existence of minimisers for these functions and explore algorithms for finding them.

What carries the argument

Polydisc spaces, products of closed balls over a non-Archimedean field, that retain hierarchical structure and gain geometric features like geodesic uniqueness for optimization.

If this is right

  • Metric trees embed into polydisc spaces, allowing hierarchical data to be represented and optimized.
  • The class of functions has a piecewise polynomial description along geodesics, enabling analysis of optimization paths.
  • Existence of minimizers is guaranteed for the proposed functions.
  • Algorithms can be developed to find the minimizers due to compatibility with classical techniques from geodesic uniqueness.

Where Pith is reading between the lines

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

  • Optimization on these spaces might apply to problems involving tree-structured data in machine learning or phylogenetics.
  • The framework could be extended to other non-Archimedean geometries beyond polydiscs.
  • Universal approximation suggests these functions can model complex objectives in hierarchical settings.
  • Implementation in software libraries indicates practical computability of the optimization procedures.

Load-bearing premise

The proposed functions given by linear combinations of absolute values of polynomials admit a piecewise polynomial description along geodesics and satisfy a universal approximation property.

What would settle it

Observing a function from the proposed class on a polydisc space without a minimizer, or an algorithm that does not locate a known minimizer.

Figures

Figures reproduced from arXiv: 2606.07782 by Anthea Monod, Paul Lezeau, Yiannis Fam, Yue Ren.

Figure 1
Figure 1. Figure 1: Four mammals separated by their evolutionary distance arranged in form of a tree (left), their [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The polydisc space B 1 for K = Q2 arranged as an infinite tree. for some absolute multiradius r = (r1, . . . , rn) ∈ R n ≥0 or valuative multiradius v = (v1, . . . , vn) ∈ (R∪ {∞}) n. The polydiscs BB(a, r) and BH(a, v) (or their multiradii r and v) are rational if ri ∈ |K| = {|a| | a ∈ K} or vi ∈ val(K) = {val(a) | a ∈ K} for all i = 1, . . . , n, respectively. The n-polydisc space B n and the hyperbolic … view at source ↗
Figure 3
Figure 3. Figure 3: The metric dB (blue, top) on B 1 and dH (red, bottom) on H1 for K = Q2. Remark 15. In [BBI01, Definition 3.6.5], convexity is defined by requiring that every geodesic between two points lies in the set. In our setting, this distinction becomes relevant when geodesics are not unique. If n = 1 or 1 < p < ∞, then (B n, dp B ) and (Hn, dp H) are uniquely geodesic by Lemma 12 and conv(S) is the unique convex se… view at source ↗
Figure 4
Figure 4. Figure 4: The convex hull of four points in B 2 over Q2 from Example 17. where γ1 ∼ γ2 if and only if γ1([0, s1]) = γ2([0, s2]) for some s1, s2 > 0. We write vBB′ := γBB′ ∈ TBB n. We set TBHn = TBB n. Definition 18 is inspired by the notion of tangent directions in Berkovich spaces [Bak08] and more generally by the concept of directions in metric geometry [BBI01, Section 3.6.6]. In classical metric geometry, directi… view at source ↗
Figure 5
Figure 5. Figure 5: Connected components of B 1 \ B in Example 21. We now verify that the sets introduced above are exactly the connected components of the complement of B and thus determine the tangent directions at B. The verification splits naturally into three cases: singleton discs; irrational discs; and rational discs of positive radius. The singleton case turns out to be degenerate: a point of K ⊂ B1 has only one outgo… view at source ↗
Figure 6
Figure 6. Figure 6: Disc spaces under field extensions (all radii in valuation). [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Embeddings in Example 35. The local dependence discussed above in Proposition 34 contrasts with isometric embeddings into the full polydisc space B 1 , where additional global constraints arise and create additional difficulties: Example 36. Let Γ1 and Γ2 both be binary trees given by V (Γi) = {1, . . . , 8}, E(Γi) = {(1, 2),(2, 3),(2, 4),(3, 5),(3, 6),(4, 7),(4, 8)} and edge length dΓ1 (2, 3) = dΓ1 (2, 4)… view at source ↗
Figure 8
Figure 8. Figure 8: Values of val(f) on H1 for f from Example 41 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Absolute polynomial sum with no global minimum. As [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A portion of the DAG for B 2 over K = Q2, starting with the Gauss point at the top. Given a rational polydisc B ∈ Bn, we denote by C(B) the set of children of B, namely those rational polydiscs obtained from B by a single refinement step. More specifically, let Ci(B) be the subset consisting of children obtained by increasing the ith valuative radial coordinate. Thus C(B) = [n i=1 Ci(B), and we refer to t… view at source ↗
Figure 11
Figure 11. Figure 11: The first two levels of the cell decomposition for [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: DAG Monte-Carlo Tree Search over B 2 (Q2) after N simulations. Brighter colours correspond to larger visit counts. The depth of the final tree is truncated. corresponds to selecting a promising refinement. The key idea is to allocate computational effort adaptively, focusing on regions that appear favourable based on past evaluations while concurrently ensuring sufficient exploration of the search space. … view at source ↗
Figure 13
Figure 13. Figure 13: Loss landscape and curves for |f| = |x 2 − 1| over Q2. Optimizer Center c Radius r Loss f(B(c, 2 −r )) Time (s) Greedy Descent 1 20 4.77 × 10−7 0.077 MCTS 1 20 4.77 × 10−7 0.027 DOO 1 20 1.53 × 10−5 0.018 [PITH_FULL_IMAGE:figures/full_fig_p043_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The search trees for the minimisation of [PITH_FULL_IMAGE:figures/full_fig_p044_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Average ranks and mean final loss of the optimisers across several polynomial minimisation tasks [PITH_FULL_IMAGE:figures/full_fig_p045_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Loss landscape and curves of P x∈X|(z − x)(z − 2x)(z − 4x)| over Q2. minimise the loss function ℓ(x) = Xm i=1 |fi(x)| over the n-dimensional polydisc space B n. For each experiment, we generate multiple random problem instances by sampling polynomial coefficients uniformly from p-adic numbers with valuations between 0 and 8, and initialize all optimisers at the Gauss point BH(1, 0). We vary the prime p ∈ … view at source ↗
Figure 17
Figure 17. Figure 17: Average ranks and mean final loss of the optimisers across several absolute polynomial sum [PITH_FULL_IMAGE:figures/full_fig_p046_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Average ranks and mean final loss of the optimisers across several polynomial interpolation tasks [PITH_FULL_IMAGE:figures/full_fig_p047_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Average ranks and mean final loss of the optimisers across several function learning tasks [PITH_FULL_IMAGE:figures/full_fig_p047_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Effect of branching (C 1 vs. C 2 ) on optimiser performance [PITH_FULL_IMAGE:figures/full_fig_p049_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Average ranks of optimisers across all experiments [PITH_FULL_IMAGE:figures/full_fig_p049_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Effect of number of simulations on MCTS and DAG–MCTS performance [PITH_FULL_IMAGE:figures/full_fig_p050_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Function learning experiment performance [PITH_FULL_IMAGE:figures/full_fig_p050_23.png] view at source ↗
read the original abstract

We propose a new framework for optimisation over non-Archimedean spaces inspired by Berkovich geometry. Specifically, we introduce polydisc spaces, which consists of products of closed balls over a non-Archimedean field. These spaces retain the rigid hierarchical structure of the non-Archimedean field whilst acquiring many desirable geometric features absent from it. We show that metric trees embed naturally into these spaces, demonstrating their capacity to represent hierarchical data. We study their metric geometry, establishing properties such as geodesic uniqueness, confirming their comaptibility with classical optimisation techniques. We further propose a class of real-valued functions given by linear combinations of absolute values of polynomials. These functions admit a piecewise polynomial description along geodesics and satisfy a universal approximation property. We formulate a theory of optimisation on polydisc spaces: we prove existence of minimisers and explore algorithms for finding them. We provide an accompanying open-source Julia library implementing the core objects and optimisation procedures introduced.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper proposes a new framework for optimization over non-Archimedean polydisc spaces inspired by Berkovich geometry. It defines polydisc spaces as products of closed balls over a non-Archimedean field, shows that metric trees embed naturally into them, establishes metric geometry properties including geodesic uniqueness, introduces a class of real-valued functions given by linear combinations of absolute values of polynomials that admit piecewise polynomial descriptions along geodesics and satisfy a universal approximation property, proves existence of minimizers for this class, explores algorithms for finding them, and provides an accompanying open-source Julia library implementing the objects and procedures.

Significance. If the central claims hold, the work formulates a coherent optimization theory on a new class of spaces that combine hierarchical structure with useful metric properties, potentially enabling new approaches to optimization on hierarchical or tree-like data. The provision of an open-source Julia library implementing the core objects and procedures is a concrete strength supporting reproducibility.

minor comments (3)
  1. [Abstract] Abstract: 'polydisc spaces, which consists of products' contains a subject-verb agreement error and should read 'which consist of products'.
  2. [Abstract] Abstract: 'comaptibility with classical optimisation techniques' contains a spelling error and should read 'compatibility'.
  3. The claim that the functions 'admit a piecewise polynomial description along geodesics' is central to the optimization theory but would benefit from an explicit statement of the degree or number of pieces in the main text (e.g., near the definition of the function class).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its potential significance for optimization on hierarchical data, and recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces polydisc spaces as products of closed balls over a non-Archimedean field and defines a class of real-valued functions as linear combinations of absolute values of polynomials. It then proves metric properties (geodesic uniqueness, tree embeddings) and optimization results (existence of minimisers) directly from these definitions. No step reduces a claimed prediction or theorem to a fitted parameter, self-citation chain, or definitional tautology. The universal approximation property and piecewise polynomial description are derived properties of the new function class, not inputs renamed as outputs. The framework is self-contained against external benchmarks with no load-bearing self-citations or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central constructions rest on the definition of polydisc spaces as products of closed balls and the choice of the function class; no explicit free parameters or invented entities with external evidence are stated in the abstract.

axioms (1)
  • standard math Standard properties of non-Archimedean fields and Berkovich geometry hold as background.
    Invoked implicitly to define the spaces and their metric features.
invented entities (1)
  • polydisc spaces no independent evidence
    purpose: Provide geometric structure retaining hierarchical features while enabling classical optimization techniques.
    Newly defined as products of closed balls over non-Archimedean fields.

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