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Diagonal and Kronecker preconditioners get exact geometric cost certificates

2026-07-09 17:21 UTC pith:PXLVNNNO

load-bearing objection Sound geometric framework for structured preconditioner certificates; core theorems are correct but practical relevance is entirely deferred to future work. the 3 major comments →

arxiv 2607.07204 v1 pith:PXLVNNNO submitted 2026-07-08 math.OC cs.LG

Restricted Dynamic Geometric Complexity: Certificates for Structured Preconditioning

Zavier Li This is my paper
classification math.OC cs.LG MSC 90C2290C2590C3015A4253C22
keywords affine-invariant metricpreconditioningcondition numberKronecker productsemidefinite programmingRiemannian optimizationlinear matrix inequalitygeodesic submanifold
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks a precise question: when you restrict a preconditioner's metric to a structured family (diagonal, block-diagonal, or Kronecker-factored), how much extra geometric deformation does that restriction cost, relative to allowing an arbitrary positive-definite metric? The author frames this as an intrinsic path-distance problem on the cone of positive-definite matrices, using the affine-invariant Riemannian metric. The central object is restricted dynamic geometric complexity, written D_{K,F}(S_0; H): the shortest affine-invariant path length from an initial relative metric S_0 to the set of metrics whose condition number (relative to a fixed Hessian H) falls below a target K, with the path constrained to stay inside the family F. The paper proves that for diagonal and block-diagonal families, the question of whether any such metric exists reduces exactly to a linear matrix inequality feasibility problem (finding a structured matrix E with E ⪯ H ⪯ KE), and that infeasibility can be certified by semidefinite programming dual witnesses. For Kronecker-factored families, the paper proves the family forms a closed, totally geodesic submanifold of the positive-definite cone, so projection onto it is unique and characterized by partial-trace normal equations, with an Armijo-step solver that converges to the projection. Hessian-relative reachability for Kronecker metrics is shown to be equivalent to a Loewner sandwich condition (B⊗A ⪯ H ⪯ K·B⊗A). An exact two-dimensional diagonal formula, a Kronecker expression threshold, and a fixed-eigenbasis linear feasibility subproblem are also derived. The paper then layers on diagnostic interfaces for curvature-proxy error, stochastic estimation, continuous-flow suboptimality, and discretization, factoring the total gap between an ideal benchmark and a real algorithm into a product of four ratios (expression, estimation, flow, discretization).

Core claim

The core discovery is that structured preconditioner reachability — the question of whether a diagonal, block, or Kronecker metric can bring the condition number of G^{-1}H below a target K — admits exact semidefinite or geometric certificate formulations, and that the cost of the structural restriction can be measured as an intrinsic affine-invariant path distance. For diagonal and block families, reachability is precisely LMI feasibility; for Kronecker families, the family is a closed totally geodesic submanifold with unique projection and partial-trace normal equations; and Hessian-relative Kronecker reachability is equivalent to a Loewner sandwich B⊗A ⪯ H ⪯ K(B⊗A). These reductions turn'

What carries the argument

The affine-invariant Riemannian metric on the positive-definite cone, the relative metric S = H^{-1/2}GH^{-1/2}, linear matrix inequality feasibility (E ⪯ H ⪯ KE with E in the structural family), semidefinite programming duality for infeasibility certificates, the partial-trace normal equations Tr_B(R*) = 0 and Tr_A(R*) = 0 characterizing Kronecker projection, the Kronecker Loewner sandwich condition for Hessian-relative reachability, and the four-ratio chain factorization (expression × estimation × flow × discretization).

Load-bearing premise

The entire framework is built on a fixed finite-dimensional quadratic model with a constant positive-definite Hessian. Real neural network training involves nonconvex, stochastic, time-varying curvature, and the paper's extension layers for proxies, stochasticity, and discretization are explicitly described as diagnostic interfaces rather than complete characterizations.

What would settle it

If one could exhibit a positive-definite Hessian H and a diagonal or block family where the LMI E ⪯ H ⪯ KE is feasible but no diagonal/block metric actually achieves condition number ≤ K (or vice versa), the core reachability equivalence of Theorems 2 and 4 would fail. Similarly, if the Kronecker family were shown not to be totally geodesic under the affine-invariant metric, Theorem 6's unique projection guarantee would collapse.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Diagonal preconditioner design can be certified or refuted by solving a semidefinite feasibility problem, giving a principled yes/no answer to whether a coordinatewise method can reach a target condition number for a given Hessian.
  • The Kronecker projection normal equations provide a computable stopping certificate for Riemannian solvers that project onto Kronecker-factored metric families, enabling certified convergence rather than heuristic stopping.
  • The expression threshold K*_K(H) gives the exact best achievable condition number for a Kronecker metric against a fixed Hessian, separating structural expressivity limits from algorithmic or estimation errors.
  • The four-ratio factorization localizes which assumption (family restriction, proxy accuracy, flow suboptimality, or discretization) is responsible for the gap between an ideal geometric benchmark and a real optimizer's performance.

Where Pith is reading between the lines

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

  • If one could extend the LMI reachability characterization beyond fixed Hessians to trajectory-dependent Hessians along a nonconvex optimization path, the framework could potentially certify or refute whether a given adaptive optimizer's structural restriction is sufficient for a specific learning problem, not just a single quadratic.
  • The Kronecker Loewner sandwich condition could serve as a diagnostic tool for deciding when K-FAC-style methods are structurally adequate for a given layer's curvature, versus when the Kronecker assumption is itself the bottleneck — though computing this for large neural network layers would require efficient approximate sandwich tests.
  • The ratio factorization suggests a natural experimental protocol: measure each ratio separately on a real optimizer run to identify whether the dominant performance bottleneck is structural (family too small), statistical (proxy too noisy), dynamical (flow too slow), or implementational (step size too large).

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 8 minor

Summary. This paper develops the concept of restricted dynamic geometric complexity (RDGC), an intrinsic affine-invariant path distance measuring the geometric cost of reaching a target condition-number class when the preconditioning metric is constrained to a structured family (diagonal, block, Kronecker, low-rank). The full positive-definite benchmark (Theorem 1) is recorded as a reference: the minimum affine-invariant deformation needed to reduce condition number when arbitrary metrics are allowed. The main results show that diagonal and block reachability reduce to LMI feasibility (Theorems 2, 4), an exact 2D diagonal formula is derived (Theorem 3), the Kronecker family is a closed totally geodesic submanifold with unique projection characterized by partial-trace normal equations (Theorem 6), and Hessian-relative Kronecker reachability is equivalent to a Loewner sandwich condition (Theorem 8). Extension layers for proxy curvature, stochastic complexity, flows, discretization, and low-rank families are presented as diagnostic interfaces. The proofs in Appendix A are clean applications of standard SPD geometry, eigenvalue majorization, and SDP duality.

Significance. The paper provides a coherent geometric language for comparing the expressive limits of structured preconditioner families. The LMI characterizations for diagonal and block reachability connect the framework to the classical scaling and equilibration literature while embedding it in an affine-invariant path-distance benchmark. The Kronecker projection theorem (Theorem 6) with its partial-trace normal equations, residual certificates, and Armijo convergence guarantee is a substantive contribution. The Hessian-relative Loewner sandwich (Theorem 8) and the candidate certificate (Proposition 10) provide a practical route to finite upper bounds. The exact chain factorization of expression, estimation, flow, and discretization ratios (Theorem 9) is a useful accounting identity for localizing which assumption is responsible for observed length increases. The repository of deterministic synthetic workflows checking diagonal expression gaps, block primal/dual certificates, Kronecker spectral width, and Hessian-relative candidate certificates adds reproducibility value. The paper is honest about the limitations of the quadratic model and the diagnostic status of the extension layers.

major comments (3)
  1. §5.1–§5.2 and the novelty table in §1: The diagonal and block LMI reachability results (Theorems 2, 4) are clean but closely related to classical optimal scaling and condition-number minimization results (Bauer 1963, Shapiro 1982/1985, Maréchal & Ye 2009, Lu & Pong 2011, Qu et al. 2025). The novelty table claims the increment is 'certificate-ready LMI specialization inside an affine-invariant restricted path-distance benchmark.' This is a reasonable framing, but the paper would benefit from a more explicit comparison with at least one of these prior formulations to clarify what is genuinely new beyond re-deriving known reachability conditions in new notation. For instance, the relationship between the LMI E ⪯ H ⪯ KE and the scaling conditions in Shapiro (1982) or Qu et al. (2025) should be stated concretely. This is load-bearing for the novelty claim but does not affect correctness.
  2. §6.3, Theorem 7 vs. Theorem 8: The paper presents two distinct Kronecker target sets: the self-conditioned target K_{m,n,K} = K_{m,n} ∩ C_K (Theorem 7) and the Hessian-relative target R_{K,K}(H) = {G ∈ K_{m,n} : κ(G⁻¹H) ≤ K} (Theorem 8). The text notes these 'coincide only under additional compatibility assumptions' but does not specify what those assumptions are. Since the Hessian-relative target is the one relevant for preconditioning, the reader is left uncertain about the practical role of Theorem 7's bounds. A brief statement of the compatibility conditions under which the two targets coincide, or a clearer statement that Theorem 7 is purely an auxiliary diagnostic, would strengthen the section.
  3. §8, Theorem 9 and Proposition 17: The exact chain factorization is an accounting identity, which the paper honestly states. However, Proposition 17 item 2 claims R_est ≥ 1 whenever 'the proxy certificate problem is no easier than the true restricted problem,' with the checkable special case Ĥ = H, Ŝ₀ = S₀, and K̂ ≤ K. This special case is trivially true by monotonicity (Proposition 1). The more interesting case where Ĥ ≠ H is left as a hypothesis. The paper should clarify whether any nontrivial sufficient condition for R_est ≥ 1 can be stated, or else explicitly acknowledge that the estimation ratio is only a diagnostic outside the trivial case. This affects how the framework is 'operational.'
minor comments (8)
  1. §3.1: The affine-invariant distance formula is standard but the notation ‖log(S₀^{-1/2} S₁ S₀^{-1/2})‖_F could benefit from a brief reminder that this is the matrix logarithm, not the scalar log, for readers less familiar with SPD geometry.
  2. §4.1, item 4: The phrase 'absolutely continuous curves in the specified component or quotient chart' is slightly vague for the stratified low-rank case. A reference to the stratified-length theory or a note that this case requires separate treatment would be cleaner.
  3. §5.2, Theorem 3: The gauge choice det D(u) = 1 is stated but the relationship between this gauge and the general scale-quotient in Remark 1 could be made more explicit. A reader might wonder whether the formula is gauge-dependent.
  4. §6.1, Remark 3: The status of the spectral surrogate is well-explained, but a forward reference to Proposition 8 (where it becomes exact) earlier in the section would help the reader navigate.
  5. §6.3, Proposition 11: The subdifferential ∂ω(Q*) is defined in terms of extremal eigenspaces, but the notation E_max and E_min is introduced only in the proof. Defining these in the proposition statement would avoid ambiguity.
  6. §9.3: The reproducibility workflows are described in prose. A table or structured listing of the scripts, their inputs, and expected outputs would improve inspectability.
  7. References: The self-citation to Li (2026) 'Optimization geometrodynamics' is an arXiv preprint. If this is by the same author, the relationship should be transparently stated. If it is not yet published, the dependency of the present paper on that framework (for Theorem 1) should be noted. A DOI or stable link should be included.
  8. §A, Proof of Theorem 6: The closedness argument for K_{m,n} uses a determinant gauge on A_k and eigenvalue bounds. This is correct but could be stated more concisely by noting that K_{m,n} = exp(L_{m,n}) and L_{m,n} is a closed linear subspace, so K_{m,n} is closed in S^n_{++}.

Circularity Check

0 steps flagged

No significant circularity; one minor self-citation that is not load-bearing

full rationale

The paper's central mathematical claims are self-contained and independently proved in Appendix A using standard tools from SPD geometry (Bhatia 2007), semidefinite programming (Boyd & Vandenberghe 2004), and matrix manifold optimization (Absil et al. 2008; Boumal 2023). The full SPD benchmark (Theorem 1) is re-proved internally via the standard eigenvalue majorization lower bound, so it does not depend circularly on the cited prior work (Li, 2026). The LMI characterizations (Theorems 2, 4) follow from generalized Rayleigh quotient arguments. The Kronecker projection theorem (Theorem 6) is proved from first principles by showing log(K_{m,n}) = L_{m,n} is a linear subspace and that ambient geodesics between Kronecker products remain Kronecker. The Hessian-relative Loewner sandwich (Theorem 8) follows from the similarity κ(G⁻¹H) = κ(H^{-1/2}GH^{-1/2}). The chain factorization (Theorem 9) is explicitly an accounting identity (multiply and divide by intermediate quantities), not a derivation claiming to produce new information. The self-citation to Li (2026) provides context and notation but is not load-bearing for any proved result. The extension layers are honestly labeled as diagnostic interfaces. No step in the derivation chain reduces to its own inputs by construction, and no 'prediction' is equivalent to a fitted parameter. The framework is a benchmark/certificate language that succeeds on its own terms.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 3 invented entities

No free parameters are fitted to data. The framework is purely mathematical with no empirical constants. The axioms are standard results from SPD geometry and SDP theory, plus the domain assumption that the quadratic model is the right abstraction. The invented entities are well-defined mathematical objects with proved properties, not postulated physical entities.

axioms (6)
  • standard math Affine-invariant Riemannian metric on S^d_{++} is the natural distance for comparing metric evolution
    Section 3.1; standard result from Bhatia (2007), Pennec et al. (2006). Used throughout as the path-length functional.
  • standard math Eigenvalue majorization lower bound: d_AI(S_0, S_1) ≥ ||y - z||_2 for log-spectra y, z
    Appendix A, proof of Theorem 1; cited to Bhatia (2007). Load-bearing for the full SPD benchmark formula.
  • domain assumption The quadratic model f(θ) = ½θᵀHθ with fixed H ≻ 0 captures the relevant preconditioning geometry
    Section 3; the entire framework is built on this model. Section 9.2 acknowledges the nonquadratic extension is a time-dependent control problem.
  • standard math SDP duality provides valid infeasibility certificates for the LMI E ⪯ H ⪯ KE
    Section 5.4, Proposition 3; standard semidefinite programming duality (Vandenberghe & Boyd 1996).
  • standard math The SPD cone with affine-invariant metric is a Hadamard manifold, so projection onto closed geodesically convex subsets is unique
    Appendix A, proof of Theorem 6; standard result in matrix manifold optimization (Boumal 2023). Load-bearing for Kronecker projection uniqueness.
  • domain assumption Realizable metric families can be given smooth embedded submanifold or quotient-space structure with well-defined path length
    Section 4.1, working requirement 3; assumed for the definition of D_{K,F} to be meaningful. Stated as a working assumption, not proved in general.
invented entities (3)
  • Restricted dynamic geometric complexity D_{K,F}(S_0; H) independent evidence
    purpose: Central object measuring intrinsic path distance to condition-number target within a restricted metric family
    Well-defined mathematical object; its properties (monotonicity, LMI reduction, Kronecker projection) are proved, not postulated. Falsifiable: if the LMI is infeasible, D = +∞.
  • Expression gap Gap_expr = D_{K,F} / D_K independent evidence
    purpose: Ratio measuring how much structural restriction costs relative to full SPD benchmark
    Derived quantity, not postulated. Computed in closed form for 2D diagonal case (Theorem 3).
  • Kronecker expression threshold K*_K(H) independent evidence
    purpose: Minimum condition number achievable by any Kronecker metric for a given Hessian H
    Defined as min over Kronecker family of κ(G⁻¹H); existence proved via compactness argument in Corollary 5 proof.

pith-pipeline@v1.1.0-glm · 28484 in / 3056 out tokens · 567051 ms · 2026-07-09T17:21:24.028417+00:00 · methodology

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read the original abstract

Optimization geometrodynamics views optimizer state as evolving geometry. Its full positive-definite quadratic benchmark gives the least affine-invariant deformation needed to reduce condition number when arbitrary metrics are allowed. This paper records that benchmark in the present notation and develops restricted dynamic geometric complexity: an intrinsic certificate distance for reaching a target condition-number class when the metric is restricted to a specified family. The main proved results are monotonicity and submanifold-distance principles, diagonal and block reachability as linear matrix inequality feasibility problems, an exact two-dimensional diagonal complexity formula, and affine-invariant Kronecker projection theorems with normal equations, computable mismatch certificates, Armijo solver convergence, auxiliary self-conditioned K-target bounds, and Hessian-relative candidate certificates through an exact Kronecker Loewner-sandwich reachability condition, including a Kronecker expression threshold and a fixed-basis exact subproblem. Low-rank spectral models, curvature-proxy inflation, stochastic restricted complexity, discrete geometric length, and expression--estimation--flow--discretization accounting are presented as diagnostic interfaces rather than full optimizer characterizations. The resulting language turns structural preconditioner questions into geometric distance, reachability, and certificate problems. The repository includes deterministic toy and synthetic workflows that check diagonal expression gaps, block primal/dual certificates, Kronecker spectral width, and Hessian-relative Kronecker candidate certificates on small quadratic instances, together with low-rank spectral monotonicity.

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