A Spectral Phase Admissibility Certificate for Complex Linear Maps

Snigdha Chandan Khilar

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A Spectral Phase Admissibility Certificate for Complex Linear Maps

Snigdha Chandan Khilar

arXiv:2609.02911v1Jul 13, 2026physics.gen-phcs.CL

TL;DR

Existing machine-learning spectral constraints largely regulate magnitude or positivity, leaving collective spectral phase largely unaddressed. This paper introduces differentiable KSW certificates, with SP exactly matching exterior-power admissibility decisions at far lower cost than exponential enumeration.

Problem
Machine-learning spectral constraints mainly regulate magnitude or positive definiteness, leaving the usefulness of collective spectral phase constraints unresolved.
Method
The paper develops three differentiable phase-only certificates—Det, subset-product envelope SP, and full KSW—with increasing certification strength.
Results
The O(d^3) SP certificate returns the identical accept/reject decisions as exponential subset enumeration across tested sizes.
Takeaways & Limitations
SP is the appropriate certificate for exterior-power well-posedness, while Det is too weak and KSW is unnecessarily restrictive for that objective.
Takeaways & Limitations
The certificate applies only to objectives that consume spectral-product phase, remaining inert for magnitude-based objectives such as normalizing-flow likelihoods.

Abstract

from arXiv · show

The paper imports the Kontsevich Segal Witten criterion from quantum gravity into machine learning to evaluate complex linear maps Standard techniques analyze magnitude or positive definiteness whereas this method exclusively limits the collective phase of a spectrum The researchers create three distinct differentiable certificates comprising a determinant sector a subset product envelope and the full criterion The subset envelope prevents all exterior power eigenvalues from touching the negative real axis This constraint precisely matches the accept or reject choices of an exponential minor enumeration while reducing processing expenses drastically The team provides a differentiable enforcement application via a Schur parameterization The document also identifies crucial boundaries regarding where this system works The constraint cannot balance deep linear propagation since restricting the phase budget damages eigenvector conditioning Furthermore the technique remains completely blind to magnitude based targets like normalizing flow likelihoods Thus researchers must restrict this tool specifically to models that process the argument of a spectral product

I. INTRODUCTION · II. BACKGROUND: THE KSW CRITERION

The paper introduces the KSW admissibility criterion as a machine-learning tool that constrains collective spectral phase rather than magnitude or positive definiteness. Its subset-product interpretation provides the basis for translating an exponential family of convergence conditions into differentiable certificates.

I. INTRODUCTION: The paper examines the Kontsevich–Segal–Witten (KSW) admissibility criterion for complex metrics as a previously unused machine-learning criterion, [2].The motivation follows successful neural architectures derived from physical structures, including diffusion models, Hopfield networks, attention, and neural ODEs.
I. INTRODUCTION: KSW constrains the collective phase of a spectrum, capping total spectral angle so eigenvalues and their products avoid the negative real axis.This keeps determinants and their logarithms single-valued, distinguishing the criterion from magnitude-focused stability tools.
I. INTRODUCTION: The paper establishes three differentiable O(d3) certificates—Det, subset-product envelope SP, and full KSW—with Det ≤ SP ≤ KSW ≤ 2SP.The hierarchy formalizes distinct levels of phase-based certification.
II. BACKGROUND: THE KSW CRITERION: For a diagonal complex metric, each p-form mode corresponds to a subset of lowered and raised eigenvalue directions, producing a Gaussian coefficient determined by those choices.The background derives the relevant mode structure from the p-form kinetic action.
II. BACKGROUND: THE KSW CRITERION: Requiring convergence for all 2^d sign patterns yields the KSW criterion, a single scalar envelope certifying an exponential family of conditions.This motivates targeting quantities formed from products over spectral subsets in the machine-learning translation.

III. THEORY: A CERTIFICATE HIERARCHY

The paper defines three phase-only spectral certificates with increasing strength: Det, SP, and KSW. SP exactly characterizes admissibility across all exterior-power sectors, while KSW is equivalent to convergence of every p-form sector.

Scope and separation: All three certificates ignore eigenvalue magnitudes entirely and constrain only collective spectral phase.The gap SP < π ≤ KSW is nonvacuous: every subset product can be admissible while a ratio sector diverges.
KSW certificate: KSW(M) < π if and only if every p-form sector converges.KSW is the total phase budget of the spectrum, whereas SP is the largest phase attainable by a subset product.
Subset-product certificate: SP(M) < π holds if and only if every subset product has phase magnitude below π.Because exterior-power eigenvalues are exactly subset products, this prevents branch-cut crossings for every exterior-power log determinant and multi-scale log-volume.
Certificate hierarchy: Det < π controls det M alone, SP < π exactly controls every exterior-power eigenvalue, and KSW < π controls all p-form ratio sectors.These certificates are computable from one eigendecomposition, with Det(M) ≤ SP(M) ≤ KSW(M) ≤ 2SP(M).

A. Distinctness from magnitude and positivity

KSW, SP, and Det constrain spectral phase rather than magnitude, remaining unchanged under positive rescaling of eigenvalues. Their phase criterion is also distinct from positive-definiteness: neither condition implies the other.

A. Distinctness from magnitude and positivity: KSW, SP, and Det are invariant under λ_i 7→r_iλ_i for r_i > 0, so they are independent of singular values and spectral radius.Magnitude-based constraints leave these phase functionals unconstrained, while eigenvalue magnitudes and singular values can vary arbitrarily.
A. Distinctness from magnitude and positivity: Positive-definiteness, requiring Re λ_i > 0 for every eigenvalue, neither implies nor is implied by KSW < π.Three eigenvalues with a_i = 0.4π have KSW = 1.2π > π despite positive real parts, whereas phases 0.9π and −0.05π give KSW = 0.95π < π with one negative real part.
A. Distinctness from magnitude and positivity: The distinction between KSW and positive-definiteness is empirically confirmed in Sec. V-A.

IV. ALGORITHM · A. A differentiable admissible layer

The admissible layer parameterizes complex maps in Schur form, exposing eigenphases and converting the certificates into differentiable closed-form functions without a forward-pass eigensolver. A phase-projection algorithm then rescales eigenphases whenever a chosen certificate exceeds its budget.

A. A differentiable admissible layer: Schur-form parameterization exposes eigenphases directly while avoiding differentiation through a general ill-conditioned eigendecomposition.The map uses Q = exp(A−A^H) as a unitary factor and an upper-triangular T with diagonal r_i e^{iθ_i}, positive magnitudes, and free strictly-upper entries.
A. A differentiable admissible layer: Because W and T are unitarily similar and T is triangular, the certificates become closed-form functions of the eigenphases.The supplied derivation states that the determinant certificate depends on the product of phase terms.
A. A differentiable admissible layer: The resulting phase-based computation requires no eigensolver in the forward pass.This makes the certificate evaluation directly differentiable with respect to the exposed phase parameters.
A. A differentiable admissible layer: Algorithm 1 projects eigenphases onto an admissibility ball for the determinant, subset-product, or KSW certificate.It takes eigenphases θ, a certificate C ∈ {Det, SP, KSW}, and a budget b < π.
A. A differentiable admissible layer: The projection computes positive and negative phase sums and selects a certificate-specific violation value.The algorithm uses |Σ+ − Σ−|, max(Σ+, Σ−), or Σ+ + Σ− for Det, SP, and KSW, respectively.
A. A differentiable admissible layer: When the violation exceeds the budget, the algorithm rescales every eigenphase by b/v using a 1-homogeneous rescaling.If v ≤ b, the phase vector is returned without the stated rescaling step.

B. Enforcement

The method enforces a phase budget b < π either exactly through radial projection or softly through a one-sided loss penalty. Because Det, SP, and KSW are 1-homogeneous in θ, one scaling step restores feasibility.

B. Enforcement: Radial projection enforces b < π exactly and without additional parameters.This procedure is given as Algorithm 1.
B. Enforcement: A one-sided penalty max(0, C(θ) − b) provides a soft alternative for C ∈ {Det, SP, KSW}.The penalty is preferred when the constraint should remain soft.
B. Enforcement: A single scaling restores feasibility because all three functionals are 1-homogeneous in θ.

V. EXPERIMENTS … C. Computational cost

Experiments validate the certificate hierarchy and its exact agreement with brute-force admissibility tests, while showing that the polynomial-time certificate sharply reduces computational cost. Phase-only trials also demonstrate discrimination unavailable to magnitude-based criteria.

V. EXPERIMENTS: All quantities are computed in double precision, and each theoretical claim is checked against exponential brute force before further use.The paper provides accompanying code for the experiments.
A. Distinctness from existing constraints: Over 2×10^4 unit-magnitude trials at d = 4, spectral radius < 1 accepts 0% while KSW < π accepts 4.4%, isolating phase discrimination.Because all eigenvalue magnitudes equal one, magnitude-based criteria are blind in this experiment.
A. Distinctness from existing constraints: The two Sec. III-A positivity witnesses are reproduced exactly, including a state with Re λ_i > 0 everywhere but KSW = 1.2π that is rejected.This confirms that positivity of individual real parts does not guarantee the collective phase criterion.
B. The hierarchy, verified against brute force: Across 1500 random spectra at d = 8, closed forms match exhaustive maxima, the ordering Det ≤ SP ≤ KSW ≤ 2SP always holds, and SP < π exactly matches subset-product admissibility.The gap SP < π ≤ KSW occurs for 287/1500 ≈19% of spectra, showing that the exterior-power tower can remain admissible while a p-form ratio sector diverges.
C. Computational cost: At d = 18, the O(d^3) certificate returns the same accept/reject decision as exponential enumeration across all tested sizes.The certificate uses eigendecomposition plus an O(d) reduction, whereas direct admissibility checking enumerates 2^d subset products.
B. The hierarchy, verified against brute force: Fig. 1 depicts the nesting Full KSW ⊆ subset-product SP ⊆ determinant-sector regions and marks a spectrum admissible under SP but violating full KSW.The visualization represents certificates as regions in the (Σ+, Σ−) plane, where Σ+ and Σ− are total positive and negative spectral phases.
C. Computational cost: 1.2 s versus 0.27 ms per decision (∼4500×) shows the certificate is substantially faster than exponential enumeration, with the projected gap exceeding 10^9× at d = 40.Fig. 2 reports identical accept/reject decisions for the O(d^3) certificate and O(d 2^d) enumeration; under Schur parameterization, the certificate becomes O(d).

D. Application: a tower-dependent objective

The application tests spectral guards on a tower-dependent objective whose multi-scale log-volume readout is discontinuous at subset-product branch crossings. Under a mixed phase target, subset-product protection prevents crossings while determinant-only protection fails and the full criterion sacrifices fit.

Objective: The objective reads out principal complex logarithms of subset products, so crossing ±π causes a 2π jump and discontinuity.The learnable spectrum is trained toward d=8 with four phases at +0.55π and four at −0.55π.
Results: The determinant guard is inactive near zero and permits a subset crossing, showing that guarding det alone is unsafe for the tower.The positive phases sum to 2.2π even while the total phase is approximately zero, fooling the determinant.
Results: Brute-force minor projection matches SP byte-for-byte but requires exponential computation.This establishes SP’s equivalence to exhaustive minor enforcement while retaining its practical computational advantage.

E. Negative result: not a propagation stabilizer · VI. SCOPE AND APPLICABILITY

The phase-only certificate cannot stabilize iterated linear propagation and may worsen transient and gradient amplification through eigenvector conditioning. Its applicability is limited to objectives that consume the argument of a spectral product, not magnitude-only spectral objectives.

E. Negative result: not a propagation stabilizer: Because the certificate is phase-only, it cannot control magnitude growth in an iterated map x_t+1 = Wx_t.The limitation follows from phase-only dependence, while propagation behavior depends on singular values and eigenvector conditioning.
VI. SCOPE AND APPLICABILITY: The certificate is inert for objectives that read the spectrum only through eigenvalue magnitudes, a limitation established as a theorem.Such objectives are unchanged because the functionals Det, SP, and KSW depend on eigenvalue arguments, which magnitude-only losses do not use.
VI. SCOPE AND APPLICABILITY: Normalizing-flow log-likelihoods and positive-definite-kernel log-determinants are unchanged by the certificates and can be neither improved nor harmed.These objectives depend on spectral magnitudes, including |det J| for normalizing flows and positive real spectra for kernels.
E. Negative result: not a propagation stabilizer: At fixed unit spectral radius, tightening the phase budget clusters eigenvalues and worsens eigenvector conditioning κ(V), increasing transient and gradient amplification under iteration.The certificate cannot control magnitude growth, which is governed by singular values and eigenvector conditioning.
VI. SCOPE AND APPLICABILITY: For positive-definite spectra, eigenvalue arguments are zero, so phase-based certificates provide no leverage over the corresponding magnitude objectives.The real positive spectrum removes the argument information on which Det, SP, and KSW depend.
VI. SCOPE AND APPLICABILITY: The certificate matters only when a loss consumes the argument of a spectral product whose imaginary part enters the objective.This defines the applicable regime for explicit complex log-volume objectives.
VI. SCOPE AND APPLICABILITY: A survey identifies normalizing flows,, Gaussian processes, determinantal point processes, and Gaussian graphical models,, as phase-blind objectives.Complex-valued networks likewise place their loss on output magnitude [14], making them phase-blind under Proposition 6.

VII. RELATED WORK · VIII. DISCUSSION AND LIMITATIONS

The certificate complements magnitude- and positivity-based spectral controls by targeting spectral phase, offering a cheap differentiable criterion for exterior-power well-posedness. Its claims are deliberately narrow: the method is determinantal rather than dynamical and has limited applicability.

VII. RELATED WORK: Unlike spectral normalization, Parseval and orthogonal networks, unitary or complex recurrent maps,,, and antisymmetric or Lipschitz recurrent networks,, the certificate constrains spectral phase rather than magnitude or norm preservation.The cited alternatives constrain singular values, fix |λ| = 1, or use skew-symmetric generators for norm preservation.
VII. RELATED WORK: The certificate is presented as a single, cheap, differentiable condition for exterior-power well-posedness, imported from complex-metric admissibility, [2], [3] and described as new to machine learning.The paper characterizes it as the middle member SP of an exact and cheap hierarchy.
VII. RELATED WORK: Log-determinant and determinant-positivity objectives in Gaussian processes, determinantal point processes,, and normalizing flows,, are phase-blind, making the certificate complementary rather than competing.The paper attributes this distinction to Proposition 6.
VIII. DISCUSSION AND LIMITATIONS: The KSW criterion ports faithfully to a spectral-phase certificate that is provably distinct from existing magnitude and positivity constraints.This establishes the certificate as a separate constraint family rather than a reformulation of prior controls.
VIII. DISCUSSION AND LIMITATIONS: The certificate heads an exact, cheap hierarchy whose middle member SP is identified as the correct certificate for exterior-power well-posedness.The hierarchy claim and SP’s role are stated as central conclusions of the discussion.
VIII. DISCUSSION AND LIMITATIONS: The method’s scope is determinantal rather than dynamical, and the paper emphasizes that applicability is narrow but precisely delimited.The discussion presents this narrow applicability as a boundary on the present claims.