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Linear Independence of Polynomial Compositions with Bounded Input Degree
Xiaotian Chang, Yacoub Hendi
TL;DR
The paper addresses whether general polynomial activations preserve linear independence for distinct, nonconstant inputs when the activation degree is sufficiently large [KMMM26, Conjecture 1.2]. It proves this for bounded input degree with a threshold depending only on the number of inputs, using bounded-degree relation loci and their scheme-theoretic images. The result also yields an identifiability theorem for bias-free polynomial neural networks.
Problem
For general polynomial activations of sufficiently large degree, it was open whether compositions of pairwise distinct, nonconstant inputs are linearly independent [KMMM26, Conjecture 1.2].
Source: §1 · p. 1Method
The proof studies bounded-degree relations through an incidence locus and its scheme-theoretic image in the space of composing polynomials.
Source: §2 · p. 4, §5 · p. 11Results
For each input-degree bound M, a nonempty open set of activations with r ≥ A_k gives linearly independent compositions for pairwise distinct, nonconstant inputs of degree at most M.
Source: §1 · p. 2Takeaways & Limitations
The theorem yields an identifiability result for bias-free polynomial neural networks with generic activations.
Source: Application to identifiability of PNNs · p. 3The open set may depend on M, and the proof does not establish one open set that works across unbounded input degrees.
Source: §5 · p. 12
Abstract
from arXiv · showhide
We prove the weak composition-independence conjectures of Kohn, Marchetti, Massarenti and Mella. The proof uses deformations of rational curves and a single kernel of the linearized equation. Riemann-Roch gives a contradiction from tangent fields with controlled poles. By settling these conjectures, we prove that a neural network with a general polynomial activation function of sufficiently large degree is identifiable.
1. Introduction
The paper proves weak composition-independence conjectures for general polynomial activations, with a degree threshold depending on the number of inputs rather than the input-degree bound. It also extends the result to bounded-joint-degree rational inputs and derives an identifiability result for bias-free polynomial neural networks.
1. Introduction
- The theorem addresses general activations, where inputs need only be distinct, unlike the pairwise-nonproportional condition for powers used in monomial-network identifiability [FRWY24] [Cra25].The prior bounded-input-degree results required the composition-degree lower bound to depend on the input-degree bound [KMMM26].Source: §1 · p. 1
- For r ≥ A_k, a general activation polynomial from a nonempty open set makes compositions of pairwise distinct, nonconstant inputs of degree at most M linearly independent.The theorem applies over characteristic-zero field extensions and also gives the origin-passing conclusion.Source: §1 · p. 2
- The open set may depend on M, but the lower bound A_k depends only on the number of inputs, proving the weak conjectures [KMMM26, Conjectures 2.6 and 3.2].The result does not assert a single open set across unbounded input degrees.Source: §1 · pp. 1–2
- The theorem extends to pairwise distinct, nonconstant rational inputs of joint degree at most M, over every field extension L/K as stated in Theorem 1.3.For rational inputs, joint degree measures the generated map rather than separately chosen numerator–denominator pairs.Source: §1 · p. 2
- The proof assumes a dense locus of bounded-degree relations and uses a dominating family to construct one kernel bundle whose Euler characteristic is controlled by Riemann–Roch.The bound is up to an effective divisor controlled by a scalar moment.Source: §1 · pp. 2–3
Application to identifiability of PNNs.
- For generic polynomial activations, a bias-free network’s weights are identifiable exactly when the network is not a subnetwork; the result also covers layer-specific activations meeting degree thresholds.The layer-specific condition is r_i ≥ A_2d_i for each activation.Source: Application to identifiability of PNNs · pp. 3–4
2. Families of Relations
The section models bounded-degree relations as a finite-type incidence, then studies a dominating family through deformations of a rational curve that lift variations of the composing polynomial.
2. Families of Relations
- A finite-type incidence parametrizes relations using distinct rational inputs of joint degree at most M and nonzero relation coefficients.Its image is the bad locus, and the reverse inclusion is obtained by completing shorter tuples with distinct translates.Source: §2 · p. 4
- A dominating family of relations yields a rational curve whose map to the incidence fiber lifts every composing-polynomial coefficient direction.Smoothness of the family projection at the chosen point makes its tangent map surjective; global sections then provide a splitting of the pulled-back tangent sequence.Source: §2 · pp. 5–6
- The incidence uses a basepoint-free homogeneous representation of the inputs, with the common degree D giving each input a pole divisor bounded by E.For polynomial inputs of maximum degree D, E is D[∞].Source: §2 · p. 6
3. The Linearized Equation
The linearized equation defines a kernel bundle of tangent directions compatible with the chosen scalar variations. A controlled failure of surjectivity bounds the divisor that enters the Riemann–Roch estimate.
3. The Linearized Equation
- The tangent sequence of the hypersurface pulls back to an exact sequence with quotient O_C(rD), while the scalar projection selects the subspace used to define B.Twisting the pulled-back Euler sequence and the kernel sequence by O_C(−1) is the setup for the Riemann–Roch calculation.Source: §3 · p. 7
- The kernel bundle B consists of tangent sections whose scalar components lie in the chosen subspace W, and its defining sequence maps to O_C(rD)(−∆).The restricted normal map can fail to be surjective; ∆ records that failure and makes B a vector bundle.Source: §3 · p. 7
- The failure divisor satisfies τ ≤ (s − 1)D, and it vanishes when the scalar-variation space has dimension s − 1.The bound follows from a nonzero scalar moment and the resulting divisor estimate; in the full scalar space, the restricted map is the full surjective normal map.Source: §3 · p. 8
4. Tangent Fields and Riemann-Roch
Explicit tangent fields span the zero-scalar tangent directions, and their pole bounds control the Riemann–Roch comparison. Projecting these fields into the kernel bundle yields a generically spanning family and a torsion cokernel.
4. Tangent Fields and Riemann-Roch
- On distinct affine coordinates, explicit fields span the tangent space to the incidence fiber, with no inverse of node differences appearing.The first families use weighted evaluations and Vandermonde identities; the remaining coefficient variation is handled by division by the monic node polynomial.Source: §4 · pp. 8–9
- The weighted-evaluation relation and Vandermonde argument establish spanning, while Riemann–Roch compares the resulting bundle sections after twisting by O_C(−1).The torsion cokernel's length is preserved by twisting.Source: §4 · pp. 9–10
- The fields have controlled pole bounds, and h selected fields have total pole bound at most sh + es.The resulting sequence has torsion cokernel; the selected fields together with the source differential form a generic basis of the bundle.Source: §4 · pp. 9–10
5. Proof of the Theorems
The proof makes the bad locus proper, yielding a nonempty open set of composing polynomials with the separation property. It transfers this result across characteristic-zero fields and to bounded-degree univariate and multivariate inputs, while retaining a distinction between a uniform degree threshold and a single open set for all input degrees.
5. Proof of the Theorems
- The two cases in the incidence argument are impossible, so the closure of the bad image is proper and defines a nonempty open set of composing polynomials.The incidence and its scheme-theoretic image are defined over Q.Source: §5 · p. 11
- For every field extension L/K, no relation occurs over the base-changed open set when the composing polynomial lies in Uk,r,M(K).A relation over L would give an L-point of an incidence that is empty over that open set.Source: §5 · p. 11
- Input translations preserve joint degree, distinctness, and nonconstancy, while translating a composing polynomial to a root places it in the origin-zero slice.The resulting intersection is a nonempty relative open with rational points over characteristic-zero coefficient fields.Source: §5 · pp. 11–12
- For univariate inputs of degrees at most M, the generated map has joint degree at most M; multivariate inputs reduce to univariate ones by restriction along a suitable line.The restriction preserves positive degrees and nonzero differences, so a relation among the original compositions would restrict to one among the univariate compositions.Source: §5 · p. 12
- The proof's degree threshold is independent of M, with r ≥ Ak meeting the stated hypotheses, but the open set is still allowed to depend on M.The sequence As is increasing and satisfies As ≥ s.Source: §5 · pp. 11–12
- The result does not bound the degree of a relation at an exceptional composing polynomial or provide one open set valid for all input degrees.Properness for each finite M does not imply properness of the closure of their union.Source: §5 · p. 12