Unverified
2026
Represent each feature as belonging to one of three \(\mathbb{Z}_3\) charge sectors and constrain every linear and multiplicative operation to obey charge addition modulo 3. Add invariant cubic gates such as \(x_1x_2x_3\) or \(x_q^3\), which can express the same phase-insensitive interaction selected by the paper's three-photon drive. This should improve data efficiency and exact cyclic-augmentation consistency when the task has a genuine ternary symmetry.
Useful5/10
Difficulty4/10
Novelty6/10
Unverified
2026
Replace a standard two-layer multiplicative interaction block with auxiliary positive features X whose neighboring products generate two coupled feature grids x and y. Add the Y-diamond recurrence as either a hard recurrent update or a differentiable consistency loss, forcing local interactions to obey the same compatibility structure as an SL2/Y-frieze.
Useful5/10
Difficulty5/10
Novelty7/10
Unverified
2026
Use the paper's explicit tree support pattern as a cheap certificate that a sparse neural linear map contains a nearly singular submatrix. During mask construction or rewiring, penalize root-row-child configurations with many disjoint child branches, or increase overlap and row degree locally when such a configuration is detected. The goal is to prevent sparse MLP, projection, or MoE expert matrices from developing directions that are almost annihilated by the layer.
Useful5/10
Difficulty6/10
Novelty8/10
Unverified
2026
Replace an unconstrained order-k weight tensor with a sum of components that are symmetric only within selected contiguous index blocks. This preserves interactions between blocks while tying parameters under within-block permutations, providing a tunable middle ground between a fully dense tensor and a fully symmetric tensor.
Useful5/10
Difficulty5/10
Novelty7/10
Unverified
2026
Add a fractional oscillation penalty to scalar functions produced by a neural network on an ordered grid. Unlike a derivative penalty, this remains meaningful for nonsmooth or nowhere-differentiable outputs and interpolates between total-variation-like behavior and Sobolev-like smoothness.
Useful5/10
Difficulty3/10
Novelty6/10
Unverified
2026
Insert a rearrangement operation on scalar feature maps sampled along an ordered coordinate such as time, spatial position, or a neural-field input grid. The operation sorts values into non-increasing order, preserving the empirical histogram exactly while provably not increasing the Riesz fractional variation in the ideal one-dimensional continuous setting.
Useful5/10
Difficulty5/10
Novelty7/10
Unverified
2026
Split the trainable state into an explicit scalar scale coordinate and a residual perturbation, then update them with separate time scales. Penalize residuals according to their distance from the scale-dependent core, so the optimizer cannot obtain apparent progress by destabilizing the scale mode. The method is a neural optimization analogue of the paper's modulation argument, not a direct consequence of the geometric singularity theorem.
Useful5/10
Difficulty5/10
Novelty7/10
Unverified
2026
Use the logarithmic exhaustion as a geometry for bounded hidden representations rather than only as a parameter constraint. A representation approaching the boundary receives an increasingly large metric, making ordinary Euclidean motion expensive and discouraging brittle saturation while preserving a bounded intrinsic gradient for the boundary coordinate.
Useful5/10
Difficulty6/10
Novelty7/10
Unverified
2026
For two monotone prediction heads receiving binary features, penalize cases where their covariance is smaller than the sharp degree-weighted collision of their Fourier spectra. This discourages uncontrolled agreement on high-order interaction patterns while preserving low-order shared structure, and can be used either as a constraint or as a diagnostic for monotone multi-task models.
Useful5/10
Difficulty5/10
Novelty7/10
Unverified
2026
Parameterize candidate eigenfunctions with a neural network, project them into a finite spectral trial space, and compute Ritz eigenvalues from the resulting Galerkin matrices. Train against the paper's rigorous lower-bound transform rather than trusting the raw Ritz values, producing a certificate that the predicted eigenvalues do not underestimate the exact eigenvalues under the projection-error assumptions.
Useful5/10
Difficulty5/10
Novelty8/10
Unverified
2026
Add an inverse-capacitary-distance penalty to coordinate-network outputs near complex forbidden sets, rather than using only Euclidean distance-to-boundary weighting. The penalty is theoretically compatible with the network's spatial Dirichlet energy: it suppresses large values near obstacles while the gradient penalty controls the weighted singularity, even when the obstacle is thin, perforated, or fractal-like.
Useful5/10
Difficulty6/10
Novelty8/10
Unverified
2026
Add a bank of quadratic features encoding tangent contact with the reciprocal manifold x1 x2 = 1, rather than forcing a generic MLP to discover this interaction from arbitrary monomials. For positive bounded feature pairs, each feature is nonnegative and becomes exactly zero at a selected reciprocal operating point. The module can be used either as an input feature expansion or as a regularizer encouraging learned gates and scales to follow a reciprocal geometry.
Useful5/10
Difficulty3/10
Novelty7/10
Unverified
2026
Replace ordinary summation of several matrix-valued residual branches by a concave spectral aggregation: form the branch sum, take its absolute value, and apply a nonnegative concave function to singular values. The paper's transfer theorem predicts that the sharp Schatten-norm amplification constant is no worse than the corresponding linear Lee-type constant, while square-root, logarithmic, and capped maps suppress dominant singular directions.
Useful5/10
Difficulty6/10
Novelty8/10
Unverified
2026
For a complex-valued neural predictor, penalize violations of positive semidefiniteness of the Nevanlinna-Pick matrix on minibatch inputs. Unlike pointwise output clipping, this couples all examples and directly enforces compatibility with a bounded analytic interpolant of prescribed norm $M$.
Useful5/10
Difficulty4/10
Novelty7/10
Unverified
2026
Replace an ordinary input-convex potential with a potential whose Hessian is encouraged to be symmetric positive definite and symplectic. Add a curvature penalty based on the scalar curvature of the Hessian metric, together with a theorem-derived interior target proportional to the inverse squared distance to the domain boundary. This should suppress pathological third-derivative oscillations while preserving nonquadratic structure near boundaries.
Useful5/10
Difficulty7/10
Novelty8/10
Unverified
2026
Parameterize a nonnegative neural penalty or energy function as a sum of weighted power-mean differences applied to polynomial features of the network representation. Each atom is globally nonnegative by the power-mean inequality, so the learned penalty cannot become negative or destabilize constrained training, while the cone can represent polynomials outside SOS-plus-nonnegative-circuit certificates.
Useful5/10
Difficulty5/10
Novelty8/10
Unverified
2026
Add a two-output anti-collapse regularizer based on the determinant of the Jacobian Gram matrix, together with a penalty against proportional highest-degree coefficient tensors. The paper's inequality predicts that preserving coefficient non-proportionality prevents the output distribution from concentrating on thin curves or tiny regions, potentially improving coverage of a two-dimensional latent or generative output.
Useful4/10
Difficulty5/10
Novelty6/10
Unverified
2026
Tie neural parameters across feature channels according to the Schur multiplicity pattern of the fermionic coinvariant representation. In the two-fermion case, use one learned parameter block for each Schur degree instead of independently parameterizing every ordered pair of fermionic channels, eliminating redundant copies while preserving the relevant GL2 channel symmetry.
Useful4/10
Difficulty5/10
Novelty7/10
Unverified
2026
Replace a monomial polynomial feature block by a fixed Narayana basis transformation. If the input polynomial has nonnegative coefficients and only real roots, the transformed polynomial is guaranteed to have only real roots as well, giving a certified stability-preserving coordinate change for polynomial neural networks.
Useful4/10
Difficulty5/10
Novelty9/10
Unverified
2026
Insert a fixed polynomial mixer before an MLP or retrieval index for tuples of discrete features. The mixer maps n+2 bounded scalar feature codes to one or several expanded scalar codes, and the paper's theorem guarantees that its image cannot collapse below order |A|^n when all coordinates come from a finite alphabet A. Use multiple independent permutations or coefficient choices to obtain a vector representation while retaining the deterministic algebraic structure.
Useful4/10
Difficulty4/10
Novelty7/10