Unverified
2026
Add a spectral regularizer to a linear state-space or recurrent layer that controls the overlap between its controllable and observable state directions. The regularizer uses the paper's identity to monitor eigenvalues of (I+PQ)^{-1}, equivalently the squared canonical correlations between reachable and observable subspaces, and penalizes degenerate or overly concentrated spectra.
Useful5/10
Difficulty5/10
Novelty6/10
Unverified
2026
For a coordinate network representing a field near a boundary or interface, factor the prediction as u(x)=h(x)v(x), where h is a known fractional-Hardy ground-state profile, and regularize v with a weighted nonlocal difference energy. Add the corresponding critical Hardy penalty to the loss so that the network spends capacity on the nonsingular residual v instead of relearning the boundary singularity.
Useful5/10
Difficulty5/10
Novelty7/10
Unverified
2026
Regularize a network using the weak-L^p tail of scale-normalized feature differences between an input and sampled perturbations, instead of averaging all pairwise differences with an ordinary L^p penalty. The weak norm emphasizes persistent high local sensitivities while being less dominated by a single extreme pair than a hard maximum.
Useful5/10
Difficulty4/10
Novelty6/10
Unverified
2026
Regularize learned low-dimensional embeddings or MoE prototypes with an aggregation-diffusion energy. The attractive term encourages compact, semantically coherent groups, while porous-medium diffusion creates density-dependent pressure that prevents points from collapsing into singular clusters.
Useful5/10
Difficulty5/10
Novelty6/10
Unverified
2026
Regularize the spatial curvature of a scalar-output image network using the paper's Burkholder integrand instead of an isotropic squared-Hessian norm. The energy is nonconvex pointwise but quasiconvex on symmetric Hessians, so compactly supported Hessian perturbations cannot lower the total energy relative to an affine field; this may suppress oscillatory curvature while allowing sharper anisotropic transitions than quadratic smoothing.
Useful5/10
Difficulty4/10
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
Use the hysteresis threshold as a regularizer for attractor diversity. Estimate how many initial states converge to each fixed point and select thresholds that maximize basin entropy or penalize domination by one attractor, reducing attractor collapse in discrete recurrent classifiers and memory modules.
Useful5/10
Difficulty4/10
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 spectral regularizer to a learned graph or sparse attention adjacency that penalizes violation of the paper's energy floor. The regularizer discourages adjacency matrices that retain many edges but collapse into a low-dimensional spectral structure, which may reduce graph-message-passing diversity and worsen oversmoothing.
Useful5/10
Difficulty5/10
Novelty5/10
Unverified
2026
Represent intermediate feature maps on a periodic rectangular grid and regularize each individual Fourier eigenspace so that its spatial energy cannot collapse almost entirely outside a chosen observation region. The target lower bound is derived from the paper's quantitative rectangular estimate and is applied only to narrow Fourier shells, where the feature map is analogous to a degenerate Laplacian eigenfunction.
Useful5/10
Difficulty5/10
Novelty7/10
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
For a neural scalar field defined on the vertices of a mesh or graph, generate several random level interfaces by adding continuous perturbations and thresholding the field. Penalize the deviation between the empirical mean Euler characteristic of these interfaces and the value predicted from the host complex's f-vector, encouraging decision boundaries with stable global topology.
Useful4/10
Difficulty6/10
Novelty7/10
Unverified
2026
Add a centro-affine Dirichlet penalty to a neural module whose inputs or outputs lie on a sphere, such as normalized embeddings or attention directions. The penalty measures intrinsic variation under an unconditional convex-body metric while projecting out the constant and coordinate-affine modes excluded by the theorem.
Useful4/10
Difficulty6/10
Novelty7/10
Unverified
2026
Add a structured boundary-like operation to an MoE router that rapidly mixes expert probabilities toward a learned distribution while preserving predefined expert groups. The operation is a rank-one stochastic kernel, so it costs linear rather than quadratic work in the number of experts and can act as a controlled reset when routing becomes concentrated.
Useful4/10
Difficulty3/10
Novelty6/10
Unverified
2026
Treat a spatial feature map or lattice-indexed embedding as a function on a d-dimensional discrete grid and penalize excessive concentration near a chosen anchor using the inverse-radial Hardy weight. Calibrate the penalty with the theorem's high-dimensional scaling 2^ell d^ell instead of selecting an arbitrary spatial L2 coefficient.
Useful4/10
Difficulty3/10
Novelty7/10
Unverified
2026
Convert an attention or MoE routing affinity matrix into a soft graph and constrain its K_r-density relative to its observed K_s-density. The regularizer penalizes pathological affinity patterns in which moderate s-way coactivation is accompanied by an implausibly low or unstable r-way coactivation.
Useful4/10
Difficulty5/10
Novelty7/10
Unverified
2026
Augment spatial training examples by replacing a compact active region with several separated components while preserving its exact value histogram, total active area, and amplitude. The augmentation probes the nonlinear interaction between diffusion-like receptive fields and threshold activations, which the paper shows can make fragmented and compact inputs evolve in opposite directions despite identical distributions.
Useful4/10
Difficulty4/10
Novelty7/10