Every idea extracted from recent arXiv mathematics papers — verified and unverified. Click an idea to open its full card; badges show the empirical verdict.
Use an online estimate of the positive feedback gain among logits, routing probabilities, and representations to adjust the softmax temperature. Increase temperature when the estimated cyclic gain approaches the instability regime, preventing exponential amplification and router collapse without globally weakening all layers.
Replace unconstrained residual updates with blocks whose Jacobian is monitored through a Davis–Wielandt shell. The shell simultaneously measures directional dissipation and non-normal amplification, yielding a per-block step-size or residual-scale bound that is stronger than checking only the largest eigenvalue or spectral norm.
Compress each hidden layer by retaining directions that are simultaneously reachable from the observed input distribution and observable at the network output. Unlike PCA or SVD, the retained subspace is weighted by downstream task sensitivity, so high-variance but output-irrelevant directions can be removed while low-variance predictive directions are preserved.
Initialize each row of a neural weight matrix as a stationary correlated Gaussian process instead of using independent entries, but constrain its correlation tail to remain on the finite-fourth-moment side of the transition. This creates controllable structured spectra while avoiding the heavy-edge regime predicted for correlations slower than \(t^{-1/2}\).
Insert a fixed DPSS/prolate projection before an expensive neural block, retaining exactly the modes whose time-frequency concentration eigenvalues exceed a target threshold. Use the paper's tail-quantile formula to choose the projection rank from sequence length, effective bandwidth, and tolerated energy loss, then optionally learn a small correction in the retained coordinates. Unlike a Fourier truncation, the basis is optimized for simultaneous localization in the finite input window and the…
Use the evolving singular spectrum of the represented matrix W_t=U_tV_t^{\top} to modulate one common, gauge-equivariant learning rate. Slow the shared update when spectral mass accumulates outside the intended low-rank subspace, preventing adaptive dynamics from amplifying nuisance tail directions while retaining the shared-rate structure needed for low-rank recovery.
Construct a robust covariance estimate of layer activations by replacing each feature with its empirical Gaussian normal score before eigendecomposition, then applying coordinate-wise nonlinear eigenvalue shrinkage rather than multiplying all eigenvalues by one scalar. Use the cleaned covariance to whiten activations or precondition updates to the associated linear layer. This targets unstable directions caused by small batches, heavy-tailed activations, and rare outliers while retaining…
Regularize the local recurrent Jacobian by its spectral radius rather than imposing the overly conservative operator-norm condition $\|J\|_2<1$. This permits useful non-normal updates with transient amplification while explicitly pushing the asymptotic dynamics toward a stable fixed point.
Regularize the end-to-end Jacobian singular-value distribution of a deep network toward the explicit free small-loss law generated by independently mixed projection-like layers. The target controls several gradient-spectrum moments, including the predicted fraction of nearly preserved directions, instead of controlling only the average gradient norm.