Machine-learning ideas tagged Initialization in the ML taxonomy of the Math2NN corpus.
Replace Cox-de Boor evaluation of each cubic B-spline edge activation with its fixed truncated-power expansion. Normalize each scalar edge input to a bounded knot coordinate, evaluate the five shifted cubic positive-part terms in parallel, and contract them with the learned spline coefficients inside one fused kernel.
Replace penalty-based orthogonality training for an \(n\times k\) weight or feature matrix \(X\) with a projected spectral flow driven by a symmetric matrix \(A\), such as a minibatch covariance or task-derived curvature estimate. The update rotates the subspace toward the top or bottom eigenspaces while preserving \(X^{\top}X=I_k\) through QR or Cayley retraction, avoiding the ill-conditioning caused by large orthogonality penalties.
Use a bounded function of the response to form a supervised, label-weighted covariance of the input and initialize the first neural layer from its leading outlier eigenspace. For vector-valued responses, use a matrix-valued response preprocessing map so several label statistics are combined in one lifted spectral estimator.
Replace uniform collocation for a fixed random-feature neural PDE solver with sampling from the leverage-score density of the operator-applied features. Whiten the retained residual feature space before solving for output coefficients, so the sampled least-squares matrix has an identity-like expected Gram rather than inheriting severe anisotropy from the differential operator. The same construction can be used for a linearized neural network by treating Jacobian features as the trial functions.
Replace the usual isotropic Gaussian random Fourier features with a frequency distribution matched to the expected spectral regularity of the target function. For coordinate fields, operator-learning maps, or PDE solution surrogates, this should place more features where the target Fourier energy lies and improve approximation at the same feature count. Stabilize the resulting feature matrix with whitening or ridge regression because spectral accuracy can create severe ill-conditioning.
Use a spectral eigenvalue-counting function to bracket each target mode before neural optimization. The network then solves only within an interval containing exactly one eigenfrequency, preventing optimization from repeatedly collapsing to the lowest mode or jumping between modes.
Train two parameter replicas with symmetric coupling, treating one replica as a prepared thermalization packet for the other. Estimate the slow local Hessian direction and initialize or periodically reset the packet so that the coupled state has zero projection onto that mode; the target should then relax according to the next-slowest mode rather than the original bottleneck.
Constrain a recurrent interaction matrix to a directed cycle, or initialize it near a cyclic block structure, and certify that the intended unstable or oscillatory mode survives independent gain perturbations. The cyclic topology makes the full network characteristic equation exactly reducible to one scalar loop equation. A robustness penalty can then preserve long-horizon oscillations under quantization, dropout-like gain errors, pruning, or hardware variation.
Replace a dense Haar or Gaussian random projection with a streamed product of random two-coordinate rotations followed by coordinate subsampling. The transform is exactly orthogonal before subsampling, requires only a list of rotation triples, and the paper's pseudo-mixing result predicts that degree-two statistics relevant to norm preservation and Johnson–Lindenstrauss embeddings become Haar-like after only O(n polylog(n)) rotations.
Replace a conventional whitening transform with a constrained whitening layer that minimizes cross-channel covariance while requiring every output channel to remain correlated with its designated input channel by at least a threshold \(\rho_{\min}\). The layer exploits the orthogonal freedom in whitening to find a rotation that preserves channel identity instead of arbitrarily mixing features. It can be inserted before an MLP, convolution, or attention projection and compared directly against…
Initialize latent coordinate-frame parameters analytically from two temporally separated neural predictions instead of starting joint optimization from arbitrary translation and orientation. This removes the continuous gauge before backpropagation and should prevent EKF-like or gradient-based failures caused by large yaw and position initialization errors.
Choose an initialization that may have worse initial loss but has a smaller projection onto the slow modes of the subsequent training dynamics. Under the same optimizer, data order, and learning rate, this initialization should overtake a lower-loss baseline after a predictable crossing time, analogous to the paper's reversal of relaxation ordering.
Tune a recurrent neural reservoir to the operating regime where an input driver produces both a strong hidden-state response and a large discrepancy between driven and innate entropy-production rates. This replaces recurrent-gain selection based only on spectral radius with a measurable non-equilibrium screening criterion. The proposed score should peak near the gain that gives the best downstream prediction accuracy, while weakly driven and excessively unstable regimes should score poorly.
Construct the sparse transition matrix and input projection of a recurrent or state-space layer so that every hidden-state row is covered by a matching in the controllability core. This prevents hidden directions from becoming unreachable from the input sequence, especially in multi-input systems and across a distribution of transition matrices or task conditions.
For each frozen weight tensor, append a second tensor of identically shaped zero weights and assign trainable scores to both the real and dummy edges. Select a fixed number of candidates by top-k score in the doubled space; real edges selected by the competition remain active, while selected dummy edges consume the quota without changing the network. The resulting number of active original edges is learned rather than imposed by a separate layerwise sparsity search.
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…
Grow a neural network by appending a trainable block together with an analytically initialized inverse block, so the newly added depth is exactly the identity at insertion time. After insertion, untie and optimize the two blocks independently; this preserves the current function while providing additional trainable degrees of freedom. For architectures with one expensive mixing operation followed by cheap channelwise blocks, the same construction can increase depth without repeatedly paying for…
Replace an unconstrained recurrent matrix by a structured asymmetric circulant coupling whose Fourier modes have analytically known complex eigenvalues. A selected nonzero mode becomes a rotating attractor, providing a phase-coded recurrent state that can preserve information through oscillatory dynamics without requiring the optimizer to discover a stable spectral structure from scratch. A weak input projection and optional mode-selection loss can use the attractor as a nonlinear memory…
Construct a weight-tied transformer loop in which the recurrent state receives a bounded diagonal carry plus a learned block increment, rather than applying a residual identity inside the learned increment. Parameterize the carry so every channel is strictly below one, allowing many recurrent iterations without the state explosion observed with an unconstrained carry.
Initialize a recurrent or state-space transition matrix with weak Wigner noise plus a shared cumulative-sum correlation structure. Increasing the correlation strength produces recurrent eigenmodes one at a time at analytically predicted BBP thresholds, yielding a controllable hierarchy of short- and long-memory modes. The matrix should then be globally rescaled or constrained so that all active modes remain inside the desired stability radius.
Use the paper's finite-dimensional second-moment equations to compute the stationary covariance induced by a Markov-switched recurrent layer before training, then whiten or scale each mode's hidden state using that covariance. This can prevent mode-specific saturation and eliminate a long burn-in period in long-context RNNs and state-space models.
Replace unconstrained token-mixing logits by a symmetric zero-row-sum response matrix generated from positive conductances on a small auxiliary electrical network. The resulting mixer has conservation and positivity structure, while circular minors have a prescribed sign pattern associated with positive grove measurements. This is especially suitable for graph neural networks and attention variants that need stable global diffusion rather than arbitrary dense affinities.
Replace the unconstrained transition of a recurrent or state-space neural network with a DMDc-initialized linear latent transition plus a learned nonlinear residual. Estimate the transition from a short warm-up dataset using Hankel delay coordinates, retain eigenmodes with decay rates near the unit circle for long-term memory, and let the neural residual model dynamics not explained by the linear backbone. This should make long-horizon prediction and slowly varying signals easier to learn while…
Replace an unconstrained overcomplete linear measurement or embedding matrix by one trained to remain well-conditioned after deletion of a prescribed number of rows. The objective explicitly targets the smallest singular value over sampled row subsets, preventing a layer from relying on fragile combinations of features that disappear under channel, sensor, token, or measurement erasures.