Attention variants, state-space and recurrent cells, normalization and token-mixing schemes — each tested against the standard block it replaces.
Replace a recurrent token-mixing operation with a finite-state carrier scan over binary or quantized token features. The local transition table is constrained to conserve a scalar token weight and to be bijective, making the mixer reversible, constant-memory, and less prone to activation drift than a generic recurrent layer.
Represent each k-element object by a vector in dimension \(r=\binom{n-2(k-s)}{s}\), and use a PSD Gram matrix to encode the rule that pairs with intersection smaller than s have zero similarity while pairs with intersection at least s have nonzero similarity. Insert this representation into set encoders, graph neural networks, or overlap-aware attention instead of allocating one feature for every s-subset.
Replace a dense recurrent transition matrix with a periodic CMV-style product of alternating local 2x2 unitary cores. The transition is exactly norm-preserving, has O(n) trainable parameters under periodic tying, and can be applied through local factor operations rather than stored as an n-by-n matrix. Use turnover refactorization when changing the ordering or boundary connection of cores, enabling a compact cyclic unitary state-space layer.
Replace a dense weight matrix by a cross approximation built from selected rows and columns rather than by a conventional truncated SVD. Periodically refresh the selected indices using residual leverage scores, warm-starting from the previous factorization so that the compressed layer can track weight changes during fine-tuning.
Replace a generic recurrent transition with two coupled unitary transitions that share one block column and differ by a sign on the other block column. Each transition preserves hidden-state norm exactly, while the structured difference gives a controlled two-path recurrent architecture for long-context modeling.
Construct a sparse attention or message-passing graph as a gain graph, where every directed edge carries a relative frame label and reverse edges carry the inverse label. Grow the graph using the paper's 2-extension operation: replace two old edges by a new vertex connected to their four endpoints, while preserving the relative gain products. The resulting mask is intended to preserve global information flow under controlled sparsity and to avoid isolated components and brittle bridges commonly…
Add a fixed or weakly parameterized residual mixer whose interaction between sequence positions at distance \(r\) is proportional to \(1/(r\log^2 r)\). Instead of truncating the kernel at a short radius, represent its heavy tail with dyadic distance bands and compute each band using prefix sums or block pooling, giving every token access to arbitrarily distant context at roughly \(O(L\log L)\) cost.
Represent every mesh interface degree of freedom by one feature copy per incident cell, and apply local neural blocks directly to these cell tensors. Enforce inter-cell consistency with valence-weighted averaging only after selected layers or hierarchy transitions, avoiding repeated construction of a global sparse graph or assembled feature vector. This is suited to adaptive quadtrees, octrees, and finite-element neural operators.
Use the expander decoder as a structured sparse-coding dictionary and replace dense OMP correlation steps with edge-wise gather-and-reduce operations. This is useful when codes must be inferred iteratively, including interpretable feature extraction, sparse retrieval, or an inference-time latent selector that cannot rely entirely on an amortized encoder.
Replace a dense channel or token-mixing matrix with a product of positive bidiagonal factors, so information propagates through a controlled sequence of local couplings rather than arbitrary signed interactions. Initialize the factors from the paper's barycentric-subdivision factorization, then learn positive diagonal and off-diagonal parameters; the resulting map is structured, parameter-efficient, and constrained to remain totally positive.
Regularize an intermediate neural representation according to its estimated low-dimensional separability capacity instead of its ambient feature width. Learn feature gates or subspace assignments, estimate the union of active supports, and penalize representations whose Cover capacity exceeds a task-dependent target.
Replace a discrete or one-hot recurrent state table with a low-dimensional vector memory whose event embeddings are orthogonal whenever the corresponding events are mutually exclusive in an input exclusivity graph. The module uses continuous state vectors and can therefore target dimension \(d=\xi(G)\), whereas a discrete state encoding is lower-bounded by \(N\geq\chi(G)\). This should be tested on graph-defined formal-language recognition tasks, where the graph is known and the claimed…
Replace dense attention between grid-arranged tokens by a deterministic block-sparse pattern generated from modular permutations. In each block, connect row token i to column token p(i)=2i modulo B; because i, i-p(i), and i+p(i) are injective modulo B when gcd(B,6)=1, the pattern avoids repeated horizontal, vertical, and diagonal projections. Use shifted permutations across heads to increase receptive-field coverage while retaining structured sparsity.
Construct a latent layer whose node states are small positive-definite matrices and whose local updates follow a weighted cluster exchange relation rather than an unconstrained affine map. The update is reversible when the old state is retained, while noncommuting matrix products preserve relational structure that scalar cluster variables cannot represent.
Use the attention probability distribution over an ordered context to choose contiguous token groups whose pooled attention masses have entropy as close as possible to a prescribed upper budget R. Replace the corresponding key/value vectors by one weighted representative per group, preserving token order and reducing the KV-cache length from n to m. Unlike unconstrained token merging, the entropy constraint gives a direct control knob over how concentrated or diffuse the retained attention…
Replace dense token-to-token attention by a learned binary relation generated from a small number of hierarchical predicates, while rejecting masks that contain a fixed K_{t,t} biclique. The paper's incidence bound predicts near-linear active edges for these structured relations, giving sparse attention with a measurable worst-case complexity target rather than relying only on average sparsity.
Replace dense grid tokens or global spectral features with coefficients of compactly supported kernels centered on a nested hierarchy of spatial points. Encode an input field into coarse-to-fine coefficients, apply a neural map to those coefficients, and decode the predicted coefficients at arbitrary query locations; the contribution from each level provides an explicit multiscale output decomposition.
Replace raw braid-generator sequences by sequences of positive simple Garside factors obtained from the left-greedy normal form. Because powers of \(\Delta\) lie in the Hilden subgroup, they can be removed while preserving the relevant double-coset presentation, reducing non-uniqueness and often shortening the sequence. Feed the resulting factor tokens to a Transformer or sequence classifier, and train it to be invariant to inserted removable \(\Delta\)-powers.
Represent candidate two-dimensional attention windows as dyadic rectangles and penalize local regions where many deeply embedded windows overlap. Use complementary horizontal and vertical depth exponents rather than independently penalizing one coordinate. The resulting router should reduce pathological concentration of sparse attention computation while preserving access to multiscale context.
Replace independently sampled sparse connectivity masks with an algebraically generated codebook in which the symmetric difference of any two distinct masks is connected and spanning on a bipartite layer graph. A d-bit state selects one of up to 2^d masks, enabling cheap mask switching without isolated units or disconnected communication components.
Prune higher-order tensor or polynomial interactions while preserving an exchange-closed support instead of independently retaining the largest weights. The resulting sparse interaction pattern avoids arbitrary holes and retains structured substitutions between coordinates, potentially improving parameter efficiency and robustness after pruning.
Replace a dense or ad hoc sparse attention pattern with a circulant mask generated by the paper's carry word c(m,k). Every query attends to exactly m of k relative positions, and the selected positions are prefix-balanced, avoiding the large gaps and collisions produced by random sparsification. Use several Farey-related slopes across heads or layers to combine local, medium-range, and long-range coverage.
Build a fixed sparse token-mixing layer on n=qr cyclic positions whose directed edges are all offsets 1 through r, but execute the edges through r Hamiltonian path scans plus one prescribed chain. The paths preserve exact coverage of the circulant connectivity while exposing long sequential traversals that can be fused into custom kernels, recurrent scans, or state-space updates.
Treat a finite shallow network as a discrete signed measure over ridge atoms and grow or prune neurons according to their contribution to total variation. This turns width selection into an atomic approximation procedure: add neurons correlated with the current residual and remove coefficients that consume budget without contributing materially.