Solves: Scalability

Machine-learning ideas tagged Scalability in the Solves taxonomy of the Math2NN corpus.

Unverified 2026

Finite-Difference Conditional Neural Operator

Replace a tensor-product network over a low-dimensional state and a large distribution embedding with a neural operator that consumes the distribution vector once and outputs values on a finite-difference grid in the low-dimensional state. Train it with the governing PDE residual, explicit boundary residuals, and optional signed shape constraints, allowing the network to preserve numerical structure that a generic MLP would learn only implicitly.

Useful6/10
Difficulty5/10
Novelty5/10
Paper: Mastering Stochastic OLG Models in Continuous Time arXiv:2608.11134
Unverified 2026

Reversal-Assisted MoE Routing

Replace one-shot top-k expert assignment with a capacity-constrained stochastic routing process in which tokens have a temporary routing direction and can reverse it at rate gamma. Tokens preferentially move through short vacancy clusters, while reversals break persistent directed congestion and should delay or eliminate expert-level jams. This creates a tunable routing phase diagram rather than relying only on an auxiliary load-balancing loss.

Useful6/10
Difficulty6/10
Novelty8/10
Paper: Jamming transition in an active exclusion process arXiv:2608.11041
Unverified 2026

Thresholded Hidden-State Restart Gate

Replace an unconstrained recurrent reset gate with a threshold policy over hidden-state age and a scalar degradation score. The model continues its recurrence while the estimated cost of retaining the state is below the cost of restarting, then resets and reinitializes the state when the threshold is crossed. This should reduce long-horizon hidden-state drift while using only one scalar gate per sequence position.

Useful6/10
Difficulty4/10
Novelty7/10
Paper: Threshold Structure of Optimal Policies in Restart POMDPs arXiv:2608.10936
Unverified 2026

Successive Orthogonal Innovation Blocks

Add a neural feature, adapter, or expert block only through the component of its outputs that is orthogonal to the span of all previously installed blocks. Quotient coefficient directions that produce nearly identical outputs with an SVD or pseudoinverse, so the new block contributes intrinsic representational dimensions instead of duplicating old features. The expected benefit is a smaller effective architecture and better-conditioned block expansion.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: Successive Schur-Riesz Analysis for Approximation arXiv:2608.10757
Unverified 2026

Unit-circle-root Toeplitz mixer

Replace a freely learned finite impulse-response mixing kernel with a matrix polynomial whose roots are constrained to the unit circle. The resulting block-Toeplitz operator has an explicitly positive semidefinite spectral construction, while increasing the polynomial degree gives a systematic capacity knob for approximating matrix-valued frequency responses.

Useful6/10
Difficulty6/10
Novelty7/10
Paper: Pure matrix states on block Toeplitz matrices arXiv:2608.10701
Unverified 2026

Tree-Hall Collision-Free Router

Replace independent top-k routing by a tree-structured hypergraph assignment layer. Each candidate route is a singleton or pair of resources, and the router selects exactly q_e routes for every tree edge e while ensuring that no resource is consumed twice. This removes capacity collisions before expert computation instead of repairing them with token dropping or load-balancing penalties.

Useful6/10
Difficulty7/10
Novelty7/10
Paper: A Necessary and Sufficient Hall Condition for Hypergraphs arXiv:2608.10193
Unverified 2026

Spectral-Guided Discounted Decentralized Optimizer

Replace a stationary federated optimizer with a decentralized optimizer whose target distribution explicitly forgets old streaming samples. Each round performs only K consensus-gradient iterations, with K selected from the mixing contraction so that the communication budget matches the temporal volatility of the objective. The method should react faster to distribution shifts while limiting disagreement and bias caused by heterogeneous clients.

Useful6/10
Difficulty5/10
Novelty5/10
Paper: Distributed Optimization with Streaming Data: A Temporal Weighting Perspective arXiv:2608.09565
Unverified 2026

Erdelyi-Kober Log-Scale Mixer

Replace generic cross-scale mixing with a fixed-shape or lightly parameterized Erdelyi-Kober fractional convolution over logarithmic scale. The fractional order controls how strongly nearby scales are emphasized, while the exponential tail parameter controls the receptive field over distant scales, providing an interpretable alternative to dense cross-scale attention.

Useful6/10
Difficulty4/10
Novelty8/10
Paper: Boundedness of Erdélyi--Kober Integrals and Mellin Fractional Integrals on Weighted Lebesgue Spaces arXiv:2608.09401
Unverified 2026

Single-Node Observable Leaky-RNN

Construct a sparse recurrent network with positive edge weights and Leaky-ReLU updates so that one selected hidden node, observed over a finite time window, contains enough information to reconstruct the full hidden state. Add an auxiliary decoder from the observed trajectory to the initial state or current state, and use graph rewiring or edge-growth until every hidden node has a directed path to the sensor within the observation horizon.

Useful6/10
Difficulty5/10
Novelty8/10
Paper: On the Observability and Controllability of Leaky-ReLU Networks arXiv:2608.09059
Unverified 2026

Balanced KL projection for MoE routing

Use the shared-marginal KL projection to turn token-to-expert routing into a low-rank, exactly balanced assignment rather than relying only on an auxiliary load-balancing penalty. Tokens retain normalized routing distributions while the shared latent marginal enforces consistent aggregate usage across two independently learned routing factors.

Useful6/10
Difficulty6/10
Novelty6/10
Paper: Exact Rank-Space KL Projection for Shared-Marginal Low-Rank Factors: Application to Doubly Stochastic Clustering arXiv:2608.08642
Unverified 2026

Vector-Balanced MoE Routing

Replace count-only MoE load balancing with greedy balancing of aggregate token-feature vectors. A token is assigned to the expert for which adding its feature vector produces the smallest increase in that expert's squared aggregate norm, encouraging experts to receive complementary semantic mixtures rather than identical token counts.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: Max-$k$-Cut via Node Features arXiv:2608.08499
Unverified 2026

Minimax-balanced progressive MoE splitting

Grow a mixture-of-experts layer by splitting one expert into two children while conserving its routing mass, and choose the split ratio to minimize the worst imbalance over all intermediate expert counts. Use the paper's sharp threshold as a hard design target: with n experts, some intermediate stage must have capacity ratio at least D_n = 2^{1-1/\lceil n/2\rceil}, so schedules substantially below this are impossible rather than merely difficult to discover. Initialize child router logits with…

Useful6/10
Difficulty5/10
Novelty8/10
Paper: Optimal Finite Interval Discrepancy via Binary Refinement arXiv:2608.08431
Unverified 2026

Granularity-Aware Feasible Routing

Replace a continuous allocation or routing decision with a lattice-valued decision whose unit size is explicitly normalized by total capacity. Round allocations downward rather than to the nearest lattice point, preserving per-example capacity feasibility, and train or evaluate against the resulting granularity ratio rather than treating discretization as an implementation detail.

Useful6/10
Difficulty5/10
Novelty7/10
Paper: Bid Lattices and the Value of Flexibility:A Granularity Ratio for Capacity Markets arXiv:2608.08371
Unverified 2026

Two-query SPSA Langevin sampler for black-box neural energies

Replace backpropagated gradients in a Langevin sampler with a simultaneous-perturbation estimate obtained from two noisy evaluations of a neural energy or simulator-defined negative log-density. This enables posterior or latent-space sampling when the energy contains nondifferentiable code, stochastic simulation, discrete operations, or an inaccessible neural-network gradient.

Useful6/10
Difficulty5/10
Novelty8/10
Paper: Zeroth-Order Langevin Monte Carlo via SPSA under Noisy Function Measurements arXiv:2608.07837
Unverified 2026

Minimal-Channel Harmonic Controller

Use a low-rank controller that observes and actuates only the graph's harmonic coordinates rather than all edge features. For a graph with first Betti number beta_1 = dim ker(B), a beta_1-dimensional cycle basis is sufficient to represent the entire harmonic sector, yielding a compact recurrent memory or adapter for circulation-dependent graph dynamics.

Useful6/10
Difficulty6/10
Novelty8/10
Paper: From a Scalar Parabolic Oscillator to Topological Thermostats: Selective Feeback Control of Harmonic Flow Modes arXiv:2608.07768
Unverified 2026

Parallel Phase Oscillator SSM

Replace real diagonal state-space channels with complex damped oscillators whose hidden states encode both amplitude and phase. Train with parallel causal convolution and deploy with the equivalent one-step recurrence, allowing the same layer to support efficient batched training and low-memory streaming inference.

Useful6/10
Difficulty5/10
Novelty4/10
Paper: Phase State Space Models: Parallel, Surrogate-Free Training of Spiking Networks arXiv:2608.07754
Unverified 2026

Dynamic Hyperedge Token Mixer

Replace dense token-to-token attention in selected layers with communication through a small number of multi-token hyperedges. Each hyperedge aggregates its incident token states and broadcasts the resulting message back to those tokens, allowing higher-order interactions while reducing the number of pairwise links. Reconstruct hyperedges periodically from cumulative token displacement so stable tokens retain useful groups while rapidly changing tokens are regrouped.

Useful6/10
Difficulty6/10
Novelty5/10
Paper: HPSO: Particle Swarm Optimization with Hypergraph-Based Topology arXiv:2608.07587
Unverified 2026

Wasserstein-Gated Cached Equilibrium Adapter

Add a small constrained equilibrium layer whose response depends on the current neural state and recent exogenous history, then cache responses keyed by a learned history embedding. For a new history, reuse a cached response only when an empirical Wasserstein distance to the cached history is below a threshold; otherwise run a few inner optimization iterations. The paper's local Holder and trajectory-stability results motivate graceful degradation rather than catastrophic errors for nearby…

Useful6/10
Difficulty5/10
Novelty6/10
Paper: Stability of Differential Stochastic Variational Inequalities with History-Dependent Responses and Transfer Learning arXiv:2608.06923
Unverified 2026

Submodular Max-Delay Batch Timer

Replace a fixed maximum-wait timeout in dynamic neural inference batching with a service-aware timer. The timer estimates the marginal reduction in computation or communication cost from adding another request and delays service only while that saving justifies the additional oldest-request delay.

Useful6/10
Difficulty4/10
Novelty6/10
Paper: Online Multi-Level Aggregation with Per-Batch Maximum Delay arXiv:2608.06796
Unverified 2026

Buffered partition-of-unity gating

Replace an unconstrained spatial gate or interpolation kernel by a compactly supported function whose translates under a lattice exactly sum to one. Impose zero products between translates under a second lattice, so active gates do not collide; thresholding a positive superlevel set then provides a nonzero separation margin and predictable sparse computation.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: Tilings, packings, and the existence of Schwartz-class Gabor windows arXiv:2608.06679
Unverified 2026

Spectral-gap-aware randomized synchronization

Replace fixed-period federated averaging or distributed all-reduce with a Bernoulli communication decision whose probability is selected from estimated network connectivity and optimization conditioning. Local workers continue making corrected updates between communication events, while the contraction theorem exposes when communication is worth its cost.

Useful6/10
Difficulty5/10
Novelty6/10
Paper: Theoretical Foundations of Communication-Efficient, Robust, and Practical Distributed and Federated Optimization arXiv:2608.06563
Unverified 2026

Random-Batch Interaction Layer

Replace an all-pairs interaction or attention-like message field with an unbiased estimator obtained from a uniformly random partition into batches. Keep the partition fixed for \(\tau\) integration steps, then redraw it; the paper's error law predicts that too-long reuse causes a quadratic-in-\(\tau\) error accumulation, while finite-population error decreases as \(N^{-1}\).

Useful6/10
Difficulty4/10
Novelty5/10
Paper: Mean field error estimate of the random batch method for vortex blob dynamics for the 2D Navier--Stokes Equation arXiv:2608.06533
Unverified 2026

First-Passage Budgeted Adaptive Computation

Represent stochastic layer execution, branching, retries, and early exit as a finite continuous-time Markov chain, with the completed-prediction state absorbing. Learn transition rates jointly with neural-network weights, but use MFPT sensitivities to allocate rate changes according to their available control budget rather than allowing one routing edge to dominate halting-time control.

Useful6/10
Difficulty6/10
Novelty7/10
Paper: A Universal Control Budget for First-Passage Kinetics arXiv:2608.06368
Unverified 2026

Intersection Euler Interaction Token

Compute a compact multiscale interaction signature between colored point clouds and append it to a point-cloud or multimodal neural network as a learned interaction token. The signature captures separated, overlapping, and higher-order enclosing configurations while remaining invariant to rigid transformations.

Useful6/10
Difficulty5/10
Novelty7/10
Paper: The Intersection Euler Characteristic Profile: Euler Calculus and Stability for Topological Interaction of Ball Unions arXiv:2608.06180