arXiv subjects

Meher Chaitanya

Publications and source records attributed to Meher Chaitanya.

5 recordsLinked to original sources

arXiv2026

Does Muon Need Fine-Grained Spectral Shaping?

Muon combines current and past gradients into matrix momentum. For $M=UΣV^\top$, the idealized polar update $Q=UV^\top$ gives every singular direction the same weight. We refer to this as the flat profile. Several recent optimizers replace this flat profile with fine-grained spectral maps that give each direction its own gain. We ask how much of this spectral detail a Muon update needs. Our spectral diagnostics show that approximately $94$--$97\%$ of measured singular modes lie below an estimated noise edge, yet collectively align positively with a reference gradient. We introduce BulkBoost, a two-band spectral reweighting framework with fixed-rank and noise-calibrated variants. The latter uses split-minibatch gradient differences to calibrate a Marchenko--Pastur reference edge for Muon's Nesterov input, separating the bulk below the edge from the spikes above it. Both variants increase the bulk's relative weight through one shared gain while preserving the Frobenius norm of each matrix's unreweighted direction. For a fixed partition, our theory gives the first-order condition under which moving weight toward the bulk lowers the loss. It also quantifies the fraction of the maximal first-order improvement rate, over all per-mode reallocations, that two bands can capture. Across 30 continued-pretraining settings spanning Pythia-14M to 410M and six corpora, two-band reweighting is competitive with the fine-grained power-law profile of Freon and outperforms Spectra. Measured against Muon's flat profile, Freon reduces final loss by $0.022\%$ of the pre-adaptation loss on average, whereas the two-band variants achieve reductions of $0.073$--$0.147\%$. These observations suggest that useful departures from the flat profile are surprisingly low-dimensional: a single bulk-to-spike gain captures at least as much benefit as the fine-grained spectral profiles.

cs.AI↗

arXiv2026

CAST: Canonical Approximate Schur Tree for Approximate Cholesky on Graphs

Graph-data workloads such as diffusion estimation, ranking, semi-supervised learning, and network optimization often solve many Laplacian or symmetric diagonally dominant M-matrix (SDDM) systems with the same coefficient matrix. Approximate Cholesky preconditioners eliminate vertices one at a time and store the resulting sparse approximate factorization, the \emph{factor}, whose construction cost is amortized across these solves. But eliminating a vertex, the \emph{pivot}, creates a dense Schur-complement clique among its $d$ active neighbors. We introduce CAST (Canonical Approximate Schur Tree), which replaces this clique with a weighted random spanning tree sampled directly from it. Every realization is connected and contains exactly d-1 edges, while reweighting each selected edge by the reciprocal of its tree-inclusion probability makes the update unbiased. The distribution is independent of the ordering of the pivot neighbors, and we prove that its leverage-score marginals minimize the largest normalized reweighted-edge contribution among unbiased inverse-marginal one-tree estimators. We also introduce CAST-$ρ$, which replaces each pivot neighbor with $ρ$ copies, each carrying a $1/ρ$ share of that neighbor's incident weight, samples a weighted random spanning tree on the expanded clique, and contracts the copies back to the original neighborhood. The resulting update remains unbiased and connected, can be sampled exactly in $O(ρd)$ time, and satisfies a $1/ρ$ bound on the second moment of the normalized local Schur error. Increasing $ρ$ therefore reduces certified local sampling variability, but may increase construction cost and downstream fill. Empirically, we observe that CAST-1 is the faster default, whereas CAST-2 is preferable when its additional edge contributions remain inexpensive.

stat.ML↗

arXiv2026

Thresholded Local Hyper-Flow Diffusion

Local Hyper-Flow Diffusion (HFD) gives an edge-size-independent Cheeger-type guarantee for seeded clustering in general submodular hypergraphs, but existing HFD solvers do not keep intermediate computation local at every iteration. We introduce Thresholded Local HFD (TL-HFD), a first-order method that maintains an active region around the seeds, performs projected subgradient updates on that region and its immediate boundary, and expands via thresholded (top-k) boundary activation. We prove that the local update is exact: the degree-preconditioned projected subgradient step restricted to the active region and its boundary coincides with the unrestricted global update. We establish finite-time dual suboptimality for both exact and thresholded updates, treating the latter as inexact projected subgradient steps with explicit skipped-boundary error. We further derive an additive activated-volume bound controlled by realized local subgradient norms and the minimum boundary-push among newly activated vertices, and translate approximate dual optimality with localized support into a robust sweep-cut guarantee for early-stopped iterates. For general submodular cut-costs, each iteration is local in the scanned region and oracle-sensitive in the hyperedge primitive. Empirically, TL-HFD often matches or improves over HFD while activating less volume, with the largest gains on noisy instances where diffusion tends to absorb non-target vertices.

cs.LG↗

arXiv2026

Graph Cascades: Contagion-Based Mesoscopic Rewiring for Structure-Aware Graph Machine Learning

We introduce Graph Cascades, a mesoscopic rewiring strategy for Graph Neural Networks (GNNs) and Graph Transformers (GTs) that captures intermediate-scale graph structure beyond purely local edges or fully global attention. Using contagion-based diffusion processes, Graph Cascades constructs, in O(|V|+|E|) time, an auxiliary graph where node pairs supported by repeated multi-hop reinforcement are promoted to direct neighbors. We theoretically characterize when reinforcement-based rewiring helps: sufficient conditions under which reinforcement-based edge selection is more label-aligned than direct adjacency, an SBM witness in which two-hop reinforcement is perfectly homophilic, and a formalization of mesoscopic connectivity via graph effective resistance. Empirically, across node-classification benchmarks, Graph Cascades improves multiple GNN and sparse-GT backbones, with the most reliable gains observed on heterophilic and moderate- to high-degree homophilic graphs. The theoretical conditions also identify regimes where mesoscopic rewiring is unlikely to be beneficial -- low-degree regular graphs and graphs with structural bottlenecks -- and these predictions match the observed failures. We additionally observe tight correlations between performance and structural properties in the rewired graphs.

cs.LG↗

arXiv2023

Tight Sampling in Unbounded Networks

The default approach to deal with the enormous size and limited accessibility of many Web and social media networks is to sample one or more subnetworks from a conceptually unbounded unknown network. Clearly, the extracted subnetworks will crucially depend on the sampling scheme. Motivated by studies of homophily and opinion formation, we propose a variant of snowball sampling designed to prioritize inclusion of entire cohesive communities rather than any kind of representativeness, breadth, or depth of coverage. The method is illustrated on a concrete example, and experiments on synthetic networks suggest that it behaves as desired.

cs.SI↗