arXiv subjects

Krishna Harish

Publications and source records attributed to Krishna Harish.

3 recordsLinked to original sources

arXiv2026

Optimal Training-Time Scaling in Gradual Adaptation

In gradual adaptation, how should the training time on each task change as the number of intermediate tasks increases? We study this question for overparameterized linear regression tasks that change smoothly and share a zero-loss solution. With $N$ tasks and training time $s_N$ on each, the final learning progress converges to a continuum curve when $Ns_N\to\tau$. The limiting progress is $\Theta(\tau)$ for small $\tau$ and $\Theta(\tau^{-1})$ for large $\tau$, so both very short and very long training produce little progress. It follows that optimal per-task training times scale as $s_N^\star=\Theta(N^{-1})$, equivalently $Ns_N^\star=\Theta(1)$. Experiments on gradually rotated MNIST and a natural Yearbook time shift are consistent with less per-task training as the path is divided more finely.

cs.LG↗

arXiv2026

Warm-Start Interior-Point Methods for Online Second-Order Cone Programming

We analyze the computational complexity of solving a sequence of related second-order cone programs (SOCPs) whose right-hand-side data $b_t$ varies between rounds. The standard primal-dual interior-point algorithm solves each round at cost $\tilde{O}(n^{2.5}\log(1/\epsilon))$ from a cold start. We show that when the per-round perturbation $\|b_t - b_{t-1}\|_2$ is bounded by a problem-specific threshold $\delta$, Newton's method warm-started at the previous round's solution $x^*_{t-1}$ converges to $x^*_t$ to accuracy $\epsilon$ in $O(\log\log(1/\epsilon))$ iterations. Over $T$ rounds the total cost is $\tilde{O}(n^{2.5}\log(1/\epsilon) + T n^2 \log\log(1/\epsilon))$, compared to $\tilde{O}(T n^{2.5}\log(1/\epsilon))$ for cold start at each round; the per-round speedup for large $T$ is $\Theta(\sqrt{n}\,\log(1/\epsilon)/\log\log(1/\epsilon))$. The argument combines an infinitesimal local-norm sensitivity bound on the central-path optimum, a self-concordant finite-difference corollary, and the standard quadratic-convergence basin of Newton's method on a self-concordant barrier. The local-norm formulation circumvents the rank-deficiency issues of Euclidean sensitivity bounds for fat constraint matrices. A multi-seed experiment on bounded SOCPs with $n=50$, $p=100$ confirms a 30-70x per-round speedup across the predicted regime.

cs.DS↗

arXiv2026

Equivariant Cellular Sheaves for Molecular Electronic Structure: Bridging Sheaf Cohomology and E(3)-Equivariant Hamiltonian Learning

Equivariant message-passing networks are the standard model for molecular property and interatomic-potential prediction, and recent work predicts the electronic Hamiltonian itself in an E(3)-equivariant way. Separately, topological deep learning has extended graph networks to cellular sheaves. Our central observation is structural: in a localized atomic-orbital basis, the molecular single-particle Hamiltonian, after a constant shift that makes it positive semidefinite, is the Laplacian of a cellular sheaf on a regular cell complex built from the molecule. Making the restriction maps O(3)-steerable two-center kernels from bond geometry recovers the Slater-Koster form as a special case and yields an E(3)- and permutation-equivariant operator. Three consequences follow. First, the zeroth sheaf cohomology H^0 = ker L is a topological invariant equal to the non-bonding (zero-mode) orbitals, recovering the classical alternant non-bonding-orbital count as a lower bound. Second, the Hodge 1-Laplacian lets higher cells (rings) carry cycle and delocalization information through H^1. Third, the model strictly generalizes E(3)-equivariant message-passing networks and CW networks, and inherits the anti-oversmoothing of non-trivial sheaf diffusion. We prove equivariance, expressivity, and cohomological-correspondence results for the Equivariant Cellular Sheaf Networks, and validate them numerically: the Hamiltonian-to-sheaf embedding is exact to machine precision, the cohomology dimension reproduces non-bonding-orbital counts across eleven conjugated molecules, the sheaf Laplacian is O(3)-equivariant to machine precision, and the equivariant model attains lower error and rotation generalization on a directional electronic target. Our contribution is this sheaf-theoretic formalization and its invariants, not equivariant Hamiltonian prediction itself.

cs.LG↗