arXiv subjects
Publications and source records attributed to Divit Rawal.
We give a short proof that the majority vote of three independent consistent classifiers is an optimal learner in the realizable PAC setting. This proves optimality for the simplest voting scheme, while simplifying both the algorithmic structure and the probabilistic analysis of previous voting learners, including the algorithm of S. Hanneke and the analysis of bagging by K. Green Larsen.
We study denoising score matching (DSM) when data are drawn from an embedded manifold $M \subset \mathbb{R}^D$. We show that under ambient Gaussian corruption, the target has variance that diverges as the noise scale decreases and correct for it by regressing against the conditional expectation given the nearest point projection on the manifold: the $L^2$-optimal Rao-Blackwellized target. We then compute the small-noise expansion of this target and show that it recovers the true intrinsic Riemannian score to first order, with a second-order bias from a Tweedie term and two geometric terms dependent on how the manifold is embedded in ambient space: a curvature operator acting on the intrinsic score, and an additive drift generated by the spatial variation of the embedding's second fundamental form. On hyperspheres, we derive a simplified formula and show that both geometric terms vanish exactly on $S^2$, offering a theoretical explanation for why ambient DSM performs comparably to intrinsic methods on real Earth science spherical data in prior work.
In deep networks with small initialization, training exhibits long plateaus separated by sharp feature-acquisition transitions. Whereas shallow nonlinear networks and deep linear networks are well studied, extending these analyses to deep nonlinear networks remains challenging. We derive an exact identity for the imbalance of Frobenius norms of layer weight matrices that holds for any smooth activation and any differentiable loss and use this to classify activation functions into four universality classes. On the permutation-symmetric submanifold, the identity combines with an approximate balance law to reduce the full matrix flow to a scalar ODE, giving a critical-depth escape time law $\tau_\star = \Theta(\varepsilon^{-(r-2)})$ governed by the number $r$ of layers at the bottleneck scale rather than the total depth $L$. We find that this same $r-2$ exponent is recovered under He-normal initialization with $r$ bottleneck layers rescaled by $\varepsilon$, where the symmetry manifold is preserved by the flow but not attracting. We find close agreement between our theory and numerical simulations.
Applications ranging from nuclear safeguards to dark matter detection require accurate predictions of neutron yields and energy spectra produced by ($\alpha$,n) reactions. Legacy tools like SOURCES-4C remain widely used despite significant limitations, including outdated nuclear data, missing target nuclides, and restricted accessibility. Here, we present ALPHANSO, an open-source Python package for calculating ($\alpha$,n) neutron source terms. ALPHANSO incorporates modern nuclear data libraries and formats covering all naturally occurring target nuclides and provides a transparent, modular framework for updating or extending the data as new evaluations are released. Comparison with an updated version of SOURCES-4A, NeuCBOT, and experimental measurements across a range of elements and materials shows that ALPHANSO reproduces neutron yields and spectra in good agreement with experimental data and state-of-the-art ($\alpha$,n) calculations. These results demonstrate that ALPHANSO is a reliable, accessible, and modern alternative to legacy ($\alpha$,n) source term codes such as SOURCES-4C. Its open-source design and modular data handling make it readily extensible to future evaluated nuclear data and low-background applications.
We prove an exponential separation in sample complexity between Euclidean and hyperbolic representations for learning on hierarchical data under standard Lipschitz regularization. For depth-$R$ hierarchies with branching factor $m$, we first establish a geometric obstruction for Euclidean space: any bounded-radius embedding forces volumetric collapse, mapping exponentially many tree-distant points to nearby locations. This necessitates Lipschitz constants scaling as $\exp(\Omega(R))$ to realize even simple hierarchical targets, yielding exponential sample complexity under capacity control. We then show this obstruction vanishes in hyperbolic space: constant-distortion hyperbolic embeddings admit $O(1)$-Lipschitz realizability, enabling learning with $n = O(mR \log m)$ samples. A matching $\Omega(mR \log m)$ lower bound via Fano's inequality establishes that hyperbolic representations achieve the information-theoretic optimum. We also show a geometry-independent bottleneck: any rank-$k$ prediction space captures only $O(k)$ canonical hierarchical contrasts.