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Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent
arXiv:2606.02596v1 Announce Type: new Abstract: The curvature exponent $\alpha$ in $h_k \propto \sigma_k^\alpha$ -- governing how Hessian eigenvalues scale with gradient singular values -- varies systematically across layer types ($\alpha \approx 2$ for convolutions, $\approx 1$ for transformer attention, $< 1$ for MLP up-projections). We prove the Spectral Alignment Decomposition: $\alpha = 2 + d\log\Phi_k / d\log\sigma_k$, where $\Phi_k$ measures alignment between Kronecker factor...
Faster PBWT prefix-array access via batching
arXiv:2605.15819v3 Announce Type: replace Abstract: The positional Burrows-Wheeler Transform (PBWT) is commonly used to store haplotype panels compactly in such a way that, given a query haplotype, we can quickly find the set maximal exact matches (SMEMs) between the query and the haplotypes in a panel. There are generally two steps in this process: first we find the maximal substrings of the query that occur in the same positions in haplotypes in the panel and then, for each such substring,...
Parallel Metric Skiplists and Nearest Neighbor Search
Announce Type: new Abstract: The metric skip-list is a data structure designed for efficient nearest and $k$-nearest neighbor search in metric spaces. For many real-world datasets with reasonable distributions - specifically, those with a constant expansion rate - it supports $\tilde{O}(n)$ construction time and $O(k\log n)$ query time, where $n$ is the input size and $k$ is the number of nearest neighbors in queries. Notably, unlike alternative approaches, it does not require a bounded...
An Upper Bound on Grothendieck's Constant
Announce Type: cross Abstract: We show that Grothendieck's real constant $K_G$ can be upper bounded by projecting vectors onto a random plane through the origin and thresholding a degree five Hermite polynomial. This resolves a conjecture of Braverman-Makarychev-Makarychev-Naor from 2011, who required an extra randomization step in their rounding scheme and proved $K_G<\frac{\pi}{2\log(1+\sqrt{2})}-10^{-500}$. As a corollary of our result, we prove the bound...
The Grothendieck Constant is Less Than $\frac{\pi}{2 \log (1+ \sqrt{2})} - 10^{-5}$
Computer Science > Data Structures and Algorithms [Submitted on 2 Jun 2026] Title:The Grothendieck Constant is Less Than $\fracπ{2 \log (1+ \sqrt{2})} - 10^{-5}$ View PDF HTML (experimental)Abstract:We prove that the Grothendieck constant $K_G < $\frac{\pi}{2 \log (1+ \sqrt{2})} - 10^{-5}$. This improves on the work of braverman et.
The Cascade Log: Reference-Stable Windowing over Tiered Append Sequences
Announce Type: new Abstract: A long-running append-mostly sequence, such as an edit log, event store, or versioned working set, is usually tiered into a bounded hot stratum and colder folded summaries. This saves memory but breaks stable references: a handle minted while a record is hot may later be resolved after the record has moved into a digest, after it has been superseded, or while a fold is in flight. We define the resulting cross-tier anomalies--dangling, stale, corrupt, and...
Show HN: Lowfat – pluggable CLI filter that saved 91.8% of my LLM tokens
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ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes
arXiv:2603.05406v2 Announce Type: replace Abstract: The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and parameterized complexity settings. Parameterized by treewidth $k$, OMM has long been known to be solvable on triangulations of $3$-manifolds in $2^{O(k^2)} n^{O(1)}$ time and in FPT time for triangulations of...
Deterministic Distance Approximation in MPC via Improved Hitting Sets
Announce Type: new Abstract: In this paper, we provide the first deterministic algorithms with sublogarithmic round complexity for spanners and approximate shortest paths in various MPC models. Moreover, we significantly improve upon the state of the art in the deterministic Congested Clique. In particular, we obtain the following four results on undirected graphs: 1.
Discrete Incremental Voting: New Bounds for General Graphs and Expanders
arXiv:2606.06381v1 Announce Type: new Abstract: We analyze the discrete incremental voting process (DIV) introduced by Cooper, Radzik, and Shiraga [OPODIS '23]. In this process, we consider a set $V$ of $n$ nodes connected in an undirected graph $G = (V, E)$ where each node has an integer opinion. In one step a randomly selected node interacts with its randomly selected neighbor and changes its opinion by $1$ in the direction of the neighbour's opinion.