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9 results for permutahedron

Paper introduces differentiable sorting and ranking with O(nlogn)O(n \log n) time complexity.

problem Non-differentiability of sorting and ranking operations in machine learning.
method Differentiable proxies constructed as projections onto the permutahedron and reduction to isotonic optimization.
result First differentiable sorting and ranking operators with O(nlogn)O(n \log n) time and O(n)O(n) space complexity.

Study on likelihood functions, associative equations, and Frobenius manifolds.

problem Maximum likelihood estimation and associativity equations in statistical models.
method Analyzes the cone of concentration matrices, log-likelihood function, and Frobenius manifolds.
result Maximum likelihood degree is indexed by components of Frobenius residuals.

A few years ago Kramer and Laubenbacher introduced a discrete notion of homotopy for simplicial complexes. In this paper, we compute the discrete fundamental group of the order complex of the Boolean lattice. As it turns out, it is equivalent to computing the discrete homotopy group of the 1-skeleton of the permutahedr…

2007-11-06abs ↗pdf ↗

Proposes differentiable and sparse top-k operators for neural networks.

problem Discontinuity of top-k operator makes it unsuitable for end-to-end training with backpropagation.
method Formulates top-k as a linear program over permutahedron, introduces p-norm regularization, and uses isotonic optimization.
result Successfully applied to neural network pruning, fine-tuning, and routing.

The paper explores trading off consistency and dimensionality in convex surrogates for multiclass classification.

problem Designing consistent surrogate losses for multiclass classification with high-dimensional outcomes.
method Investigates embedding outcomes into convex polytopes and examining consistency under low-noise assumptions.
result Consistency can be achieved with less than n1n-1 dimensions, but hallucination occurs for some distributions.