Dynamic systems linked to infinite permutation matrices.
problem Dynamic equivalence of control systems.
method Association of infinite permutation matrices.
result Relationship between dynamic equivalences and permutation matrices.
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48 results for “infinite permutation matrices”
AutoShuffleNet learns permutation matrices in CNNs for improved accuracy.
problem Manual design of channel shuffling in ShuffleNet.
method Learning permutation matrices via an exact Lipschitz continuous penalty in deep learning.
result Improved classification accuracies on CIFAR-10 and ImageNet datasets.
The paper studies stochastic optimization on matrices and its limits as dimensions grow.
problem Optimizing functions on large symmetric matrices using stochastic gradient descent.
method Deterministic limits of random curves on matrices, using graphons and stochastic differential equations.
result The limit is a gradient flow on graphons, extending classical McKean-Vlasov limits.
On meandric permutationsmath.AT
We found a way to code meanders and show they are idempotent.
problem Understanding and coding meandric permutations.
method We established a bijection between meanders and Gauss diagrams, and used this to construct matrices that are idempotent.
result Meandric permutations are idempotent over the field GF(2).
Many matching, tracking, sorting, and ranking problems require probabilistic reasoning about possible permutations, a set that grows factorially with dimension. Combinatorial optimization algorithms may enable efficient point estimation, but fully Bayesian inference poses a severe challenge in this high-dimensional, di…
The paper models financial correlation matrices using permutation invariant Gaussian models and predicts market anomalies.
problem Modeling and predicting financial correlation matrices from high-frequency data.
method Constructing permutation invariant Gaussian matrix models with 4 parameters, using graph theory and polynomial functions.
result The permutation invariant Gaussian matrix model predicts the expectation values of cubic and quartic polynomials with strong evidence of fit.
New method connects neural networks to diagrammatic algebra.
problem Constructing permutation equivariant neural networks.
method Schur-Weyl duality between symmetric group and partition algebra.
result Simple diagrammatic method for calculating weight matrices.
Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
π-GNN learns soft permutations for graph representations, improving graph classification and regression.
problem Limitations of MPNNs in graph neural networks.
method Proposes π-GNN, which learns a soft permutation matrix for each graph, projecting graphs into a common vector space.
result π-GNN achieves performance competitive with state-of-the-art models on graph classification and regression tasks.
Many problems at the intersection of combinatorics and computer science require solving for a permutation that optimally matches, ranks, or sorts some data. These problems usually have a task-specific, often non-differentiable objective function that data-driven algorithms can use as a learning signal. In this paper, w…
Kaleidoscope matrices improve model quality and inference speed.
problem Choosing structured linear transformations for efficiency and accuracy.
method Introduce kaleidoscope matrices that can capture any structured matrix with near-optimal space and time complexity. Learn these matrices automatically within end-to-end pipelines.
result Kaleidoscope matrices can improve model quality and inference speed.
Generates valid Euclidean distance matrices for molecular structures.
problem Generating point clouds in arbitrary rotations and translations is challenging.
method Developed a neural network architecture that produces valid Euclidean distance matrices invariant to rotations and translations.
result The architecture can generate molecular structures in a one-shot fashion by producing Euclidean distance matrices with a three-dimensional embedding.
Semi-direct products of finite groups have permutation representations that are constructed from the permutation representations of their constituents. One can envision these in a metaphoric sense in which a rope is made from a bundle of threads. In this way, subgroups and quotients are easily visualized. The general i…
Paper recovers multi-subspace matrices from permuted data.
problem Recovering a multi-subspace matrix from permuted data with corrupted columns.
method Four-stage algorithm pipeline: outlier identification, subspace reconstruction, outlier classification, unsupervised sensing.
result The pipeline provides theoretical guarantees for reliable multi-subspace matrix recovery.
Derives formulae for general permutation equivariant layers and presents a second order graph variational encoder.
problem Tackles the limitation of previous equivariant neural networks by considering permutations of matrices.
method Derives formulae for general permutation equivariant layers, including matrix permutations. Presents a second order graph variational encoder.
result Latent distribution of equivariant generative models must be exchangeable.
Motivated by the algebraic open-closed string models, we introduce and discuss an infinite-dimensional counterpart of the open-closed Hurwitz theory describing branching coverings generated both by the compact oriented surfaces and by the foam surfaces. We manifestly construct the corresponding infinite-dimensional equ…
Graph alignment problem solved with convex relaxations for correlated matrices.
problem Recovering hidden vertex permutations from correlated Gaussian matrices.
method Convex relaxations of the quadratic assignment problem over doubly stochastic matrices.
result The solution of the convex relaxation concentrates around the ground-truth permutation matrix for certain correlation parameters.
NeuralSort optimizes sorting networks using continuous relaxations.
problem Non-differentiability of sorting operator hinders gradient-based optimization.
method Continuous relaxation of sorting operator to unimodal row-stochastic matrices, enabling gradient-based optimization.
result Gradient-based stochastic optimization over permutations is achieved.
Given a matrix the seriation problem consists in permuting its rows in such way that all its columns have the same shape, for example, they are monotone increasing. We propose a statistical approach to this problem where the matrix of interest is observed with noise and study the corresponding minimax rate of estimatio…
Estimation of Monge Matricesmath.ST
Study on estimating Monge matrices with statistical methods.
problem Estimating Monge matrices with additive noise.
method Viewing structure as a shape constraint, establishing minimax rates, proposing efficient estimators.
result Established minimax rates of estimation for Monge and pre-Monge matrices.
We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming that the underlying matrix has low rank, or is well-approximated by a low rank matr…
New algorithm finds sparse matrices on Stiefel manifold for optimisation.
problem Finding sparse matrices on Stiefel manifold for optimisation.
method Modified Orthogonal Iteration algorithm for sparse global optimality.
result Proposed method finds globally optimal sparse Stiefel matrices.
PiNet learns graph representations invariant to node permutations.
problem Graph classification and representation learning invariant to node permutations.
method Differentiable node attention pooling, permutation invariant graph neural network.
result Significant accuracy improvement in isomorphic graph classification with limited training data.
UPCA solves data matrix completion with permuted columns.
problem Data matrix completion with permuted columns.
method Algebraic geometry and two-stage algorithm.
result UPCA recovers the ground-truth matrix from corrupted data.
We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers via a Poisson point process with mean measure on the power set of . Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, an…
Resolving Schwartz's quadratic meander number conjecture
problem Meander number of cyclic permutations
method Constructing families of cyclic permutations
result Meander number is bounded above and below quadratically in n
Develops a fast BMF approach for binary matrices.
problem Finding patterns in binary matrices for various applications.
method MEBF (Median Expansion for Boolean Factorization) using geometric segmentation and heuristic submatrix identification.
result Superior performance in reconstruction error and computational efficiency compared to existing methods.
It is of increasing importance to develop learning methods for ranking. In contrast to many learning objectives, however, the ranking problem presents difficulties due to the fact that the space of permutations is not smooth. In this paper, we examine the class of rank-linear objective functions, which includes popular…
Transformers tend to learn more symmetric functions in sequence data.
problem Understanding inductive bias in Transformers with infinitely over-parameterized models.
method Analyzing Transformers in the Gaussian process limit, using representation theory of the symmetric group.
result Transformers are biased towards more permutation symmetric functions, and this can be quantitatively predicted.
A new bootstrapping method reduces key sizes and runtime in FHE.
problem Large plaintext evaluation in FHE increases bootstrapping complexity.
method New polynomial vector representation and monic monomial permutation matrices.
result Polynomial factor improvement in key size and constant factor in runtime.
Many applications, including rank aggregation, crowd-labeling, and graphon estimation, can be modeled in terms of a bivariate isotonic matrix with unknown permutations acting on its rows and/or columns. We consider the problem of estimating an unknown matrix in this class, based on noisy observations of (possibly, a su…
A new DNN structure reduces complexity for wireless tasks.
problem Reducing complexity in training deep neural networks for wireless tasks.
method Proposes a DNN with special structure using permutation invariant a priori information.
result The proposed DNN structure reduces training complexity and model parameters.
The crossing matrix of a braid on strands is the integer matrix with zero diagonal whose entry is the algebraic number (positive minus negative) of crossings by strand over strand . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N…
This work tackles Bayesian neural networks by addressing loss landscape symmetries.
problem Understanding and optimizing the loss landscape of Bayesian neural networks.
method The approach involves extending marginalized loss barrier formalism to BNNs, proposing a matching algorithm to search for linearly connected solutions using permutation matrices and combinatorial optimization.
result Nearly zero marginalized loss barriers for linearly connected solutions were found.
This paper tackles fitting multilevel low rank matrices by addressing three problems.
problem Fitting a given matrix by an MLR matrix in the Frobenius norm.
method Factor fitting, rank allocation, and hierarchical partitioning.
result The proposed methods can fit a given matrix by an MLR matrix in the Frobenius norm.
In the modern age, rankings data is ubiquitous and it is useful for a variety of applications such as recommender systems, multi-object tracking and preference learning. However, most rankings data encountered in the real world is incomplete, which prevents the direct application of existing modelling tools for complet…
A new method computes Teichmüller polynomials from integer permutations.
problem Computing Teichmüller polynomials for fibered 3-manifolds.
method Using integer permutations to characterize pseudo-Anosov homeomorphisms and train tracks.
result Direct implementation of McMullen's algorithm for Teichmüller polynomials.
Matching correlated VAR time series databases by recovering matching permutations.
problem Matching perturbed and permuted correlated VAR time series.
method Probabilistic framework modeling, maximum likelihood estimator (MLE), linear assignment, convex relaxations.
result Recovery guarantees for perfect or partial recovery of matching permutations, thresholds for .
Deep kernel processesstat.ML
Deep kernel processes unify various models using Gram matrices and kernel functions.
problem Unified representation of various deep learning models.
method Defining deep kernel processes with progressively transformed Gram matrices and sampling from inverse Wishart distributions.
result Deep Gaussian processes, BNNs, infinite BNNs, and infinite BNNs with bottlenecks can all be written as deep kernel processes.
We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots code…
Extends metrics for SPD matrices to infinite dimensions.
problem Lack of generalized forms for Riemannian metrics.
method Unitized Hilbert-Schmidt operators and extended Mahalanobis norm.
result Improved performance in high-dimensional comparisons.
The paper explores how low-degree polynomials can detect shuffled linear regression models.
problem Detecting multivariate shuffled linear regression models from independent Gaussian random matrices.
method Investigates the effectiveness of low-degree polynomial algorithms for distinguishing the model from independent Gaussian random matrices.
result Establishes a phase transition phenomenon in the performance of low-degree polynomial algorithms for distinguishing the model.
The paper introduces fixed-point centralities for networks and graphons.
problem Defining network centralities for networks and graphons.
method Fixed-point centralities defined via permutation equivariant mappings and graphons.
result Variation bounds of fixed-point centralities under mild assumptions.
Permutation-valued features arise in a variety of applications, either in a direct way when preferences are elicited over a collection of items, or an indirect way in which numerical ratings are converted to a ranking. To date, there has been relatively limited study of regression, classification, and testing problems …
Volume-preserving neural networks prevent gradient issues.
problem Vanishing and exploding gradients in deep neural networks.
method A new neural network architecture with volume-preserving sublayers.
result Volume-preserving neural networks maintain gradient stability.
New invariants derived from random matrices for words in free groups.
problem Defining and understanding new topological invariants for words in free groups.
method Defining and analyzing invariants from -random matrices and permutations.
result Presented new topological, combinatorial, and algebraic invariants of words.
Study groups acting on trees with specific local actions, proving cohomology vanishing or infinite.
problem Understanding bounded cohomology of groups with prescribed local actions.
method Proving vanishing or infinite bounded cohomology based on the 2-transitivity of .
result Vanishing or infinite bounded cohomology depending on 's 2-transitivity.
Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an ex…