Cheap permutation tests speed up distribution testing without sacrificing accuracy.
problem Efficiently testing distribution differences and independence.
method Group datapoints into bins and permute only these bins, using stored sufficient statistics.
result Cheap permutation tests maintain the accuracy and optimality of standard tests but are significantly faster.
C-OPH improves One Permutation Hashing by using a shorter circulant permutation.
problem Improving the accuracy of One Permutation Hashing (OPH) for Jaccard similarity estimation.
method Develops a new densification method using a shorter circulant permutation.
result Achieves the smallest estimation variance for Jaccard similarity.
Random permutations can offer faster convergence than with-replacement sampling for some functions.
problem Understanding when and how random permutations outperform with-replacement sampling in SGD convergence.
method Analyzing convergence rates for different function classes (1D strongly convex, general strongly convex, quadratic strongly convex).
result The optimal convergence gap between random and permutation-based SGD varies from exponential to nonexistent, depending on the function class.
Representations of sets are challenging to learn because operations on sets should be permutation-invariant. To this end, we propose a Permutation-Optimisation module that learns how to permute a set end-to-end. The permuted set can be further processed to learn a permutation-invariant representation of that set, avoid…
Permutations linked to knots and links, with unknots counted by Schröder numbers.
problem Understanding permutations as knots and links.
method Using grid diagrams and Bennequin's inequality.
result Permutations corresponding to unknots and links are counted by Schröder numbers.
Regularizes RNNs to be invariant to input order.
problem Making RNNs invariant to input order.
method Stochastic regularization to enforce permutation invariance.
result Improves model performance on permutation invariant tasks.
We tackle permutation in linear regression with a new inference framework.
problem Statistical investigation of permutation in linear regression models.
method Localization step followed by conditional Monte Carlo test and coefficient inference.
result Valid statistical inference procedures for permutation and regression coefficients.
Permutability of surface transforms yields discrete analogs.
problem Discretization of smooth surfaces with specific properties.
method Permutability of transforms of smooth surfaces.
result Discrete surfaces with discrete analogs of original properties.
A new permutation method improves two-sample testing power.
problem Two-sample testing with improved power and validity.
method Structured block-restricted cross-swaps.
result Block-restricted permutations achieve higher power than full permutations.
Recently, the method of b-bit minwise hashing has been applied to large-scale linear learning and sublinear time near-neighbor search. The major drawback of minwise hashing is the expensive preprocessing cost, as the method requires applying (e.g.,) k=200 to 500 permutations on the data. The testing time can also be ex…
New link topology connects permutation discrepancies to Diaconis-Graham inequalities.
problem Characterize permutations for which Diaconis-Graham inequalities hold with equality.
method Relate permutation discrepancies to the Euler characteristic of their associated links.
result Permutation discrepancies are directly related to the Euler characteristic of their associated links.
We consider a simple and overarching representation for permutation-invariant functions of sequences (or multiset functions). Our approach, which we call Janossy pooling, expresses a permutation-invariant function as the average of a permutation-sensitive function applied to all reorderings of the input sequence. This …
ShuffleNet is a state-of-the-art light weight convolutional neural network architecture. Its basic operations include group, channel-wise convolution and channel shuffling. However, channel shuffling is manually designed empirically. Mathematically, shuffling is a multiplication by a permutation matrix. In this paper, …
We introduce and study the writhe of a permutation, a circular variant of the well-known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non-Ga…
New sampling methods improve Shapley value estimation for machine learning models.
problem Approximating Shapley values for non-trivial models is computationally challenging.
method Investigates new quadrature techniques and quasi-Monte Carlo methods for permutation sampling.
result Significant improvements in Shapley value estimates over existing methods.
4-Legendrian permutation racks can't distinguish knots but recover classical invariants.
problem Distinguishing Legendrian knots using permutation racks.
method Study of 4-Legendrian racks and their effectiveness.
result 4-Legendrian permutation racks cannot distinguish knots but recover classical invariants.
This work refines claims about neural network connectivity, showing that simultaneous linear connectivity is possible under certain conditions.
problem Neural networks' loss landscapes are non-convex due to permutation symmetries, leading to high loss barriers between permuted networks.
method The authors introduce and analyze three claims of increasing strength regarding the connectivity of neural networks, focusing on permutations that align networks.
result The authors provide evidence that strong linear connectivity may be possible under certain conditions, specifically when interpolating among three networks of increasing width.
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.
Distributions over permutations arise in applications ranging from multi-object tracking to ranking of instances. The difficulty of dealing with these distributions is caused by the size of their domain, which is factorial in the number of considered entities (n!). It makes the direct definition of a multinomial dist…
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
A new knot invariant uses permutations to extend Jones polynomials.
problem Extending Jones polynomials to classical and virtual knots and links.
method Colorings by permutations of a finite set to define new knot invariants.
result Established properties and computed polynomials for small cases.
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…
A new method reduces computational costs for testing RF variable importance measures.
problem Testing variable importance measures from random forests is computationally expensive and challenging.
method Sequential permutation testing and sequential p-value estimation to reduce computational costs.
result Theoretical properties of sequential tests are confirmed, maintaining type-I error and high power.
A new method reduces memory requirements for sorting high-dimensional data.
problem Efficiently sorting and organizing high-dimensional data with low memory usage.
method Iteratively shuffles N indices and applies SoftSort optimization steps.
result Significantly improves sorting quality for multidimensional data.
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.
Enhances GNNs by capturing node relationships, outperforming 2-WL test.
problem Inability of conventional GNNs to fully capture node relationships due to permutation invariance.
method Develops permutation-sensitive aggregation mechanism using permutation groups.
result Proves superior expressivity compared to 2-WL test and not less than 3-WL test.
Permutation-equivariant neural networks improve auction mechanisms by reducing regret and sample complexity.
problem Designing optimal auction mechanisms that balance revenue and bidders' regret.
method Introduced permutation-equivariant neural networks to auction mechanisms.
result Permutation-equivariant neural networks decrease expected ex-post regret and improve model generalizability.
New tests detect high-order interactions without permutations.
problem Scalability issues in kernel-based tests for high-order interactions.
method Permutation-free high-order tests using V-statistics and cross-centring.
result Tests yield standard normal distribution under null hypothesis.
C-MinHash reduces the number of permutations needed for MinHash from thousands to just two.
problem Approximating Jaccard similarity in large binary datasets using many permutations.
method Initial permutation followed by circulant shifting of a second permutation to generate hashes.
result C-MinHash achieves unbiased Jaccard similarity estimation with uniformly smaller variance.
Bayesian optimization method for permutations accelerates combinatorial search.
problem Optimizing expensive-to-evaluate objectives on permutation problems.
method LAW2ORDER, a batch Bayesian optimization method based on the acquisition weighted kernel.
result LAW2ORDER achieves sublinear batch cumulative regret, demonstrating accelerated search.
New algorithm learns permutations mixtures with optimal sample complexity.
problem Learning mixtures of permutations in high-dimensional settings.
method Combining groups of pairwise comparisons and combinatorial method of moments.
result Optimal sample complexity proportional to log(n) for high-dimensional data.
In regression analysis of multivariate data, it is tacitly assumed that response and predictor variables in each observed response-predictor pair correspond to the same entity or unit. In this paper, we consider the situation of "permuted data" in which this basic correspondence has been lost. Several recent papers hav…
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…
The permutation symmetry of neurons in each layer of a deep neural network gives rise not only to multiple equivalent global minima of the loss function, but also to first-order saddle points located on the path between the global minima. In a network of d−1 hidden layers with nk neurons in layers $k = 1, \ldots, …
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…
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.
CPI overcomes limitations of permutation importance by providing accurate variable selection.
problem Misidentification of unimportant variables in complex models due to covariate correlations.
method Developed a model agnostic and computationally lean Conditional Permutation Importance (CPI) approach.
result CPI provides accurate type-I error control and more parsimonious variable selection.
Improves modeling of sets with permutation invariant densities.
problem Challenges in calculating trace limit practicality of current methods.
method Proposes an alternative approach to define permutation equivariant transformations with closed form trace.
result Improves both training and final performance.
Signed-permutation coordinate transport improves model alignment across checkpoints.
problem Improper alignment of coordinate-indexed objects across model checkpoints.
method Introduces sign-marginalized Hungarian matching and coordinate-preserving transport.
result Recovering signed-permutation gauge improves coordinate alignment and model performance.
We propose new positive definite kernels for permutations. First we introduce a weighted version of the Kendall kernel, which allows to weight unequally the contributions of different item pairs in the permutations depending on their ranks. Like the Kendall kernel, we show that the weighted version is invariant to rela…
A new tensor ring mixture model improves density estimation efficiency.
problem Efficient probability density estimation in statistical machine learning.
method Tensor ring decomposition with mixture model for adaptive weights.
result Enhanced expressive capability and flexibility in density estimation.
QWO speeds up causal discovery in LiGAMs by O(n2).
problem Scalability issues in permutation-based causal discovery methods for LiGAMs.
method Constructing a specific DAG for a permutation and searching over permutations to minimize edge count.
result QWO achieves a speed-up of O(n2) compared to state-of-the-art methods. TRIP detects unreliable feature importance scores in random forests.
problem Unreliable feature importance scores in random forests due to model extrapolation.
method Develops TRIP (Test for Reliable Interpretation via Permutation) to detect unreliable permutation feature importance scores.
result TRIP reliably detects unreliable permutation feature importance scores in high-dimensional settings.
Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We v…
Sample efficiency and scalability to a large number of agents are two important goals for multi-agent reinforcement learning systems. Recent works got us closer to those goals, addressing non-stationarity of the environment from a single agent's perspective by utilizing a deep net critic which depends on all observatio…
SMP model preserves proximity and permutation in graph neural networks.
problem Challenges in graph mining, such as community and leader finding.
method Stochastic Message Passing (SMP) model that maintains proximity and permutation-equivariance.
result SMP model effectively preserves node proximities and permutation-equivariance.
LMC loss barrier decreases to zero with large network width.
problem Understanding the LMC phenomenon in neural networks.
method Fine-grained analysis of LMC for two-layer ReLU networks.
result LMC loss barrier decreases to zero at a rate of O(m^-1/2) for large network width.
We describe a simple, efficient, permutation based procedure for selecting the penalty parameter in the LASSO. The procedure, which is intended for applications where variable selection is the primary focus, can be applied in a variety of structural settings, including generalized linear models. We briefly discuss conn…