We introduce a new method to handle permutations efficiently using variational inference.
problem Efficient probabilistic reasoning about permutations in high-dimensional spaces.
method We reparameterize the Birkhoff polytope to enable variational inference over permutations.
result Our method enables efficient and accurate Bayesian inference over permutations.
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.
We discuss the Ribaucour transformation of Legendre maps in Lie sphere geometry. In this context, we give a simple conceptual proof of Bianchi's original Permutability Theorem and its generalisation by Dajczer--Tojeiro. We go on to formulate and prove a higher dimensional version of the Permutability Theorem. It is sho…
Permutation invariant network learns Wasserstein metrics.
problem Understanding the space of probability measures and comparing distributions.
method Permutation invariant network mapping samples to a low-dimensional space.
result Network can generalize to compute distances between unseen densities and learn moments.
We consider the problem of counting and of listing topologically inequivalent "planar" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options wh…
Study on fixed points of random permutations with surface group constraints.
problem Understanding fixed points of random permutations with surface group constraints.
method Computing expected number of fixed points using word maps and surface group constraints.
result The expected number of fixed points is bounded by O(1/dimχ) for shortest representatives. New neural network learns set-equivariant functions efficiently.
problem Efficiently implementing and learning set-equivariant functions.
method Gated recurrent network applied iteratively to all entities and syncs with population progression.
result SWARM mappings achieve state-of-the-art performance in various applications.
Generative model for set-valued data using permutation invariant flows.
problem Modeling set-valued data with conditional generative models.
method Conditional generative probabilistic model using continuous normalizing flows with permutation equivariant dynamics.
result Significantly outperforms non-permutation invariant baselines in log likelihood and domain-specific metrics.
We study a notion of a Lipschitz, permutation-invariant "centroid" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).
We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers N via a Poisson point process with mean measure Λ on the power set of N. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, an…
RapidPT accelerates permutation testing in neuroimaging by reducing runtime.
problem Significant computational burden in voxel-wise analysis.
method Exploits low-rank structure of permutation testing matrix for efficient recovery.
result Achieves substantial speedups (1.5x - 1000x) over existing methods.
Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of …
CR and conformal maps coincide on certain stratified groups.
problem Understanding mappings on stratified groups with CR structures.
method Analyzing conformal, CR, and anti-CR maps on specific Lie groups.
result Conformal maps coincide with CR and anti-CR maps on these groups.
New framework reduces factorisation model run time by exploiting sparse vector geometry.
problem Inefficient inner product computations over large sparse vectors in real-time applications.
method Geometry-aware permutation maps on a tessellated unit sphere for sparse vector embeddings.
result Significant reduction in run time with minimal accuracy loss.
A formula for triangle area in Deep Sets form.
problem Finding a polynomial formula for triangle area in Deep Sets form.
method Expressing area as a permutation-invariant function and finding a suitable Deep Sets form.
result Explicit polynomial formula for triangle area in Deep Sets form.
A new Bayesian multinomial regression model using permuted and augmented stick-breaking.
problem Modeling categorical response variables given covariates.
method Permuted and augmented stick-breaking (paSB) construction.
result Transforms multinomial regression into regression of stick-specific binary variables.
Generates mapping class groups with specific order elements.
problem Generating mapping class groups with elements of fixed finite order.
method Proves generation of specific order elements in mapping class groups and related groups.
result Can generate mapping class groups with 3 elements of order k and 4 elements of order 5 for sufficiently large genus. Study hyperplanes in abelian groups and their signatures for manifold identification.
problem Identifying manifolds based on their homology groups and coordinate hyperplanes.
method Investigates isomorphisms preserving coordinate hyperplanes in products of cyclic groups.
result Recovering coordinate hyperplanes from their union and applying to manifold identification.
New translation equivariant neural processes improve spatio-temporal data modeling.
problem Improving posterior prediction maps for spatio-temporal data.
method Introduced translation equivariant transformers within neural processes.
result TE-TNPs outperform non-equivariant TNPs and other baselines.
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.
New model improves image scene graph labeling.
problem Understanding complex scenes with multiple inter-related objects.
method Permutation-invariant structured prediction using deep learning.
result Achieves state-of-the-art results on Visual Genome benchmark.
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.
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.
The possibilities for new or unusual kinds of topological, locally linear periodic maps of non-prime order on closed, simply connected 4-manifolds with positive definite intersection pairings are explored. On the one hand, certain permutation representations on homology are ruled out under appropriate hypotheses. On th…
Unified framework connects different neural network models.
problem Understanding the geometry of neural network loss landscapes.
method Unified framework capturing four symmetry classes.
result First discovery of low- and zero-barrier linear interpolation paths.
Model learns set representations through optimized permutations.
problem Challenges in learning set representations due to permutation-invariance.
method Proposes a Permutation-Optimisation module to learn set permutations.
result Achieves state-of-the-art results on various set learning tasks.
Transformers can approximate any sequence-to-sequence function, surprising given their complexity.
problem Understanding the expressive power of Transformer models for sequence-to-sequence functions.
method Established that Transformers are universal approximators of continuous permutation equivariant sequence-to-sequence functions with compact support, and extended this to arbitrary functions using positional encodings.
result Transformers are universal approximators of arbitrary continuous sequence-to-sequence functions on a compact domain.
For any smooth compact manifold W of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k co…
In this paper, we calculate the p-torsion of the Farrell cohomology for low genus pure mapping class groups with punctures, where p is an odd prime. Here, `low genus' means g=1,2,3; and `pure mapping class groups with punctures' means the mapping class groups with any number of punctures, where the punctures are not al…
A rack of order n is a binary operation $\rack$ on a set X of cardinality n, such that right multiplication is an automorphism. More precisely, $(X,\rack)$ is a rack provided that the map $x\mapsto x\rack y$ is a bijection for all y∈X, and $(x\rack y)\rack z=(x\rack z)\rack (y\rack z)$ for all x,y,z∈X. …
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.
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.
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.
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.
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.
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.
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.
Janossy pooling averages permutation-sensitive functions over all sequences to create invariant functions.
problem Creating deep, invariant functions for variable-size inputs.
method Janossy pooling: average permutation-sensitive functions over all reorderings.
result Improved performance over state-of-the-art methods.
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.
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.
A structured prediction method for ranking labels.
problem Solving label ranking problems as structured output regression.
method Two-step approach: regression in feature space followed by pre-image solving.
result Efficiency on real-world datasets for partial and complete rankings.
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.
ABI adapts to graph data for fast, scalable inference.
problem Challenges in inference on graph-structured data.
method Amortized Bayesian Inference (ABI) framework for graph data.
result ABI successfully addresses challenges in graph data inference.
We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a simil…
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.
The paper uses permutation representations to visualize group extensions and subgroups.
problem Visualizing and understanding group extensions and subgroups.
method Developing metaphoric rope-thread diagrams to represent semi-direct products and their constituents.
result Injective homomorphisms into semi-direct products are established.
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n,n,n]: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. I…