arXiv research

A new method computes Teichmüller polynomials from integer permutations.

Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alterna…

K. Ding studied a class of Schubert varieties X_λin type A partial flag manifolds, corresponding to integer partitions λand in bijection with dominant permutations. He observed that the Schubert cell structure of X_λis indexed by maximal rook placements on the Ferrers board B_λ, and that the integral cohomology groups …

We compute the integer cohomology rings of the ``polygon spaces'' introduced in [Hausmann,Klyachko,Kapovich-Millson]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory of Gröbner bases. Since we do not inver…

Let $M$ be a cusped finite-volume hyperbolic three-manifold with isometry group $G$. Then $G$ induces a $k$-transitive action by permutation on the cusps of $M$ for some integer $k\ge 0$. Generically $G$ is trivial and $k=0$, but $k>0$ does occur in special cases. We show examples with $k=1,2,4$. An interesting questio…

The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N…

Cheap permutation tests speed up distribution testing without sacrificing accuracy.

C-OPH improves One Permutation Hashing by using a shorter circulant permutation.

Random permutations can offer faster convergence than with-replacement sampling for some functions.

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.

Regularizes RNNs to be invariant to input order.

We tackle permutation in linear regression with a new inference framework.

Permutability of surface transforms yields discrete analogs.

A new permutation method improves two-sample testing power.

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.

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 …

We present an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation z0^n + z1^n + z2^n + z3^n = 0 in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Our tessellation…

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.

4-Legendrian permutation racks can't distinguish knots but recover classical invariants.

This work refines claims about neural network connectivity, showing that simultaneous linear connectivity is possible under certain conditions.

Paper recovers multi-subspace matrices from permuted data.

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

A new knot invariant uses permutations to extend Jones polynomials.

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.

A new method reduces memory requirements for sorting high-dimensional data.

Derives formulae for general permutation equivariant layers and presents a second order graph variational encoder.

Enhances GNNs by capturing node relationships, outperforming 2-WL test.

Permutation-equivariant neural networks improve auction mechanisms by reducing regret and sample complexity.

New tests detect high-order interactions without permutations.

C-MinHash reduces the number of permutations needed for MinHash from thousands to just two.

Bayesian optimization method for permutations accelerates combinatorial search.

New algorithm learns permutations mixtures with optimal sample complexity.

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 $n_k$ 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.

CPI overcomes limitations of permutation importance by providing accurate variable selection.

Improves modeling of sets with permutation invariant densities.

Signed-permutation coordinate transport improves model alignment across checkpoints.

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.