Unknown Permutation Research Articles

Statistical and Computational Efficiency for Smooth Tensor Estimation with Unknown Permutations

We consider the problem of structured tensor denoising in the presence of unknown permutations. Such data problems arise commonly in recommendation systems, neuroimaging, community detection, and multiway comparison applications. Here, we develop a general family of smooth tensor models up to arbitrary index permutations; the model incorporates the popular tensor block models and Lipschitz hypergraphon models as special cases. We show that a constrained least-squares estimator in the block-wise polynomial family achieves the minimax error bound. A phase transition phenomenon is revealed with respect to the smoothness threshold needed for optimal recovery. In particular, we find that a polynomial of degree up to ( m − 2 ) ( m + 1 ) / 2 is sufficient for accurate recovery of order-m tensors, whereas higher degrees exhibit no further benefits. This phenomenon reveals the intrinsic distinction for smooth tensor estimation problems with and without unknown permutations. Furthermore, we provide an efficient polynomial-time Borda count algorithm that provably achieves the optimal rate under monotonicity assumptions. The efficacy of our procedure is demonstrated through both simulations and Chicago crime data analysis. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Faster algorithms for the alignment of sparse correlated Erdős–Rényi random graphs

Abstract The correlated Erdős–Rényi random graph ensemble is a probability law on pairs of graphs with n vertices, parametrized by their average degree λ and their correlation coefficient s. It can be used as a benchmark for the graph alignment problem, in which the labels of the vertices of one of the graphs are reshuffled by an unknown permutation; the goal is to infer this permutation and thus properly match the pairs of vertices in both graphs. A series of recent works has unveiled the role of Otter’s constant α (that controls the exponential rate of growth of the number of unlabeled rooted trees as a function of their sizes) in this problem: for s > α and large enough λ, it is possible to recover in a time polynomial in n a positive fraction of the hidden permutation. The exponent of this polynomial growth is, however, quite large and depends on the other parameters, which limits the range of applications of the algorithm. In this work, we present a family of faster algorithms for this task, show through numerical simulations that their accuracy is only slightly reduced with respect to the original one and conjecture that they undergo, in the large λ limit, phase transitions at modified Otter’s thresholds α ^ > α , with α ^ related to the enumeration of a restricted family of trees.

RPM: Robust Anonymity at Scale

This work presents RPM, a scalable anonymous communication protocol suite using secure multiparty computation (MPC) with the offline-online model. We generate random, unknown permutation matrices in a secret-shared fashion and achieve improved (online) performance and the lightest communication and computation overhead for the clients compared to the state of art robust anonymous communication protocols. Using square-lattice shuffling, we make our protocol scale well as the number of clients increases. We provide three protocol variants, each targeting different input volumes and MPC frameworks/libraries. Besides, due to the modular design, our protocols can be easily generalized to support more MPC functionalities and security properties as they get developed. We also illustrate how to generalize our protocols to support two-way anonymous communication and secure sorting. We have implemented our protocols using the MP-SPDZ library suit and the benchmark illustrates that our protocols achieve unprecedented online phase performance with practical offline phases.

Power-conjugate equations in symmetric groups

First we consider the solutions of the general “cubic” equation (with ) in the symmetric group . In certain cases this equation can be rewritten as or as , where depends on the αi ’s and the new unknown permutation is a product of x (or ) and one of the permutations . Using combinatorial arguments and some basic number theoretical facts, we obtain results about the solutions of the so-called power-conjugate equation in , where is an integer exponent. A divisibility condition involving the type of α provide solutions with and a further condition gives the complete list of solutions. Some other divisibility assumptions concerning the type of α ensure that the solutions of are exactly the solutions of in the centralizer of α. Slightly stronger assumptions provide a complete answer to the question, when our equation has only the trivial solution y = 1.

Learn to Predict Sets Using Feed-Forward Neural Networks.

This paper addresses the task of set prediction using deep feed-forward neural networks. A set is a collection of elements which is invariant under permutation and the size of a set is not fixed in advance. Many real-world problems, such as image tagging and object detection, have outputs that are naturally expressed as sets of entities. This creates a challenge for traditional deep neural networks which naturally deal with structured outputs such as vectors, matrices or tensors. We present a novel approach for learning to predict sets with unknown permutation and cardinality using deep neural networks. In our formulation we define a likelihood for a set distribution represented by a) two discrete distributions defining the set cardinally and permutation variables, and b) a joint distribution over set elements with a fixed cardinality. Depending on the problem under consideration, we define different training models for set prediction using deep neural networks. We demonstrate the validity of our set formulations on relevant vision problems such as: 1) multi-label image classification where we outperform the other competing methods on the PASCAL VOC and MS COCO datasets, 2) object detection, for which our formulation outperforms popular state-of-the-art detectors, and 3) a complex CAPTCHA test, where we observe that, surprisingly, our set-based network acquired the ability of mimicking arithmetics without any rules being coded.

The Benefits of Diversity: Permutation Recovery in Unlabeled Sensing From Multiple Measurement Vectors

In1 “Unlabeled Sensing”, one observes a set of linear measurements of an underlying signal with incomplete or missing information about their ordering, which can be modeled in terms of an unknown permutation. Previous work on the case of a single noisy measurement vector has exposed two main challenges: 1) a high requirement concerning the signal-to-noise ratio (snr), i.e., approximately of the order of n5, and 2) a massive computational burden in light of NP-hardness in general. In this paper, we study the case of multiple noisy measurement vectors (MMVs) resulting from a common permutation and investigate to what extent the number of MMVs m facilitates permutation recovery by “borrowing strength”. The above two challenges have at least partially been resolved within our work. First, we show that a large stable rank of the signal significantly reduces the required snr which can drop from a polynomial in n for m = 1 to a constant for m = Ω(logn), where m denotes the number of MMVs and n denotes the number of measurements per MV. This bound is shown to be sharp and is associated with a phase transition phenomenon. Second, we propose computational methods for recovering the unknown permutation. For the “oracle case” with known signal, the maximum likelihood (ML) estimator reduces to a linear assignment problem whose global optimum can be obtained efficiently. If both the signal and the permutation are unknown, the problem becomes a quadratic assignment problem; while such a problem is generally NP-hard and hence poses a significant challenge, we propose to tackle it via projected gradient descent with a non-convex constraint set, and establish a monotonic descent property of this scheme. Numerical experiments based on the proposed computational approach confirm the tightness of our theoretical analysis.

Graph matching beyond perfectly-overlapping Erd\u0151s\u2013R\xe9nyi random graphs

Graph matching is a fruitful area in terms of both algorithms and theories. Given two graphs G_1 = (V_1, E_1) and G_2 = (V_2, E_2), where V_1 and V_2 are the same or largely overlapped upon an unknown permutation pi ^*, graph matching is to seek the correct mapping pi ^*. In this paper, we exploit the degree information, which was previously used only in noiseless graphs and perfectly-overlapping Erdős–Rényi random graphs matching. We are concerned with graph matching of partially-overlapping graphs and stochastic block models, which are more useful in tackling real-life problems. We propose the edge exploited degree profile graph matching method and two refined variations. We conduct a thorough analysis of our proposed methods’ performances in a range of challenging scenarios, including coauthorship data set and a zebrafish neuron activity data set. Our methods are proved to be numerically superior than the state-of-the-art methods. The algorithms are implemented in the R (A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2020) package GMPro (GMPro: graph matching with degree profiles, 2020).

Isotonic regression with unknown permutations: Statistics, computation and adaptation

Motivated by models for multiway comparison data, we consider the problem of estimating a coordinatewise isotonic function on the domain [0,1]d from noisy observations collected on a uniform lattice, but where the design points have been permuted along each dimension. While the univariate and bivariate versions of this problem have received significant attention, our focus is on the multivariate case d≥3. We study both the minimax risk of estimation (in empirical L2 loss) and the fundamental limits of adaptation (quantified by the adaptivity index) to a family of piecewise constant functions. We provide a computationally efficient Mirsky partition estimator that is minimax optimal while also achieving the smallest adaptivity index possible for polynomial time procedures. Thus, from a worst-case perspective and in sharp contrast to the bivariate case, the latent permutations in the model do not introduce significant computational difficulties over and above vanilla isotonic regression. On the other hand, the fundamental limits of adaptation are significantly different with and without unknown permutations: Assuming a hardness conjecture from average-case complexity theory, a statistical-computational gap manifests in the former case. In a complementary direction, we show that natural modifications of existing estimators fail to satisfy at least one of the desiderata of optimal worst-case statistical performance, computational efficiency and fast adaptation. Along the way to showing our results, we improve adaptation results in the special case d=2 and establish some properties of estimators for vanilla isotonic regression, both of which may be of independent interest.

Estimation of Large Block Structured Covariance Matrices: Application to ‘Multi-Omic’ Approaches to Study Seed Quality

Abstract Motivated by an application in high-throughput genomics and metabolomics, we propose a novel and fully data-driven approach for estimating large block structured sparse covariance matrices in the case where the number of variables is much larger than the number of samples without limiting ourselves to block diagonal matrices. Our approach consists in approximating such a covariance matrix by the sum of a low-rank sparse matrix and a diagonal matrix. Our methodology also can deal with matrices for which the block structure appears only if the columns and rows are permuted according to an unknown permutation. Our technique is implemented in the R package BlockCov which is available from the Comprehensive R Archive Network (CRAN) and from GitHub. In order to illustrate the statistical and numerical performance of our package some numerical experiments are provided as well as a thorough comparison with alternative methods. Finally, our approach is applied to the use of ‘multi-omic’ approaches for studying seed quality.

Detection by Block- and Band-Permuted Data

Unlabeled detection is an important problem in large sensor networks faced with big-data applications. The focus of this paper is on decision problems in which the data vector received at the fusion center (FC), tasked to perform the final inference, undergoes an unknown permutation. Two representative permutation models are considered, whose structure is imposed by practical considerations. The first is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> -block permutation model, where the observations are split into different blocks of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> , and independent data permutations affect each block. The second is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$r$</tex-math></inline-formula> -banded permutation model, in which each sample of the vector received by the FC lies at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$r$</tex-math></inline-formula> positions ahead or behind its original location. To solve these challenging detection tasks, three detectors for both permutation models are proposed. The first is inspired by the uniformly most powerful invariant principle; the second relies on a distribution-averaged strategy; and the third is designed according the generalized likelihood ratio approach. The performance of these detectors is investigated by computer experiments, and their relative merits are discussed.

R-Local Unlabeled Sensing: A Novel Graph Matching Approach for Multiview Unlabeled Sensing Under Local Permutations

Unlabeled sensing is a linear inverse problem where the measurements are scrambled under an unknown permutation leading to loss of correspondence between the measurements and the rows of the sensing matrix. Motivated by practical tasks such as mobile sensor networks, target tracking and the pose and correspondence estimation between point clouds, we study a special case of this problem restricting the class of permutations to be local and allowing for multiple views. In this setting, namely unlabeled multi-view sensing with local permutation, previous results and algorithms are not directly applicable. In this paper, we propose a computationally efficient algorithm that creatively exploits the machinery of graph alignment and Gromov-Wasserstein alignment and leverages the multiple views to estimate the local permutations. Simulation results on synthetic data sets indicate that the proposed algorithm is scalable and applicable to the challenging regimes of low to moderate SNR.

Towards optimal estimation of bivariate isotonic matrices with unknown permutations

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 subset of) its entries. We design and analyze polynomial-time algorithms that improve upon the state of the art in two distinct metrics, showing, in particular, that minimax optimal, computationally efficient estimation is achievable in certain settings. Along the way, we prove matching upper and lower bounds on the minimax radii of certain cone testing problems, which may be of independent interest. (A corollary of Theorem 3.5 of this paper was presented at the Conference on Learning Theory (COLT) 2018, and a statement of this result appears in the abstract (In Proceedings of the 31st Conference On Learning Theory (2018) 2037–2042 PMLR).)

Self-supervised representation learning by predicting visual permutations

We propose a self-supervised learning method to uncover the spatial or temporal structure of visual data by identifying the position of a patch within an image or the position of a video frame over time, which is related to Jigsaw puzzle reassembly problem in previous works. A Jigsaw puzzle can be seen as a shuffled sequence, which is generated by shuffling image patches or video frames according to an unknown permutation. The task of predicting the visual permutations can be used to train a learning system to capture structural information which is important for semantic-level tasks, such as object recognition and action recognition. To this end, we propose a multi-task learning framework where a group of principal tasks aims to predict the index of each sample in the original sequence, and a group of auxiliary tasks aims to predict the spatial or temporal relation of adjacent samples in the shuffled sequence. Our scheme can handle the whole space of permutations and is fairly scalable, and it is also generic to solve many problems such as self-supervised representation learning, relative attributes, and learning to rank. Our method achieves state-of-the-art performance on the STL-10 benchmarks for unsupervised representation learning, and it is competitive with state-of-the-art performance on UCF-101 and HMDB-51 as a pretraining method for action recognition. In addition, we apply the proposed method on age comparison task to prove it is generic to solve ranking problems.

On the Fundamental Limits of Coded Data Shuffling for Distributed Machine Learning

We consider the data shuffling problem in a distributed learning system, in which a master node is connected to a set of worker nodes, via a shared link, in order to communicate a set of files to the worker nodes. The master node has access to a database of files. In every shuffling iteration, each worker node processes a new subset of files, and has excess storage to partially cache the remaining files, assuming the cached files are uncoded. The caches of the worker nodes are updated every iteration, and they should be designed to satisfy any possible unknown permutation of the files in subsequent iterations. For this problem, we characterize the exact load-memory trade-off for worst-case shuffling by deriving the minimum communication load for a given storage capacity per worker node. As a byproduct, the exact load-memory trade-off for any shuffling is characterized when the number of files is equal to the number of worker nodes. We propose a novel deterministic coded shuffling scheme, which improves the state of the art, by exploiting the cache memories to create coded functions that can be decoded by several worker nodes. Then, we prove the optimality of our proposed scheme by deriving a matching lower bound and showing that the placement phase of the proposed coded shuffling scheme is optimal over all shuffles.

Permutations Unlabeled Beyond Sampling Unknown

A recent unlabeled sampling result by Unnikrishnan, Haghighatshoar and Vetterli states that with probability one over iid Gaussian matrices $A$, any $x$ can be uniquely recovered from an unknown permutation of $y = A x$ as soon as $A$ has at least twice as many rows as columns. We show that this condition on $A$ implies something much stronger: that an unknown vector $x$ can be recovered from measurements $y = T A x$, when the unknown $T$ belongs to an arbitrary set of invertible, diagonalizable linear transformations $\mathcal{T}$. The set $\mathcal{T}$ can be finite or countably infinite. When it is the set of $m \times m$ permutation matrices, we have the classical unlabeled sampling problem. We show that for almost all $A$ with at least twice as many rows as columns, all $x$ can be recovered either uniquely, or up to a scale depending on $\mathcal{T}$, and that the condition on the size of $A$ is necessary. Our proof is based on vector space geometry. Specializing to permutations we obtain a simplified proof of the uniqueness result of Unnikrishnan, Haghighatshoar and Vetterli. In this letter we are only concerned with uniqueness; stability and algorithms are left for future work.

Visual Permutation Learning.

We present a principled approach to uncover the structure of visual data by solving a deep learning task coined visual permutation learning. The goal of this task is to find the permutation that recovers the structure of data from shuffled versions of it. In the case of natural images, this task boils down to recovering the original image from patches shuffled by an unknown permutation matrix. Permutation matrices are discrete, thereby posing difficulties for gradient-based optimization methods. To this end, we resort to a continuous approximation using doubly-stochastic matrices and formulate a novel bi-level optimization problem on such matrices that learns to recover the permutation. Unfortunately, such a scheme leads to expensive gradient computations. We circumvent this issue by further proposing a computationally cheap scheme for generating doubly stochastic matrices based on Sinkhorn iterations. To implement our approach we propose DeepPermNet, an end-to-end CNN model for this task. The utility of DeepPermNet is demonstrated on three challenging computer vision problems, namely, relative attributes learning, supervised learning-to-rank, and self-supervised representation learning. Our results show state-of-the-art performance on the Public Figures and OSR benchmarks for relative attributes learning, chronological and interestingness image ranking for supervised learning-to-rank, and competitive results in the classification and segmentation tasks of the PASCAL VOC dataset for self-supervised representation learning.

Linear Regression With Shuffled Data: Statistical and Computational Limits of Permutation Recovery

Consider a noisy linear observation model with an unknown permutation, based on observing $y = \Pi ^{*} A x^{*} + w$ , where $x^{*} \in {\mathbb {R}} ^{d}$ is an unknown vector, $\Pi ^{*}$ is an unknown $n \times n$ permutation matrix, and $w \in {\mathbb {R}} ^{n}$ is additive Gaussian noise. We analyze the problem of permutation recovery in a random design setting in which the entries of matrix $A$ are drawn independently from a standard Gaussian distribution and establish sharp conditions on the signal-to-noise ratio, sample size $n$ , and dimension $d$ under which $\Pi ^{*}$ is exactly and approximately recoverable. On the computational front, we show that the maximum likelihood estimate of $\Pi ^{*}$ is NP-hard to compute for general $d$ , while also providing a polynomial time algorithm when $d =1$ .

The linear programming relaxation permutation symmetry group of an orthogonal array defining integer linear program

There is always a natural embedding of $S_{s}\wr S_{k}$ into the linear programming (LP) relaxation permutation symmetry group of an orthogonal array integer linear programming (ILP) formulation with equality constraints. The point of this paper is to prove that in the $2$-level, strength-$1$ case the LP relaxation permutation symmetry group of this formulation is isomorphic to $S_{2}\wr S_{k}$ for all $k$, and in the $2$-level, strength-$2$ case it is isomorphic to $S_{2}^{k}\rtimes S_{k+1}$ for $k\geqslant 4$. The strength-$2$ result reveals previously unknown permutation symmetries that cannot be captured by the natural embedding of $S_{2}\wr S_{k}$. We also conjecture a complete characterization of the LP relaxation permutation symmetry group of the ILP formulation.Supplementary materials are available with this article.

Sparse Modeling of Intrinsic Correspondences

AbstractWe present a novel sparse modeling approach to non‐rigid shape matching using only the ability to detect repeatable regions. As the input to our algorithm, we are given only two sets of regions in two shapes; no descriptors are provided so the correspondence between the regions is not know, nor we know how many regions correspond in the two shapes. We show that even with such scarce information, it is possible to establish very accurate correspondence between the shapes by using methods from the field of sparse modeling, being this, the first non‐trivial use of sparse models in shape correspondence. We formulate the problem of permuted sparse coding, in which we solve simultaneously for an unknown permutation ordering the regions on two shapes and for an unknown correspondence in functional representation. We also propose a robust variant capable of handling incomplete matches. Numerically, the problem is solved efficiently by alternating the solution of a linear assignment and a sparse coding problem. The proposed methods are evaluated qualitatively and quantitatively on standard benchmarks containing both synthetic and scanned objects.

Symmetric functions of qubits in an unknown basis

Consider an $n$-qubit computational basis state corresponding to a bit-string $x$, which has had an unknown local unitary applied to each qubit, and whose qubits have been reordered by an unknown permutation. We show that, given such a state with Hamming weight $|x|\ensuremath{\le}\ensuremath{\lfloor}n/2\ensuremath{\rfloor}$, it is possible to reconstruct $|x|$ with success-probability $1\ensuremath{-}|x|/(n\ensuremath{-}|x|+1)$, and thus to compute any symmetric function of $x$. We give explicit algorithms for computing whether or not $|x|\ensuremath{\ge}t$ for some $t$, and for computing the parity of $x$, and show that these are essentially optimal. These results can be seen as generalizations of the swap test for comparing quantum states.