In "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 \emph{signal-to-noise ratio} ($\snr$), i.e., approximately of the order of $n^{5}$, and 2) a massive computational burden in light of NP-hardness in general.
In this paper, we study the case of \emph{multiple} noisy measurement vectors (MMVs) resulting from a \emph{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 $m = Ω(\log n)$, 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 a computational scheme for recovering the unknown permutation in practice. For the "oracle case" with the known signal, the maximum likelihood (ML) estimator reduces to a linear assignment problem whose global optimum can be obtained efficiently. For the case in which both the signal and permutation are unknown, the problem is reformulated as a bi-convex optimization problem with an auxiliary variable, which can be solved by the Alternating Direction Method of Multipliers (ADMM). Numerical experiments based on the proposed computational scheme confirm the tightness of our theoretical analysis.
In several applications, including imaging of deformable objects while in motion, simultaneous localization and mapping, and unlabeled sensing, we encounter the problem of recovering a signal that is measured subject to unknown permutations. In this paper we take a fresh look at this problem through the lens of optimal transport (OT). In particular, we recognize that in most practical applications the unknown permutations are not arbitrary but some are more likely to occur than others. We exploit this by introducing a regularization function that promotes the more likely permutations in the so…
NTT Computer and Data Science Laboratories（NTT计算机与数据科学实验室）；NTT Communication Science Laboratories（NTT通信科学实验室）
We propose a few-shot learning method for unsupervised feature selection, which is a task to select a subset of relevant features in unlabeled data. Existing methods usually require many instances for feature selection. However, sufficient instances are often unavailable in practice. The proposed method can select a subset of relevant features in a target task given a few unlabeled target instances by training with unlabeled instances in multiple source tasks. Our model consists of a feature selector and decoder. The feature selector outputs a subset of relevant features taking a few unlabeled…
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 ap…
In this paper, we study the recovery of a signal from a set of noisy linear projections (measurements), when such projections are unlabeled, that is, the correspondence between the measurements and the set of projection vectors (i.e., the rows of the measurement matrix) is not known a priori. We consider a special case of unlabeled sensing referred to as Unlabeled Ordered Sampling (UOS) where the ordering of the measurements is preserved. We identify a natural duality between this problem and classical Compressed Sensing (CS), where we show that the unknown support (location of nonzero element…
We consider the problem of recovering an invertible $n \times n$ matrix $A$ and a sparse $n \times p$ random matrix $X$ based on the observation of $Y = AX$ (up to a scaling and permutation of columns of $A$ and rows of $X$). Using only elementary tools from the theory of empirical processes we show that a version of the Er-SpUD algorithm by Spielman, Wang and Wright with high probability recovers $A$ and $X$ exactly, provided that $p \ge Cn\log n$, which is optimal up to the constant $C$.
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. Wh…
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Georgia State University（佐治亚州立大学）；Shenzhen Research Institute of Big Data（深圳大数据研究院）
We study a matrix recovery problem with unknown correspondence: given the observation matrix $M_o=[A,\tilde P B]$, where $\tilde P$ is an unknown permutation matrix, we aim to recover the underlying matrix $M=[A,B]$. Such problem commonly arises in many applications where heterogeneous data are utilized and the correspondence among them are unknown, e.g., due to privacy concerns. We show that it is possible to recover $M$ via solving a nuclear norm minimization problem under a proper low-rank condition on $M$, with provable non-asymptotic error bound for the recovery of $M$. We propose an algo…
We give unique recovery guarantees for matrices of bounded rank that have undergone permutations of their entries. We even do this for a more general matrix structure that we call ladder matrices. We use methods and results of commutative algebra and algebraic geometry, for which we include a preparation as needed for a general audience.
In this paper, we study the problem of sparse mixed linear regression on an unlabeled dataset that is generated from linear measurements from two different regression parameter vectors. Since the data is unlabeled, our task is not only to figure out a good approximation of the regression parameter vectors but also to label the dataset correctly. In its original form, this problem is NP-hard. The most popular algorithms to solve this problem (such as Expectation-Maximization) have a tendency to stuck at local minima. We provide a novel invex relaxation for this intractable problem which leads t…
The unlabeled sensing problem is to solve a noisy linear system of equations under unknown permutation of the measurements. We study a particular case of the problem where the permutations are restricted to be r-local, i.e. the permutation matrix is block diagonal with r x r blocks. Assuming a Gaussian measurement matrix, we argue that the r-local permutation model is more challenging compared to a recent sparse permutation model. We propose a proximal alternating minimization algorithm for the general unlabeled sensing problem that provably converges to a first order stationary point. Applied…
We address the recovery of sparse vectors in an overcomplete, linear and noisy multiple measurement framework, where the measurement matrix is known upto a permutation of its rows. We derive sparse Bayesian learning (SBL) based updates for joint recovery of the unknown sparse vector and the sensing order, represented using a permutation matrix. We model the sparse matrix using multiple uncorrelated and correlated vectors, and in particular, we use the first order AR model for the correlated sparse vectors. We propose the Permutation-MSBL and a Kalman filtering based Permutation-KSBL algorithm …
Group synchronization asks to recover group elements from their pairwise measurements. It has found numerous applications across various scientific disciplines. In this work, we focus on orthogonal and permutation group synchronization which are widely used in computer vision such as object matching and structure from motion. Among many available approaches, the spectral methods have enjoyed great popularity due to their efficiency and convenience. We will study the performance guarantees of the spectral methods in solving these two synchronization problems by investigating how well the comput…
Signal amplitude estimation and detection from unlabeled quantized binary samples are studied, assuming that the order of the time indexes is completely unknown. First, maximum likelihood (ML) estimators are utilized to estimate both the permutation matrix and unknown signal amplitude under arbitrary, but known signal shape and quantizer thresholds. Sufficient conditions are provided under which an ML estimator can be found in polynomial time and an alternating maximization algorithm is proposed to solve the general problem via good initial estimates. In addition, the statistical identifiabili…
This paper considers the sparse recovery with shuffled labels, i.e., $\by = \bPitrue \bX \bbetatrue + \bw$, where $\by \in \RR^n$, $\bPi\in \RR^{n\times n}$, $\bX\in \RR^{n\times p}$, $\bbetatrue\in \RR^p$, $\bw \in \RR^n$ denote the sensing result, the unknown permutation matrix, the design matrix, the sparse signal, and the additive noise, respectively. Our goal is to reconstruct both the permutation matrix $\bPitrue$ and the sparse signal $\bbetatrue$. We investigate this problem from both the statistical and computational aspects. From the statistical aspect, we first establish the minimax…
We study the problem of solving a linear sensing system when the observations are unlabeled. Specifically we seek a solution to a linear system of equations y = Ax when the order of the observations in the vector y is unknown. Focusing on the setting in which A is a random matrix with i.i.d. entries, we show that if the sensing matrix A admits an oversampling ratio of 2 or higher, then with probability 1 it is possible to recover x exactly without the knowledge of the order of the observations in y. Furthermore, if x is of dimension K, then any 2K entries of y are sufficient to recover x. This…
Consider a noisy linear observation model with an unknown permutation, based on observing $y = Π^* A x^* + w$, where $x^* \in \mathbb{R}^d$ is an unknown vector, $Π^*$ 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 the matrix $A$ are drawn i.i.d. from a standard Gaussian distribution, and establish sharp conditions on the SNR, sample size $n$, and dimension $d$ under which $Π^*$ is exactly and approximately recoverable. On the computational fron…
A recent line of research termed unlabeled sensing and shuffled linear regression has been exploring under great generality the recovery of signals from subsampled and permuted measurements; a challenging problem in diverse fields of data science and machine learning. In this paper we introduce an abstraction of this problem which we call homomorphic sensing. Given a linear subspace and a finite set of linear transformations we develop an algebraic theory which establishes conditions guaranteeing that points in the subspace are uniquely determined from their homomorphic image under some transf…
The efficient sparse coding and reconstruction of signal vectors via linear observations has received a tremendous amount of attention over the last decade. In this context, the automated learning of a suitable basis or overcomplete dictionary from training data sets of certain signal classes for use in sparse representations has turned out to be of particular importance regarding practical signal processing applications. Most popular dictionary learning algorithms involve NP-hard sparse recovery problems in each iteration, which may give some indication about the complexity of dictionary lear…
A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as "permute-and-add" network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their "degree" which is the number of …
This paper addresses the problem of simultaneous signal recovery and dictionary learning based on compressive measurements. Multiple signals are analyzed jointly, with multiple sensing matrices, under the assumption that the unknown signals come from a union of a small number of disjoint subspaces. This problem is important, for instance, in image inpainting applications, in which the multiple signals are constituted by (incomplete) image patches taken from the overall image. This work extends standard dictionary learning and block-sparse dictionary optimization, by considering compressive mea…
The problem of aligning Erdös-Rényi random graphs is a noisy, average-case version of the graph isomorphism problem, in which a pair of correlated random graphs is observed through a random permutation of their vertices. We study a polynomial time message-passing algorithm devised to solve the inference problem of partially recovering the hidden permutation, in the sparse regime with constant average degrees. We perform extensive numerical simulations to determine the range of parameters in which this algorithm achieves partial recovery. We also introduce a generalized ensemble of correlated r…
In this paper, we study the security of a compressed sensing (CS) based cryptosystem called a sparse one-time sensing (S-OTS) cryptosystem, which encrypts a plaintext with a sparse measurement matrix. To construct the secret matrix and renew it at each encryption, a bipolar keystream and a random permutation pattern are employed as cryptographic primitives, which can be obtained by a keystream generator of stream ciphers. With a small number of nonzero elements in the measurement matrix, the S-OTS cryptosystem achieves efficient CS encryption in terms of memory and computational cost. In secur…
