In this article, we consider a particular data structure consisting of a union of several nested low-rank subspaces with missing data entries. Given the rank of each subspace, we treat the data completion problem, i.e., to estimate the missing entries. Starting from the case of two-dimensional data, i.e., matrices, we show that the union of nested subspaces data structure leads to a structured decomposition U = XY where the factor Y has blocks of zeros that are determined by the rank values. Moreover, for high-dimensional data, i.e., tensors, we show that a similar structured CP decomposition also exists, U = Σ l=1 r a 1 l ⊗ a 2 l ⊗ ··· ⊗ a d l , where A d = [a d 1 ···a d r ] contains blocks of zeros determined by the rank values. Based on such structured decompositions, we develop efficient alternating minimization algorithms for both matrix and tensor completions, by enforcing the above structures in each iteration including the initialization. Compared with naive approaches where either the additional rank constraints are ignored, or data completion is performed part by part, the proposed structured alternating minimization approaches exhibit faster convergence and higher recovery accuracy.

•A geometrical sampling pattern is obtained that guarantees the retrieval of the low-rank multi-view data in certain ways.•An analysis is proposed that incorporates several rank constraints of the multi-view data simultaneously.•Combinatorial analysis is proposed to find the sampling rate that guarantees the obtained geometrical sampling pattern hold with high probability.

In this paper, we analyze the fundamental conditions for low-rank tensor completion given the separation or tensor-train (TT) rank, i.e., ranks of TT unfoldings. We exploit the algebraic structure of the TT decomposition to obtain the deterministic necessary and sufficient conditions on the locations of the samples to ensure finite completability. Specifically, we propose an algebraic geometric analysis on the TT manifold that can incorporate the whole rank vector simultaneously in contrast to the existing approach based on the Grassmannian manifold that can only incorporate one rank component. Our proposed technique characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern and the TT decomposition, which is instrumental to obtaining the deterministic condition on the sampling pattern for finite completability. In addition, based on the proposed analysis, assuming that the entries of the tensor are sampled independently with probability p, we derive a lower bound on the sampling probability p, or equivalently, the number of sampled entries that ensures finite completability with high probability. Moreover, we also provide the deterministic and probabilistic conditions for unique completability.

We consider the problem of clustering and completing a set of tensors with missing data that are drawn from a union of low-rank tensor spaces. In the clustering problem, given a partially sampled tensor data that is composed of a number of subtensors, each chosen from one of a certain number of unknown tensor spaces, we need to group the subtensors that belong to the same tensor space. We provide a geometrical analysis on the sampling pattern and subsequently derive the sampling rate that guarantees the correct clustering under some assumptions with high probability. Moreover, we investigate the fundamental conditions for finite/unique completability for the union of tensor spaces completion problem. Both deterministic and probabilistic conditions on the sampling pattern to ensure finite/unique completability are obtained. For both the clustering and completion problems, our tensor analysis provides significantly better bound than the bound given by the matrix analysis applied to any unfolding of the tensor data.

This paper is concerned with investigating the fundamental conditions on the locations of the sampled entries, i.e., sampling pattern, for finite completability of a matrix that represents the union of several subspaces with given ranks. In contrast with the existing analysis on Grassmannian manifold for the conventional matrix completion, we propose a geometric analysis on the manifold structure for the union of several subspaces to incorporate all given rank constraints simultaneously. In order to obtain the deterministic conditions on the sampling pattern, we characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern, which is closely related to finite completion. We also give a probabilistic condition in terms of the number of samples per column, i.e., the sampling probability, which leads to finite completability with high probability. Furthermore, using the proposed geometric analysis for finite completability, we characterize sufficient conditions on the sampling pattern that ensure there exists only one completion for the sampled data.

Newton's method is a widely applicable and empirically efficient method for finding the solution to a set of equations. The recently developed algebraic geometry analyses provide information-theoretic bounds on the sampling rate to ensure the existence of a unique completion. A remained open question from these works is to retrieve the sampled data when the sampling rate is very close to the mentioned information-theoretic bounds. This paper is concerned with proposing algorithms to retrieve the sampled data when the sampling rate is too small and close to the mentioned information-theoretic bounds. Hence, we propose a new approach for recovering a partially sampled low-rank matrix or tensor when the number of samples is only slightly more than the dimension of the corresponding manifold, by solving a set of polynomial equations using Newton's method. In particular, we consider low-rank matrix completion, matrix sensing, and tensor completion. Each observed entry contributes one polynomial equation in terms of the factors in the rank factorization of the data. By exploiting the specific structures of the resulting set of polynomial equations, we analytically characterize the convergence regions of the Newton's method for matrix completion and matrix sensing. Through extensive numerical results, we show that the proposed approach outperforms the well-known methods such as nuclear norm minimization and alternating minimization in terms of the success rate of data recovery (noiseless case) and peak signal-to-noise ratio (noisy case), especially when the sampling rate is very low.

We investigate the fundamental conditions on the sampling pattern, i.e., locations of the sampled entries, for finite completability of a low-rank tensor given some components of its Tucker rank. In order to find the deterministic necessary and sufficient conditions, we propose an algebraic geometric analysis on the Tucker manifold, which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the Grassmannian manifold. This analysis characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern, which is closely related to finite completability of the sampled tensor, where finite completability simply means that the number of possible completions of the sampled tensor is finite. Probabilistic conditions are then studied and a lower bound on the sampling probability is given, which guarantees that the proposed deterministic conditions on the sampling patterns for finite completability hold with high probability. Furthermore, using the proposed geometric approach for finite completability, we propose a sufficient condition on the sampling pattern that ensures there exists exactly one completion of the sampled tensor.

We derive fundamental conditions for clustering a union of low-rank subspaces with missing data. In particular, given an incomplete matrix, assuming its columns are drawn from K different subspaces with different dimensions, the subspace clustering problem is to cluster the columns that belong to the same subspace. We derive a lower bound on the number of columns from each subspace such that the columns can be clustered correctly with high probability. The analysis focuses on the subspace with the lowest dimension and is a generalization of the corresponding results in [18] that assumes the subspaces are independent and with the same dimension. (C) 2019 Elsevier B.V. All rights reserved.

In this paper, we characterize the deterministic conditions on the locations of the sampled entries, i.e., sampling pattern, that are equivalent to finite completabil-ity of a matrix that represents the union of several low-rank subspaces. To this end, in contrast with the existing analysis on Grassmannian manifold for conventional matrix completion problem, we propose a geometric analysis on the manifold structure for the union of several subspaces to incorporate all given rank constraints simultaneously. Then, using the developed tools for this analysis, we also derive a sufficient condition on the sampling pattern that ensures there exists only one completion of the sampled data.

This paper studies the degrees of freedom (DoF) of two-way $2\times 2\times 2$ MIMO interference networks, a class of two-way four-unicast networks. We first provide upper and lower bounds on the sum DoF of general two-way $2\times 2\times 2$ MIMO interference networks with any number of antennas in each node. We then investigate the special case where all user nodes have $M$ antennas and relay nodes have $N$ antennas, and provide the simplified bounds on the sum DoF. We show that our proposed achievable rate for this special case is higher than the achievable rates recently proposed in [1] . Moreover, for this special case, we obtain the exact DoF $=4M$ when $N \geq 2M$ , and DoF $=4N$ when $M \leq 2N$ .

In this letter, we study the deterministic sampling patterns for the completion of low-rank matrix, when corrupted with a sparse noise, also known as robust matrix completion. We extend the recent results on the deterministic sampling patterns in the absence of noise based on the geometric analysis on the Grassmannian manifold. A special case where each column has a certain number of noisy entries is considered, where our probabilistic analysis performs very efficiently. Furthermore, assuming that the rank of the original matrix is not given, we provide an analysis to determine if the rank of a valid completion is indeed the actual rank of the data corrupted with sparse noise by verifying some conditions.

In this paper, we consider the problem of completing a sampled matrix U = [U 1 |U 2 ] given the ranks of U, U 1 , and U 2 which is known as the multi-view data completion problem. We characterize the deterministic conditions on the locations of the sampled entries that is equivalent (necessary and sufficient) to finite completability of the sampled matrix. To this end, in contrast with the existing analysis on Grassmannian manifold for a single-view matrix, i.e., conventional matrix completion, we propose a geometric analysis on the manifold structure for multi-view data to incorporate more than one rank constraint. Then, using the proposed geometric analysis, we propose sufficient conditions on the sampling pattern, under which there exists only one completion (unique completability) given the three rank constraints.

We consider the multi-view data completion problem, i.e., to complete a matrix $\mathbf{U}=[\mathbf{U}_1|\mathbf{U}_2]$ where the ranks of $\mathbf{U},\mathbf{U}_1$, and $\mathbf{U}_2$ are given. In particular, we investigate the fundamental conditions on the sampling pattern, i.e., locations of the sampled entries for finite completability of such a multi-view data given the corresponding rank constraints. In contrast with the existing analysis on Grassmannian manifold for a single-view matrix, i.e., conventional matrix completion, we propose a geometric analysis on the manifold structure for multi-view data to incorporate more than one rank constraint. We provide a deterministic necessary and sufficient condition on the sampling pattern for finite completability. We also give a probabilistic condition in terms of the number of samples per column that guarantees finite completability with high probability. Finally, using the developed tools, we derive the deterministic and probabilistic guarantees for unique completability.

In this paper, we characterize the deterministic conditions on the locations of the sampled entries, which are equivalent (necessary and sufficient) to finite completability of a tensor given some components of its Tucker rank. In order to derive this characterization, we propose an algebraic geometric analysis on the Tucker manifold, which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the Grassmannian manifold. Then, using the developed tools for this analysis, we also derive a sufficient condition on the sampling pattern that ensures there exists only one completion for the sampled tensor (unique completability).

In this paper, we analyze the fundamental conditions for low-rank tensor completion given the separation or tensor-train (TT) rank, i.e., ranks of unfoldings. We exploit the algebraic structure of the TT decomposition to obtain the deterministic necessary and sufficient conditions on the locations of the samples to ensure finite completability. Specifically, we propose an algebraic geometric analysis on the TT manifold that can incorporate the whole rank vector simultaneously in contrast to the existing approach based on the Grassmannian manifold that can only incorporate one rank component. Our proposed technique characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern and the TT decomposition, which is instrumental to obtaining the deterministic condition on the sampling pattern for finite completability. In addition, based on the proposed analysis, assuming that the entries of the tensor are sampled independently with probability $p$, we derive a lower bound on the sampling probability $p$, or equivalently, the number of sampled entries that ensures finite completability with high probability. Moreover, we also provide the deterministic and probabilistic conditions for unique completability.

We consider the problem of low canonical polyadic (CP) rank tensor completion. A completion is a tensor whose entries agree with the observed entries and its rank matches the given CP rank. We analyze the manifold structure corresponding to the tensors with the given rank and define a set of polynomials based on the sampling pattern and CP decomposition. Then, we show that finite completability of the sampled tensor is equivalent to having a certain number of algebraically independent polynomials among the defined polynomials. Our proposed approach results in characterizing the maximum number of algebraically independent polynomials in terms of a simple geometric structure of the sampling pattern, and therefore we obtain the deterministic necessary and sufficient condition on the sampling pattern for finite completability of the sampled tensor. Moreover, assuming that the entries of the tensor are sampled independently with probability p and using the mentioned deterministic analysis, we propose a combinatorial method to derive a lower bound on the sampling probability p, or equivalently, the number of sampled entries that guarantees finite completability with high probability. We also show that the existing result for the matrix completion problem can be used to obtain a loose lower bound on the sampling probability p. In addition, we obtain deterministic and probabilistic conditions for unique completability. It is seen that the number of samples required for finite or unique completability obtained by the proposed analysis on the CP manifold is orders-of-magnitude lower than that is obtained by the existing analysis on the Grassmannian manifold.

The unit commitment (UC) problem aims to find an optimal schedule of generating units subject to demand and operating constraints for an electricity grid. The majority of existing algorithms for the UC problem rely on solving a series of convex relaxations by means of branch-and-bound and cutting-planning methods. The objective of this paper is to obtain a convex model of polynomial size for practical instances of the UC problem. To this end, we develop a convex conic relaxation of the UC problem, referred to as a strengthened semidefinite program (SDP) relaxation. This approach is based on first deriving certain valid quadratic constraints and then relaxing them to linear matrix inequalities. These valid inequalities are obtained by the multiplication of the linear constraints of the UC problem, such as the flow constraints of two different lines. The performance of the proposed convex relaxation is evaluated on several hard instances of the UC problem. For most of the instances, globally optimal integer solutions are obtained by solving a single convex problem. For the cases where the strengthened SDP does not give rise to a global integer solution, we incorporate other valid inequalities. The major benefit of the proposed method compared to the existing techniques is threefold: (i) the proposed formulation is a single convex model with polynomial size and, hence, its global minimum can be found efficiently using well-established first-and second-order methods by starting from any arbitrary initial state, (ii) unlike heuristic methods and local-search algorithms that return local minima whose closeness to a global solution cannot be measured efficiently, the proposed formulation aims at obtaining global minima, (iii) the proposed convex model can be used in convex-hull pricing to minimize uplift payments made to generating units in energy markets. The proposed technique is extensively tested on IEEE 9-bus, IEEE 14-bus, IEEE 30-bus, IEEE 57-bus, IEEE 118-bus, and IEEE 300-bus systems with different settings and over various time horizons.

Minimizing the nuclear norm of a matrix has been shown to be very efficient in reconstructing a low-rank sampled matrix. Furthermore, minimizing the sum of nuclear norms of matricizations of a tensor has been shown to be very efficient in recovering a low-Tucker-rank sampled tensor. In this paper, we propose to recover a low-TT-rank sampled tensor by minimizing a weighted sum of nuclear norms of unfoldings of the tensor. We provide numerical results to show that our proposed method requires significantly less number of samples to recover to the original tensor in comparison with simply minimizing the sum of nuclear norms since the structure of the unfoldings in the TT tensor model is fundamentally different from that of matricizations in the Tucker tensor model.

We exploit the recent algebraic geometry analyses that study the fundamental conditions on the locations of the sampled entries for finite completability of low-rank sampled tensor to treat the problem of CP-rank approximation for a partially sampled tensor. Particularly, the goal is to approximate the unknown rank based on the locations of the sampled entries, i.e., the sampling pattern, and the rank of an arbitrary given completion. First we provide an upper bound on the unknown CP-rank with probability one assuming that the sampling pattern satisfies the proposed combinatorial properties. However, the proposed combinatorial properties may be hard to verify. Hence, we also provide probabilistic versions of such bounds that hold with high probability assuming that the sampling probability is above a threshold, i.e., we provide the sampling probability that results the sampling pattern satisfies the proposed combinatorial properties with high probability. In addition, these upper bounds can be exactly equal to the unknown CP-rank given the lowest-rank completion. To illustrate how tight our proposed upper bounds are, we have provided some numerical results for the case of two-way tensor, i.e., matrix, in which we applied the nuclear norm minimization to find a low-rank completion of the sampled data and observe that the proposed upper bound is almost equal to the true unknown rank.

Recently, fundamental conditions on the sampling patterns have been obtained for finite completability of low-rank matrices or tensors given the corresponding ranks. In this paper, we consider the scenario where the rank is not given and we aim to approximate the unknown rank based on the location of sampled entries and some given completion. We consider a number of data models, including single-view matrix, multi-view matrix, CP tensor, tensor-train tensor and Tucker tensor. For each of these data models, we provide an upper bound on the rank when an arbitrary low-rank completion is given. We characterize these bounds both deterministically, i.e., with probability one given that the sampling pattern satisfies certain combinatorial properties, and probabilistically, i.e., with high probability given that the sampling probability is above some threshold. Moreover, for both single-view matrix and CP tensor, we are able to show that the obtained upper bound is exactly equal to the unknown rank if the lowest-rank completion is given. Furthermore, we provide numerical experiments for the case of single-view matrix, where we use nuclear norm minimization to find a low-rank completion of the sampled data and we observe that in most of the cases the proposed upper bound on the rank is equal to the true rank.