我们证明了低秩矩阵可以有效地恢复从少量的测量，从某一矩阵组的轨道采样。作为一种特殊情况，我们的理论对相位恢复问题进行了说明。在这里，任务是恢复一个向量，只给出它的内积的振幅与少量的向量从轨道。在数学的许多领域中，这个群的变体以不同的名字出现。在编码理论和量子信息中，它是复克利福德群;在时频分析中，它是振子群;在数学物理中，它是亚群。它提供了一个特别小的和高度结构化的轨道，包括和推广离散傅立叶基：虽然傅立叶向量具有常数模量的系数和线性依赖于其指数的相位，但所述轨道中的向量具有二次依赖的相位。在量子信息中，轨道被广泛使用，并被称为稳定态的集合。我们认为，由于其丰富的几何结构和接近最佳的恢复性能，稳定状态形成一个理想的模型，相位恢复的结构化测量。我们的结果适用于$m\geq C κ_r r d \log(d)$测量，其中过采样因子k在$κ_r=1$和$κ_r = r^2$之间变化，取决于轨道。重建对加性噪声和偏离低秩假设都是稳定的。如果感兴趣的矩阵是另外半正定的，则可以通过简单的约束最小二乘回归来执行重建。我们的证明方法可以适用于其他群体的轨道。
英文摘要：
We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. In quantum information, the orbit is used extensively and is known as the set of stabilizer states. We argue that due to their rich geometric structure and their near-optimal recovery properties, stabilizer states form an ideal model for structured measurements for phase retrieval. Our results hold for $m\geq C κ_r r d \log(d)$ measurements, where the oversampling factor k varies between $κ_r=1$ and $κ_r = r^2$ depending on the orbit. The reconstruction is stable towards both additive noise and deviations from the assumption of low rank. If the matrices of interest are in addition positive semidefinite, reconstruction may be performed by a simple constrained least squares regression. Our proof methods could be adapted to cover orbits of other groups.
In this paper, we consider the matrix recovery from rank-one projection measurements proposed in [Cai and Zhang, Ann. Statist., 43(2015), 102-138], via nonconvex minimization. We establish a sufficient identifiability condition, which can guarantee the exact recovery of low-rank matrix via Schatten-$p$ minimization $\min_{X}\|X\|_{S_p}^p$ for $0<p<1$ under affine constraint, and stable recovery of low-rank matrix under $\ell_q$ constraint and Dantzig selector constraint. Our condition is also sufficient to guarantee low-rank matrix recovery via least $q$ minimization $\min_{X}\|\mathcal{A}(X)-…
In this paper we investigate the reconstruction conditions of nuclear norm minimization for low-rank matrix recovery. We obtain sufficient conditions $δ_{tr}<t/(4-t)$ with $0<t<4/3$ to guarantee the robust reconstruction $(z\neq0)$ or exact reconstruction $(z=0)$ of all rank $r$ matrices $X\in\mathbb{R}^{m\times n}$ from $b=\mathcal{A}(X)+z$ via nuclear norm minimization. Furthermore, we not only show that when $t=1$, the upper bound of $δ_r<1/3$ is the same as the result of Cai and Zhang \cite{Cai and Zhang}, but also demonstrate that the gained upper bounds concerning the recovery error are …
Tremendous efforts have been made to study the theoretical and algorithmic aspects of sparse recovery and low-rank matrix recovery. This paper fills a theoretical gap in matrix recovery: the optimal sample complexity for stable recovery without constants or log factors. We treat sparsity, low-rankness, and potentially other parsimonious structures within the same framework: constraint sets that have small covering numbers or Minkowski dimensions. We consider three types of random measurement matrices (unstructured, rank-1, and symmetric rank-1 matrices), following probability distributions tha…
We prove new results about the robustness of well-known convex noise-blind optimization formulations for the reconstruction of low-rank matrices from underdetermined linear measurements. Our results are applicable for symmetric rank-one measurements as used in a formulation of the phase retrieval problem. We obtain these results by establishing that with high probability rank-one measurement operators defined by i.i.d. Gaussian vectors exhibit the so-called Schatten-1 quotient property, which corresponds to a lower bound for the inradius of their image of the nuclear norm (Schatten-1) unit bal…
Tremendous efforts have been made to study the theoretical and algorithmic aspects of sparse recovery and low-rank matrix recovery. This paper fills a theoretical gap in matrix recovery: the optimal sample complexity for stable recovery without constants or log factors. We treat sparsity, low-rankness, and potentially other parsimonious structures within the same framework: constraint sets that have small covering numbers or Minkowski dimensions. We consider three types of random measurement matrices (unstructured, rank-1, and symmetric rank-1 matrices), following probability distributions tha…
In this paper, we bring forward a completely perturbed nonconvex Schatten $p$-minimization to address a model of completely perturbed low-rank matrix recovery. The paper that based on the restricted isometry property generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, gives the restricted isometry property condition that guarantees the recovery of low-rank matrix and the corresponding reconstruction error bound. In particular, the analysis of the result reveals that in the case that $p$ decreases $0$ and $a>1$ for the complete pert…
We propose in this paper a novel, information-theoretic method, called MaxEnt, for efficient data acquisition for low-rank matrix recovery. This proposed method has important applications to a wide range of problems, including image processing and text document indexing. Fundamental to our design approach is the so-called maximum entropy principle, which states that the measurement masks which maximize the entropy of observations, also maximize the information gain on the unknown matrix $\mathbf{X}$. Coupled with a low-rank stochastic model for $\mathbf{X}$, such a principle (i) reveals novel …
Matrix sensing is the problem of reconstructing a low-rank matrix from a few linear measurements. In many applications such as collaborative filtering, the famous Netflix prize problem, and seismic data interpolation, there exists some prior information about the column and row spaces of the ground-truth low-rank matrix. In this paper, we exploit this prior information by proposing a weighted optimization problem where its objective function promotes both rank and prior subspace information. Using the recent results in conic integral geometry, we obtain the unique optimal weights that minimize…
We study Sigma-Delta quantization methods coupled with appropriate reconstruction algorithms for digitizing randomly sampled low-rank matrices. We show that the reconstruction error associated with our methods decays polynomially with the oversampling factor, and we leverage our results to obtain root-exponential accuracy by optimizing over the choice of quantization scheme. Additionally, we show that a random encoding scheme, applied to the quantized measurements, yields a near-optimal exponential bit-rate. As an added benefit, our schemes are robust both to noise and to deviations from the l…
Previous work regarding low-rank matrix recovery has concentrated on the scenarios in which the matrix is noise-free and the measurements are corrupted by noise. However, in practical application, the matrix itself is usually perturbed by random noise preceding to measurement. This paper concisely investigates this scenario and evidences that, for most measurement schemes utilized in compressed sensing, the two models are equivalent with the central distinctness that the noise associated with (\ref{eq.3}) is larger by a factor to $mn/M$, where $m,~n$ are the dimension of the matrix and $M$ is …
Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is the nuclear norm minimization. In [12,14,15], the authors established the necessary and sufficient null space conditions for nuclear norm minimization to recover every possible low-rank matrix with rank at most r (the strong null space condition). In addition, in [12], Oymak et …
In this work, combined calibration and DoA estimation is approached as an extension of the formulation for the Single Measurement Vector (SMV) model of self-calibration to the Multiple Measurement Model (MMV) case. By taking advantage of multiple snapshots, a modified nuclear norm minimization problem is proposed to recover a low-rank larger dimension matrix. We also give the definition of a linear operator for the MMV model, and give its corresponding matrix representation to generate a variant of a convex optimization problem. In order to mitigate the computational complexity of the approach…
In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based spa…
Many problems in statistics and machine learning require the reconstruction of a rank-one signal matrix from noisy data. Enforcing additional prior information on the rank-one component is often key to guaranteeing good recovery performance. One such prior on the low-rank component is sparsity, giving rise to the sparse principal component analysis problem. Unfortunately, there is strong evidence that this problem suffers from a computational-to-statistical gap, which may be fundamental. In this work, we study an alternative prior where the low-rank component is in the range of a trained gener…
