Rank-one Projections Research Articles

Parsimonious Tensor Dimension Reduction

Tensor data is emerging in many scientific applications, such as multi-tissue transcriptomics. In such cases, the covariates for each individual are no longer a vector. To apply traditional vector-based methods to this type of data, we need to either do the vectorization or analyze data marginally, which suffers a significant information loss. We propose a novel parsimonious tensor dimension reduction (pTDR) approach to directly link the response and tensor covariate through an unknown function g. In pTDR, the response variable, continuous or discrete, depends on K rank-one projections of the covariates, with the projections estimated via a sequential iterative dimension reduction algorithm. We further propose an asymptotic sequential statistical test to select the correct number of rank-one tensors. In contrast to the classic low-rank tensor regression, pTDR model is not restricted to the linear relationship between response and covariates. We apply pTDR to two modern genomic studies. We find that the gene expression of multiple tissues has a stronger association with aging and obesity than was apparent using previous approaches. Numerical results demonstrate the advantages of pTDR over competitors in terms of prediction accuracy and computing efficiency. Our software is publicly available on GitHub (https://github.com/BioAlgs/pTDR).

Interferometric Lensless Imaging: Rank-One Projections of Image Frequencies With Speckle Illuminations

Interferometric Lensless Imaging: Rank-One Projections of Image Frequencies With Speckle Illuminations

Phase retrieval for affine groups over prime fields

We study phase retrieval for group frames arising from permutation representations, focusing on the action of the affine group of a finite field. We investigate various versions of the phase retrieval problem, including conjugate phase retrieval, sign retrieval, and matrix recovery. Our main result establishes that the canonical irreducible representation of the affine group Zp⋊Zp⁎ (with p prime), acting on the vectors in Cp with zero-sum, has the strongest retrieval property, allowing to reconstruct matrices from scalar products with a group orbit consisting of rank-one projections. We explicitly characterize the generating vectors that ensure this property, provide a linear matrix recovery algorithm and explicit examples of vectors that allow matrix recovery. We also comment on more general permutation representations.

Iteratively consistent one-bit phase retrieval

We introduce the ICQPhase algorithm for iteratively consistent quantized phase retrieval. ICQPhase addresses the problem of recovering a signal [Formula: see text] or [Formula: see text] from one-bit quantized phaseless measurements [Formula: see text], [Formula: see text], where [Formula: see text] are orthogonal projections and where [Formula: see text] is a one-bit scalar quantizer. We numerically validate the performance of ICQPhase and demonstrate that it experimentally achieves mean squared error of order [Formula: see text] in a variety of settings. We provide a theoretical analysis of the algorithm for the case of rank-one projections of signals in two-dimensional real space and prove that the mean squared error is of order [Formula: see text].

Rank computation in Euclidean Jordan algebras

Euclidean Jordan algebras are the abstract foundation for symmetric cone optimization. Every element in a Euclidean Jordan algebra has a complete spectral decomposition analogous to and subsuming that of a real symmetric matrix into rank-one projections. This general spectral decomposition stems from the element's likewise-analogous characteristic polynomial whose degree (they all have the same degree) is called the rank of the algebra. As a prerequisite for the spectral decomposition, we derive an algorithm that computes the rank of a Euclidean Jordan algebra and allows us to construct the characteristic polynomials of its elements. The ultimate goal of this work is to support a generic computational framework for solving symmetric cone optimization problems in Jordan-algebraic terms.

Three-component coupled nonlinear Schrödinger system in a multimode optical fiber: Darboux transformation induced via a rank-two projection matrix

In this paper, the investigation is on a three-component coupled nonlinear Schrödinger (NLS) system which governs the wave evolution in a multimode optical fiber. We construct a Darboux transformation (DT) induced via a rank-two projection matrix, and then derive an (N,m)-generalized DT and the Nth-order solution, where the positive integers N and m denote the numbers of iterative times and distinct spectral parameters, respectively. Focusing on the Nth-order solution on the nonzero–zero–zero background, we derive two kinds of waves which could not be derived by the DT induced via a rank-one projection matrix, that is, the so-called fundamental nonlinear wave (N=1) and degenerate fundamental nonlinear wave (N=2 and m=1). Via the asymptotic analysis, we find that the fundamental nonlinear wave is the nonlinear superposition of two dark–bright–bright solitons and a breather/Kuznetsov–Ma​ breather/rogue wave; and the degenerate fundamental nonlinear wave is the nonlinear superposition of four dark–bright–bright solitons and two breathers/two Kuznetsov–Ma​ breathers/a second-order rogue wave. Since such phenomena are not admitted for the one-component NLS equation and two-component coupled NLS system, they are more useful to understand the three-component coupled NLS system than what the latter two models admit. For other three-component coupled systems, more phenomena may be expected when the rank of projection matrix used to construct a DT is two rather than one, because our study presents an example.

A geometric approach to Wigner‐type theorems

Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

Low-Rank Matrix Estimation from Rank-One Projections by Unlifted Convex Optimization

We study an estimator with a convex formulation for recovery of low-rank matrices from rank-one projections. Using initial estimates of the factors of the target $d_1\times d_2$ matrix of rank-$r$,...

Rank-One Network: An Effective Framework for Image Restoration.

The principal rank-one (RO) components of an image represent the self-similarity of the image, which is an important property for image restoration. However, the RO components of a corrupted image could be decimated by the procedure of image denoising. We suggest that the RO property should be utilized and the decimation should be avoided in image restoration. To achieve this, we propose a new framework comprised of two modules, i.e., the RO decomposition and RO reconstruction. The RO decomposition is developed to decompose a corrupted image into the RO components and residual. This is achieved by successively applying RO projections to the image or its residuals to extract the RO components. The RO projections, based on neural networks, extract the closest RO component of an image. The RO reconstruction is aimed to reconstruct the important information, respectively from the RO components and residual, as well as to restore the image from this reconstructed information. Experimental results on four tasks, i.e., noise-free image super-resolution (SR), realistic image SR, gray-scale image denoising, and color image denoising, show that the method is effective and efficient for image restoration, and it delivers superior performance for realistic image SR and color image denoising. Our source code is available online.

The Distance from a Rank Projection to the Nilpotent Operators on

AbstractBuilding on MacDonald’s formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$ is $\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where Q is a projection of rank $n-1$ and $T\in \mathbb {M}_n(\mathbb {C})$ is a nilpotent of minimal distance to Q. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.

Moment Maps and Galois Orbits in Quantum Information Theory

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $\mathbb{C}^d$. The resulting equations are interpreted in terms of moment maps by focusing attention on the orbit of a cyclic subgroup and the maximal torus in $\mathrm{U}(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.

Frame potential for finite-dimensional Banach spaces

We define the frame potential for a Schauder frame on a finite dimensional Banach space as the square of the 2-summing norm of the frame operator. As is the case for frames for Hilbert spaces, we prove that the frame potential can be used to characterize finite unit norm tight frames (FUNTFs) for finite dimensional Banach spaces. We prove the existence of FUNTFs for a variety of spaces, and in particular that every n-dimensional complex Banach space with a 1-unconditional basis has a FUNTF of N vectors for every N≥n. However, many interesting results on FUNTFs and sums of rank-one projections for Hilbert spaces remain unknown for Banach spaces and we conclude the paper with multiple open questions.

Iterative hard thresholding for low-rank recovery from rank-one projections

A novel algorithm for the recovery of low-rank matrices acquired via compressive linear measurements is proposed and analyzed. The algorithm, a variation on the iterative hard thresholding algorithm for low-rank recovery, is designed to succeed in situations where the standard rank-restricted isometry property fails, e.g. in case of subexponential unstructured measurements or of subgaussian rank-one measurements. The stability and robustness of the algorithm are established based on distinctive matrix-analytic ingredients and its performance is substantiated numerically.

A notion of symmetry witness related to Wigner’s theorem on symmetry transformations

A symmetry witness is a subset of the space of selfadjoint trace class operators that allows one to ascertain whether a linear map acting in that space is a symmetry transformation. This notion arises from a certain type of linear preserver problems. Precisely, a symmetry witness is a suitable set which is invariant with respect to an injective linear map in the Banach space of selfadjoint trace class operators where the quantum states live if and only if this map acts as a symmetry transformation. In particular, by a linear version of Wigner’s classical theorem, the set of pure states — the rank-one projections — is a symmetry witness. Linearity entails that the usual assumption of preservation of the transition probability between pure states becomes superfluous. This result extends to every set of projections of a fixed (finite) rank, with some suitable constraint on this rank. One then obtains a classification of the sets of projections of a fixed rank that are symmetry witnesses. These symmetry witnesses are projectable. Namely, formulating the mentioned result in terms of quantum states, the sets of ‘uniform’ density operators of a suitable fixed rank are symmetry witnesses as well.

Symmetry witnesses

A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigner’s theorem, the set of pure states—the rank-one projections—is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the sets of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable; i.e. reasoning in terms of quantum states, the sets of ‘uniform’ density operators of corresponding fixed rank are symmetry witnesses too.

Wigner's theorem on Grassmann spaces

Wigner's celebrated theorem, which is particularly important in the mathematical foundations of quantum mechanics, states that every bijective transformation on the set of all rank-one projections of a complex Hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator. This vital theorem has been generalised in various ways by several scientists. In 2001, Molnár provided a natural generalisation, namely, he provided a characterisation of (not necessarily bijective) maps which act on the Grassmann space of all rank-n projections and leave the system of Jordan principal angles invariant (see [17] and [20]). In this paper we give a very natural joint generalisation of Wigner's and Molnár's theorems, namely, we prove a characterisation of all (not necessarily bijective) transformations on the Grassmann space which fix the quantity TrPQ (i.e. the sum of the squares of cosines of principal angles) for every pair of rank-n projections P and Q.

Linear algebraic proof of Wigner theorem and its consequences

Abstract We present new proof of non-bijective Wigner theorem on symmetries of quantum systems using only basic linear algebra. It is based on showing that any non-zero Jordan ∗-homomorphism between matrix algebras preserving rank-one projections is implemented by either a unitary or an anitiunitary map. As a new application we extend hitherto known results on preservers of quantum relative entropy to infinite quantum systems.

A Finite Algorithm to Compute Rank-1 Tensor Approximations

We propose a noniterative algorithm, called SeROAP, 1 1 Sequential Rank-One Approximation and Projection.

Video Denoising Based on Riemannian Manifold Similarity and Rank-One Projection

In this paper, we combine two powerful tools to handle the video denoising problem: one is an effective video denoising method based on Riemannian Manifold Similarity, and the other is a Rank-One Projection matrix completion based on video denoising method. Similarly, in our algorithm, a noisy video is processed in block-wise manner for each processed block. we form a 3D data array that we call “group” by stacking together blocks which is found similar to the currently processed one. “Collaborative filtering” exploits the correlation between grouped blocks and the corresponding highly sparse representation of the true signal in the transform domain. By employ Rank-One Projection matrix completion method in our framework, our technique is also robust to different types of noise. Experiments demonstrate that our techniques produce state-of-the-art results for video denoising applications.

ROP: Matrix recovery via rank-one projections

Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is adaptive to the rank and robust against small perturbations. Both upper and lower bounds for the estimation accuracy under the Frobenius norm loss are obtained. The proposed estimator is shown to be rate-optimal under certain conditions. The estimator is easy to implement via convex programming and performs well numerically. The techniques and main results developed in the paper also have implications to other related statistical problems. An application to estimation of spiked covariance matrices from one-dimensional random projections is considered. The results demonstrate that it is still possible to accurately estimate the covariance matrix of a high-dimensional distribution based only on one-dimensional projections.