Low Dimensional Linear Space Research Articles

A Cayley–Bacharach theorem and plane configurations

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the Cayley–Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.

Matrix and tensor rigidity and Lp-approximation

We apply methods originated in Complexity theory to some problems of Approximation. We notice that the construction of Alman and Williams that disproves the rigidity of Walsh-Hadamard matrices, provides good ℓp-approximation for p<2. Hence the first n functions of Walsh system can be approximated by a low-dimensional linear space:dn1−δ({w1,…,wn},Lp[0,1])⩽n−δ,1≤p<2, where δ=δ(p)>0. We do not know if this is possible for the trigonometric system. We show that the algebraic method of Alon–Frankl–Rödl for bounding the number of low-signum-rank matrices, works for tensors: almost all signum-tensors have large signum-rank and can't be ℓ1-approximated by low-rank tensors. This implies lower bounds for Θm — the error of m-term approximation of multivariate functions by sums of tensor products u1(x1)⋯ud(xd). For the set of trigonometric polynomials with spectrum in ∏j=1d[−nj,nj] and of norm ‖t‖∞⩽1, and m⩽c(d)∏nj/max⁡{nj}, we haveΘm(T(n1,…,nd)∞,L1[−π,π]d)⩾c1(d)>0. Sharp bounds follow for classes of dominated mixed smoothness in the case 2⩽p⩽∞,1⩽q⩽2:Θm(Wp(r,r,…,r),Lq[0,1]d)≍m−rdd−1.

Resolution Enhancement for Large-Scale Real Beam Mapping Based on Adaptive Low-Rank Approximation

Recently, a variety of super-resolution (SR) methods have been devoted to enhancing the angular resolution of real beam mapping (RBM) imagery in modern microwave remote sensing applications. When addressing large-scale datasets, however, they suffer from notably high computational complexity due to high-dimensional matrix inversion, multiplication, or singular value decomposition (SVD). To overcome this limitation, this article presents a low-complexity SR strategy based on adaptive low-rank approximation (LRA). Our underlying idea is first to construct a random matrix sketching to sample the raw echo measurements and restore the surface map of reflectivity in a low-dimensional linear space. The resulting low-complexity strategy enables substantial computational complexity reduction for a group of SR methods, at the cost of introducing a manually adjusted LRA parameter. Using the Fourier transform-based antenna analysis method, we further reveal that the LRA parameter that ensures support resolution improvement can be determined by a closed-form function of the aperture length, the wavelength, and the field of view, allowing for adaptively and efficiently selecting the optimal LRA parameter that well balances the tradeoff between LRA error and computational efficiency. We use both simulated and real datasets to demonstrate that the proposed LRA-based SR strategy can provide significant speedup without performance loss.

Data-Driven model order reduction for problems with parameter-dependent jump-discontinuities

We propose a data-driven model order reduction (MOR) technique for parametrized partial differential equations that exhibit parameter-dependent jump-discontinuities. Such problems have poor-approximability in a linear space and therefore, are challenging for standard MOR techniques. We build upon the methodology of approximating the map between the parameter domain and the expansion coefficients of the reduced basis via regression. The online stage queries the regression model for the expansion coefficients and recovers a reduced approximation for the solution. We propose to apply this technique to a transformed solution that results from composing the solution with a spatial transform. Unlike the (untransformed) solution, it is sufficiently regular along the parameter domain and thus, is well-approximable in a low-dimensional linear space. To recover an approximation for the (untransformed) solution, we propose an online efficient regression-based technique that approximates the inverse of the spatial transform. Our method features a decoupled online and offline stage, and benchmark problems involving hyperbolic and parabolic equations demonstrate its effectiveness.

Changing the Geometry of Representations: α-Embeddings for NLP Tasks.

Word embeddings based on a conditional model are commonly used in Natural Language Processing (NLP) tasks to embed the words of a dictionary in a low dimensional linear space. Their computation is based on the maximization of the likelihood of a conditional probability distribution for each word of the dictionary. These distributions form a Riemannian statistical manifold, where word embeddings can be interpreted as vectors in the tangent space of a specific reference measure on the manifold. A novel family of word embeddings, called -embeddings have been recently introduced as deriving from the geometrical deformation of the simplex of probabilities through a parameter , using notions from Information Geometry. After introducing the -embeddings, we show how the deformation of the simplex, controlled by , provides an extra handle to increase the performances of several intrinsic and extrinsic tasks in NLP. We test the -embeddings on different tasks with models of increasing complexity, showing that the advantages associated with the use of -embeddings are present also for models with a large number of parameters. Finally, we show that tuning allows for higher performances compared to the use of larger models in which additionally a transformation of the embeddings is learned during training, as experimentally verified in attention models.

Nonparametric registration to low-dimensional function spaces

Registration aims to decompose amplitude and phase variation of samples of curves. Phase variation is captured by warping functions which monotonically transform the domains. Resulting registered curves should then only exhibit amplitude variation. Most existing methods assume that all sample functions exhibit a typical sequence of shape features like peaks or valleys, and registration focuses on aligning these features. A more general perspective is adopted which goes beyond feature alignment. A registration method is introduced where warping functions are defined in such a way that the resulting registered curves span a low dimensional linear function space. The approach may be used as a tool for analyzing any type of functional data satisfying a structural regularity condition called bounded shape variation. Problems of identifiability are discussed in detail, and connections to established registration procedures are analyzed. The method is applied to real and simulated data.

Fast Single-Image Super-Resolution Via Tangent Space Learning of High-Resolution-Patch Manifold

Manifold assumption has been used in example-based super-resolution (SR) reconstruction from a single frame. Previous manifold-based SR approaches (more generally example-based SR) mainly focus on analyzing the co-occurrence properties of low-resolution and high-resolution patches. This paper develops a novel single-image SR approach based on linear approximation of the high-resolution-patch space using a sparse subspace clustering algorithm. The approach exploits the underlying high-resolution patches nonlinear space by considering it as a low-dimensional manifold in a high-dimensional Euclidean space, and by considering each training high-resolution-patch as a sample from the manifold. We utilize the sparse subspace clustering algorithm to create the set of low-dimensional linear spaces that are considered, approximately, as tangent spaces at the high-resolution samples. Furthermore, we consider and analyze each tangent space as one point in a Grassmann manifold, which helps to compute geodesic pairwise distances among these tangent spaces. An optimal subset of these tangent spaces is then selected using a min-max algorithm. The optimal subset reduces the computational cost in comparison with using the full set of tangent spaces while still preserving the quality of the high-resolution image reconstruction. In addition, we perform hierarchical clustering on the optimal subset based on the geodesic distance, which helps to further achieve much faster SR algorithm. We also analytically prove the validity of the geodesic distance based clustering under the proposed framework. A comparison of the obtained results with other related methods in both high-resolution image quality and computational complexity clearly indicates the viability of the proposed framework.

Real-time 3D eyelids tracking from semantic edges

State-of-the-art real-time face tracking systems still lack the ability to realistically portray subtle details of various aspects of the face, particularly the region surrounding the eyes. To improve this situation, we propose a technique to reconstruct the 3D shape and motion of eyelids in real time. By combining these results with the full facial expression and gaze direction, our system generates complete face tracking sequences with more detailed eye regions than existing solutions in real-time. To achieve this goal, we propose a generative eyelid model which decomposes eyelid variation into two low-dimensional linear spaces which efficiently represent the shape and motion of eyelids. Then, we modify a holistically-nested DNN model to jointly perform semantic eyelid edge detection and identification on images. Next, we correspond vertices of the eyelid model to 2D image edges, and employ polynomial curve fitting and a search scheme to handle incorrect and partial edge detections. Finally, we use the correspondences in a 3D-to-2D edge fitting scheme to reconstruct eyelid shape and pose. By integrating our fast fitting method into a face tracking system, the estimated eyelid results are seamlessly fused with the face and eyeball results in real time. Experiments show that our technique applies to different human races, eyelid shapes, and eyelid motions, and is robust to changes in head pose, expression and gaze direction.

Universal Manifold Embedding for Geometrically Deformed Functions

Assume we have a set of observations (for example, images) of different objects, each undergoing a different geometric deformation, yet all the deformations belong to the same family. As a result of the action of these deformations, the set of different observations on each object is generally a manifold in the ambient space of observations. In this paper we show that in those cases where the set of deformations admits a finite-dimensional representation, there is a mapping from the space of observations to a low-dimensional linear space. The manifold corresponding to each object is mapped to a distinct linear subspace of Euclidean space. The dimension of the subspace is the same as that of the manifold. This mapping, which we call universal manifold embedding, enables the estimation of geometric deformations using the classical linear theory. The universal manifold embedding further enables the representation of the object classification and detection problems in a linear subspace matching framework. The embedding of the space of observations depends on the deformation model, and is independent of the specific observed object; hence, it is universal. We study two cases of this embedding: that of elastic deformations of 1-D signals, and the case of affine deformations of $n$ -dimensional signals.

Linear Estimation of Time-Warped Signals

We introduce a novel methodology for estimating the time-axis deformation between two observations on a time-warped signal. Since the problem of estimating the warping function is nonlinear, existing methods iteratively minimize some metric between the observation and a hypothesized deformed template. Assuming the family of possible deformations the signal may undergo admits a finite-dimensional representation, we show that there is a nonlinear mapping from the space of observations to a low-dimensional linear space, such that in this space the problem of estimating the parametric model of the warping function is solved by a linear system of equations. We call the family of estimators derived based on this representation, linear warping estimators (LWE). The new representation of the problem enables an analytic analysis of the behavior of the solution in the presence of model mismatches, which is prohibitive when iterative methods are employed. The ability to achieve this major simplification both in the solution and in analyzing its performance results from the representation of the problem in a new coordinate system which is natural to the properties of the problem, instead of representing it in the standard coordinate system imposed by the sampling mechanism. The proposed solution is unique and exact, as it provides a closed-form expression for evaluating each of the parameters of the warping model using only measurements of the amplitude information of the observed and reference signals. The solution is applicable to any elastic warping regardless of its magnitude. We analyze the behavior of the LWE in the presence of noise and obtain a minimum variance unbiased estimator for the model parameters, by finding an optimal set of nonlinear operators for mapping the original problem into a low-dimensional linear space.

Feature constrained compressed sensing CT image reconstruction from incomplete data via robust principal component analysis of the database

In computed tomography (CT), incomplete data problems such as limited angle projections often cause artifacts in the reconstruction results. Additional prior knowledge of the image has shown the potential for better results, such as a prior image constrained compressed sensing algorithm. While a pre-full-scan of the same patient is not always available, massive well-reconstructed images of different patients can be easily obtained from clinical multi-slice helical CTs. In this paper, a feature constrained compressed sensing (FCCS) image reconstruction algorithm was proposed to improve the image quality by using the prior knowledge extracted from the clinical database. The database consists of instances which are similar to the target image but not necessarily the same. Robust principal component analysis is employed to retrieve features of the training images to sparsify the target image. The features form a low-dimensional linear space and a constraint on the distance between the image and the space is used. A bi-criterion convex program which combines the feature constraint and total variation constraint is proposed for the reconstruction procedure and a flexible method is adopted for a good solution. Numerical simulations on both the phantom and real clinical patient images were taken to validate our algorithm. Promising results are shown for limited angle problems.

Subspace video stabilization

We present a robust and efficient approach to video stabilization that achieves high-quality camera motion for a wide range of videos. In this article, we focus on the problem of transforming a set of input 2D motion trajectories so that they are both smooth and resemble visually plausible views of the imaged scene; our key insight is that we can achieve this goal by enforcing subspace constraints on feature trajectories while smoothing them. Our approach assembles tracked features in the video into a trajectory matrix, factors it into two low-rank matrices, and performs filtering or curve fitting in a low-dimensional linear space. In order to process long videos, we propose a moving factorization that is both efficient and streamable. Our experiments confirm that our approach can efficiently provide stabilization results comparable with prior 3D methods in cases where those methods succeed, but also provides smooth camera motions in cases where such approaches often fail, such as videos that lack parallax. The presented approach offers the first method that both achieves high-quality video stabilization and is practical enough for consumer applications.

General solution of lattices for Cartesian lattice Bhatanagar-Gross-Krook models

We give the general solutions of lattices, i.e., velocity sets and weights, for the lattice Bhatanagar-Gross-Krook (LBGK) models on two- and three-dimensional Cartesian grids. The solutions define the necessary and sufficient conditions so that the resulting LBGK model can accurately capture the dynamics of the moments retained in the distribution function. In the parameter space of the weights, the general solutions form low-dimensional linear spaces from which minimal velocity sets are identified for the degrees of precision that are most relevant to the construction of high-order LBGK models. All well-known LBGK lattices are found to be special cases of the given general solutions.

Acquiring linear subspaces for face recognition under variable lighting

Previous work has demonstrated that the image variation of many objects (human faces in particular) under variable lighting can be effectively modeled by low-dimensional linear spaces, even when there are multiple light sources and shadowing. Basis images spanning this space are usually obtained in one of three ways: A large set of images of the object under different lighting conditions is acquired, and principal component analysis (PCA) is used to estimate a subspace. Alternatively, synthetic images are rendered from a 3D model (perhaps reconstructed from images) under point sources and, again, PCA is used to estimate a subspace. Finally, images rendered from a 3D model under diffuse lighting based on spherical harmonics are directly used as basis images. In this paper, we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low-dimensional linear space and that this subspace is close to those acquired by the other methods. More specifically, there exist configurations of k point light source directions, with k typically ranging from 5 to 9, such that, by taking k images of an object under these single sources, the resulting subspace is an effective representation for recognition under a wide range of lighting conditions. Since the subspace is generated directly from real images, potentially complex and/or brittle intermediate steps such as 3D reconstruction can be completely avoided; nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse (harmonic) light fields. We validate the use of subspaces constructed in this fashion within the context of face recognition.