Affine Subspace Research Articles

On a cube and subspace projections

We consider the arrangement of vertices of a unit multidimensional cube, an affine subspace, and its orthogonal projections onto coordinate subspaces. Upper and lower bounds on the subspace dimension are given under which some orthogonal projection always preserves the incidence relation between the subspace and cube vertices. Some oblique projections are also considered. Moreover, a brief review of the history of the development of multidimensional descriptive geometry is given. Analytic and synthetic methods in geometry diverged since the 17th century. Although both synthesis and analysis are tangled, from this time forth many geometers as well as engineers keep up a nice distinction. One can find references to the idea of higher-dimensional spaces in the 18th-century works, but proper development has been since the middle of the 19th century. Soon such works have appeared in Russian. Next, mathematicians generalized their theories to many dimensions. Our new results are obtained by both analytic and synthetic methods. They illustrate the complexity of pseudo-Boolean programming problems because reducing the problem dimension by orthogonal projection meets obstacles in the worst case.

Effective Lower Bounds on the Matrix Rank and Their Applications

We propose an efficiently verifiable lower bound on the rank of a sparse fully indecomposable square matrix that contains two non-zero entries in each row and each column. The rank of this matrix is equal to its order or differs from it by one. Bases of a special type are constructed in the spaces of quadratic forms in a fixed number of variables. The existence of these bases allows us to substantiate a heuristic algorithm for recognizing whether a given affine subspace passes through a vertex of a multidimensional unit cube. In the worst case, the algorithm may output a computation denial warning; however, for the general subspace of sufficiently small dimension, it correctly rejects the input. The algorithm is implemented in Python. The running time of its implementation is estimated in the process of testing.

Affine Subspace Robust Low-Rank Self-Representation: from Matrix to Tensor.

Low-rank self-representation based subspace learning has confirmed its great effectiveness in a broad range of applications. Nevertheless, existing studies mainly focus on exploring the global linear subspace structure, and cannot commendably handle the case where the samples approximately (i.e., the samples contain data errors) lie in several more general affine subspaces. To overcome this drawback, in this paper, we innovatively propose to introduce affine and nonnegative constraints into low-rank self-representation learning. While simple enough, we provide their underlying theoretical insight from a geometric perspective. The union of two constraints geometrically restricts each sample to be expressed as a convex combination of other samples in the same subspace. In this way, when exploring the global affine subspace structure, we can also consider the specific local distribution of data in each subspace. To comprehensively demonstrate the benefits of introducing two constraints, we instantiate three low-rank self-representation methods ranging from single-view low-rank matrix learning to multi-view low-rank tensor learning. We carefully design the solution algorithms to efficiently optimize the proposed three approaches. Extensive experiments are conducted on three typical tasks, including single-view subspace clustering, multi-view subspace clustering, and multi-view semi-supervised classification. The notably superior experimental results powerfully verify the effectiveness of our proposals.

Bicycling geodesics are Kirchhoff rods

A bicycle path is a pair of trajectories in , the ‘front’ and ‘back’ tracks, traced out by the endpoints of a moving line segment of fixed length (the ‘bicycle frame’) and tangent to the back track. Bicycle geodesics are bicycle paths whose front track’s length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily of Kirchhoff rods, a class of curves introduced in 1859 by Kirchhoff, generalizing the planar elastic curves of Bernoulli and Euler.

Affine Subspace Concentration Conditions

We define a new notion of affine subspace concentration conditions for lattice polytopes, and prove that they hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class $c_1(\mathcal{T}_X)$ on Fano toric varieties.

Subspace coverings with multiplicities

AbstractWe study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$ . The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and Füredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k\geq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-\left\lfloor{\frac{k}{2^{n-d}}}\right\rfloor$ , while for $n \gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$ , and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.

Alternating projection‐based iterative learning control for discrete‐time systems with non‐uniform trial lengths

AbstractThis article develops a novel framework for iterative learning control (ILC) design of discrete‐time systems with non‐uniform trial lengths by using the method of alternating projections. In contrast to existing results for the non‐uniform trial length problem, this article uses the Hilbert space setting and hence the linear discrete‐time system dynamics with non‐uniform trial lengths can be represented by multiple affine subspaces (or linear varieties). Motivated by the successive projection design between two closed convex sets, the considered ILC problem can be transformed into alternating projections between multiple sets, then the Hilbert space setting is used to establish key systems theoretic properties. Moreover, an optimal ILC design is developed for systems with non‐uniform trial lengths, which is also extended to the case of input constraints. A numerical case study is given to illustrate the applicability of the new design.

Learning spatio-temporal discriminative model for affine subspace based visual object tracking

Discriminative correlation filters (DCF) with powerful feature descriptors have proven to be very effective for advanced visual object tracking approaches. However, due to the fixed capacity in achieving discriminative learning, existing DCF trackers perform the filter training on a single template extracted by convolutional neural networks (CNN) or hand-crafted descriptors. Such single template learning cannot provide powerful discriminative filters with guaranteed validity under appearance variation. To pinpoint the structural relevance of spatio-temporal appearance to the filtering system, we propose a new tracking algorithm that incorporates the construction of the Grassmannian manifold learning in the DCF formulation. Our method constructs the model appearance within an online updated affine subspace. It enables joint discriminative learning in the origin and basis of the subspace, achieving enhanced discrimination and interpretability of the learned filters. In addition, to improve tracking efficiency, we adaptively integrate online incremental learning to update the obtained manifold. To this end, specific spatio-temporal appearance patterns are dynamically learned during tracking, highlighting relevant variations and alleviating the performance degrading impact of less discriminative representations from a single template. The experimental results obtained on several well-known datasets, i.e., OTB2013, OTB2015, UAV123, and VOT2018, demonstrate the merits of the proposed method and its superiority over the state-of-the-art trackers.

Combinatorial Bernoulli factories

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter p∈[0,1] means the algorithm does not know p, but has sample access to independent draws of a Bernoulli random variable with mean equal to p. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector x∈P for a given polytope P⊂[0,1]n, output a randomized vertex such that the expected value of the i-th coordinate is exactly equal to xi. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix [xij] and asked to sample a matching such that the probability of each edge (i,j) be present in the matching is exactly equal to xij. We show that a polytope P admits a Bernoulli factory if and and only if P is the intersection of [0,1]n with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on P. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the k-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.1 1 A preliminary conference version of this work has appeared in the proceeding of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC’21) [24]. The current version presents all the missing proofs and technical details, as well as new results, alternative proofs, and more explanations.

Invariant affine subspaces

We study the problem: whether a given operator on a Banach space has an invariant affine subspace. We examine the existence or non-existence of such affine subspaces. Moreover, we give some applications.

Conic Linear Optimization for Computer-Assisted Proofs

From a mathematical perspective, optimization is the science of proving inequalities. In this sense, computational optimization is a method for computer-assisted proofs. Conic (linear) optimization is the problem of minimizing a linear functional over the intersection of a convex cone with an affine subspace of a topological vector space. For many cones this problem is computationally tractable, and as a result there is a growing number of computer-assisted proofs using conic optimization in discrete geometry, (extremal) graph theory, numerical analysis, and other fields, the most famous example perhaps being the proof of the Kepler Conjecture. The aim of this workshop was to bring researchers from these diverse fields together to work towards expanding the current scope of conic optimization as a method of generating proofs, and to identify problems and challenges to work on together.

Affine subspaces of antisymmetric matrices with constant rank

For every n ∈ N and every field K, let A ( n , K ) be the vector space of the antisymmetric ( n × n ) -matrices over K. We say that an affine subspace S of A ( n , K ) has constant rank r if every matrix of S has rank r. Define A antisym K ( n ; r ) = { S | S affine subspace of A ( n , K ) of constant rank r } , a antisym K ( n ; r ) = max { dim ⁡ S ∣ S ∈ A antisym K ( n ; r ) } . In this paper, we prove the following formulas: for n ≥ 2 r + 2 a antisym R ( n ; 2 r ) = ( n − r − 1 ) r ; for n = 2r a antisym R ( n ; 2 r ) = r ( r − 1 ) ; for n = 2r + 1 a antisym R ( n ; 2 r ) = r ( r + 1 ) .

A generalization of a theorem of Rothschild and van Lint

A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function f over F2n has the same absolute value, namely |fˆ(α)|=1/2k for every α in the Fourier support of f, then f must be the indicator function of some affine subspace of dimension n−k. In this paper we slightly generalize their result. Our main result shows that, roughly speaking, Boolean functions whose Fourier coefficients take values in the set {−2/2k,−1/2k,0,1/2k,2/2k} are indicator functions of two disjoint affine subspaces of dimension n−k or four disjoint affine subspaces of dimension n−k−1. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of F2n when the doubling constant of the subset is small.

Rapid antenna optimization with restricted sensitivity updates by automated dominant direction identification

Meticulous tuning of geometry parameters turns pivotal in improving performance of antenna systems. It is more and more often realized using formal optimization methods, which is demonstrably the most efficient way of handling multiple design variables, objectives, and constraints. Although in some cases a need for launching global search arises, a typical design scenario only requires local optimization, especially when a decent initial design can be rendered using engineering experience or parametric studies. At the same time, antenna optimization is typically conducted using full-wave electromagnetic (EM) simulations, which entails considerable computational expenses. In this paper, we introduce a novel procedure for expedited antenna tuning. Its fundamental mechanism is to restrict the antenna response sensitivity updates to the selected dominant directions within the parameter space, determined based on the problem-specific knowledge, in particular, the estimated changes of antenna characteristics when moving across one-dimensional affine subspaces spanned by these directions. Thus, the said dominant directions affect the most the responses of the antenna structure under design as assessed using the introduced metrics. The decision making process concerning the number of directions to be used relies on quantification of the aggregated system response variability metrics. The proposed approach is demonstrated by means of several antenna structures and benchmarked against conventional trust-region algorithm, but also its accelerated versions. The results indicate considerable (up to over 60%) speedup over the reference procedure without noticeable quality degradation.

Computationally Relaxed Unscented Kalman Filter.

Advanced robotics and autonomous vehicles rely on filtering and sensor fusion techniques to a large extent. These mobile applications need to handle the computations onboard at high rates while the computing capacities are limited. Therefore, any improvement that lowers the CPU time of the filtering leads to more accurate control or longer battery operation. This article introduces a generic computational relaxation for the unscented transformation (UT) that is the key operation of the Unscented Kalman filter-based applications. The central idea behind the relaxation is to pull out the linear part of the filtering model and avoid the calculations for the kernel of the nonlinear part. The practical merit of the proposed relaxation is demonstrated through a simultaneous localization and mapping (SLAM) implementation that underpins the superior performance of the algorithm in the practically relevant cases, where the nonlinear dependencies influence only an affine subspace of the image space. The numerical examples show that the computational demand can be mitigated below 50% without decreasing the accuracy of the approximation. The method described in this article is implemented and published as an open-source C ++ library RelaxedUnscentedTransformation on GitHub.

Affine subspace concentration conditions for centered polytopes

AbstractRecently, K.‐Y. Wu introduced affine subspace concentration conditions for the cone volumes of polytopes and proved that the cone volumes of centered, reflexive, smooth lattice polytopes satisfy these conditions. We extend the result to arbitrary centered polytopes.

Generalized Gearhart-Koshy acceleration for the Kaczmarz method

The Kaczmarz method is an iterative numerical method for solving large and sparse rectangular systems of linear equations. Gearhart, Koshy and Tam have developed an acceleration technique for the Kaczmarz method that minimizes the distance to the desired solution in the direction of a full Kaczmarz step. The present paper generalizes this technique to an acceleration scheme that minimizes the Euclidean norm error over an affine subspace spanned by a number of previous iterates and one additional cycle of the Kaczmarz method. The key challenge is to find a formulation in which all parameters of the least-squares problem defining the unique minimizer are known, and to solve this problem efficiently. When only a single Kaczmarz cycle is considered, the proposed affine search is more effective than the Gearhart-Koshy/Tam line-search, which in turn is more effective than the underlying Kaczmarz method. A numerical experiment from the context of computerized tomography suggests that the proposed affine search has the potential to outperform the the Gearhart-Koshy/Tam line-search and the underlying Kaczmarz method in terms of the computational cost that is needed to achieve a given error tolerance.

Linear-time approximation scheme for k-means clustering of axis-parallel affine subspaces

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with Δ>0 unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of n axis-parallel affine subspaces of dimension Δ that yields an (1+ϵ)-approximate solution in O(nd) time. The constants hidden behind O(⋅) depend only on Δ,ϵ and k. This improves the O(n2d)-time algorithm by Eiben et al. (2021) [7] by a factor of n.

On Lipschitz Geometry at infinity of complex analytic sets

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein’s problem. Moser’s Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex parametric version of Moser’s Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $$\mathbb {C}^n$$ which is Lipschitz regular at infinity must be an affine linear subspace of $$\mathbb {C}^n$$ . In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if X and Y are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then X is a complex algebraic set if and only if Y is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here.

Generalization of the Subset Sum Problem and Cubic Forms

A new algorithm is proposed for deciding whether a system of linear equations has a binary solution over a field of zero characteristic. The algorithm is efficient under a certain constraint on the system of equations. This is a special case of an integer programming problem. In the extended version of the subset sum problem, the weight can be positive or negative. The problem under consideration is equivalent to the analysis of solution existence for several instances of this problem simultaneously. New sufficient conditions are found under which the computational complexity of almost all instances of this problem is polynomial. In fact, the algorithm checks the existence of a cubic hypersurface that passes through each vertex of the unit cube, but does not intersect a given affine subspace. Several heuristic algorithms for solving this problem have been known previously. However, the new methods expand the solution possibilities. Although only the solution existence problem is considered in detail, binary search allows one to find a solution, if any.