D-dimensional Subspace Research Articles

Maximal subspace averages

We consider maximal operators associated to singular averages along finite subsets Σ of the Grassmannian Gr(d,n) of d-dimensional subspaces of Rn. The well studied d=1 case corresponds to the directional maximal function with respect to arbitrary finite subsets of Gr(1,n)=Sn−1. We provide a systematic study of all cases 1≤d<n and prove essentially sharp L2(Rn) bounds for the maximal subspace averaging operator in terms of the cardinality of Σ, with no assumption on the structure of Σ. In the codimension 1 case, that is n=d+1, we prove the precise critical weak (2,2)-bound.Drawing on the analogy between maximal subspace averages and (d,n)-Nikodym maximal averages, we also formulate the appropriate maximal Nikodym conjecture for general 1<d<n by providing examples that determine the critical Lp-space for the (d,n)-Nikodym problem. Unlike the d=1 case, the maximal Kakeya and Nikodym problems are shown not to be equivalent when d>1. In this context, we prove the best possible L2(Rn)-bound for the (d,n)-Nikodym maximal function for all combinations of dimension and codimension.Our estimates rely on Fourier analytic almost orthogonality principles, combined with polynomial partitioning, but we also use spatial analysis based on the precise calculation of intersections of d-dimensional plates in Rn.

Grassmann graphs, degenerate DAHA, and non-symmetric dual q-Hahn polynomials

We discuss the Grassmann graph Jq(N,D) with N≥2D, having as vertices the D-dimensional subspaces of an N-dimensional vector space over the finite field Fq. This graph is distance-regular with diameter D; to avoid trivialities we assume D≥3. Fix a pair of a Delsarte clique C of Jq(N,D) and a vertex x in C. We construct a 2D-dimensional irreducible module W for the Terwilliger algebra T of Jq(N,D) associated with the pair x, C. We show that W is an irreducible module for the confluent Cherednik algebra HV and describe how the T-action on W is related to the HV-action on W. Using the HV-module W, we define non-symmetric dual q-Hahn polynomials and prove their recurrence and orthogonality relations from a combinatorial viewpoint.

The lattice of subspaces of a vector space over a finite field

For finite m and q we study the lattice mathbf {L}(mathbf {V})=(L(mathbf {V}),+,cap ,{vec {0}},V) of subspaces of an m-dimensional vector space mathbf {V} over a field mathbf {K} of cardinality q. We present formulas for the number of d-dimensional subspaces of mathbf {V}, for the number of complements of a subspace and for the number of e-dimensional subspaces including a given d-dimensional subspace. It was shown in Eckmann and Zabey (Helv Phys Acta 42:420–424, 1969) that mathbf {L}(mathbf {V}) possesses an orthocomplementation only in case m=2 and {{,mathrm{char},}}mathbf {K}ne 2. Hence, only in this case mathbf {L}(mathbf {V}) can be considered as an orthomodular lattice. On the contrary, we show that a complementation ' on mathbf {L}(mathbf {V}) can be chosen in such a way that (L(mathbf {V}),+,cap ,{}') is both weakly orthomodular and dually weakly orthomodular. Moreover, we show that (L(mathbf {V}),+,cap ,{}^perp ,{vec {0}},V) is paraorthomodular in the sense of Giuntini et al. (Stud Log 104:1145–1177, 2016).

Quantum effective potential from discarded degrees of freedom

I obtain the quantum correction ΔVeff=(ħ2/8m)[(1−4ξd+1d)(S′)2+2(1−4ξ)S″] that appears in the effective potential whenever a compact d-dimensional subspace (of volume ∝exp⁡[S(x)]) is discarded from the configuration space of a nonrelativistic particle of mass m and curvature coupling parameter ξ. This correction gives rise to a force −〈ΔVeff′〉 that pushes the expectation value 〈x〉 off its classical trajectory. Because ΔVeff does not depend on the details of the discarded subspace, these results constitute a generic model of the quantum effect of discarded variables with maximum entropy/information capacity S(x).

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Let Gr(d; n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V n. A submanifold X Gr(d; n) gives rise to a differential system ⊂(X) that governs d-dimensional submanifolds of V n whose Gaussian image is contained in X. Systems of the form Σ(X) appear in numerous applications in continuum mechanics, theory of integrable systems, general relativity and differential geometry. They include such wellknown examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, Plebansky's heavenly equations, and so on. In this paper we concentrate on the particularly interesting case of this construction where X is a fourfold in Gr(3; 5). Our main goal is to investigate differential-geometric and integrability aspects of the corresponding systems Σ(X). We demonstrate the equivalence of several approaches to dispersionless integrability such as • the method of hydrodynamic reductions, • the method of dispersionless Lax pairs, • integrability on solutions, based on the requirement that the characteristic variety of system Σ(X) defines an Einstein-Weyl geometry on every solution, • integrability on equation, meaning integrability (in twistor-theoretic sense) of the canonical GL(2;R) structure induced on a fourfold X ⊂ Gr(3; 5). All these seemingly different approaches lead to one and the same class of integrable systems Σ(X). We prove that the moduli space of such systems is 6-dimensional. We give a complete description of linearisable systems (the corresponding fourfold X is a linear section of Gr(3; 5)) and linearly degenerate systems (the corresponding fourfold X is the image of a quadratic map P4 99K Gr(3; 5)). The fourfolds corresponding to `generic' integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.

Integrable Systems in Four Dimensions Associated with Six-Folds in Gr(4, 6)

AbstractLet Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a six-fold in Gr(4, 6). The corresponding system Σ(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems Σ(X). These naturally fall into two subclasses.• Systems of Monge–Ampère type. The corresponding six-folds X are codimension 2 linear sections of the Plücker embedding Gr(4, 6)$ \hookrightarrow \mathbb{P}^{14}$.• General linearly degenerate systems. The corresponding six-folds X are the images of quadratic maps $\mathbb{P}^{6}\dashrightarrow \ $Gr(4, 6) given by a version of the classical construction of Chasles.We prove that integrability is equivalent to the requirement that the characteristic variety of system Σ(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.

Robust principal component analysis based on trimming around affine subspaces

Principal Component Analysis (PCA) is a widely used technique for reducing dimensionality of multivariate data. The principal component subspace is defined as the affine subspace of a given dimension d giving the best fit to the data. However, PCA suffers from a well-known lack of robustness. As a robust alternative, one can resort to an impartial trimming based approach. Here one searches for the best subsample containing a proportion 1 − α of the observations, with 0 < α < 1, and the best d-dimensional affine subspace fitting this subsample, yielding the trimmed principal component subspace. A population version will be given and existence of a solution to both the sample and population problem will be proven. Moreover, under mild conditions, the solutions of the sample problem are consistent toward the solutions of the population problem. The robustness of the method is studied

Optimal quantum state identification with qudit-encoded unknown states

We consider the problem of optimally identifying the state of a probe qudit, prepared with given prior probability in a pure state belonging to a finite set of possible states which together span a D-dimensional subspace of the d-dimensional Hilbert space the qudit is defined in.It is assumed that we do not know some or all of the states in the set, but for each unknown state we are given a reference qudit into which this state is encoded. We show that from the measurement for optimal state identification with d=D one can readily determine the optimal figure of merit for qudits with d>D, without solving a new optimization problem. This result is applied to the minimum-error identification and to the optimal unambiguous identification of two qudit states with d > 2, where either one or both of the states are unknown, and also to the optimal unambiguous identification of N equiprobable linearly independent unknown pure qudit states with d > N. In all cases the optimal figure of merit, averaged over the unknown states, increases with growing dimensionality d of the qudits.

Counting d-dimensional hyperinvariant subspaces of a Weyr matrix

A characterization of hyperinvariant subspaces is given in terms of Weyr characteristic. Using this characterization we compute the number of the d-dimensional hyperinvariant subspaces.

Robust subspace recovery by Tyler's M-estimator

This paper considers the problem of robust subspace recovery: given a set of N points in RD, if many lie in a d-dimensional subspace, then can we recover the underlying subspace? We show that Tyler’s M-estimator can be used to recover the underlying subspace, if the percentage of the inliers is larger than d/D and the data points lie in general position. Empirically, Tyler’s M-estimator compares favorably with other convex subspace recovery algorithms in both simulations and experiments on real data sets.

$${l_p}$$ l p -Recovery of the Most Significant Subspace Among Multiple Subspaces with Outliers

We assume data sampled from a mixture of d-dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the lp-averaged distances of data points from d-dimensional subspaces, where p>0. Unlike other lp minimization problems, this minimization is non-convex for all p>0 and thus requires different methods for its analysis. We show that if 0<p<=1, then for any fraction of outliers the most significant subspace can be recovered by lp minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces the most significant subspace can be nearly recovered by lp minimization for any 0<p<=1 with an error proportional to the noise level. On the other hand, if p>1 and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.

A flag representation for finite collections of subspaces of mixed dimensions

Given a finite set of subspaces of Rn, perhaps of differing dimensions, we describe a flag of vector spaces (i.e. a nested sequence of vector spaces) that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation. The utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions. When the set of subspaces all have the same dimension d, the flag mean is related to several commonly used subspace representations. For instance, the d-dimensional subspace in the flag corresponds to the extrinsic manifold mean. When the set of subspaces is both well clustered and equidimensional of dimension d, then the d-dimensional component of the flag provides an approximation to the Karcher mean. An intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of Multiset Canonical Correlation Analysis. Two examples utilizing the Carnegie Mellon University Pose, Illumination, and Expression Database (CMU-PIE) serve as visual illustrations of the algorithm.

Discrete Tomography determination of bounded sets in [formula omitted

This paper deals with the uniqueness problem for bounded sets in Discrete Tomography, namely to decide when an unknown finite subset of points in a multidimensional grid is uniquely determined by its X-rays corresponding to a given set of lattice directions.In Brunetti et al. (2013) the authors completely settled the problem in dimension two, by proving that whole families of suitable sets of four directions can be found, for which they provided a complete characterization.In this paper we examine the problem in higher dimensions. We show that d+1 represents the minimal number of directions we need in Zn (n≥d≥3) to distinguish all the subsets of a finite grid by their X-rays in these directions, under the requirement that such directions span a d-dimensional subspace of Zn. It turns out that the situation differs from the planar case, where less than four directions are never sufficient for uniqueness. Moreover we characterize the minimal sets of directions of uniqueness by providing a necessary and sufficient condition. As a consequence, results in Brunetti et al. (2013) can be easily extended to the n-dimensional case.

Invariant Measure Under the Affine Group Over

A rational polyhedron$P\subseteq {\mathbb{R^n}}$ is a finite union of simplexes in ${\mathbb{R^n}}$ with rational vertices. P is said to be $\mathbb Z$-homeomorphic to the rational polyhedron $Q\subseteq {\mathbb{R^{\it m}}}$ if there is a piecewise linear homeomorphism η of P onto Q such that each linear piece of η and η−1 has integer coefficients. When n=m, $\mathbb Z$-homeomorphism amounts to continuous $\mathcal{G}_n$-equidissectability, where $\mathcal{G}_n=GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$ is the affine group over the integers, i.e., the group of all affinities on $\mathbb{R^{n}}$ that leave the lattice $\mathbb Z^{n}$ invariant. $\mathcal{G}_n$ yields a geometry on the set of rational polyhedra. For each d=0,1,2,. . ., we define a rational measure λd on the set of rational polyhedra, and show that any two $\mathbb Z$-homeomorphic rational polyhedra $$P\subseteq {\mathbb{R^n}}$$ and $Q\subseteq {\mathbb{R^{\it m}}}$ satisfy $\lambda_d(P)=\lambda_d(Q)$. $\lambda_n(P)$ coincides with the n-dimensional Lebesgue measure of P. If 0 ≤ dim P=d &lt; n then λd(P)&gt;0. For rational d-simplexes T lying in the same d-dimensional affine subspace of ${\mathbb{R^{\it n}}, $\lambda_d(T)$$ is proportional to the d-dimensional Hausdorff measure of T. We characterize λd among all unimodular invariant valuations.

Incidence matrices of finite attenuated spaces and class dimension of association schemes

Let d,k,n be integers with 1≤d<k≤n−d. In Kantor (1972), Kantor proved that the incidence matrix of d-dimensional subspaces versus k-dimensional subspaces of an n-dimensional vector space has full row rank over the real number field R. In this paper, we generalize Kantor’s result to the attenuated space A(n+ℓ;Fq) and show that the incidence matrix of d-dimensional subspaces versus k-dimensional subspaces of A(n+ℓ;Fq) also has full row rank over R. As an application, we obtain upper bounds for the class dimension of association schemes based on attenuated spaces.

Local behavior of traces of Besov functions: Prevalent results

Let 1⩽d<D and (p,q,s) satisfying 0<p<∞, 0<q⩽∞, 0<s−d/p<∞. In this article we study the global and local regularity properties of traces, on affine subsets of RD, of functions belonging to the Besov space Bp,qs(RD). Given a d-dimensional subspace H⊂RD, for almost all functions in Bp,qs(RD) (in the sense of prevalence), we are able to compute the singularity spectrum of the traces fa of f on affine subspaces of the form a+H, for Lebesgue-almost every a∈RD−d. In particular, we prove that for Lebesgue-almost every a∈RD−d, these traces fa are more regular than what could be expected from standard trace theorems, and that fa enjoys a multifractal behavior.

LEAST SQUARES APPROXIMATIONS OF MEASURES VIA GEOMETRIC CONDITION NUMBERS

For a probability measure on a real separable Hilbert space, we are interested in "volume-based" approximations of the d-dimensional least squares error of it, i.e., least squares error with respect to a best fit d-dimensional affine subspace. Such approximations are given by averaging real-valued multivariate functions which are typically scalings of squared (d+1)-volumes of (d+1)-simplices. Specifically, we show that such averages are comparable to the square of the d-dimensional least squares error of that measure, where the comparison depends on a simple quantitative geometric property of it. This result is a higher dimensional generalization of the elementary fact that the double integral of the squared distances between points is proportional to the variance of measure. We relate our work to two recent algorithms, one for clustering affine subspaces and the other for Monte-Carlo SVD based on volume sampling.

Robust recovery of multiple subspaces by geometric lp minimization

We assume i.i.d. data sampled from a mixture distribution with K components along fixed d-dimensional linear subspaces and an additional outlier component. For p>0, we study the simultaneous recovery of the K fixed subspaces by minimizing the l_p-averaged distances of the sampled data points from any K subspaces. Under some conditions, we show that if $0<p\leq1$, then all underlying subspaces can be precisely recovered by l_p minimization with overwhelming probability. On the other hand, if K>1 and p>1, then the underlying subspaces cannot be recovered or even nearly recovered by l_p minimization. The results of this paper partially explain the successes and failures of the basic approach of l_p energy minimization for modeling data by multiple subspaces.

INFLATING BALLS IS NP-HARD

A collection [Formula: see text] of balls in ℝd is δ-inflatable if it is isometric to the intersection [Formula: see text] of some d-dimensional affine subspace E with a collection [Formula: see text] of (d + δ)-dimensional balls that are disjoint and have equal radius. We give a quadratic-time algorithm to recognize 1-inflatable collections of balls in any fixed dimension, and show that recognizing δ-inflatable collections of d-dimensional balls is NP-hard for δ ≥ 2 and d ≥ 3 if the balls' centers and radii are given by numbers of the form [Formula: see text] where a, …, e are integers.

Recent results on designs with classical parameters

We report on recent results concerning designs with the same parameters as the classical geometric designs PGd(n, q) formed by the points and d-dimensional subspaces of the n-dimensional projective space PG(n, q) over the field GF(q) with q elements, where 1 ≤ d ≤ n−1. The corresponding case of designs with the same parameters as the classical geometric designs AGd(n, q) formed by the points and d-dimensional subspaces of the n-dimensional affine space AG(n, q) will also be discussed, albeit in less detail.