Affine Subspace Research Articles

Lower Bounds on Stabilizer Rank

The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩=∑j=1rcj|φj⟩ for cj∈C and stabilizer states φj. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n-th tensor power of single-qubit magic states.We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(n) of Bravyi, Smith and Smolin \cite{BSS16}. Further, we prove that for a sufficiently small constant δ, the stabilizer rank of any state which is δ-close to those states is Ω(n/log⁡n). This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F2n, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.

Stabilizer rank and higher-order Fourier analysis

We establish a link between stabilizer states, stabilizer rank, and higher-order Fourier analysis – a still-developing area of mathematics that grew out of Gowers's celebrated Fourier-analytic proof of Szemerédi's theorem \cite{gowers1998new}. We observe that n-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of Fpn where p is the dimension of the qudit) which are fundamental objects in higher-order Fourier analysis. This allows us to import tools from this theory to analyze the stabilizer rank of quantum states. Quite recently, in \cite{peleg2021lower} it was shown that the n-qubit magic state has stabilizer rank Ω(n). Here we show that the qudit analog of the n-qubit magic state has stabilizer rank Ω(n), generalizing their result to qudits of any prime dimension. Our proof techniques use explicitly tools from higher-order Fourier analysis. We believe this example motivates the further exploration of applications of higher-order Fourier analysis in quantum information theory.

On the rank of an A$A$‐hypergeometric D$D$‐module versus the normalized volume of A$A$

The rank of an $A$-hypergeometric $D$-module $M_A(\beta)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $\beta\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when $\beta$ lies outside the exceptional arrangement $\mathcal{E}(A)$, an affine subspace arrangement of codimension at least two. If $\beta\in \mathcal{E}(A)$ is simple, we prove that $d-1$ is a tight upper bound for the ratio $\mathrm{rank}(M_A(\beta))/\mathrm{vol}(A)$ for any $d\geq 3$. We also prove that the set of parameters $\beta$ such that this ratio is at least $2$ is an affine subspace arrangement of codimension at least $3$.

Online Acoustic System Identification Exploiting Kalman Filtering and an Adaptive Impulse Response Subspace Model

We introduce a novel algorithm for online estimation of Acoustic Impulse Responses (AIRs) which allows for fast convergence by exploiting prior knowledge about the fundamental structure of AIRs. The proposed method assumes that the variability of AIRs of an acoustic scene is confined to a low-dimensional manifold which is embedded in a high-dimensional space of possible AIR estimates. We discuss various approaches which exploit a training data set of AIRs, e.g., high-accuracy AIR estimates from the acoustic scene, to learn a local affine subspace approximation of the AIR manifold. The model is motivated by the idea of describing the generally nonlinear AIR manifold locally by tangential hyperplanes and its validity is verified for simulated data. Subsequently, we describe how the manifold assumption can be used to enhance online AIR estimates by projecting them onto an adaptively estimated subspace. Motivated by the assumption of manifolds being locally Euclidean, the parameters determining the adaptive subspace are learned from the nearest neighbor AIR training samples to the current AIR estimate. To assess the proximity of training data AIRs to the current AIR estimate, we introduce a probabilistic extension of the Euclidean distance which improves the performance for applications with non-white excitation signals. Furthermore, we describe how model imperfections can be tackled by a soft projection of the AIR estimates. The proposed algorithm exhibits significantly faster convergence properties in comparison to a high-performance state-of-the-art algorithm. Furthermore, we show an improved steady-state performance for speech-excited system identification scenarios suffering from high-level interfering noise and nonunique solutions.

Limiting Spherical Integrals of Bounded Continuous Functions

We use nonstandard analysis to study the limiting behavior of spherical integrals in terms of a Gaussian integral. Peterson and Sengupta proved that if a Gaussian measure $$\mu $$ has full support on a finite-dimensional Euclidean space, then the expected value of a bounded measurable function on that domain can be expressed as a limit of integrals over spheres $$S^{n-1}(\sqrt{n})$$ intersected with certain affine subspaces of $${\mathbb {R}}^n$$ . This allows one to realize the Gaussian Radon transform of such functions as a limit of spherical integrals. We study such limits in terms of Loeb integrals over a single hyperfinite-dimensional sphere. This nonstandard geometric approach generalizes the known limiting result for bounded continuous functions to the case when the Gaussian measure is not necessarily fully supported. We also present an asymptotic linear algebra result needed in the above proof.

On the Symmetric Lamination Convex and Quasiconvex Hull for the Coplanar $n$-Well Problem in Two Dimensions

We study some particular cases of the $n$ -well problem in two-dimensional linear elasticity. Assuming that every well in $\mathcal{U}\subset \mathbb{R}^{2\times 2}_{\text{sym}}$ belong to the same two-dimensional affine subspace, we characterize the symmetric lamination convex hull $L^{e}(\mathcal{U})$ for any number of wells in terms of the symmetric lamination convex hull of all three-well subsets contained in $\mathcal{U}$ . For a family of four-well sets where two pairs of wells are rank-one compatible, we show that the symmetric lamination convex and quasiconvex hulls coincide, but are strictly contained in its convex hull $C(\mathcal{U})$ . We extend this result to some particular configurations of $n$ wells. Most of the proofs are constructive, and we also present explicit examples.

Guided neighborhood affine subspace embedding for feature matching

Feature matching, which refers to determining reliable correspondences between two sets of feature points, is a fundamental component of numerous visual tasks. This paper proposes a novel method, termed as guided neighborhood affine subspace embedding (NASE), to eliminate false matches from the given tentative feature matches. Its essential philosophy is to preserve the underlying intrinsic manifold of potential true matches. Specifically, we aim to approximate the manifold of an inlier with an affine subspace fitted on its neighbors by imposing a motion-consistency constraint. Considering that the “corresponding manifold” of inliers may be biased by gross outliers, we introduce a density-based seed point selection strategy for neighborhood refinement. Based on the above two strategies, we further formulate the general feature matching problem into a mathematical optimization model and deduce a closed-form solution with linearithmic time complexity (i.e., O(NlogN)) for mismatch removal. Additionally, we devise a multi-scale strategy for neighborhood construction, making our method more robust to various degradations. Extensive experiments on general feature matching, fundamental matrix estimation, and loop closure detection demonstrate the clear superiority of NASE over the state-of-the-arts.

Weights of exact threshold functions

We consider Boolean exact threshold functions defined by linear equations and, more generally, polynomials of degree . We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close. In the linear case in particular they are almost matching. This quantity is the same as the maximum magnitude of the integer coefficients of linear equations required to express every possible intersection of a hyperplane in and the Boolean cube or, in the general case, intersections of hypersurfaces of degree in and . In the process we construct new families of ill-conditioned matrices. We further stratify the problem (in the linear case) in terms of the dimension of the affine subspace spanned by the solutions and give upper and lower bounds in this case as well. There is a substantial gap between these bounds, a challenge for future work.

Best approximation mappings in Hilbert spaces

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional spaces was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of a BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspaces to nonempty closed convex sets and from $${\mathbb {R}}^{n}$$ to general Hilbert spaces. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al. proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $${\mathbb {R}}^{n}$$ . We generalize their result from $${\mathbb {R}}^{n}$$ to general Hilbert spaces and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAMs with circumcenter mappings in Hilbert spaces and we report on Cegielski’s results on relationships with strongly quasinonexpansive mappings.

On a generalization of Jacobi sums

We prove an estimate for multi-variable multiplicative character sums over affine subspaces of Akn, which generalizes the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.

Representations and geometrical properties of generalized inverses over fields

In this paper, as a generalization of Urquhart's formulas, we present a full description of the sets of inner inverses and ( B , C ) -inverses over an arbitrary field. In addition, identifying the matrix-vector space with an affine space, we analyse geometrical properties of the main generalized inverse sets. We prove that the set of inner inverses, and the set of ( B , C ) -inverses, form affine subspaces and we study their dimensions. Furthermore, under some hypotheses, we prove that the set of outer inverses is not an affine subspace, but it is an affine algebraic variety. We also provide lower and upper bounds for the dimension of the outer inverse set.

Singular vectors on manifolds and fractals

We generalize Khintchine’s method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of ℝn, in particular on any analytic submanifold of ℝn of dimension ≥2 which is not contained in a proper rational affine subspace.

Geometric-algebra affine projection adaptive filter

Geometric algebra (GA) is an efficient tool to deal with hypercomplex processes due to its special data structure. In this article, we introduce the affine projection algorithm (APA) in the GA domain to provide fast convergence against hypercomplex colored signals. Following the principle of minimal disturbance and the orthogonal affine subspace theory, we formulate the criterion of designing the GA-APA as a constrained optimization problem, which can be solved by the method of Lagrange Multipliers. Then, the differentiation of the cost function is calculated using geometric calculus (the extension of GA to include differentiation) to get the update formula of the GA-APA. The stability of the algorithm is analyzed based on the mean-square deviation. To avoid ill-posed problems, the regularized GA-APA is also given in the following. The simulation results show that the proposed adaptive filters, in comparison with existing methods, achieve a better convergence performance under the condition of colored input signals.

The Abel map for surface singularities III: Elliptic germs

The present note is part of a series of articles targeting the theory of Abel maps associated with complex normal surface singularities with rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, 2019; Nagy and Némethi in Adv Math 371:20, 2020; Nagy and Némethi in Pure Appl Math Q 16(4):1123–1146, 2020). Besides the general theory, by the study of specific families we wish to show the power of this new method. Indeed, using the general theory of Abel maps applied for elliptic singularities in this note we are able to prove several key properties for elliptic singularities (e.g. the statements of the next paragraph), which by ‘old’ techniques were not reachable. If ({widetilde{X}},E)rightarrow (X,o) is the resolution of a complex normal surface singularity and c_1:{mathrm{Pic}}({widetilde{X}})rightarrow H^2({widetilde{X}},{mathbb {Z}}) is the Chern class map, then {mathrm{Pic}}^{l'}({widetilde{X}}):= c_1^{-1}(l') has a (Brill–Noether type) stratification W_{l', k}:= {{mathcal {L}}in {mathrm{Pic}}^{l'}({widetilde{X}}),:, h^1({mathcal {L}})=k}. In this note we determine it for elliptic singularities together with the stratification according to the cycle of fixed components. E.g., we show that the closure of any W(l',k) is an affine subspace. For elliptic singularities we also characterize the End Curve Condition and Weak End Curve Condition in terms of the Abel map, we provide several characterization of them, and finally we show that they are equivalent.

Root systems, affine subspaces, and projections

We tackle several problems related to a finite irreducible crystallographic root system Φ in the real vector space E. In particular, we study the combinatorial structure of the subsets of Φ cut by affine subspaces of E and their projections. As byproducts, we obtain easy algebraic combinatorial proofs of refinements of Oshima's Lemma and of a result by Kostant, a partial result towards the resolution of a problem by Hopkins and Postnikov, and new enumerative results on root systems.

Statistical inference for Bures–Wasserstein barycenters

In this work we introduce the concept of Bures–Wasserstein barycenter Q∗, that is essentially a Fréchet mean of some distribution P supported on a subspace of positive semi-definite d-dimensional Hermitian operators H+(d). We allow a barycenter to be constrained to some affine subspace of H+(d), and we provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of Q∗ in both Frobenius norm and Bures–Wasserstein distance, and explain, how the obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics.

Affine Subspaces of Curvature Functions from Closed Planar Curves

Given a pair of real functions (k, f), we study the conditions they must satisfy for k+lambda f to be the curvature in the arc-length of a closed planar curve for all real lambda . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

Convergence properties of the Broyden-like method for mixed linear\u2013nonlinear systems of equations

We consider the Broyden-like method for a nonlinear mapping F:mathbb {R}^{n}rightarrow mathbb {R}^{n} that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping G:mathbb {R}^{d}rightarrow mathbb {R}^{d}, where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by frac {sqrt {5}+1}{2} if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

Sampling-based dimension reduction for subspace approximation with outliers

The subspace approximation problem with outliers, for given n points in d dimensions x1,x2,…,xn∈Rd, an integer 1≤k≤d, and an outlier parameter 0≤α≤1, is to find a k-dimensional linear subspace of Rd that minimizes the sum of squared distances to its nearest (1−α)n points. More generally, the ℓp subspace approximation problem with outliers minimizes the sum of p-th powers of distances instead of the sum of squared distances. Even the case of p=2 or robust PCA is non-trivial, and previous work requires additional assumptions on the input or generative models for it. Any multiplicative approximation algorithm for the subspace approximation problem with outliers must solve the robust subspace recovery problem, a special case in which the (1−α)n inliers in the optimal solution are promised to lie exactly on a k-dimensional linear subspace. However, robust subspace recovery is Small Set Expansion (SSE)-hard, and known algorithmic results for robust subspace recovery require strong assumptions on the input, e.g., any d outliers must be linearly independent.In this paper, we show how to extend dimension reduction techniques and bi-criteria approximations based on sampling and coresets to the problem of subspace approximation with outliers. To get around the SSE-hardness of robust subspace recovery, we assume that the squared distance error of the optimal k-dimensional subspace summed over the optimal (1−α)n inliers is at least δ times its squared-error summed over all n points, for some 0<δ≤1−α. Under this assumption, we give an efficient algorithm to find a weak coreset or a subset of poly(k/ϵ)log⁡(1/δ)log⁡log⁡(1/δ) points whose span contains a k-dimensional subspace that gives a multiplicative (1+ϵ)-approximation to the optimal solution. Our technique is based on the squared-length sampling algorithm suggested for low-rank approximation problems in the seminal work of Frieze, Kannan, and Vempala [12]. The running time of our algorithm is linear in n and d. Interestingly, our results hold even when the fraction of outliers α is large, as long as the obvious condition 0<δ≤1−α is satisfied. We show similar results for subspace approximation with ℓp error or more general M-estimator loss functions, and also give an additive approximation for the affine subspace approximation problem.

Tomography bounds for the Fourier extension operator and applications

We explore the extent to which the Fourier transform of an L^p density supported on the sphere in mathbb {R}^n can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form X(|widehat{gdsigma }|^2) and mathcal {R}(|widehat{gdsigma }|^2), where X and mathcal {R} denote the X-ray and Radon transforms respectively; here dsigma denotes Lebesgue measure on the unit sphere mathbb {S}^{n-1}, and gin L^p(mathbb {S}^{n-1}). We also identify some conjectural bounds of this type that sit between the classical Fourier restriction and Kakeya conjectures. Finally we provide some applications of such tomography bounds to the theory of weighted norm inequalities for widehat{gdsigma }, establishing some natural variants of conjectures of Stein and Mizohata–Takeuchi from the 1970s. Our approach, which has its origins in work of Planchon and Vega, exploits cancellation via Plancherel’s theorem on affine subspaces, avoiding the conventional use of wave-packet and stationary-phase methods.