Low Rank Cases Research Articles

Sample Efficient Nonparametric Regression via Low-Rank Regularization

Nonparametric regression suffers from curse of dimensionality, requiring a relatively large sample size for accurate estimation beyond the univariate case. In this article, we consider a simple method of dimension reduction in nonparametric regression via series estimation, based on the concept of low-rankness which was previously studied in parametric multivariate reduced-rank regression and matrix regression. For d > 2 , the low-rank assumption is realized via tensor regression. We establish a faster convergence rate of the estimator in the (approximate) low-rank case. Limitations of the model are also discussed. Through simulation studies and real data analysis, we compare the estimation accuracy of the proposed method with that of existing approaches. The results demonstrate that the proposed method yields estimates with lower RMSE compared to existing methods. Supplementary materials for this article are available online.

Yang-Baxter R-operators for osp superalgebras

We study Yang-Baxter equations with orthosymplectic supersymmetry. We extend a new approach of the construction of the spinor and metaplectic Rˆ-operators with orthogonal and symplectic symmetries to the supersymmetric case of orthosymplectic symmetry. In this approach the orthosymplectic Rˆ-operator is given by the ratio of two operator valued Euler Gamma-functions. We illustrate this approach by calculating such Rˆ operators in explicit form for special cases of the osp(n|2m) algebra, in particular for a few low-rank cases. We also propose a novel, simpler and more elegant, derivation of the Shankar-Witten type formula for the osp invariant Rˆ-operator and demonstrate the equivalence of the previous approach to the new one in the general case of the Rˆ-operator invariant under the action of the osp(n|2m) algebra.

False Discovery and its Control in Low Rank Estimation

SummaryModels specified by low rank matrices are ubiquitous in contemporary applications. In many of these problem domains, the row–column space structure of a low rank matrix carries information about some underlying phenomenon, and it is of interest in inferential settings to evaluate the extent to which the row–column spaces of an estimated low rank matrix signify discoveries about the phenomenon. However, in contrast with variable selection, we lack a formal framework to assess true or false discoveries in low rank estimation; in particular, the key source of difficulty is that the standard notion of a discovery is a discrete notion that is ill suited to the smooth structure underlying low rank matrices. We address this challenge via a geometric reformulation of the concept of a discovery, which then enables a natural definition in the low rank case. We describe and analyse a generalization of the stability selection method of Meinshausen and Bühlmann to control for false discoveries in low rank estimation, and we demonstrate its utility compared with previous approaches via numerical experiments.

An Improvement to Exact Reanalysis Algorithm for Local Non-topological Structural Modifications

A new direct reanalysis algorithm for local high-rank structural modifications, based on triangular factorization, has recently been developed. In this work, an improvement is proposed for further reduction of the workload so that the algorithm becomes more efficient and also suitable for low-rank modifications. Compared to the original algorithm, firstly, an alternative formula for updating the triangular factors is derived to avoid repetitive calculations for certain low-rank cases. Secondly, to maximize the efficiency, a combined algorithm is proposed that estimates the workloads of the original and alternative algorithms for each row individually before numerical calculations and selects the one with smaller workload. Numerical experiments show that compared with full factorization, the combined algorithm is significantly more efficient and expected to save up to 75% of execution time in our numerical examples. The new method can be easily implemented and applied to engineering problems, especially to local and step-by-step modification of structures.

Biderivations of the higher rank Witt algebra without anti-symmetric condition

Abstract The Witt algebra 𝔚 d of rank d(≥ 1) is the derivation algebra of Laurent polynomial algebras in d commuting variables. In this paper, all biderivations of 𝔚 d without anti-symmetric condition are determined. As an applications, commutative post-Lie algebra structures on 𝔚 d are obtained. Our conclusions recover and generalize results in the related papers on low rank or anti-symmetric cases.

Fermionic modular categories and the 16-fold way

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin topological quantum field theories at low energy. We formulate a 16-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of PSU(2)4m+2 with an eye towards a classification of the low-rank cases.

Fast Hessenberg Reduction of Some Rank Structured Matrices

We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$, where $D$ is a real or unitary $n \times n$ diagonal matrix and $U, V \in\mathbb{C}^{n \times k}$. The proposed algorithm for the real case exploits a two-stage approach by first reducing the matrix to a generalized Hessenberg form and then completing the reduction by annihilation of the unwanted subdiagonals. It is shown that the novel method requires $O(n^2k)$ arithmetic operations and is significantly faster than other reduction algorithms for rank structured matrices. The method is then extended to the unitary plus low rank case by using a block analogue of the CMV form of unitary matrices. It is shown that a block Lanczos-type procedure for the block tridiagonalization of $\Re(D)$ induces a structured reduction on $A$ in a block staircase CMV-type shape. Then, we present a numerically stable method for performing this reduction using unitary transformations and show how to generalize the subdiagonal elimina...

Hall–Littlewood polynomials and characters of affine Lie algebras

The Weyl–Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac–Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall–Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras Cn(1), A2n(2) and Dn+1(2). Through specialisation this yields generalisations for Bn(1), Cn(1), A2n−1(2), A2n(2) and Dn+1(2) of Macdonald's identities for powers of the Dedekind eta-function. These generalised eta-function identities include the Rogers–Ramanujan, Andrews–Gordon and Göllnitz–Gordon q-series as special, low-rank cases.

A Sufficient Condition for Low-Rank Recovery via Iterative Hard Thresholding Pursuit

In this pape, we propose a matrix Iterative Hard thresholding pursuit algorithm for low-rank minimization that extends Foucart's Hard Thresholding Pursuit (HTP) algorithm from the sparse vector to the low-rank matrix case. The performance guarantee is given in terms of the rank-restricted isometry property and a low-rank solotion is presented. The numerical experiments empirically demonstrate that, although the affine constraints does not satisfy the restricted isometry property in matrix completion, our algorithm also recovers the low-rank matrix from a number of uniformly sampled entries and is more efficient compared with SVT and ADMiRA.

On Multiplicities of Maximal Weights of sl ̂ ( n ) $\widehat {sl}(n)$ -Modules

We determine explicitly the maximal dominant weights for the integrable highest weight $\widehat {sl}(n)$ -modules V((k − 1)Λ 0 + Λ s ), 0 ≤ s ≤ n − 1, k ≥ 2. We give a conjecture for the number of maximal dominant weights of V(k Λ 0) and prove it in some low rank cases. We give an explicit formula in terms of lattice paths for the multiplicities of a family of maximal dominant weights of V(k Λ 0). We conjecture that these multiplicities are equal to the number of certain pattern avoiding permutations. We prove that the conjecture holds for k = 2 and give computational evidence for the validity of this conjecture for k > 2.

ON SUBGROUPS OF CLIFFORD GROUPS DEFINED BY JORDAN PAIRS OF RECTANGULAR MATRICES

AbstractSome embeddings of general linear groups into hyperbolic Clifford groups are constructed generically by using Jordan pairs of rectangular and alternating matrices over a ring. In low rank cases through exceptional isomorphisms, their direct description and relationships to some automorphisms of Clifford groups are given. Generic norms are calculated in detail, and equivariant embeddings of representation spaces are constructed.

Generalized Harish-Chandra Modules with Generic Minimal <b>t</b>-Type

We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras (g,k), we construct, via cohomological induction, the fundamental series F · (p ,E ) of generalized Harish-Chandra modules. We then use F · (p ,E )t o char- acterize any simple generalized Harish-Chandra module with generic minimal k-type. More precisely, we prove that any such simple (g,k)-module of finite type arises as the unique simple submodule of an appropriate fundamental series module F s (p ,E ) in the middle dimension s. Under the stronger assumption that k contains a semisimple regular element of g, we prove that any simple (g,k)-module with generic minimal k-type is necessarily of finite type, and hence obtain a reconstruction theorem for ac lass of simple (g,k)-modules which can a priori have infinite type. We also obtain generic gen- eral versions of some classical theorems of Harish-Chandra, such as the Harish-Chandra admissibility theorem. The paper is concluded by examples, in particular we compute the genericity condition on a k-type for any pair (g,k )w ithks� (2). Introduction. The goal of the present paper is to make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish- Chandra modules or weight modules of finite type. This work is part of the program of study of generalized Harish-Chandra modules laid out in (PZ). Let g be a semisimple Lie algebra. A simple generalized Harish-Chandra module is by definition a simple g-module with locally finite action of a reductive in g subalgebra k ⊂ g and with finite k-multiplicities. In the classical case of Harish-Chandra modules, the pair (g,k )i s in addition assumed to be symmetric. In a recent joint paper with V. Serganova (PSZ), we have constructed new families of generalized Harish-Chandra modules; however, no general classification is known beyond the case when the pair (g,k) is symmetric and the case when k is a Cartan subalgebra of g. The first case is settled in the well- known work of R. Langlands (L2), A. Knapp and the second named author (KZ), D. Vogan and the second named author (V2), (Z), A. Beilinson - J. Bernstein (BB) and I. Mirkovic (Mi); the second case is settled in a more recent breakthrough by O. Mathieu (M). Some low rank cases of certain special non-symmetric pairs (g,k) (where k is not a Cartan subalgebra) have been settled by G. Savin (Sa). In this paper, we consider simple generalized Harish-Chandra modules which have a generic minimal k-type for some arbitrary fixed reductive pair (g,k) (the precise definitions see in Section 1 below). One of our main results is the construction of as eries of (g,k)-modules, which we call the fundamental series (it generalizes the fundamental series of Harish-Chandra modules), and in addition the theorem that any simple generalized Harish-Chandra module with generic minimal k-type is a submodule of the fundamental series. We refer to the latter result as the first reconstruction theorem for generalized Harish-Chandra modules. This theorem is based on new

Infinite abelian subalgebra of W(sl( n))

A representation theoretical construction of the conservation laws of affine Toda type systems is described. The construction employs the completely degenerate representations of the extended conformal algebras W(sl( n)). The conserved charges are shown to generate an infinite-dimensional abelian subalgebra of W(sl( n)). Different characterizations of this subalgebra are obtained: As space of physical Fock space operators with dihedral symmetry, as constants of commuting flows of quantum KdV-type equations and as subalgebra of the sl( n) singlets in affine sl ( n) level-1 modules. The existence of the subalgebras is established for low-rank cases by means of an algorithmic Fock space procedure.

The genus of low rank hamiltonian groups

Any finite hamiltonian group H can be written as a direct product of three groups: the quaternion group Q, an abelian group Ar of odd order and rank r⩾0, and an abelian 2-group Zs2, s⩾0. The genus of H is computed for r=2, s=1, and, when 3 divides the order of H, for r=2, s=2, 3 and r=3, s=1, 2. New bounds are given for the other low rank cases of r=2, s⩽3 and r=3, s⩽2. Tables are given indicating the status of the genus problem for all cases of r and s and listing all unknown cases of order 1000 or less.

Special cases of a matrix exponential formula

Some low dimension or low rank cases of a formula for a product of exponentials of matrices are studied using eigenvalue considerations. Counterexamples to certain natural conjectures involving a product of matrix exponentials are also given, and comments relating to certain recently discovered exponential identities are included.

Representations of orthosymplectic superalgebras. II. Young diagrams and weight space techniques

For pt.I see ibid., vol.16, p.473 (1983). Finite-dimensional graded tensor representations of OSp(M/N) are enumerated via standard Young diagrams, and their corresponding highest weight and Kac-Dynkin labels are given. A uniform set of conditions on the diagram shape, necessary and sufficient for atypicality, are presented. Weight space techniques are used to provide a complete analysis of an atypical representations for the low-rank cases OSp(2/2), OSp(3/2) and OSp(4/2).

Extensions of Modules for Algebraic Groups

Let G be a connected semi-simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p. In this paper we show how one can obtain the extensions of any two simple G-modules if one knows the socles of all the G-modules ExtG, (L, L ') where G1 denotes the kernel of the Frobenius endomorphism on G and L as well as L' are simple G1-modules. Alternatively, we prove how all such G-extensions can be found once one knows the extensions of any two simple G-modules belonging to a specified (finite) set of simple modules with small highest weight. Such information is available in certain low rank cases. It should also be pointed out that it is possible from Lusztig's conjecture on irreducible characters [11] to derive the dimensions of Ext (L1, L2) when L1 and L2 are simple with such small highest weights. To obtain our results on Exth we need to know that there exists no G1-extension of a simple module by itself. This turns out to be somewhat

Simultaneous triangularization of matrices—low rank cases and the nonderogatory case

(1978). Simultaneous triangularization of matrices—low rank cases and the nonderogatory case. Linear and Multilinear Algebra: Vol. 6, No. 4, pp. 269-305.

Classifying spaces, steenrod operations and algebraic closure

IF BX IS a CW-space such that its loop space RBX is homotopy equivalent to a finite CW-complex, then for all sufficiently large prime numbers p, H*(BX,F,) is a graded polynomial algebra over F,, with generators in dimensions (2ni). Considerable effort has been devoted to the determination of the possibilities for the(2ni). Complete results are available only in the low rank cases, e.g. Hubbuck [ 121, while Clark’s criteria [7] remain the most useful general results. If only one prime is considered, the problem is more difficult, and even the classification of the rank two case is still incomplete, see however [9,29,22,33] for partial results. This work began as a general attack on these problems. It seemed that the best chance to obtain a complete classification was to apply some constructive methods, rather than the argument by contradiction or special cases typical of most earlier work. This optimism was supported by the special nature of all known examples, and by the way in which the Steenrod algebra action was exploited in the proof of the Adams-Mahmud theorem[l]. The classification of possible H*(BX, F,) ideally would be carried out in two steps. In all known examples (the classifying spaces of compact connected Lie groups and the exotic classifying spaces of [26, 18,8,28]), there is a common theme. The F,-cohomology of (CP”)” appears as a finite integral extension of H*(BX, F,), and the extension is as algebras over the Steenrod algebra. This suggests, by analogy with K-theory, that there should be a “splitting principle” for such H*(BX, F,). That is, one should be able to embed a polynomial algebra over F, with an unstable Steenrod algebra action in the polynomial algebra F,[t,, . . ., t.] on two-dimensional generators, at least for large p. The second phase of the program would be to classify these embeddings. In particular, if the extensions are Galois, it is known that the Galois group must be a finite generalized reflection group of the type studied by Ewing and Clark[8], and these possibilities are thus known. The second aspect of this program is completely achieved in this paper. Also, some interesting conditions which imply the “splitting principle” are discussed. However, the question remains open, and hence the classification of possible H*(BX, F,) is still incomplete. The majority of this work is devoted to the proof of step two of the program and to some of its applications. In the following .&, denotes the mod p Steenrod algebra. We prove a slightly more general result than is needed: