Set Of Linear Transformations Research Articles

Autoweighted multi-view smooth representation preserve projection for dimensionality reduction

Over the past few decades, we have witnessed a large family of algorithms that has been designed to provide different solutions to the problem of dimensionality reduction (DR). DR is an essential tool to excavate important information from the high-dimensional data by mapping the data to a low-dimensional subspace. Even though most DR algorithms can achieve satisfactory performance for many practical applications, they fail to fully consider the information from multiple views. Therefore, how to learn the subspace for high-dimensional features by utilizing the consistency and complementary properties of multi-view features is important and challenging in the present. To tackle this problem, we propose an effective multi-view DR algorithm named multi-view smooth representation preserve projection (MvSMRP2). MvSMRP2 seeks to find a set of linear transformations to project each view features into one common low-dimensional subspace where the multi-view smooth reconstructive weights are preserved as much as possible. A pair-wise-consistent scheme is designed by fully exploiting the Hilbert–Schmidt independence criterion to maximize the dependence among different views, which can obtain one common subspace and force multiple views to learn from each other jointly. Moreover, MvSMRP2 can allocate ideal weight for each view automatically according to view importance without explicit weight definition. Finally, an iterative alternating strategy is proposed to obtain the optimal solution of MvSMRP2. Plenty of experiments on various datasets show the excellent performances of the proposed method.

OPTIMAL CONTROL OF A STEP-BY-STEP TRANSMISSION OF VEHICLES

The actual problem of optimal selection of means of transport (MT) gear ratios, equipped with internal combustion engines (ICE) with a step-by-step gearbox according to the criterion of minimum fuel consumption is considered in the article. An analysis of the scientific and technical literature has showed the absence of simple and sufficiently justified algorithms for controlling the vehicle’s power plant, which is a system that consists of an ICE and an adjustable transmission. The aim of this work is the theoretical justification of the law of gear ratios shifting of a step-by-step transmission, ensuring maximum fuel efficiency for any power and speed mode of MT. The proposed method is based on set of linear transformations that connect the engine moment and engine speed with a traction force and a speed of MT. The set of gear ratios correspond to the set of one-extremal functions of specific fuel consumptions. The set of points of equal levels lines corresponding to two neighboring values of the gear ratios of the transmission forms a gear shift line. An appropriate algorithm for switching gear ratios is proposed, which ensures the most efficient operation of a power plant of MT in conditions of variable power and speed modes.

On Association Measures for Continuous Variables and Correction for Chance

This paper studies correction for chance for association measures for continuous variables. The set of linear transformations of Pearson’s product-moment correlation is used as the domain of the correction for chance function. Examples of measures in this set are Tucker’s congruence coefficient, Jobson’s coefficient, and Pearson’s correlation. An equivalence relation on the set of linear transformations is defined. The fixed points of the correction for chance function are characterized. It is shown that each linear transformation is mapped to the fixed point in its equivalence class.

Towards dimension expanders over finite fields

In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let $\mathbb{F}^n $ be the n dimensional linear space over the field $\mathbb{F}$. Find a small (ideally constant) set of linear transformations from $\mathbb{F}^n $ to itself {A i }i∈I such that for every linear subspace V ⊂ $\mathbb{F}^n $ of dimension dim(V) 0 is some constant. In other words, the dimension of the subspace spanned by {A i (V)}i∈I should be at least (1+α)·dim(V). For fields of characteristic zero Lubotzky and Zelmanov [10] completely solved the problem by exhibiting a set of matrices, of size independent of n, having the dimension expansion property. In this paper we consider the finite field version of the problem and obtain the following results. We give a constant number of matrices that expand the dimension of every subspace of dimension d<n/2 by a factor of (1+1/logn). We give a set of O 0. Our constructions are algebraic in nature and rely on expanding Cayley graphs for the group ℤ=ℤn and small-diameter Cayley graphs for the group SL2(p).

Unsupervised Data-Driven Feature Vector Normalization With Acoustic Model Adaptation for Robust Speech Recognition

In this paper, an unsupervised data-driven robust speech recognition approach is proposed based on a joint feature vector normalization and acoustic model adaptation. Feature vector normalization reduces the acoustic mismatch between training and testing conditions by mapping the feature vectors towards the training space. Model adaptation modifies the parameters of the acoustic models to match the test space. However, since neither is optimal, both approaches use an intermediate space between training and testing spaces to map either the feature vectors or acoustic models. The joint optimization of both approaches provides a common intermediate space with a better match between normalized feature vectors and adapted acoustic models. In this paper, feature vector normalization is based on a minimum mean square error (MMSE) criterion. A class dependent multi-environment model linear normalization (CD-MEMLIN) based on two classes (silence/speech) with a cross probability model (CD-MEMLIN-CPM) is used. CD-MEMLIN-CPM assumes that each class of clean and noisy spaces can be modeled with a Gaussian mixture model (GMM), training a linear transformation for each pair of Gaussians in an unsupervised data-driven training process. This feature vector normalization maps the recognition space feature vector to a normalized space. The acoustic model adaptation maps the training space to the normalized space by defining a set of linear transformations over an expanded HMM-state space, compensating for those degradations that the feature vector normalization is not able to model, like rotations. Experiments have been carried out with the Spanish SpeechDat Car database and Aurora 2 databases using both the standard Mel-frequency cepstral coefficient (MFCC) and advanced ETSI front-ends. Consistent improvements were reached for both corpora and front-ends. Using the standard MFCC front-end, a 92.08% average improvement on WER for Spanish SpeechDat Car and a 69.75% average improvement for clean condition evaluation of Aurora 2 was obtained, improving those results reached with ETSI advanced front-end (83.28% and 67.41%, respectively). Using the ETSI advanced front-end with the proposed solution, a 75.47% average improvement was obtained for the clean condition evaluation of Aurora 2 database.

Compact Multiplication Operators on Weighted Spaces of Vector-Valued Continuous Functions

In this note we characterize the compact multiplication operators Mπ on the weighted locally convex spaces CV0(X,E) of vector-valued continuous functions induced by the operator-valued mappings π : X → B(E). 0. Introduction. Let L(X,E) be a vector space of functions from a nonempty set X to a vector space E over the field C or R. Let T (E) be a set of linear transformations from E to itself. If φ : X → X and π : X → T (E) are mappings such that the weighted composite function π.f ◦ φ belongs to L(X,E), whenever f ∈ L(X,E), then the mapping taking f to π.f ◦φ is a linear transformation on L(X,E) and we denote it by Wπ,φ. In case L(X,E) is a topological vector space and the mapping Wπ,φ is continuous, we call Wπ,φ the weighted composition operator on L(X,E) induced by the symbol (π, φ). In case φ is the identity map, we call it the multiplication operator induced by π and we denote it by Mπ. For details on these operators we refer to [11]. The compact weighted composition operators on spaces of continuous functions have been studied extensively by many authors like Kamowitz [5], Feldman [3], Singh and Summers [13], Jamison and Rajagopalan [4], Takagi [14], Chan [2] and Singh and Manhas [12]. As we know, the class of weighted composition operators include the class of multiplication operators and the class of composition operators. One natural question arises: is it possible to get the behavior of the compact multiplication operators from the study of the compact weighted composition operators? In general, it is not possible since the conditions obtained earlier for a weighted composition operator to be compact is not satisfied by the identity map φ. So it motivates us to 1991 AMS Mathematics Subject Classification. Primary 47B38, 47B07, Secondary 47A56, 46E10, 46E40.

Vertices of planar curves under the action of linear transformations

The number of vertices of a smooth Jordan curve with nowhere vanishing curvature can change under the action of a nonsingular real linear transformation. We examine the bifurcation set in the space of linear transformations for the number of vertices on the image curve, showing that generally there is a codimension-one set of linear transformations making an arbitrary point into a vertex, and obtaining conditions that the point be capable of being transformed into a higher vertex. We demonstrate that there is always an open set of linear transformations such that the image curves have at least six vertices.

Counting of overlapped photon detector single pulses by analog/digital sampling and deconvolution

The application of analog/digital converters and inverse techniques for optical receiving by photon detectors, when their output single pulses are strongly overlapped, is described. It is shown that the counting of overlapped pulses can be realized by a set of linear transformations—conversion of single pulses into decayed oscillations, digitizing, averaging, Fourier deconvolution, normalizing, and digital filtering. This method does not use discriminators, being insensitive to the pulse shape. The theoretical analysis is supported by computer simulations and experimental data. It is shown that photon statistics can be saved into the retrieved optical profiles at mean counting rates exceeding 1 GHz. The retrieving error is not higher than Poisson fluctuations. This method is effective simultaneously in the both typical regimes of photon detectors—photon counting and overlapping. It can be applied in lidar remote sensing, time-resolved spectroscopy, etc.

Vector Space Projection Approach for Perfect Reconstruction Filter Banks Design

A vector space projection for the analysis and design of general M-channel multirate filter banks is proposed in this paper. In this context, the input signals are interpreted as vectors in the measurable, square-summable real vector space I 2( L ). The operation of multirate signal filtering is modeled with a set of vector space projectors which projects signals onto a family of subspaces of I 2 ( L ). The projections onto the specific subspaces are performed by a set of linear transformations. The analysis filters and the synthesis filters of the filter banks are characterized by the bases which span the projection spaces. With these vector space operations, the design of multirate filter banks can be interpreted as the construction of the bases used to span the associated subspaces of I 2 ( L ). The perfect reconstruction of multirate filter banks can then be achieved by constructing those subspaces as the direct complementary subspaces of the input signal vector space, With this vector space description, the necessary and sufficient condition for perfect reconstruction can be derived according to the direct complementary property of subspaces, Both the biorthonormal system and the orthonormal system are formulated and discussed for perfect reconstruction finite impulse response filter banks design.

A characterization of the independence - distribution - preserving covariance structure for the multivariate maximum squared - radii statistic

Necessary and sufficient conditions on the observation covariance structure and on the set of linear transformations are given for which the distribution of the multivariate maximum squared - radii statistic for detecting a single multivariate outlier is invariant from the distribution assuming the usual independence covariance structure. Thus, we extend the work of Baksalary and Puntanen (1990), who have given necessary and sufficient conditions for an independence-distribution-preserving covariance structure for Grubbs' statistic for detecting a univariate outlier. We also extend the work of Marco, Young, and Turner (1987) and Pavur and Young (1991), who have given sufficient conditions for an independence-distribution-preserving dependency structure for the multivariate squared - radii statistic.

Rational Functions Invariant under an Orthogonal Group

is a purely transcendental extension of k. Much has been written on this topic for certain groups G, particularly abelian groups or linear groups over fields of characteristic zero (see, for example, [4, 5, 6]) but, on the whole, very little is known when G is non-abelian especially over fields of positive characteristic. In this paper we answer the question affirmatively for a case of the orthogonal group over a finite field. Let Fq be the finite field of odd prime power order q. Given a non-degenerate quadratic form Qn(xv...,xn) on V = F%, the orthogonal group O(n, Qn) is the set of linear transformations a on V such that Qn{av) = Qn(v) for all v in V. While, in principle, Qn is arbitrary, in practice, as is well known, Qn can be taken to be a diagonal form and indeed can be specified uniquely when n is odd and as one of two alternative forms when n is even (see [3, §§6.3, 6.10] and below). The group O(«, Qn), a subgroup of the general linear group GLn(Fq), acts as a group of /^-automorphisms on the polynomial ring Rn = Fq[xv...,xn] and its associated quotient field Kn = FQ(x19...,xn) in a natural way. The invariant subfield A^» will be denoted by Kn and the corresponding invariant subring of Rn by /?+. Evidently, R^ = Rnft K+ and Rn is the integral closure of R+ in Kn [1, p. 68, Exercise 12]. In this situation, Huah Chu [2] has conjectured that K^ is purely transcendental in precise terms, for whose statement we introduce some notation. For a non-negative integer i and a diagonal form Qn put

Parallel algorithms for solvable permutation groups

A number of basic problems involving solvable and nilpotent permutation groups are shown to have fast parallel solutions. Testing solvability is in NC as well as, for solvable groups, finding order, testing membership, finding centralizers, finding centers, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in NC, find a central composition series, and find pointwise stabilizers of sets. The latter is applied to an instance of graph isomorphism. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to NC.

Unilateral quasi-regularizers of closed operators

Let X and Y be normed spaces and let L(X, Y) denote the set of linear transformations from X into Y, with domain D(T) and range R(T). For a given T ∈ L(X, Y) we investigate the existence and properties of a closed densely defined operator S ∈ L(Y, X) such that ST − I/D(T) (or TS − I/D(S)) is a bounded operator of finite dimensional range. These results were previously announced without proof in (2).

Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods

For a given starting point, the sequence of Newton iterates is well known to be invariant under affine transformation of the operator equation to be solved. This property, however, is not sufficiently reflected in most convergence theorems that are presently in common use. For this reason, certain affine invariant convergence theorems are given in this paper. The new theorems can be understood as refined versions of the Newton–Mysovskii theorem, the Newton–Kantorovitch theorem (including optimal error bounds) and a convergence theorem for approximate Newton processes. In addition, the concept of affine invariance associated with Newton’s method is extended to S-invariance associated with related iterative methods where S is some set of linear transformations not changing the iterates. As an application of these considerations, a new convergence theorem for a class of generalized Gauss–Newton methods is given. This theorem is S-invariant with an S containing all unitary transformations. Unlike previous competing theorems, the new one reduces to the Newton–Mysovskii theorem when the Gauss–Newton method reduces to Newton’s method.

The Inventory Calculus

Inventory calculation procedures may be viewed as a set of linear transformations or functions which map the appropriate price and quantity vectors into a corresponding set of valuation scalars. This paper examines some of these calculations in terms of elementary matrix operations. The matrix can be readily adapted to the computer, and standard programs now exist for performing matrix algebra operations. These can be used for the required accounting calculations discussed herein. First, we consider purchase transactions; each such transaction may be described by a pair of points representing quantity and unit price, respectively. The purchase quantities, by years, may be represented by a diagonal matrix having, as the elements of its main diagonal, the quantities purchased during each accounting period. Such a matrix, Q, with typical element qji, for n accounting periods would appear as follows: