Identification Of Subspaces Research Articles

Identification of dominant subspaces for model reduction of structured parametric systems

SummaryIn this paper, we discuss a novel model reduction framework for linear structured dynamical systems. The transfer functions of these systems are assumed to have a special structure, for example, coming from second‐order linear systems or time‐delay systems, and they may also have parameter dependencies. Firstly, we investigate the connection between classic interpolation‐based model reduction methods with the reachability and observability subspaces of linear structured parametric systems. We show that if enough interpolation points are taken, the projection matrices of interpolation‐based model reduction encode these subspaces. Consequently, we are able to identify the dominant reachable and observable subspaces of the underlying system. Based on this, we propose a new model reduction algorithm combining these features and leading to reduced‐order systems. Furthermore, we discuss computational aspects of the approach and its applicability to a large‐scale setting. We illustrate the efficiency of the proposed approach with several numerical large‐scale benchmark examples.

Iterated Relevance Matrix Analysis (IRMA) for the identification of class-discriminative subspaces

We introduce and investigate the iterated application of Generalized Matrix Learning Vector Quantization for the analysis of feature relevances in classification problems, as well as for the construction of class-discriminative subspaces. The suggested Iterated Relevance Matrix Analysis (IRMA) identifies a linear subspace representing the classification specific information of the considered data sets using Generalized Matrix Learning Vector Quantization (GMLVQ). By iteratively determining a new discriminative subspace while projecting out all previously identified ones, a combined subspace carrying all class-specific information can be found. This facilitates a detailed analysis of feature relevances, and enables improved low-dimensional representations and visualizations of labeled data sets. Additionally, the IRMA-based class-discriminative subspace can be used for dimensionality reduction and the training of robust classifiers with potentially improved performance.

Visual Analytics of Co-Occurrences to Discover Subspaces in Structured Data

We present an approach that shows all relevant subspaces of categorical data condensed in a single picture. We model the categorical values of the attributes as co-occurrences with data partitions generated from structured data using pattern mining. We show that these co-occurrences are a-priori , allowing us to greatly reduce the search space, effectively generating the condensed picture where conventional approaches filter out several subspaces as these are deemed insignificant. The task of identifying interesting subspaces is common but difficult due to exponential search spaces and the curse of dimensionality. One application of such a task might be identifying a cohort of patients defined by attributes such as gender, age, and diabetes type that share a common patient history, which is modeled as event sequences. Filtering the data by these attributes is common but cumbersome and often does not allow a comparison of subspaces. We contribute a powerful multi-dimensional pattern exploration approach (MDPE-approach) agnostic to the structured data type that models multiple attributes and their characteristics as co-occurrences, allowing the user to identify and compare thousands of subspaces of interest in a single picture. In our MDPE-approach, we introduce two methods to dramatically reduce the search space, outputting only the boundaries of the search space in the form of two tables. We implement the MDPE-approach in an interactive visual interface (MDPE-vis) that provides a scalable, pixel-based visualization design allowing the identification, comparison, and sense-making of subspaces in structured data. Our case studies using a gold-standard dataset and external domain experts confirm our approach’s and implementation’s applicability. A third use case sheds light on the scalability of our approach and a user study with 15 participants underlines its usefulness and power.

Non-injective gas sensor arrays: identifying undetectable composition changes

Metal-organic frameworks (MOFs) are nanoporous materials with good prospects as recognition elements for gas sensors owing to their adsorptive sensitivity and selectivity. A gravimetric, MOF-based sensor functions by measuring the mass of gas adsorbed in a MOF. Changes in the gas composition are expected to produce detectable changes in the mass of gas adsorbed in the MOF. In practical settings, multiple components of the gas adsorb into the MOF and contribute to the sensor response. As a result, there are typically many distinct gas compositions that produce the same single-sensor response. The response vector of a gas sensor array places multiple constraints on the gas composition. Still, if the number of degrees of freedom in the gas composition is greater than the number of MOFs in the sensor array, the map from gas compositions to response vectors will be non-injective (many-to-one). Here, we outline a mathematical method to determine undetectable changes in gas composition to which non-injective gas sensor arrays are unresponsive. This is important for understanding their limitations and vulnerabilities. We focus on gravimetric, MOF-based gas sensor arrays. Our method relies on a mixed-gas adsorption model in the MOFs comprising the sensor array, which gives the mass of gas adsorbed in each MOF as a function of the gas composition. The singular value decomposition of the Jacobian matrix of the adsorption model uncovers (i) the unresponsive directions and (ii) the responsive directions, ranked by sensitivity, in gas composition space. We illustrate the identification of unresponsive subspaces and ranked responsive directions for gas sensor arrays based on Co-MOF-74 and HKUST-1 aimed at quantitative sensing of CH4/N2/CO2/C2H6 mixtures relevant to natural gas sensing.

From the street to the metropolitan region: Pedestrian perspective in urban fabric analysis

The urban fabric is a fundamental small-scale component of urban form. Its quantitative analysis has so far been limited either in its geographical extent or in the diversity of components analysed. Moreover, the planning approach has traditionally privileged an aerial perspective. A new approach integrating the pedestrian point of view is proposed. Spatial analysis procedures are implemented with a twofold objective: identifying urban fabrics and studying their spatial organization within a large metropolitan area. The former is achieved through multiple fabric assessment, a three-step protocol using a network-based partition of urban space: (i) a set of skeletal streetscape indicators is implemented on each spatial unit, considering different constituents of the urban fabric; (ii) spatial patterns on the street network are identified, applying geostatistical analysis to each indicator; and (iii) spatial patterns with Bayesian clustering are recombined, allowing the identification and characterization of urban fabric types and subspaces within the city. This methodology is tested on the French Riviera metropolitan area, where nine families of urban fabric are identified. Disentangling the spatial organization of urban fabrics represents the second objective of this paper: the geographical distribution of urban fabrics is investigated, applying mathematical morphology and variography while considering network-constrained topological contiguities.

Degenerate Landau–Zener model in the presence of quantum noise

The degenerate Landau–Zener–Majorana–Stückelberg model consists of two degenerate energy levels whose energies vary with time and in the presence of an interaction which couples the states of the two levels. In the adiabatic limit, it allows for the populations transfer from states of one level to the states of the other level. The presence of an interaction with the environment influences the efficiency of the process. Nevertheless, identification of possible decoherence-free subspaces permits to engineer coupling schemes for which the effects of quantum noise can be made negligible.

Uncertainty-based design optimization approach based on cumulative distribution matching

Uncertainty-based design optimization (UBDO) is becoming increasingly important with great attention to critical system reliability and robustness. However, most traditional UBDO methods only consider low-order response moments and a limited number of aleatory uncertainties, which desiderates to be resolved. In this study, an efficient UBDO approach based on cumulative distribution matching is presented to address all response moments under mixed aleatory and epistemic uncertainties. The mathematical formulation of cumulative distribution function (CDF) matching is systematically derived including the distance metric definition and numerical computation process. Uncertainty analysis within the CDF matching optimization loop is also investigated, including the identification of active subspaces and the propagation of constructed surrogate model in reduced dimensions. Extended active subspaces are furthered discussed for mixed uncertainty analysis to obtain the bounds of CDF efficiently, instead of the time-consuming full-dimensional simulation. The proposed approach is illustrated by three typical optimization examples, i.e., response function matching, standard NASA test problem, and satellite system design in aerospace engineering. The UBDO results based on the CDF matching are discussed and compared with the probability density function matching, which exhibits higher efficiency and better convergence in tackling constrained optimization design problems subject to mixed uncertainties.

New Approaches to Direction-of-Arrival Estimation With Sensor Arrays in Unknown Nonuniform Noise

It is known that classical subspace-based direction-of-arrival (DOA) estimation algorithms are not straightforwardly applicable to scenarios with unknown spatially nonuniform noise. Among the state-of-the-art solutions, this problem is tackled by iterative subspace estimation algorithms to incorporate subspace-based approaches or nonlinear optimization routines to bypass the direct identification of subspaces. In this paper, the problem of DOA estimation in nonuniform noise is revisited by devising two computationally efficient proposals. It is proved herein that, if the signals are uncorrelated, the signal and noise subspaces can be directly obtained from the eigendecomposition of a reduced array covariance matrix. On the other hand, when the signals are correlated, the estimation of the noise covariance matrix is formulated into a rank minimization problem which can be approximately solved by semidefinite programming. In both cases, the signal and noise subspaces are easy to compute without iterations. Consequently, classical subspace-based algorithms can be employed to determine the DOAs. Numerical examples are provided to demonstrate the performance and applicability of the proposed methods.

Informed decomposition of electroencephalographic data

BackgroundBlind source separation techniques have become the de facto standard for decomposing electroencephalographic (EEG) data. These methods are poorly suited for incorporating prior information into the decomposition process. While alternative techniques to this problem, such as the use of constrained optimization techniques, have been proposed, these alternative techniques tend to only minimally satisfy the prior constraints. In addition, the experimenter must preset a number of parameters describing both this minimal limit as well as the size of the target subspaces. New methodWe propose an informed decomposition approach that builds upon the constrained optimization approaches for independent components analysis to better model and separate distinct subspaces within EEG data. We use a likelihood function to adaptively determine the optimal model size for each target subspace. ResultsUsing our method we are able to produce ordered independent subspaces that exhibit less residual mixing than those obtained with other methods. The results show an improvement in modeling specific features of the EEG space, while also showing a simultaneous reduction in the number of components needed for each model. Comparison with existing method(s)We first compare our approach to common methods in the field of EEG decomposition, such as Infomax, FastICA, PCA, JADE, and SOBI for the task of modeling and removing both EOG and EMG artifacts. We then demonstrate the utility of our approach for the more complex problem of modeling neural activity. ConclusionsBy working in a one-size-fits-all fashion current EEG decomposition methods do not adapt to the specifics of each data set and are not well designed to incorporate additional information about the decomposition problem. However, by adding specific information about the problem to the decomposition task, we improve the identification and separation of distinct subspaces within the original data and show better preservation of the remaining data.

Quantum metrology subject to spatially correlated Markovian noise: restoring the Heisenberg limit

Environmental noise can hinder the metrological capabilities of entangled states. While the use of entanglement allows for Heisenberg-limited resolution, the largest permitted by quantum mechanics, deviations from strictly unitary dynamics quickly restore the standard scaling dictated by the central limit theorem. Product and maximally entangled states become asymptotically equivalent when the noisy evolution is both local and strictly Markovian. However, temporal correlations in the noise have been shown to lift this equivalence while fully (spatially) correlated noise allows for the identification of decoherence-free subspaces. Here we analyze precision limits in the presence of noise with finite correlation length and show that there exist robust entangled state preparations which display persistent Heisenberg scaling despite the environmental decoherence, even for small correlation length. Our results emphasize the relevance of noise correlations in the study of quantum advantage and could be relevant beyond metrological applications.

Identification of decoherence-free subspaces without quantum process tomography

Characterizing a quantum process is the critical first step towards applying such a process in a quantum information protocol. Full process characterization is known to be extremely resource-intensive, motivating the search for more efficient ways to extract salient information about the process. An example is the identification of ``decoherence-free subspaces,'' in which computation or communications may be carried out, immune to the principal sources of decoherence in the system. Here we propose and demonstrate a protocol which enables one to directly identify a decoherence-free subspace (DFS) without carrying out a full reconstruction. Our protocol offers an up-to-quadratic speedup over standard process tomography. In this paper, we experimentally identify the DFS of a two-qubit process with 32 measurements rather than the usual 256, characterize the robustness and efficiency of the protocol, and discuss its extension to higher-dimensional systems.

BSS and ICA in Neuroinformatics: From Current Practices to Open Challenges

We give a general overview of the use and possible misuse of blind source separation (BSS) and independent component analysis (ICA) in the context of neuroinformatics data processing. A clear emphasis is given to the analysis of electrophysiological recordings, as well as to functional magnetic resonance images (fMRI). Two illustrative examples include the identification and removal of artefacts in both kinds of data, and the analysis of a simple fMRI. A second part of the paper addresses a set of currently open challenges in signal processing. These include the identification and analysis of independent subspaces, the study of networks of functional brain activity, and the analysis of single-trial event-related data.

Non-combinatorial estimation of independent autoregressive sources

Identification of mixed independent subspaces is thought to suffer from combinatorial explosion of two kinds: the minimization of mutual information between the estimated subspaces and the search for the optimal number and dimensions of the subspaces. Here we show that independent autoregressive process analysis, under certain conditions, can avoid this problem using a two-phase estimation process. We illustrate the solution by computer demonstration.

Interpretable dimension reduction

The analysis of high-dimensional data often begins with the identification of lower dimensional subspaces. Principal component analysis is a dimension reduction technique that identifies linear combinations of variables along which most variation occurs or which best “reconstruct” the original variables. For example, many temperature readings may be taken in a production process when in fact there are just a few underlying variables driving the process. A problem with principal components is that the linear combinations can seem quite arbitrary. To make them more interpretable, we introduce two classes of constraints. In the first, coefficients are constrained to equal a small number of values (homogeneity constraint). The second constraint attempts to set as many coefficients to zero as possible (sparsity constraint). The resultant interpretable directions are either calculated to be close to the original principal component directions, or calculated in a stepwise manner that may make the components more orthogonal. A small dataset on characteristics of cars is used to introduce the techniques. A more substantial data mining application is also given, illustrating the ability of the procedure to scale to a very large number of variables.

A general method to obtain vibrational symmetry adapted bases in a local scheme

A general approach to obtain symmetry adapted bases from a local set of states is presented. The approach is based on the identification of the invariant subspaces which, when projected by means of the eigenfunction method developed by Chen (1989, Group Representation Theory for Physicists Singapore, World Scientific), allow the generation of a symmetry adapted basis. The symmetrized functions so obtained are further taken as a basis to diagonalize simultaneously a set of normal number operators, which provides a set of normal states expanded in terms of the symmetry adapted local basis. In this approach the normal number operators are generated implicitly from the one quantum space through a tensorial formalism. Although the normal operators are defined in a harmonic basis, the locality of the basis allows the approach to be extended to anharmonic functions. This approach has the additional advantage of allowing the elimination of the spurious states, a common problem in a local coordinate representation. An important advantage of this symmetrization method is that it allows generation of a code to analyse any molecular system with a minimum set of input data.

ICEAGE: Interactive Clustering and Exploration of Large and High-Dimensional Geodata

The unprecedented large size and high dimensionality of existing geographic datasets make the complex patterns that potentially lurk in the data hard to find. Clustering is one of the most important techniques for geographic knowledge discovery. However, existing clustering methods have two severe drawbacks for this purpose. First, spatial clustering methods focus on the specific characteristics of distributions in 2- or 3-D space, while general-purpose high-dimensional clustering methods have limited power in recognizing spatial patterns that involve neighbors. Second, clustering methods in general are not geared toward allowing the human-computer interaction needed to effectively tease-out complex patterns. In the current paper, an approach is proposed to open up the “black box” of the clustering process for easy understanding, steering, focusing and interpretation, and thus to support an effective exploration of large and high dimensional geographic data. The proposed approach involves building a hierarchical spatial cluster structure within the high-dimensional feature space, and using this combined space for discovering multi-dimensional (combined spatial and non-spatial) patterns with efficient computational clustering methods and highly interactive visualization techniques. More specifically, this includes the integration of: (1) a hierarchical spatial clustering method to generate a 1-D spatial cluster ordering that preserves the hierarchical cluster structure, and (2) a density- and grid-based technique to effectively support the interactive identification of interesting subspaces and subsequent searching for clusters in each subspace. The implementation of the proposed approach is in a fully open and interactive manner supported by various visualization techniques.

Components of spaces of representations and stable triples

We consider the moduli spaces of representations of the fundamental group of a surface of genus g⩾2 in the Lie groups SU(2, 2) and Sp(4, R ) . It is well known that there is a characteristic number, d, of such a representation, satisfying the inequality | d|⩽2 g−2. This allows one to write the moduli space as a union of subspaces indexed by d, each of which is a union of connected components. The main result of this paper is that the subspaces corresponding to d=±(2 g−2) are connected in the case of representations in SU(2, 2) , while they break up into 3·2 2 g +2 g−4 connected components in the case of representations in Sp(4, R ) . We obtain our results using the interpretation of the moduli space of representations as a moduli space of Higgs bundles, and an important step is an identification of certain subspaces as moduli spaces of stable triples, as studied by Bradlow and Garcı́a-Prada.

On the identification of symmetry-forbidden spin subspaces for configurations employing nonorthogonal orbitals

The implications of orbital symmetry for a configuration's spin function are considered from the fairly general viewpoint required for multiconfiguration wave functions employing nonorthogonal orbitals. A general procedure is derived for the imposition of the spin constraints that may be necessary to ensure a given configuration, or a set of configurations as a whole, exhibit required symmetry properties, thus preventing symmetry contamination of the overall electronic wave function. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 24–31, 2000

Formation of wrinkles in outwardly propagating flames.

Numerical integrations of the partial differential equation proposed by Filyand, Sivashinsky, and Frankel [Physica D 72, 110 (1994)] to describe the dynamics of outward accelerating flames are presented. The computational results reported by Filyand, Sivashinsky, and Frankel are confirmed: as time increases, a repetitive formation of cusps, as well as a rapid (power-law) expansion of the mean flame radius, are observed. However, the identification of invariant subspaces for the equation shows that even when the initial condition belongs to such subspaces, numerical round-off errors are responsible for excursions of the solution outside these subspaces. In Fourier space, this corresponds to the generation of spurious Fourier modes that grow as time increases. This computational error is controlled here by a filter that forces the solution, at each time step, to stay inside the invariant subspaces. The results of our filtered simulations are very similar to those resulting from unfiltered integrations, showing that both the formation of cusps and the rapid acceleration of the flame front are independent of the growth of spurious Fourier modes. The connection between such dynamics and exact pole solutions of the equation (in which the number of poles is fixed) is investigated. It is found that the latter are unstable and the more complicated (stable) dynamics consists of successive instabilities through which the flame front closely follows a (2N+1)-pole solution before approaching a (2N+3)-pole solution. These migrations are responsible for both the formation of new cusps and the rapid power-law acceleration of the mean front.