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## Learning Algorithms for Human–Machine Interfaces

Danziger, Zachary; Fishbach, Alon; Mussa-Ivaldi, Ferdinando A.
Tipo: Artigo de Revista Científica
EN
Relevância na Pesquisa
26.25%
The goal of this study is to create and examine machine learning algorithms that adapt in a controlled and cadenced way to foster a harmonious learning environment between the user and the controlled device. To evaluate these algorithms, we have developed a simple experimental framework. Subjects wear an instrumented data glove that records finger motions. The high-dimensional glove signals remotely control the joint angles of a simulated planar two-link arm on a computer screen, which is used to acquire targets. A machine learning algorithm was applied to adaptively change the transformation between finger motion and the simulated robot arm. This algorithm was either LMS gradient descent or the Moore–Penrose (MP) pseudoinverse transformation. Both algorithms modified the glove-to-joint angle map so as to reduce the endpoint errors measured in past performance. The MP group performed worse than the control group (subjects not exposed to any machine learning), while the LMS group outperformed the control subjects. However, the LMS subjects failed to achieve better generalization than the control subjects, and after extensive training converged to the same level of performance as the control subjects. These results highlight the limitations of coadaptive learning using only endpoint error reduction.

## Accurate solution of structured least squares problems via rank-revealing decompositions

Castro González, Nieves; Ceballos Cañón, Johan Armando; Martínez Dopico, Froilán C.; Molera, Juan M.
Fonte: Society for Industrial and Applied Mathematics Publicador: Society for Industrial and Applied Mathematics
Tipo: info:eu-repo/semantics/publishedVersion; info:eu-repo/semantics/article
Relevância na Pesquisa
66.72%
Least squares problems min(x) parallel to b - Ax parallel to(2) where the matrix A is an element of C-mXn (m >= n) has some particular structure arise frequently in applications. Polynomial data fitting is a well-known instance of problems that yield highly structured matrices, but many other examples exist. Very often, structured matrices have huge condition numbers kappa(2)(A) = parallel to A parallel to(2) parallel to A(dagger)parallel to(2) (A(dagger) is the Moore-Penrose pseudoinverse of A) and therefore standard algorithms fail to compute accurate minimum 2-norm solutions of least squares problems. In this work, we introduce a framework that allows us to compute minimum 2-norm solutions of many classes of structured least squares problems accurately, i.e., with errors parallel to(x) over cap (0) - x(0)parallel to(2)/parallel to x(0)parallel to(2) = O(u), where u is the unit roundoff, independently of the magnitude of kappa(2)(A) for most vectors b. The cost of these accurate computations is O(n(2)m) flops, i.e., roughly the same cost as standard algorithms for least squares problems. The approach in this work relies in computing first an accurate rank-revealing decomposition of A, an idea that has been widely used in recent decades to compute...

## A higher Boltzmann distribution

Catanzaro, Michael J.; Chernyak, Vladimir Y.; Klein, John R.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.25%
We characterize the classical Boltzmann distribution as the unique solution to a certain combinatorial Hodge theory problem in homological degree zero on a finite graph. By substituting for the graph a CW complex of dimension d, we are able to define, by direct analogy, a higher dimensional Boltzmann distribution as a certain (d-1)-cycle on the real cellular chain complex which is characterized by appropriate constraints. We then give an explicit summation formula for this cycle. Lastly, we explain how this circle of ideas relates to the authors' Higher Kirchhoff Network Theorem.; Comment: Fixed a sign mistake in the definition of the master operator and in the statement of the summation formula for the Moore-Penrose pseudoinverse

## Optimal linear Glauber model

Sahoo, Shaon; Ganguly, Soumya Kanti
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.25%
Contrary to the actual nonlinear Glauber model (NLGM), the linear Glauber model (LGM) is exactly solvable, although the detailed balance condition is not generally satisfied. This motivates us to address the issue of writing the transition rate ($w_j$) in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least. The advantage of this work is that, by studying the LGM analytically, we will be able to anticipate how the kinetic properties of an arbitrary Ising system depend on the temperature and the coupling constants. The analytical expressions for the optimal values of the parameters involved in the linear $w_j$ are obtained using a simple Moore-Penrose pseudoinverse matrix. This approach is quite general, in principle applicable to any system and can reproduce the exact results for one dimensional Ising system. In the continuum limit, we get a linear time-dependent Ginzburg-Landau (TDGL) equation from the Glauber's microscopic model of non-conservative dynamics. We analyze the critical and dynamic properties of the model, and show that most of the important results obtained in different studies can be reproduced by our new mathematical approach. We will also show in this paper that the effect of magnetic field can easily be studied within our approach; in particular...

## Regularized Computation of Approximate Pseudoinverse of Matrices Using Low-Rank Tensor Train Decompositions

Lee, Namgil; Cichocki, Andrzej
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.46%
We propose a new method for low-rank approximation of Moore-Penrose pseudoinverses (MPPs) of large-scale matrices using tensor networks. The computed pseudoinverses can be useful for solving or preconditioning large-scale overdetermined or underdetermined systems of linear equations. The computation is performed efficiently and stably based on the modified alternating least squares (MALS) scheme using low-rank tensor train (TT) decomposition. The large-scale optimization problem is reduced to sequential smaller-scale problems for which any standard and stable algorithms can be applied. Regularization technique is further introduced in order to alleviate ill-posedness and obtain low-rank solutions. Numerical simulation results illustrate that the pseudoinverses of a wide class of nonsquare or nonsymmetric matrices admit good approximate low-rank TT approximations. It is demonstrated that the computational cost of the proposed method is only logarithmic in the matrix size given that the TT-ranks of a data matrix and its approximate pseudoinverse are bounded. It is illustrated that a strongly nonsymmetric convection-diffusion problem can be efficiently solved by using the preconditioners computed by the proposed method.; Comment: 26 pages

## Fast Computation of Moore-Penrose Inverse Matrices

Courrieu, Pierre
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.65%
Many neural learning algorithms require to solve large least square systems in order to obtain synaptic weights. Moore-Penrose inverse matrices allow for solving such systems, even with rank deficiency, and they provide minimum-norm vectors of synaptic weights, which contribute to the regularization of the input-output mapping. It is thus of interest to develop fast and accurate algorithms for computing Moore-Penrose inverse matrices. In this paper, an algorithm based on a full rank Cholesky factorization is proposed. The resulting pseudoinverse matrices are similar to those provided by other algorithms. However the computation time is substantially shorter, particularly for large systems.; Comment: Number of pages: 5. Typo page 26 line 3: one must read W=G^+F (instead of W=G^+W, which does not make sense!)

## A comparative study of Gaussian Graphical Model approaches for genomic data

Stifanelli, P. F.; Creanza, T. M.; Anglani, R.; Liuzzi, V. C.; Mukherjee, S.; Ancona, N.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.25%
The inference of networks of dependencies by Gaussian Graphical models on high-throughput data is an open issue in modern molecular biology. In this paper we provide a comparative study of three methods to obtain small sample and high dimension estimates of partial correlation coefficients: the Moore-Penrose pseudoinverse (PINV), residual correlation (RCM) and covariance-regularized method $(\ell_{2C})$. We first compare them on simulated datasets and we find that PINV is less stable in terms of AUC performance when the number of variables changes. The two regularized methods have comparable performances but $\ell_{2C}$ is much faster than RCM. Finally, we present the results of an application of $\ell_{2C}$ for the inference of a gene network for isoprenoid biosynthesis pathways in Arabidopsis thaliana.; Comment: 7 pages, 1 figure, RevTex4, version to appear in the proceedings of 1st International Workshop on Pattern Recognition, Proteomics, Structural Biology and Bioinformatics: PR PS BB 2011, Ravenna, Italy, 13 September 2011

## Optimal control theory with arbitrary superpositions of waveforms

Meister, Selina; Stockburger, Jürgen T.; Schmidt, Rebecca; Ankerhold, Joachim
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.25%
Standard optimal control methods perform optimization in the time domain. However, many experimental settings demand the expression of the control signal as a superposition of given waveforms, a case that cannot easily be accommodated using time-local constraints. Previous approaches [1,2] have circumvented this difficulty by performing optimization in a parameter space, using the chain rule to make a connection to the time domain. In this paper, we present an extension to Optimal Control Theory which allows gradient-based optimization for superpositions of arbitrary waveforms directly in a time-domain subspace. Its key is the use of the Moore-Penrose pseudoinverse as an efficient means of transforming between a time-local and waveform-based descriptions. To illustrate this optimization technique, we study the parametrically driven harmonic oscillator as model system and reduce its energy, considering both Hamiltonian dynamics and stochastic dynamics under the influence of a thermal reservoir. We demonstrate the viability and efficiency of the method for these test cases and find significant advantages in the case of waveforms which do not form an orthogonal basis.; Comment: 16 pages, 6 figures

## The Moore-Penrose Pseudoinverse. A Tutorial Review of the Theory

Barata, J. C. A.; Hussein, M. S.
Tipo: Artigo de Revista Científica
This paper studies the set of $n\times n$ matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and furthermore is isomorphic to the set of arbitrary $(n-1)\times (n-1)$ matrices. The Moore-Penrose pseudoinverse corresponds with the true inverse, (when it exists), in this lower dimension and an explicit representation of this pseudoinverse in terms of the lower dimensional space is given. This analysis is then extended to non-square matrices with all row or all column sums equal to zero.; Comment: 9 Pages