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Transfer Learning for BCIs
Published in Chang S. Nam, Anton Nijholt, Fabien Lotte, Brain–Computer Interfaces Handbook, 2018
Vinay Jayaram, Karl-Heinz Fiebig, Jan Peters, Moritz Grosse-Wentrup
Tangent space approximation refers to projecting data onto a space tangent to the manifold at a certain point, such that the projected data are well-described by Euclidean distances. This allows classifiers that are normally used with Euclidean spaces—LDA, SVM, logistic regression—to be used. However, it requires computing a point from which to determine the tangent space, which may be problematic if the new task is far enough away on the manifold for the assumptions of this method to break down.
Implications of causality for quantum biology – I: topology change
Published in Molecular Physics, 2018
The QDMEs used to describe the evolution of QMSTs can be derived from the condition a differential manifold evolve causally. By the RAZH theorem, this implies tangent spaces have a Minkowski metric. If the physics of the underlying manifold to the tangent manifold is to be independent of arbitrary position, then the tangent spaces must be affine. As the tangent spaces are Minkowski and affine, their associated symmetry group is the Poincare group of Lorentz transformations and translations of the Minkowski space. We use the double cover of the Poincare group in order to have the complete set of even and odd dimensional representations. The (double cover of) Poincare group has associated to it a pair of operators that commute with all elements of the group. These are the linear translation squared P2 and the square of the helical translations, the Pauli–Lubanski operator W2: these operators are its casimirs. Because they commute with the group elements, these two operators provide quantum numbers useful for describing the states of the QMST. Since they mutually commute, they give simultaneously specifiable eigenvectors and eigenvalues: where describes simultaneous rotation and translation – helical translation. Here Wμ is the Pauli–Lubanski (pseudo-) vector. It determines the dimension, dim = 2|σ|max + 1, of the representation of the Clifford algebra used below to factor the wave operator, P2 = ±□. The eigenvalues − (p0)2 and − (p0)2σ(σ + 1), which we collectively denote m2τ describe a discretisation of affine translations and spin rotations away from the group origin. The σ(σ + 1) eigenvalues arise from representations of the Lie algebra . The 2 × 2 matrix representation being the fundamental, spin1/2, representation that will be associated with the Dirac electron. Significantly, this result yields a wave equation that can be factored into Dirac-like equations describing CCF systems comprised of fermion singletons and the excitation of single-pairs, double-pairs, etc.