Eric Klassen — Page 2

arXiv2011

Registration of Functional Data Using Fisher-Rao Metric

We introduce a novel geometric framework for separating the phase and the amplitude variability in functional data of the type frequently studied in growth curve analysis. This framework uses the Fisher-Rao Riemannian metric to derive a proper distance on the quotient space of functions modulo the time-warping group. A convenient square-root velocity function (SRVF) representation transforms the Fisher-Rao metric into the standard $\ltwo$ metric, simplifying the computations. This distance is then used to define a Karcher mean template and warp the individual functions to align them with the Karcher mean template. The strength of this framework is demonstrated by deriving a consistent estimator of a signal observed under random warping, scaling, and vertical translation. These ideas are demonstrated using both simulated and real data from different application domains: the Berkeley growth study, handwritten signature curves, neuroscience spike trains, and gene expression signals. The proposed method is empirically shown to be be superior in performance to several recently published methods for functional alignment.

math.ST↗

arXiv1994

The Spectral Flow of the Odd Signature operator and Higher Massey Products

We show how to compute the spectral flow of the odd signature operator $\pm *d_{a_t}-d_{a_t}*$ along an analytic path of flat connections $a_t$ on a bundle over a closed odd-dimensional manifold in terms of Massey products in the DGLA of bundle-valued differential forms. To obtain this information, we set up a sequence of cochain complexes $\{\calg^*_n,δ_n\}$, for $n=0,1,2,\ldots$ and Hermitian forms $$Q_n:\calg_n\times\calg_n\ra \bbbC$$ whose signatures determine the spectral flow through $t=0$. The complexes and Hermitian forms are constructed using Massey products.

dg-ga↗