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"Hilbert spaces"
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EQUIVALENCE OF DISTANCE-BASED AND RKHS-BASED STATISTICS IN HYPOTHESIS TESTING
We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. In the case where the energy distance is computed with a semimetric of negative type, a positive definite kernel, termed distance kernel, may be defined such that the MMD corresponds exactly to the energy distance. Conversely, for any positive definite kernel, we can interpret the MMD as energy distance with respect to some negative-type semimetric. This equivalence readily extends to distance covariance using kernels on the product space. We determine the class of probability distributions for which the test statistics are consistent against all alternatives. Finally, we investigate the performance of the family of distance kernels in two-sample and independence tests: we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.
Journal Article
NONPARAMETRIC STOCHASTIC APPROXIMATION WITH LARGE STEP-SIZES
We consider the random-design least-squares regression problem within the reproducing kernel Hubert space (RKHS) framework. Given a stream of independent and identically distributed input/output data, we aim to learn a regression function within an RKHS ℋ, even if the optimal predictor (i.e., the conditional expectation) is not in ℋ. In a stochastic approximation framework where the estimator is updated after each observation, we show that the averaged unregularized least-mean-square algorithm (a form of stochastic gradient descent), given a sufficient large step-size, attains optimal rates of convergence for a variety of regimes for the smoothnesses of the optimal prediction function and the functions in ℋ. Our results apply as well in the usual finite-dimensional setting of parametric least-squares regression, showing adaptivity of our estimator to the spectral decay of the covariance matrix of the covariates.
Journal Article
Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are
given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the
Airy
In this paper, we employ the Brownian Gibbs property to make a close
comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially
growing moment bound on Radon-Nikodym derivatives.
We also determine the value of a natural exponent describing in Brownian last
passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common
endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness
tending to zero.
To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on
probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property.
Several results in this article
play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which
geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.
eBook
Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
Transfer operators such as the Perron–Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We propose
kernel transfer operators
, which extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings. The proposed numerical methods to compute empirical estimates of these kernel transfer operators subsume existing data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernel-based approaches is that they can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we provide elementary results on eigendecompositions of finite-rank RKHS operators. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis.
Journal Article
Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions
The aim of this article is to propose a new definition of fuzzy fractional derivative, so-called fuzzy conformable. To this end, we discussed fuzzy conformable fractional integral softly. Meanwhile, uniqueness, existence, and other properties of solutions of certain fuzzy conformable fractional differential equations under strongly generalized differentiability are also utilized. Furthermore, all needed requirements for characterizing solutions by equivalent systems of crisp conformable fractional differential equations are debated. In this orientation, modern trend and new computational algorithm in terms of analytic and approximate conformable solutions are proposed. Finally, the reproducing kernel Hilbert space method in the conformable emotion is constructed side by side with numerical results, tabulated data, and graphical representations.
Journal Article
Quasi Gelfand Triples
We generalize the notion of Gelfand triples (also called Banach-Gelfand triples or rigged Hilbert spaces) by dropping the necessity of a continuous embedding. This means in our setting we lack of a chain inclusion. We replace the continuous embedding by a closed embedding of a dense subspace. This notion will be called
quasi Gelfand triple
. These triples appear naturally, when we regard the boundary spaces of spatially multidimensional differential operators, e.g., the Maxwell operator. We will show that there is a smallest space where we can continuously embed the entire triple. Moreover, we will show density results for intersections of members of the quasi Gelfand triple. Finally, we show that every quasi Gelfand triple can be decomposed into two “ordinary” Gelfand triples.
Journal Article
Complex interpolation between Hilbert, Banach and operator spaces
Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces X satisfying the following property: there is a function \\varepsilon\\to \\Delta_X(\\varepsilon) tending to zero with \\varepsilon0 such that every operator T\\colon \\ L_2\\to L_2 with \\|T\\|\\le \\varepsilon that is simultaneously contractive (i.e., of norm \\le 1) on L_1 and on L_\\infty must be of norm \\le \\Delta_X(\\varepsilon) on L_2(X). The author shows that \\Delta_X(\\varepsilon) \\in O(\\varepsilon^\\alpha) for some \\alpha0 if X is isomorphic to a quotient of a subspace of an ultraproduct of \\theta-Hilbertian spaces for some \\theta0 (see Corollary 6.7), where \\theta-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).
eBook
PARTIAL DISTANCE CORRELATION WITH METHODS FOR DISSIMILARITIES
Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions and applications of distance correlation have been discussed in the recent literature, but the problem of defining the partial distance correlation has remained an open question of considerable interest. The problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space. For the definition of partial distance correlation, we introduce a new Hubert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hubert space, and develop and implement a test for zero partial distance correlation. Our intermediate results provide an unbiased estimator of squared distance covariance, and a neat solution to the problem of distance correlation for dissimilarities rather than distances.
Journal Article
ONLINE ESTIMATION OF THE GEOMETRIC MEDIAN IN HILBERT SPACES: NONASYMPTOTIC CONFIDENCE BALLS
Estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with very large samples of high dimensional data. The collected data may be contaminated by noise so that robust location indicators, such as the geometric median, may be preferred to the mean. In this context, an estimator of the geometric median based on a fast and efficient averaged nonlinear stochastic gradient algorithm has been developed by [Bernoulli 19 (2013) 18-43]. This work aims at studying more precisely the nonasymptotic behavior of this nonlinear algorithm by giving nonasymptotic confidence balls in general separable Hubert spaces. This new result is based on the derivation of improved L² rates of convergence as well as an exponential inequality for the nearly martingale terms of the recursive nonlinear Robbins-Monro algorithm.
Journal Article