Differential-geometrical methods in statistics

By Shun-ichi Amari. 1985

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Citations in the corpus

Revisiting Natural Gradient for Deep Networks (2013, methodology): “Natural gradient can be traced back to Amari’s work on information geometry (Amari, 1985) and its application to various neural networks (Amari et al. , 1992; Amari, 1997), though a more in depth introduction can be found in Amari (1998); Park et al. (2000); Arnold et al. (2011).”
Dropout Inference in Bayesian Neural Networks with Alpha-divergences (2017, methodology): “…include variational inference (VI) (Jordan et al., 1999) and expectation propagation (EP) (Minka, 2001), where these two algorithms are special cases of power EP (Minka, 2004) that minimises Amari’s α-divergence (Amari, 1985) Dα[p||q] in a local way: Dα[p||q] = 1 α(1− α) ( 1− ∫ p(ω)αq(ω)1−αdω ) .”
Black-box $\alpha$-divergence Minimization (2015, methodology): “The α -divergence (Amari, 1985) measures the “similarity” between two distributions, and in this paper we adopt a more convenient form 1 (Zhu & Rohwer, 1995): The following examples with different α values are interesting special cases: For the ﬁrst two limiting cases KL[ p || q ] denotes the…”
Neural Learning in Structured Parameter Spaces - Natural Riemannian Gradient (1996, background): “Information geometry (Amari, 1985) proves that the Fisher information G is the only invariant metric to be introduced in the space S = {1o} of the parameters of probability distributions.”
Statistical Theory of Learning Curves under Entropic Loss Criterion (1993, methodology): “This approach was successfully applied to neural networks (Baum and Haussler 1989). Haussler et al. (1988) studied the convergence rate of general learning curves by relaxing algorithmic constraints.”
Dependence, Correlation and Gaussianity in Independent Component Analysis (2003, background): “Exponential families play a central role in statistics as well as in information geometry where they behave to some extent like 'flat manifolds', (see Amari, 1985).”
Complex-Valued Autoencoders (2011, methodology): “Methods from differential geometry has been applied effectively to statistical machine learning in previous studies by (Amari, 1990; Amari & Nagaoka, 2007) and others.”
Asymptotics and the theory of inference (2003, background): “Amari (1985) shows that the asymptotic power of the likelihood ratio test is largest over a wide range of alternatives.”
Robust parameter estimation with a small bias against heavy contamination (2008, background): “The Pythagorean relation is known in terms of the KL-divergence related to an exponential family and maximum likelihood estimation [1], the KL-divergence related to the predictive distribution [7], the extended KL-divergence related to AdaBoost [8], the Bregman divergence related to boosting [13], a”
Covariance and Fisher information in quantum mechanics (2001, background): “On the one hand standard quantum mechanics is a statistical theory, on the other hand, there is a so-called geometrical approach to mathematical statistics [1, 4].”
Information Theory and Statistics: A Tutorial (2004, background): “The remarkable analogy of certain information theoretic concepts and results to geometric ones, instrumental in this Chapter and later on, has a profound background in a differential geometric structure of probability distributions, beyond the scope of this tutorial, see C̆encov [49], Amari [1].”
Geometry of Quantum States: Index (2007, methodology): “Amari [30] contains a highly recommended book length discussion of statistical geometry.”