Search ResultsShowing 1-19 of 19
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arXiv:2102.01616 (Published 2021-02-02)
Divergence of an integral of a process with small ball estimate
The paper contains sufficient conditions on the function $f$ and the stochastic process $X$ that supply the rate of divergence of the integral functional $\int_0^Tf(X_t)^2dt$ at the rate $T^{1-\epsilon}$ as $T\to\infty$ for every $\epsilon>0$. These conditions include so called small ball estimates which are discussed in detail. Statistical applications are provided.
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arXiv:2011.08441 (Published 2020-11-15)
Formulation and properties of a divergence used to compare probability measures without absolute continuity and its application to uncertainty quantification
Comments: 132 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1911.07422This thesis develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. We include examples of computation and approximation of the divergence, and its applications in uncertainty quantification in discrete models and Gauss-Markov models.
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arXiv:2011.00629 (Published 2020-11-01)
Distances between probability distributions of different dimensions
Comments: 17 pagesComparing probability distributions is an indispensable and ubiquitous task in machine learning and statistics. The most common way to compare a pair of Borel probability measures is to compute a metric between them, and by far the most widely used notions of metric are the Wasserstein metric and the total variation metric. The next most common way is to compute a divergence between them, and in this case almost every known divergences such as those of Kullback-Leibler, Jensen-Shannon, R\'enyi, and many more, are special cases of the $f$-divergence. Nevertheless these metrics and divergences may only be computed, in fact, are only defined, when the pair of probability measures are on spaces of the same dimension. How would one measure, say, a Wasserstein distance between the uniform distribution on the interval $[-1,1]$ and a Gaussian distribution on $\mathbb{R}^3$? We will show that various common notions of metrics and divergences can be extended in a completely natural manner to Borel probability measures defined on spaces of different dimensions, e.g., one on $\mathbb{R}^m$ and another on $\mathbb{R}^n$ where $m, n$ are distinct, so as to give a meaningful answer to the previous question.
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arXiv:2009.04157 (Published 2020-09-09)
On Perfect Obfuscation: Local Information Geometry Analysis
We consider the problem of privacy-preserving data release for a specific utility task under perfect obfuscation constraint. We establish the necessary and sufficient condition to extract features of the original data that carry as much information about a utility attribute as possible, while not revealing any information about the sensitive attribute. This problem formulation generalizes both the information bottleneck and privacy funnel problems. We adopt a local information geometry analysis that provides useful insight into information coupling and trajectory construction of spherical perturbation of probability mass functions. This analysis allows us to construct the modal decomposition of the joint distributions, divergence transfer matrices, and mutual information. By decomposing the mutual information into orthogonal modes, we obtain the locally sufficient statistics for inferences about the utility attribute, while satisfying perfect obfuscation constraint. Furthermore, we develop the notion of perfect obfuscation based on $\chi^2$-divergence and Kullback-Leibler divergence in the Euclidean information geometry.
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arXiv:2003.08671 (Published 2020-03-19)
Divergence of non-random fluctuation for Euclidean First-passage percolation
Comments: 10 pages, 2 figuresCategories: math.PRIn this paper, we discuss non-random fluctuation in euclidean first-passage percolations and show that it diverges for any dimension and direction.
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arXiv:1912.01439 (Published 2019-12-01)
Generalization Error Bounds Via Rényi-, $f$-Divergences and Maximal Leakage
Comments: arXiv admin note: text overlap with arXiv:1903.01777In this work, the probability of an event under some joint distribution is bounded by measuring it with the product of the marginals instead (which is typically easier to analyze) together with a measure of the dependence between the two random variables. These results find applications in adaptive data analysis, where multiple dependencies are introduced and in learning theory, where they can be employed to bound the generalization error of a learning algorithm. Bounds are given in terms of $\alpha-$Divergence, Sibson's Mutual Information and $f-$Divergence. A case of particular interest is the Maximal Leakage (or Sibson's Mutual Information of order infinity) since this measure is robust to post-processing and composes adaptively. This bound can also be seen as a generalization of classical bounds, such as Hoeffding's and McDiarmid's inequalities, to the case of dependent random variables.
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arXiv:1911.07422 (Published 2019-11-18)
Formulation and properties of a divergence used to compare probability measures without absolute continuity
Comments: 46 pagesThis paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. Also included are examples of computation and approximation of the divergence, and the demonstration of properties that are useful when one quantifies model uncertainty.
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arXiv:1903.03917 (Published 2019-03-10)
Iterates of Conditional Expectations: Convergence and Divergence
Categories: math.PRIn this paper, we investigate the convergence and divergence of iterates of conditional expectation operators. We show that if $(\Omega,\cal{F},P)$ is a non-atomic probability space, then divergent iterates of conditional expectations involving 3 or 4 sub-$\sigma$-fields of $\cal{F}$ can be constructed for a large class of random variables in $L^2(\Omega,\cal{F},P)$. This settles in the negative a long-open conjecture. On the other hand, we show that if $(\Omega,\cal{F},P)$ is a purely atomic probability space, then iterates of conditional expectations involving any finite set of sub-$\sigma$-fields of $\cal{F}$ must converge for all random variables in $L^1(\Omega,\cal{F},P)$.
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arXiv:1804.09533 (Published 2018-04-25)
Divergence of non-random fluctuation in First Passage Percolation
Comments: 9 page, 2 figures, an early version of these results appeared in Sections 1.2 and 7 of arXiv:1706.03493We study non-random fluctuation in the first passage percolation on $\mathbb{Z}^d$ and show that it diverges for any dimension. We also prove the divergence of the non-random shape fluctuation, which was conjectured in [Yu Zhang. The divergence of fluctuations for shape in first passage percolation. {\em Probab. Theory. Related. Fields.} 136(2) 298--320, 2006].
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arXiv:1802.07780 (Published 2018-02-21)
Proving ergodicity via divergence of ergodic sums
Comments: 22 pages, 0 figuresA classical fact in ergodic theory is that ergodicity is equivalent to almost everywhere divergence of ergodic sums of all nonnegative integrable functions which are not identically zero. We show two methods, one in the measure preserving case and one in the nonsingular case, which enable one to prove this criteria by checking it on a dense collection of functions and then extending it to all nonnegative functions. The first method is then used in a new proof of a folklore criterion for ergodicity of Poisson suspensions which does not make any reference to Fock spaces. The second method which involves the double tail relation is used to show that a large class of nonsingular Bernoulli and inhomogeneous Markov shifts are ergodic if and only if they are conservative. In the last section we discuss an extension of the Bernoulli shift result to other countable groups including $\mathbb{Z}^{d},\ d\geq 2$ and discrete Heisenberg groups.
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arXiv:1706.03493 (Published 2017-06-12)
Divergence of shape fluctuation in First Passage Percolation
Comments: 9 pages, 4 figuresWe study the shape fluctuation in First Passage Percolation on $\Z^d$ and show that it diverges for any dimension $d\ge 2$. Moreover we will find that the order of the fluctuation is at least $\log{t}$ where $t$ is the typical diameter of the shape. Our results extends the result of [Yu Zhang. The divergence of fluctuations for shape in first passage percolation. Probab. Theory Related Fields 136 298--320, 2006] for general distributions. Our methods might be applicable to other models, such as Howard--Newman's model.
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arXiv:1606.08504 (Published 2016-06-27)
Mixed $f$-divergence for multiple pairs of measures
Comments: arXiv admin note: substantial text overlap with arXiv:1304.6792Categories: math.PRIn this paper, the concept of the classical $f$-divergence for a pair of measures is extended to the mixed $f$-divergence for multiple pairs of measures. The mixed $f$-divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed $f$-divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov-Fenchel type inequality and an isoperimetric inequality for the mixed $f$-divergence are proved.
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arXiv:1606.02155 (Published 2016-06-06)
Orlicz addition for measures and an optimization problem for the $f$-divergence
In this paper, the Orlicz addition of measures is proposed and an interpretation of the $f$-divergence is provided based on a linear Orlicz addition of two measures. Fundamental inequalities, such as, a dual functional Orlicz-Brunn-Minkowski inequality, are established. We also investigate an optimization problem for the $f$-divergence and establish functional affine isoperimetric inequalities for the dual functional Orlicz affine and geominimal surface areas of measures.
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arXiv:1409.6951 (Published 2014-09-24)
On divergence of expectations of the Feynman-Kac type with singular potentials
Motivated by the work of Baras-Goldstein (1984), we discuss when expectations of the Feynman-Kac type with singular potentials are divergent. Underlying processes are Brownian motion and $\alpha$-stable process. In connection with the work of Ishige-Ishiwata (2012) concerned with the heat equation in the half-space with a singular potential on the boundary, we also discuss the same problem in the half-space for the case of Brownian motion.
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Divergence of the correlation length for critical planar FK percolation with $1\le q\le4$ via parafermionic observables
Comments: 26 pagesKeywords: critical planar fk percolation, correlation length, parafermionic observables, divergence, article gathersTags: journal articleParafermionic observables were introduced by Smirnov for planar FK percolation in order to study the critical phase $(p,q)=(p_c(q),q)$. This article gathers several known properties of these observables. Some of these properties are used to prove the divergence of the correlation length when approaching the critical point for FK percolation when $1\le q\le 4$. A crucial step is to consider FK percolation on the universal cover of the punctured plane. We also mention several conjectures on FK percolation with arbitrary cluster-weight $q>0$.
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A class of even walks and divergence of high moments of large Wigner random matrices
Comments: version 2: minor changes; formulas (1.4) and (2.2) correctedSubjects: 15A52We study high moments of truncated Wigner nxn random matrices by using their representation as the sums over the set W of weighted even closed walks. We construct the subset W' of W such that the corresponding sum diverges in the limit of large n and the number of moments proportional to n^{2/3} for any truncation of the order n^{1/6+epsilon}, epsilon>0 provided the probability distribution of the matrix elements is such that its twelfth moment does not exist. This allows us to put forward a hypothesis that the finiteness of the twelfth moment represents the necessary condition for the universal upper bound of the high moments of large Wigner random matrices.
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arXiv:0710.4483 (Published 2007-10-24)
Correction to "The divergence of Banach space valued random variables on Wiener space", Prob. Th. Rel. Fields 132, 291-320 (2005)
Categories: math.PRAs a result of some mistakes discovered in the paper mentioned in the title, Corollaries 3.5 and 3.17a) are withdrawn and a new proof is provided for Proposition 3.14, under the added assumption that the second dual of the underlying Banach space Y possesses the Radon-Nykodim property.
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Divergence of a stationary random vector field can be always positive (a Weiss' phenomenon)
Comments: 6 pages. The phenomenon is not new, -- published by B. Weiss in 1997Categories: math.PRThe divergence of a stationary random vector field at a given point is usually a centered (that is, zero mean) random variable. Strangely enough, it can be equal to 1 almost surely. This fact is another form of a phenomenon disclosed by B. Weiss in 1997.
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arXiv:math/0501095 (Published 2005-01-06)
The divergence of fluctuations for the shape on first passage percolation
Consider the first passage percolation model on ${\bf Z}^d$ for $d\geq 2$. In this model we assign independently to each edge the value zero with probability $p$ and the value one with probability $1-p$. We denote by $T({\bf 0}, v)$ the passage time from the origin to $v$ for $v\in {\bf R}^d$ and $$B(t)=\{v\in {\bf R}^d: T({\bf 0}, v)\leq t\}{and} G(t)=\{v\in {\bf R}^d: ET({\bf 0}, v)\leq t\}.$$ It is well known that if $p < p_c$, there exists a compact shape $B_d\subset {\bf R}^d$ such that for all $\epsilon >0$ $$t B_d(1-\epsilon) \subset {B(t)} \subset tB_d(1+\epsilon){and} G(t)(1-{\epsilon}) \subset {B(t)} \subset G(t)(1+{\epsilon}) {eventually w.p.1.}$$ We denote the fluctuations of $B(t)$ from $tB_d$ and $G(t)$ by &&F(B(t), tB_d)=\inf \{l:tB_d(1-{l\over t})\subset B(t)\subset tB_d(1+{l\over t})\} && F(B(t), G(t))=\inf\{l:G(t)(1-{l\over t})\subset B(t)\subset G(t)(1+{l\over t})\}. The means of the fluctuations $E[F(B(t), tB_d]$ and $E[F(B(t), G(t))]$ have been conjectured ranging from divergence to non-divergence for large $d\geq 2$ by physicists. In this paper, we show that for all $d\geq 2$ with a high probability, the fluctuations $F(B(t), G(t))$ and $F(B(t), tB_d)$ diverge with a rate of at least $C \log t$ for some constant $C$. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting.