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32
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary ..."
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Cited by 93 (10 self)
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A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that
On choosing and bounding probability metrics
 Internat. Statist. Rev. (2002
"... Abstract. When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can prov ..."
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Cited by 84 (2 self)
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Abstract. When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric. Abrégé. Le choix de métrique de probabilité est une décision très importante lorsqu’on étudie la convergence des mesures. Nous vous fournissons avec un sommaire de plusieurs métriques/distances de probabilité couramment utilisées par des statisticiens(nes) at par des probabilistes, ainsi que certains nouveaux résultats qui se rapportent à leurs bornes. Avoir connaissance d’autres métriques peut vous fournir avec un moyen de dériver des bornes pour une autre métrique dans un problème appliqué. Le fait de prendre en considération plusieurs métriques vous permettra d’approcher des problèmes d’une manière différente. Ainsi, nous vous démontrons que les taux de convergence peuvent dépendre de façon importante sur votre choix de métrique. Il est donc important de tout considérer lorsqu’on doit choisir une métrique. 1.
Effective strong dimension in algorithmic information and computational complexity
 SIAM Journal on Computing
, 2004
"... The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded exten ..."
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Cited by 79 (29 self)
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The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomialspace, polynomialtime, and finitestate dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual
Equivalence of Measures of Complexity Classes
"... The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases ..."
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Cited by 70 (19 self)
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The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomialtime, truthtable reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the cointoss probability measure given by the sequence ~ fi. (2) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...
Endomorphisms Of B(H), II: Finitely Correlated States on O_n
 J. FUNCT. ANAL
, 1991
"... We identify sets of conjugacy classes of ergodic endomorphisms of B(H) where H is a fixed separable Hilbert space. They correspond to certain equivalence classes of pure states on the Cuntz algebras On where n is the Powers index. These states, called finitely correlated states, and strongly asympt ..."
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Cited by 15 (10 self)
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We identify sets of conjugacy classes of ergodic endomorphisms of B(H) where H is a fixed separable Hilbert space. They correspond to certain equivalence classes of pure states on the Cuntz algebras On where n is the Powers index. These states, called finitely correlated states, and strongly asymptotically shift invariant states, are defined and characterized. The subsets of these states defining shifts will in general be identified in [BJW], but here an interesting cross section for the conjugacy classes of shifts called diagonalizable shifts is introduced and studied.
Transient and Recurrent Spectrum
, 1981
"... We deal primarily with spectral analysis of an abstract selfadjoint operator. H, on a Hilbert space, X”. We propose a further refinement of the absolutely continuous subspace,;F”a,, into the transient absolutely continuous subspace, &‘a,. which is the closure of those cp with (cp, ej/Ho) = O(t“) ..."
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Cited by 11 (2 self)
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We deal primarily with spectral analysis of an abstract selfadjoint operator. H, on a Hilbert space, X”. We propose a further refinement of the absolutely continuous subspace,;F”a,, into the transient absolutely continuous subspace, &‘a,. which is the closure of those cp with (cp, ej/Ho) = O(t“) for all N and the recurrent absolutely continuous subspace, 2 & = ‘qc nX&. We discuss general features of this breakup. In a subsequent paper, we construct analytic almost periodic functions, V, on (03, 03) so that H =d*/dx * + V(x) on L2(co, co) has only recurrent absolutely continuous spectrum in the sense that qa, =.;Y.
Iterated function systems, representations, and Hilbert space
 International J. Math
"... In this paper, we are concerned with spectraltheoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i = 1,...,N, in some Hausdorff space Y. There is a standard construction which generally allows us to redu ..."
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Cited by 10 (9 self)
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In this paper, we are concerned with spectraltheoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i = 1,...,N, in some Hausdorff space Y. There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X ⊂ Y. Typically, some kind of contractivity property for the maps τi is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure µ available which allows us to pass the pointmaps τi to operators on the Hilbert space L2 (µ). Instead, we show that it is possible to realize the maps τi quite generally in Hilbert spaces H (X) of squaredensities on X. The elements in H (X) are equivalence classes of pairs (ϕ,µ), where ϕ is a Borel function on X, µ is a positive Borel measure on X, and ∫ X ϕ2 dµ < ∞. We say that (ϕ,µ) ∼ (ψ,ν) if there is a positive Borel measure λ such that µ < λ, ν < λ, and ϕ dµ dν = ψ dλ dλ λ a.e. on X. We prove that, under general conditions on the system (X,τi), there are isometries Si: (ϕ,µ) ↦− → ( ϕ ◦ σ,µ ◦ τ −1) i in H (X) satisfying ∑N i=1 SiS ∗ i = I = the identity operator in H (X). For the construction we assume that some mapping σ: X → X satisfies the conditions σ ◦ τi = idX, i = 1,...,N. We further prove that this representation in the Hilbert space H (X) has several universal properties.
The BurbeaRao and Bhattacharyya centroids
 IEEE Transactions on Information Theory
, 2010
"... Abstract—We study the centroid with respect to the class of informationtheoretic BurbeaRao divergences that generalize the celebrated JensenShannon divergence by measuring the nonnegative Jensen difference induced by a strictly convex and differentiable function. Although those BurbeaRao diverge ..."
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Cited by 10 (7 self)
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Abstract—We study the centroid with respect to the class of informationtheoretic BurbeaRao divergences that generalize the celebrated JensenShannon divergence by measuring the nonnegative Jensen difference induced by a strictly convex and differentiable function. Although those BurbeaRao divergences are symmetric by construction, they are not metric since they fail to satisfy the triangle inequality. We first explain how a particular symmetrization of Bregman divergences called JensenBregman distances yields exactly those BurbeaRao divergences. We then proceed by defining skew BurbeaRao divergences, and show that skew BurbeaRao divergences amount in limit cases to compute Bregman divergences. We then prove that BurbeaRao centroids can be arbitrarily finely approximated by a generic iterative concaveconvex optimization algorithm with guaranteed convergence property. In the second part of the paper, we consider the Bhattacharyya distance that is commonly used to measure overlapping degree of probability distributions. We show that Bhattacharyya distances on members of the same statistical exponential family amount to calculate a BurbeaRao divergence in disguise. Thus we get an efficient algorithm for computing the Bhattacharyya centroid of a set of parametric distributions belonging to the same exponential families, improving over former specialized methods found in the literature that were limited to univariate or “diagonal ” multivariate Gaussians. To illustrate the performance of our Bhattacharyya/BurbeaRao centroid algorithm, we present experimental performance results for kmeans and hierarchical clustering methods of Gaussian mixture models.