L. Gross, "Integration and non-linear transformations in Hilbert space," Transactions of the American Mathematical Society, Vol. 94, pp. 404--440, 1960. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
A Bayesian Framework for Regularization - Keren, Werman (1994) (1 citation) (Correct)
....it is shown how this is equiwlent to simpler, one dimensional, optimization problem. The question of how to compute such integrals s those ppering in Equation 1 which re defined over domains that re infinite dimensional hs been solved for some types of integrals in the realm of pure mthemtics [8, 14, 16, 29, 15, 7]. It ws pplied to the types of spces used in regulriztion in [12, 13] The spce M, is Hilbert spce [30] Let us recall that if U is subspce of Hilbert spce H, its orthwonal subspace, U , is defined s following 4 It is well known that for every h H T there are u I U and u2 U so that Ul u2 = ....
L. Gross. Integration and non-linear transformations in hilbert space. Trans- actions of the American Mathematical Society, 94:404-440, 1960.
A Full Bayesian Approach to Curve and Surface Reconstruction - Keren, Werman (1999) (3 citations) (Correct)
....rest of this section is dedicated to this reduction, culminating in the expression of Eq. 7) The problem of computing such integrals as those appearing in Eq. 2) which are defined over infinite dimensional domains has been solved for some types of integrals in the realm of pure mathematics [8, 11, 17 19, 41]. It was applied to the types of spaces used in regularization in [15, 16] The space M #,# is a Hilbert space [42] We will need to use the notion of an orthogonal subspace; let us recall that if U is a subspace of a Hilbert space H , its orthogonal subspace, U # , is defined as U # ....
.... ) note that these are different than the H x i ) By a change of variables this turns out to be # [g(x) f 0 (x) 2 exp( g, g) Dg # g 2 (x) exp( g, g) Dg = # # f 2 0 (x) g 2 (x) # (Dg) 42 Keren and Werman where (Dg) is the Gaussian measure induced by the inner product ( [8, 15, 19]. We need to compute # g 2 (x) Dg) In [15] it is shown that this integral equals # 2 V x # B 1 ## 2 B 2 1 # 1 V T x where, if we denote x = x n 1 , V x = H x 1 (x) H x n 1 (x) and B 1 is the (n 1) n 1) matrix defined by (B 1 ) i, j = H i (x j ) so, B ....
L. Gross, "Integration and non-linear transformations in Hilbert space," Transactions of the American Mathematical Society, Vol. 94, pp. 404--440, 1960.
A Bayesian Framework for Regularization - Keren, Werman (1994) (1 citation) (Correct)
....how this is equivalent to a simpler, one dimensional, optimization problem. The question of how to compute such integrals as those appearing in Equation 1 which are defined over domains that are infinite dimensional has been solved for some types of integrals in the realm of pure mathematics [8, 14, 16, 29, 15, 7]. It was applied to the types of spaces used in regularization in [12, 13] The space M ;oe is a Hilbert space [30] Let us recall that if U is a subspace of a Hilbert space H , its orthogonal subspace, U , is defined as following U = fh 2 H ju 2 U = u; h) 0g It is well known ....
L. Gross. Integration and non-linear transformations in hilbert space. Transactions of the American Mathematical Society, 94:404--440, 1960.
A Full Bayesian Approach to Curve and Surface Reconstruction - Keren, Werman (1999) (3 citations) (Correct)
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L. Gross, "Integration and non-linear transformations in Hilbert space," Transactions of the American Mathematical Society, Vol. 94, pp. 404--440, 1960.
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