Transformations and Hardy--Krause Variation

Abstract

Using a multivariable Faa di Bruno formula we give conditions on transformations τ : [0, 1] m → X where X is a closed and bounded subset of R d such that f • τ is of bounded variation in the sense of Hardy and Krause for all f ∈ C d (X ). We give similar conditions for f •τ to be smooth enough for scrambled net sampling to attain O(n −3/2+ ) accuracy. Some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition. Some other transformations due to Fang and Wang (1993) satisfy the first but not the second condition. We provide transformations for the simplex that makes f • τ smooth enough to fully benefit from scrambled net sampling for all f in a class of generalized polynomials. We also find sufficient conditions for the Rosenblatt-HlawkaMück transformation in R 2 and for importance sampling to be of bounded variation in the sense of Hardy and Krause.

Asymptotic normality of scrambled geometric net quadrature

“…This is shown in Figure 3. Throughout this simulation we keep n = 4 6 . The intervals shown in red fail to contain the true value of µ, and as a result it can be seen that we have desired control over the coverage of the confidence intervals.…”

“…Measure preserving mapping from the unit cube to such shapes work very well for plain Monte Carlo. Unfortunately, the composition of the integrand with the mapping may fail to have even mild smoothness properties that QMC exploits [6].…”

Analytical variation in the generalization of deep feed-forward neural networks

“…What we should take from the above bounds is the fundamental role the hypothesis space choice plays in the approximation of the true error. If H is not too complex in the VC or Rademacher sense, that is, if it does not have a large variety of functions, then their empirical loss is fairly close to the true loss for any dataset S. On the other hand, if either the VC dimension or the Rademacher complexity is infinite, then we cannot possibly expect to learn and we should look for simpler solutions 5 .…”

“…That is, it will receive a dataset S ⊆ Z n sampled from µ n as an input, and sample a hypothesis h according to the induced distribution P (H|S) in the hypothesis class H parametrized by the weight space W. Notice that H is now a random variable 4 with values in H, whose distribution is characterized by the algorithm, defining a stochastic learning map. Thus, it makes sense to address the information shared by S and the output, or in other words, to how the prior's entropy H(W ) induced by the algorithm changes when it reads the dataset 5 .…”

“…for Γ being any closed surface outside the support, n its normal and ∇ x the divergent. Because in the above equation k is seen as a constant, the integrand on the left may be calculated as follows 5 :…”