In [Paper, Algorithms 1 and 2] we need the gradient of the residuals dg(Y(ri),Yi) (here, Y(ri) is equal to X1(ri)) with respect to the base point Y(ri) in order to compute the jump conditions for the adjoint variable λ1. The residual measures the squared geodesic distance between the point Y(ri) on the fitted curve and the corresponding measurement Yi. To derive this gradient, we consider the c...

In this paper, we prove that if X is regular strongly screenable DC-like (C-scattered), then X × Y is strongly screenable for every strongly screenable space Y . We also show that the product ∏ i∈ω Yi is strongly screenable if every Yi is a regular strongly screenable DC-like space. Finally, we present that the strongly screenableness are poorly behaved with its Tychonoff products. Keywords—top...

Example 1 To illustrate the distinction between classification and regression, consider a simple, scalar signal plus noise problem. Consider Yi = θ +Wi, i = 1, . . . , n, where θ is a fixed unknown scalar parameter and the Wi are independent, zero-mean, unit variance random variables. Let Ȳ = 1/n ∑n i=1 Yi. Then, according to the Central Limit Theorem, Ȳ is distributed approximately N(θ, 1/n). ...

This is the usual introduction to least squares fit by a line when the data represents measurements where the y–component is assumed to be functionally dependent on the x–component. Given a set of samples {(xi, yi)}i=1, determine A and B so that the line y = Ax + B best fits the samples in the sense that the sum of the squared errors between the yi and the line values Axi + B is minimized. Note...

• Note for i.i.d. samples (xi, yi) ∈ R × R, i = 1, . . . , n, we can always write yi = f0(xi) + i, i = 1, . . . , n, where i, i = 1, . . . , n are i.i.d. random errors, with mean zero. Therefore we can think about the sampling distribution as follows: (xi, i), i = 1, . . . , n are i.i.d. draws from some common joint distribution, where E( i) = 0, and yi, i = 1, . . . , n are generated from the ...

Given a commutative ring R (respectively a positively graded commutative ring A = ⊕j>0Aj which is finitely generated as an A0-algebra), a bijection between the torsion classes of finite type in ModR (respectively tensor torsion classes of finite type in QGrA) and the set of all subsets Y ⊆ Spec R (respectively Y ⊆ Proj A) of the form Y = S i∈Ω Yi, with Spec R \ Yi (respectively Proj A \ Yi) qua...

Transitive signatures allow a signer to authenticate edges in a graph in such a way that anyone, given the public key and two signatures on adjacent edges (i, j) and (j, k), can compute a third signature on edge (i, k). A number of schemes have been proposed for undirected graphs, but the case of directed graphs remains an open problem. At CT-RSA 2007, Yi presented a scheme for directed trees b...

Given the value of π1100, the linear programming problem (19)-(25) in the paper has a solution if and only if the set Φ = [max{q110|1 −π1000, q111|1π1100 π1111+π1110+π1010 }, q110|1π1100 π1100+π1000 ] is not empty, which is essentially q110|1 p10|1 ≥ q111|1 p11|1 , an inequality that must be satisfied based on Assumptions 5-7. IfΦ is not empty, let T = max{q110|1−π1000, q111|1π1100 π1111+π1110+...

We completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k, 3) be a bipartite graph with bipartition X = {x1, x2, . . . , xk}, Y = {y1, y2, . . . , yk} and edges x1 y1, x1 y2, xk yk−1, xk yk , and xi yi−1, xi yi , xi yi+1 for i = 2, 3, . . . , k − 1. We completely characterize all complete bipartite graphs Kn,n ...

This paper is concerned with scattered data approximation in high dimensions: Given a data set X ⊂ Rd of N data points xi along with values yi ∈ Rd , i = 1, . . . , N , and viewing the yi as values yi = f(xi) of some unknown function f , we wish to return for any query point x ∈ Rd an approximation f̃(x) to y = f(x). Here the spatial dimension d should be thought of as large. We wish to emphasiz...