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19 Jun 2024: Theorem 1. Let D denote a distribution over X. Let f : X Y Rd denote a feature mappingsuch that supx,y ‖f(x,y)‖2 B, for some finite constant, B <. ... Let f : X Y Rd denote afeature mapping such that supx,y ‖f(x,y)‖2 B, for some finite constant,

19 Jun 2024: fairness. Our empirical analysissuggests that process fairness may be achieved with little cost to outcome fairness,but that some loss of accuracy is unavoidable. ... Consider a scenario for making some important decision. Let U denote the set of all

19 Jun 2024: of GM renormalizations, M(1) is the original GM,and each transition from M(t) to M(t1) corresponds torenormalization of some mini-bucket Bi to B̃i. ... Physical Review B, 97(4):045111, 2018. Hinton, Geoffrey E and Salakhutdinov, Ruslan R.

19 Jun 2024: 1We briefly note that some methods always return matrices. Unifying Orthogonal Monte Carlo Methods. ... xj)for all i,j [N], for some dataset {xi}Ni=1 Rd.

19 Jun 2024: 3). for some ξij [0, min(qi, qj)], where µij(a, b) = q(Xi =a, Xj = b). ... However,we have shown theoretically that in some cases it can causea significant effect.

19 Jun 2024: for some ξij [0, min(qi, qj)], where µij(a, b) = q(Xi =a, Xj = b). ... Some, such as dual approaches,may provide a helpful bound even if the optimum is notfound.

19 Jun 2024: dxtdt. = f(xt, t,θ), (1). parameterized by θ Rn and where f : Rd R Rn Rd is some nonlinear mapping definingdynamics. ... 2,‖Q′‖2,‖N′′‖2,‖Q′′‖2 D,‖b′‖2,‖b′′‖2 Db for some D,Db > 0 it holds that.

19 Jun 2024: These sets need not be disjointwhich we amend using the following observation: mul-tiple couplings are optimal iff either (a) xi = xj oryi = yj, for some i 6= j; or (b) ... v, xi〉 = 〈v, xj〉 or〈v, yi〉 = 〈v, yj〉, for some i 6= j.

19 Jun 2024: Alearning algorithm receives D as the input; then utilizes the datato select a model θ Θ that minimizes some notion of loss. ... Subgroup decomposability is astructural property of some inequality measures requiring that forany partition G of the

19 Jun 2024: Let P ={x Rm|Ax b} be a polytope for some A =[a1,. ... ak]. > Rkm, b Rk (for some k N). Thenfor v Ext(P), we have.

19 Jun 2024: Thus all 5 policies corre-spond to some variants of our GCMC mechanism.

19 Jun 2024: 142. B.2.3 Some frustrated K4 structures (treewidth 3). 145. B.3 Discussion. ... ic χi. This simplifies some methods of analysis but in any overcomplete representation,.