Thisboundisattainedif nisanevennumber, λn/2 isthatdesiredeigenvalue.Basedonthenumerical experiment, we know it ifn is an odd number, this bound cannot be attained. The ring topology is undirected, and is illustrated in Figure 1(a). The star topology is undirected, and is illustrated in Figure 1(b). Its weight matrix is generated according totheMetropolis rule,which issymmetric. The 2D-grid topology is undirected, and is illustrated in Figure 1(c).

Structural equation models (SEMs) are widely used in sciences, ranging from economics topsychology,touncovercausal relationships underlying acomplex system under consideration and estimate structural parameters of interest. We study estimation in a class of generalized SEMs where the object of interest is defined as the solution to a linear operator equation.

We set the output of each rule as the majority class (resp.

The key idea of the proof is to exploit the problem representation in terms of confusion matrices. Here we set up and discuss the example in 3.2 in more detail. In the following, whenAj is used to denote an event inside a probability, it refers to the event {Aj =1}. First step is to extract the error incurred by plugging inˆη rather than η. C.2 WeightedERM In the weighed ERM approach (referred to as cost-sensitive classification for the binary case [1]) we parametrizeh: X [K]by a function classF of functions: X RK.

Further, this choice offα,β allows us to boundkfα,βk given that the ground cost functionc is boundedonX. The proof is given in Appendix C.4. B.3 LasttermconvergenceofSD With a slight change toSD, we can claim its last term convergence: In each iteration, check if S(αt,{βi}ni=1) . For simplicity, we omit the subscript of the Sinkhorn potentialfα,β and simply usef. This implies that 2f(x) exists and is bounded from above: x X,k 2f(x)k Lf, which concludestheproof.

In this section, we provide proofs for Proposition 2.1.B. Inthe proof, we inherit the notations that weuseforprovingTheorem2.1. The instance normalization that we incorporate into the DGM is not the same as the instance normalization that is typically used in image stylization [35]. CNN-F-5 significantly improves the robustness of CNN. CNN-F achieves higher accuracy on MNIST than CNN for under both standard training and adversarial training.