Secondly, TANGO is more robust to the number of vertices (i.e., the mesh quality).

The StoNet consisted of one hidden layer withq hidden units. Extension of our proof for the mini-batch case will be discussed in Remark S2.

The bottom graph is a historical example ofunfairness: evenifthere would benobias betweenLoanand Race,redlining(i.e. the practice of refusing aloan topeople living in certain areas) would discriminate indirectly based on race [1,2,3,4]. This example also showswhysimply removing or not measuring a sensitive attribute does not suffice: not only does this ignore indirect bias, but hiding the protected attribute leads to an (additional) correlation betweenPostcodeandLoandue to confounding. InTable 1, we observethat naively removing the protected attribute only ensures FTU fairness, asshown by: GAN-PR, WGAN-GP-PR, and DECAF-PR. This is the direct result of the construction of generatorG and follows a similar argument asProposition 2of[6]. P(Xi|{Xj:(Xj Xi) E}) Given each Gi (see Eq. 2 paper) has enough capacity,G can thus express the full distribution PX(X).

Each module could be either parallel module or fusion module, which is determined by optimizing the architecture parametersαp and αf. Specifically,the learned twoarchitectures both contain eight fusion modules and four parallel modules, and the only one difference between them is the position ofthefusion andtheparallel modules. From theobservations, wecould conclude that: 1) themulti-scale information isremarkably important toimage restoration. Image restoration using very deep convolutional encoder-decoder networks with symmetric skip connections. From the top to the bottom for each image, the noise levels areσ = 30,50,70. From the left to the right are Input, BM3D[1],RED[9],WNNM[3],NLRN[6],DuRN-P [7],N3Net[10],CLEARER, andGround truth.

A.1 ProofsofSection3 In the following, the symbolh, i will be used to represent both Frobenius norm of matrices and standard Euclidean norm of vectors. For the second part, letv be an eigenvector ofA corresponding to eigenvalueλmax(A). Incase the maximum eigenvalue ofT isnonpositive, then from Lemma A.1 we see that the objectivevalue of problem(A.2)evaluated For anp preal matrixA, its spectral radiusR(A)is defined as the largest absolute value of its eigenvalues. Then the matrixI A is invertible and all entries of(I A) 1 are nonnegative. Also the spectral radius of(γ?) 1bΩ12V(β)bΩ12 is smaller than1 by the feasibility ofγ? in problem (A.5c).

The statistics are listed in Table 5, where the highest record ateach label rate ishighlighted inbold. Inparticular,whenthereare ten labeled data per class, the performance ofallmethods becomes similar.

Themeasurementmodel corresponds toy = Ax + e, where x is the true solution ande is the noise.

To better understand the(M,X) D and Assumption 1, let's take a simple example.