<p style='text-indent:20px;'>We study the deformation of input space by a trained autoencoder via Jacobians weight matrices. In doing so, we prove bounds for mean squared errors points in space, under assumptions regarding orthogonality eigenvectors. We also show that trace and product eigenvalues Jacobian matrices is good predictor on test points. This dataset independent means testing a...

with a symmetric and positive definite operator A defined on a Hilbert space V with dimV = N and its preconditioned version PCG. We derive the CG from the projection in A-inner product point of view. We shall use (·, ·) for the standard inner product and (·, ·)A for the inner product introduced by the SPD operator A. When we say ‘orthogonal’ vectors, we refer to the default (·, ·) inner product...

MacWilliams’ equivalence theorem states that Hamming isometries between linear codes extend to monomial transformations of the ambient space. One of the most elegant proofs for this result is due to K. P. Bogart et al. (1978, Inform. and Control 37, 19–22) where the invertibility of orthogonality matrices of finite vector spaces is the key step. The present paper revisits this technique in orde...

Abstract For the linear Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate error constant under maximum norm by using finite-element method (FEM). In solving optimization problem corresponding constant, in constraint condition is most difficult part process. To handle this difficulty, novel proposed combining orthogonality of space decompos...

The orthogonality relations of multivariate Krawtchouk polynomials are discussed. In case of two variables, the necessary and sufficient conditions of orthogonality is given by Grünbaum and Rahman in [SIGMA 6 (2010), 090, 12 pages]. In this study, a simple proof of the necessary and sufficient condition of orthogonality is given for a general case.

AbstractEstimating the average treatment effect (ATE) from observational data is challenging due to selection bias. Existing works mainly tackle this challenge in two ways. Some researchers propose constructing a score function that satisfies orthogonal condition, which guarantees established ATE estimator “orthogonal" be more robust. The others explore representation learning models achieve ba...

A unified electrodynamic approach to the guided wave excitation by external sources in the waveguiding structures with bianisotropic media is developed. Effect of electric, magnetic, and magneto-electric losses in such media manifests itself in the special form of eigenmode orthogonality referred to as the quasi-orthogonality relation. It reflects the existence of the cross-power flow Pkl and l...

Orthogonality of polynomials in several variableswith respect to a positiveBorelmeasure supported on an algebraic set is themain theme of this paper.As a step towards this goal quasi-orthogonality with respect to a non-zero Hermitian linear functional is studied in detail; this occupies a substantial part of the paper. Therefore necessary and sufficient conditions for quasi-orthogonality in ter...

Abstract The orthogonality time is examined for different initial states settings interacting locally with types of spin interaction: XX , Ising and anisotropic models. It shown that the number increases, consequently decreases as environment qubits increase. shortest displayed chain model, while largest model. external field increases numbers orthogonality, Dzyaloshinsky–Moriya interaction ort...