相关论文: Orthonormal integrators based on Householder and G…

This document aims to be a self-contained, mathematically precise overview of transformer architectures and algorithms (*not* results). It covers what transformers are, how they are trained, what they are used for, their key architectural…

I adapt a recently introduced method for integrating over the unitary group (S. Aubert and C.S. Lam, J.Math.Phys. 44, 6112-6131 (2003)) to the orthogonal group. I derive explicit formulas for a number of one, two and three-vector integrals,…

In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types…

This paper proposes a new framework and algorithms to address the problem of diffeomorphic registration on a general class of geometric objects that can be described as discrete distributions of local direction vectors. It builds on both…

We investigate the structure of graded commutative exponential functors. We give applications of these structure results, including computations of the homology of the symmetric groups and of extensions in the category of strict polynomial…

We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate…

New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all…

The article presents four identities containing the spatio-temporal differential operator also known as four-divergence. These equations are used to prove the invariance of wave equation under orthogonal paravector transformations.…

We examine a class of embeddings based on structured random matrices with orthogonal rows which can be applied in many machine learning applications including dimensionality reduction and kernel approximation. For both the…

We demonstrate the utility of effective Hamilonians for studying strongly correlated systems, such as quantum spin systems. After defining local relevant degrees of freedom, the numerical Contractor Renormalization (CORE) method is applied…

We consider the relations between nonstationary quantum oscillators and their stationary counterpart in view of their applicability to study particles in electromagnetic traps. We develop a consistent model of quantum oscillators with…

For a datastream, the change over a short interval is often of low rank. For high throughput information arranged in matrix format, recomputing an optimal SVD approximation after each step is typically prohibitive. Instead, incremental and…

New real structure-preserving decompositions are introduced to develop fast and robust algorithms for the (right) eigenproblem of general quaternion matrices. Under the orthogonally JRS-symplectic transformations, the Francis JRS-QR step…

Various methods of constructing an orthonomal set out of a given set of linearly independent vectors are discussed. Particular attention is paid to the Gram-Schmidt and the Schweinler-Wigner orthogonalization procedures. A new…

Langevin and Brownian simulations play a prominent role in computational research, and state of the art integration algorithms provide trajectories with different stability ranges and accuracy in reproducing statistical averages. The…

Deep learning models often rely only on a small set of features even when there is a rich set of predictive signals in the training data. This makes models brittle and sensitive to distribution shifts. In this work, we first examine vision…

We construct an algorithm for implementing the discrete wavelet transform by means of matrices in SO_2(R) for orthonormal compactly supported wavelets and matrices in SL_m(R), m > = 2, for compactly supported biorthogonal wavelets. We show…

Several laws are found for the Diffractive Deep Neural Networks (D2NN). They reveal the inner product of any two light fields in D2NN is invariant and the D2NN act as a unitary transformation for optical fields. If the output intensities of…

Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…

The Householder algorithm for the QR factorization of a tall thin n x p full-rank matrix X has the added bonus of producing a matrix M with orthonormal columns that are a basis for the orthocomplement of the column space of X. We give a…