相关论文: Fast Differentiable Matrix Square Root and Inverse…

We show how shadows can be efficiently generated in differentiable rendering of triangle meshes. Our central observation is that pre-filtered shadow mapping, a technique for approximating shadows based on rendering from the perspective of a…

A fast algorithm for inverse Cholesky factorization is proposed, to compute a triangular square-root of the estimation error covariance matrix for Vertical Bell Laboratories Layered Space-Time architecture (V-BLAST). It is then applied to…

We present a FFT-based algorithm for the computation of a polynomial's coefficients from its roots, and apply it to obtain the coefficients of interpolation polynomials, to invert Vandermondians and to evaluate the symmetric functions of a…

Volker Strassen first suggested an algorithm to multiply matrices with worst case running time less than the conventional $\mathcal{O}(n^3)$ operations in 1969. He also presented a recursive algorithm with which to invert matrices, and…

Incorporating the Forward Forward algorithm into neural network training represents a transformative shift from traditional methods, introducing a dual forward mechanism that streamlines the learning process by bypassing the complexities of…

A new O(N) algorithm based on a recursion method, in which the computational effort is proportional to the number of atoms N, is presented for calculating the inverse of an overlap matrix which is needed in electronic structure calculations…

We address the task of higher-order derivative evaluation of computer programs that contain QR decompositions and real symmetric eigenvalue decompositions. The approach is a combination of univariate Taylor polynomial arithmetic and matrix…

We study two procedures (reverse-mode and forward-mode) for computing the gradient of the validation error with respect to the hyperparameters of any iterative learning algorithm such as stochastic gradient descent. These procedures mirror…

Low-rank regularization-based deep unrolling networks have achieved remarkable success in various inverse imaging problems (IIPs). However, the singular value decomposition (SVD) is non-differentiable when duplicated singular values occur,…

In this paper, we discuss numerical methods for the eigenvalue decomposition of real symmetric matrices. While many existing methods can compute approximate eigenpairs with sufficiently small backward errors, the magnitude of the resulting…

Stochastic gradient descent (SGD) has achieved great success in training deep neural network, where the gradient is computed through back-propagation. However, the back-propagated values of different layers vary dramatically. This…

There is rising interest in differentiable rendering, which allows explicitly modeling geometric priors and constraints in optimization pipelines using first-order methods such as backpropagation. Incorporating such domain knowledge can…

Obtaining the inverse of a large symmetric positive definite matrix $\mathcal{A}\in\mathbb{R}^{p\times p}$ is a continual challenge across many mathematical disciplines. The computational complexity associated with direct methods can be…

$k$-diagonal circulant matrices and cyclic banded matrices are widely used in numerical simulations and signal processing of circular linear systems. Algorithms that directly involve or specify linear or quadratic complexity for the…

The Forward-Forward (FF) algorithm presents a compelling, bio-inspired alternative to backpropagation. However, while efficient in training, it has a computationally prohibitive inference process that requires a separate forward pass for…

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to…

Global covariance pooling in convolutional neural networks has achieved impressive improvement over the classical first-order pooling. Recent works have shown matrix square root normalization plays a central role in achieving…

Pan-sharpening algorithms utilize a panchromatic image and a multispectral image to generate a high spatial and high spectral image. However, the optimizations of the algorithms are designed with different standards. We employ a simple…

In this paper, we first present an explicit expression for the inverse\emph{} of a type of matrices. As special applications, the inverse of some matrices arising from implicit time integration techniques, such as the well-known implicit…

It is known that the multiplication of an $N \times M$ matrix with an $M \times P$ matrix can be performed using fewer multiplications than what the naive $NMP$ approach suggests. The most famous instance of this is Strassen's algorithm for…