Related papers: Generalizations of Matrix Multiplication can solve…
Large neural networks spend most computation on floating point tensor multiplications. In this work, we find that a floating point multiplier can be approximated by one integer adder with high precision. We propose the linear-complexity…
Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras. In subsequent work with Kleinberg and Szegedy, they connected this to the search for…
We consider optimization problems involving the multiplication of variable matrices to be selected from a given family, which might be a discrete set, a continuous set or a combination of both. Such nonlinear, and possibly discrete,…
Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time $O(n^{2.3755})$. Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an…
A linear inverse problem is proposed that requires the determination of multiple unknown signal vectors. Each unknown vector passes through a different system matrix and the results are added to yield a single observation vector. Given the…
Matrix multiplication is a fundamental computation in many scientific disciplines. In this paper, we show that novel fast matrix multiplication algorithms can significantly outperform vendor implementations of the classical algorithm and…
We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs…
This article introduces two Tensor Network-based iterative algorithms for the identification of high-order discrete-time nonlinear multiple-input multiple-output (MIMO) Volterra systems. The system identification problem is rewritten in…
Fast matrix multiplication can be described as searching for low-rank decompositions of the matrix--multiplication tensor. We design a neural architecture, \textsc{StrassenNet}, which reproduces the Strassen algorithm for $2\times 2$…
Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop…
In recent years, the fervent demand for computational power across various domains has prompted hardware manufacturers to introduce specialized computing hardware aimed at enhancing computational capabilities. Particularly, the utilization…
The problem of solving linear systems is one of the most fundamental problems in computer science, where given a satisfiable linear system $(A,b)$, for $A \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^n$, we wish to find a vector $x…
Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent $\omega$, is a central problem in algebraic complexity theory. The best upper bounds on $\omega$, leading…
We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our…
Although reliable long precision floating-point arithmetic libraries such as QD and MPFR/GMP are necessary to solve ill-conditioned problems in numerical simulation, long precision BLAS-level computation such as matrix multiplication has…
Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole…
We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems,…
Recently, tensor algebra have witnessed significant applications across various domains. Each operator in tensor algebra features different computational workload and precision. However, current general accelerators, such as VPU, GPGPU, and…
In prior work, Gupta et al. (SPAA 2022) presented a distributed algorithm for multiplying sparse $n \times n$ matrices, using $n$ computers. They assumed that the input matrices are uniformly sparse--there are at most $d$ non-zeros in each…
Optical computing harnesses the speed of light to perform vector-matrix operations efficiently. It leverages interference, a cornerstone of quantum computing algorithms, to enable parallel computations. In this work, we interweave quantum…