Related papers: Faster Tensor Canonicalization
Language models can largely benefit from efficient tokenization. However, they still mostly utilize the classical BPE algorithm, a simple and reliable method. This has been shown to cause such issues as under-trained tokens and sub-optimal…
In this paper, we consider the completely positive tensor decomposition problem with ideal-sparsity. First, we propose an algorithm to generate the maximal cliques of multi-hypergraphs associated with completely positive tensors. This also…
We propose an adaptive and provably accurate tensor completion approach based on combining matrix completion techniques (see, e.g., arXiv:0805.4471, arXiv:1407.3619, arXiv:1306.2979) for a small number of slices with a modified noise robust…
Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It is generalization of approximate…
This document contains notes based on lectures given by Hendrik Lenstra at the PCMI summer school 2022. There are many problems in algebraic number theory which one would like to solve algorithmically, for example computation of the maximal…
Template matching is widely used for many applications in image and signal processing and usually is time-critical. Traditional methods usually focus on how to reduce the search locations by coarse-to-fine strategy or full search combined…
The fastest known algorithm for factoring a degree $n$ univariate polynomial over a finite field $\mathbb{F}_q$ runs in time $O(n^{3/2 + o(1)}\text{polylog } q)$, and there is a reason to believe that the $3/2$ exponent represents a…
We study Sigma-Delta quantization methods coupled with appropriate reconstruction algorithms for digitizing randomly sampled low-rank matrices. We show that the reconstruction error associated with our methods decays polynomially with the…
Sparse and low rank tensor recovery has emerged as a significant area of research with applications in many fields such as computer vision. However, minimizing the $\ell_0$-norm of a vector or the rank of a matrix is NP-hard. Instead, their…
Tensor decomposition is a fundamental method used in various areas to deal with high-dimensional data. \emph{Tensor power method} (TPM) is one of the widely-used techniques in the decomposition of tensors. This paper presents a novel tensor…
We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent…
We propose a loop optimization algorithm based on nuclear norm regularization for tensor network. The key ingredient of this scheme is to introduce a rank penalty term proposed in the context of data processing. Compared to standard…
Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. Unlike binary matrix factorization based on standard arithmetic, BMF employs the Boolean OR and AND operations for the…
Many large arithmetic computations rely on tables of all primes less than $n$. For example, the fastest algorithms for computing $n!$ takes time $O(M(n\log n) + P(n))$, where $M(n)$ is the time to multiply two $n$-bit numbers, and $P(n)$ is…
The paper has been withdrawn since more effective experiments should be completed. Auto-encoders (AE) has been widely applied in different fields of machine learning. However, as a deep model, there are a large amount of learnable…
Tensor recovery has recently arisen in a lot of application fields, such as transportation, medical imaging and remote sensing. Under the assumption that signals possess sparse and/or low-rank structures, many tensor recovery methods have…
When solving large scale semidefinite programs that admit a low-rank solution, an efficient heuristic is the Burer-Monteiro factorization: instead of optimizing over the full matrix, one optimizes over its low-rank factors. This reduces the…
In this paper, we propose an efficient transformer architecture that uses reinforced positional embedding to obtain superior performance with half the number of encoder decoder layers. We demonstrate that concatenating positional encoding…
We present a new approach for solving (minimum disagreement) correlation clustering that results in sublinear algorithms with highly efficient time and space complexity for this problem. In particular, we obtain the following algorithms for…
Unlike matrix completion, tensor completion does not have an algorithm that is known to achieve the information-theoretic sample complexity rate. This paper develops a new algorithm for the special case of completion for nonnegative…