Related papers: Numerical Matrix Decomposition
The idea of decomposing a matrix into a product of structured matrices such as triangular, orthogonal, diagonal matrices is a milestone of numerical computations. In this paper, we describe six new classes of matrix decompositions,…
We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and…
Nonnegative matrix factorization (NMF) was popularized as a tool for data mining by Lee and Seung in 1999. NMF attempts to approximate a matrix with nonnegative entries by a product of two low-rank matrices, also with nonnegative entries.…
Matrix reordering is a task to permute the rows and columns of a given observed matrix such that the resulting reordered matrix shows meaningful or interpretable structural patterns. Most existing matrix reordering techniques share the…
Suppose a given observation matrix can be decomposed as the sum of a low-rank matrix and a sparse matrix (outliers), and the goal is to recover these individual components from the observed sum. Such additive decompositions have…
In this work we describe an efficient implementation of a hierarchy of algorithms for the decomposition of dense matrices over the field with two elements (GF(2)). Matrix decomposition is an essential building block for solving dense…
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However,…
Nonlinear methods such as Deep Neural Networks (DNNs) are the gold standard for various challenging machine learning problems, e.g., image classification, natural language processing or human action recognition. Although these methods…
A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work we consider the case, where the target can be represented by a decomposition of spatial…
CUR matrix decomposition computes the low rank approximation of a given matrix by using the actual rows and columns of the matrix. It has been a very useful tool for handling large matrices. One limitation with the existing algorithms for…
Recent work in the matrix completion literature has shown that prior knowledge of a matrix's row and column spaces can be successfully incorporated into reconstruction programs to substantially benefit matrix recovery. This paper proposes a…
The ability to decompose a signal in an orthonormal basis (a set of orthogonal components, each normalized to have unit length) using a fast numerical procedure rests at the heart of many signal processing methods and applications. The…
Low rank approximation is an important tool used in many applications of signal processing and machine learning. Recently, randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate…
Constrained low-rank matrix approximations have been known for decades as powerful linear dimensionality reduction techniques to be able to extract the information contained in large data sets in a relevant way. However, such low-rank…
We exploit the truncated singular value decomposition and the recently proposed circulant decomposition for an efficient first-order approximation of the multiplication of large dense matrices. A decomposition of each matrix into a sum of a…
A common approach for compressing NLP networks is to encode the embedding layer as a matrix $A\in\mathbb{R}^{n\times d}$, compute its rank-$j$ approximation $A_j$ via SVD, and then factor $A_j$ into a pair of matrices that correspond to…
Recently, a family of tractable NMF algorithms have been proposed under the assumption that the data matrix satisfies a separability condition Donoho & Stodden (2003); Arora et al. (2012). Geometrically, this condition reformulates the NMF…
One of the grand challenges of Mathematics instruction is to provide students with problems that are both accessible and have a reasonably elegant solution. Instructors commonly resort to resources like course textbooks, online-learning…
We present a new distributed representation in deep neural nets wherein the information is represented in native form as a matrix. This differs from current neural architectures that rely on vector representations. We consider matrices as…
Tensor Network (TN) decompositions have emerged as an indispensable tool in Big Data analytics owing to their ability to provide compact low-rank representations, thus alleviating the ``Curse of Dimensionality'' inherent in handling…