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The Kronecker product is an invaluable tool for data-sparse representations of large networks and matrices with countless applications in machine learning, graph theory and numerical linear algebra. In some instances, the sparsity pattern…
Matrix decomposition is a very important mathematical tool in numerical linear algebra for data processing. In this paper, we introduce a new randomized matrix decomposition algorithm, which is called randomized approximate SVD based on…
Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are…
We develop a unified and systematic framework for performing online nonnegative matrix factorization under a wide variety of important divergences. The online nature of our algorithm makes it particularly amenable to large-scale data. We…
In recent years, a number of methods have been developed for the dimension reduction and decomposition of multiple linked high-content data matrices. Typically these methods assume that just one dimension, rows or columns, is shared among…
Matrix factorization is a simple and effective solution to the recommendation problem. It has been extensively employed in the industry and has attracted much attention from the academia. However, it is unclear what the low-dimensional…
Truncated singular value decomposition (SVD), also known as the best low-rank matrix approximation, has been successfully applied to many domains such as biology, healthcare, and others, where high-dimensional datasets are prevalent. To…
The Hadamard decomposition is a powerful technique for data analysis and matrix compression, which decomposes a given matrix into the element-wise product of two or more low-rank matrices. In this paper, we develop an efficient algorithm to…
Tensor decompositions, which represent an $N$-order tensor using approximately $N$ factors of much smaller dimensions, can significantly reduce the number of parameters. This is particularly beneficial for high-order tensors, as the number…
Lossy image compression is essential for efficient transmission and storage. Traditional compression methods mainly rely on discrete cosine transform (DCT) or singular value decomposition (SVD), both of which represent image data in…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
The increased availability of the multi-view data (data on the same samples from multiple sources) has led to strong interest in models based on low-rank matrix factorizations. These models represent each data view via shared and individual…
Since many real-world data can be described from multiple views, multi-view learning has attracted considerable attention. Various methods have been proposed and successfully applied to multi-view learning, typically based on matrix…
It is well known by analysts that a concept lattice has an exponential size in the data. Thus, as soon as he works with real data, the size of the concept lattice is a fundamental problem. In this chapter, we propose to investigate factor…
We consider the problem of factorizing a structured 3-way tensor into its constituent Canonical Polyadic (CP) factors. This decomposition, which can be viewed as a generalization of singular value decomposition (SVD) for tensors, reveals…
Matrix factorization is a widely used approach for top-N recommendation and collaborative filtering. When implemented on implicit feedback data (such as clicks), a common heuristic is to upweight the observed interactions. This strategy has…
We present a generalisation of the pseudoinverse operation to pairs of matrices, as opposed to single matrices alone. We note the fact that the Singular Value Decomposition can be used to compute the ordinary Moore-Penrose pseudoinverse. We…
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…
Matrix factorization is one of the best approaches for collaborative filtering, because of its high accuracy in presenting users and items latent factors. The main disadvantages of matrix factorization are its complexity, and being very…
Recent SVD-free matrix factorization formulations have enabled rank minimization for systems with millions of rows and columns, paving the way for matrix completion in extremely large-scale applications, such as seismic data interpolation.…