Related papers: Matrices over a Hilbert space and their low-rank a…
A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications. This has both numerical…
Matrix approximations are a key element in large-scale algebraic machine learning approaches. The recently proposed method MEKA (Si et al., 2014) effectively employs two common assumptions in Hilbert spaces: the low-rank property of an…
Hessian operators arising in inverse problems governed by partial differential equations (PDEs) play a critical role in delivering efficient, dimension-independent convergence for both Newton solution of deterministic inverse problems, as…
We study private matrix analysis in the sliding window model where only the last $W$ updates to matrices are considered useful for analysis. We give first efficient $o(W)$ space differentially private algorithms for spectral approximation,…
We present quadrature schemes to calculate matrices, where the so-called modified Hilbert transformation is involved. These matrices occur as temporal parts of Galerkin finite element discretizations of parabolic or hyperbolic problems when…
A novel compressed matrix format is proposed that combines an adaptive hierarchical partitioning of the matrix with low-rank approximation. One typical application is the approximation of discretized functions on rectangular domains; the…
Frames for Hilbert spaces are interesting for mathematicians but also important for applications e.g. in signal analysis and in physics. Both in mathematics and physics it is natural to consider a full scale of spaces, and not only a single…
In this paper we will look at the connection of frames and finite dimensionality. A main focus is to present simple algorithms and make them available online. The main result is a way to 'switch' between different frames, giving an…
Low-rank matrix completion has been studied extensively under various type of categories. The problem could be categorized as noisy completion or exact completion, also active or passive completion algorithms. In this paper we focus on…
Recent progress in few-shot learning promotes a more realistic cross-domain setting, where the source and target datasets are from different domains. Due to the domain gap and disjoint label spaces between source and target datasets, their…
Dimensionality reduction is a popular approach to tackle high-dimensional data with low-dimensional nature. Subspace Restricted Isometry Property, a newly-proposed concept, has proved to be a useful tool in analyzing the effect of…
In many applications, it is of interest to approximate data, given by mxn matrix A, by a matrix B of at most rank k, which is much smaller than m and n. The best approximation is given by singular value decomposition, which is too time…
Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic…
We develop an approximation theory in Hilbert spaces that generalizes the classical theory of approximation by entire functions of exponential type. The results advance harmonic analysis on manifolds and graphs, thus facilitating data…
An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and…
Predicting unobserved entries of a partially observed matrix has found wide applicability in several areas, such as recommender systems, computational biology, and computer vision. Many scalable methods with rigorous theoretical guarantees…
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…
Parametric entities appear in many contexts, be it in optimisation, control, modelling of random quantities, or uncertainty quantification. These are all fields where reduced order models (ROMs) have a place to alleviate the computational…
In this note, we investigate how well we can reconstruct the best rank-$r$ approximation of a large matrix from a small number of its entries. We show that even if a data matrix is of full rank and cannot be approximated well by a low-rank…
The trace regression model, a direct extension of the well-studied linear regression model, allows one to map matrices to real-valued outputs. We here introduce an even more general model, namely the partial-trace regression model, a family…