Related papers: Hybrid hierarchical matrices with adaptive mixed p…
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…
We present new algorithms for the randomized construction of hierarchically semi-separable matrices, addressing several practical issues. The HSS construction algorithms use a partially matrix-free, adaptive randomized projection scheme to…
This paper presents a hierarchical low-rank decomposition algorithm assuming any matrix element can be computed in $O(1)$ time. The proposed algorithm computes rank-revealing decompositions of sub-matrices with a blocked adaptive cross…
Hierarchical forecasting problems arise when time series have a natural group structure, and predictions at multiple levels of aggregation and disaggregation across the groups are needed. In such problems, it is often desired to satisfy the…
This work proposes a hierarchical clustering algorithm for high-dimensional datasets using the cyclic space of reversible finite cellular automata. In cellular automaton (CA) based clustering, if two objects belong to the same cycle, they…
High-dimensional compositional data are prevalent in many applications. The simplex constraint poses intrinsic challenges to inferring the conditional dependence relationships among the components forming a composition, as encoded by a…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
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 present an $hr$-adaptivity framework for optimization of high-order meshes. This work extends the $r$-adaptivity method for mesh optimization by Dobrev et al., where we utilized the Target-Matrix Optimization Paradigm (TMOP) to minimize…
Hierarchical clustering is a critical task in numerous domains. Many approaches are based on heuristics and the properties of the resulting clusterings are studied post hoc. However, in several applications, there is a natural cost function…
Matrix-vector multiplication forms the basis of many iterative solution algorithms and as such is an important algorithm also for hierarchical matrices which are used to represent dense data in an optimized form by applying low-rank…
With the emergence of Artificial Intelligence, numerical algorithms are moving towards more approximate approaches. For methods such as PCA or diffusion maps, it is necessary to compute eigenvalues of a large matrix, which may also be dense…
Real-world vision based applications require fine-grained classification for various area of interest like e-commerce, mobile applications, warehouse management, etc. where reducing the severity of mistakes and improving the classification…
Conformal prediction (CP) is a powerful framework for quantifying uncertainty in machine learning models, offering reliable predictions with finite-sample coverage guarantees. When applied to classification, CP produces a prediction set of…
A new fast algebraic method for obtaining an $\mathcal{H}^2$-approximation of a matrix from its entries is presented. The main idea behind the method is based on the nested representation and the maximum-volume principle to select…
In this paper, a hierarchical Tucker low-rank (HTLR) matrix is proposed to approximate non-oscillatory kernel functions in linear complexity. The HTLR matrix is based on the hierarchical matrix, with the low-rank blocks replaced by Tucker…
Clinical decision-making demands uncertainty quantification that provides both distribution-free coverage guarantees and risk-adaptive precision, requirements that existing methods fail to jointly satisfy. We present a hybrid…
HDBSCAN is a density-based clustering algorithm that constructs a cluster hierarchy tree and then uses a specific stability measure to extract flat clusters from the tree. We show how the application of an additional threshold value can…
An adaptive isogeometric method based on $d$-variate hierarchical spline constructions can be derived by considering a refine module that preserves a certain class of admissibility between two consecutive steps of the adaptive loop [6]. In…
In this work we introduce a new multiscale artificial neural network based on the structure of $\mathcal{H}$-matrices. This network generalizes the latter to the nonlinear case by introducing a local deep neural network at each spatial…