Related papers: Tree quasi-separable matrices: a simultaneous gene…
We provide a computational framework for approximating a class of structured matrices; here, the term structure is very general, and may refer to a regular sparsity pattern (e.g., block-banded), or be more highly structured (e.g., symmetric…
Matrix representations of quantum operators are computationally complete but often obscure the structural topology of information flow within a quantum circuit \cite{nielsen2000}. In this paper, we introduce a generalized graph-theoretic…
Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the…
Decision forests induce supervised similarities through the partition structure of their trees. Yet forest proximity computation is still often treated as a quadratic operation in the number of samples, which limits scalability and…
We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination, and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which…
Pairwise ordered tree alignment are combinatorial objects that appear in RNA secondary structure comparison. However, the usual representation of tree alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce identical…
The semi-tensor product (STP) of matrices is extended to the STP of hypermatrices. Some basic properties of the STP of matrices are extended to the STP of hypermatrices. The hyperdeterminant of hypersquares is introduced. Some algebraic and…
The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format…
A new matrix product, called the semi-tensor product (STP), is briefly reviewed. The STP extends the classical matrix product to two arbitrary matrices. Under STP the set of matrices becomes a monoid (semi-group with identity). Some related…
The hierarchical SVD provides a quasi-best low rank approximation of high dimensional data in the hierarchical Tucker framework. Similar to the SVD for matrices, it provides a fundamental but expensive tool for tensor computations. In the…
Hierarchical Text Classification (HTC) is a challenging task where a document can be assigned to multiple hierarchically structured categories within a taxonomy. The majority of prior studies consider HTC as a flat multi-label…
We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states…
We derive the exact S-matrix for the scattering of particular representations of the centrally-extended psu(1|1)^2 Lie superalgebra, conjectured to be related to the massive modes of the light-cone gauge string theory on AdS_2 x S^2 x T^6.…
This paper addresses the problem of approximate MAP-MRF inference in general graphical models. Following [36], we consider a family of linear programming relaxations of the problem where each relaxation is specified by a set of nested pairs…
Starting from the local structures to study hierarchical trees is a common research method. However, the cumbersome analysis and description make the naive method challenging to adapt to the increasingly complex hierarchical tree problems.…
We propose a new tensor factorization method, called the Sparse Hierarchical-Tucker (Sparse H-Tucker), for sparse and high-order data tensors. Sparse H-Tucker is inspired by its namesake, the classical Hierarchical Tucker method, which aims…
Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in\mathcal{T}$ is…
Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices…
Connected acyclic graphs (trees) are data objects that hierarchically organize categories. Collections of trees arise in a diverse variety of fields, including evolutionary biology, public health, machine learning, social sciences and…
This paper studies the "explanation problem" for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A whose…