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We provide a first systematic treatment of so-called rectangular multispectral perturbation theory. With their paper from 2003, Hochstenbach and Plestenjak ["Backward Error, Condition Numbers, and Pseudospectra for the Multiparameter…

An important element of the $S$-matrix bootstrap program is the relationship between the modulus of an $S$-matrix element and its phase. Unitarity relates them by an integral equation. Even in the simplest case of elastic scattering, this…

In an era of unprecedented deluge of (mostly unstructured) data, graphs are proving more and more useful, across the sciences, as a flexible abstraction to capture complex relationships between complex objects. One of the main challenges…

Spectral analysis plays a crucial role in high-dimensional statistics, where determining the asymptotic distribution of various spectral statistics remains a challenging task. Due to the difficulties of deriving the analytic form, recent…

In a recent paper, an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices -- the so called `matrix properties'. In this paper we extend this algorithm to…

Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In…

Graph sparsification is a well-established technique for accelerating graph-based learning algorithms, which uses edge sampling to approximate dense graphs with sparse ones. Because the sparsification error is random and unknown, users must…

The maximum a-posteriori (MAP) perturbation framework has emerged as a useful approach for inference and learning in high dimensional complex models. By maximizing a randomly perturbed potential function, MAP perturbations generate unbiased…

We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…

We present a general framework that utilizes different efficient data structures to improve various sparsification problems involving an iterative process. We also provide insights and characterization for different iterative process, and…

This paper focusses on the sparse estimation in the situation where both the the sensing matrix and the measurement vector are corrupted by additive Gaussian noises. The performance bound of sparse estimation is analyzed and discussed in…

Many major works in social science employ matching to make causal conclusions, but different matches on the same data may produce different treatment effect estimates, even when they achieve similar balance or minimize the same loss…

The problem of attribution is concerned with identifying the parts of an input that are responsible for a model's output. An important family of attribution methods is based on measuring the effect of perturbations applied to the input. In…

We investigate the methods that simultaneously enforce sparsity and low-rank structure in a matrix as often employed for sparse phase retrieval problems or phase calibration problems in compressive sensing. We propose a new approach for…

We present improved approximation bounds for the Moore-Penrose inverses of banded matrices, where the bandedness is induced by a metric on the index set. We show that the pseudoinverse of a banded matrix can be approximated by another…

We consider the properties of the bootstrap as a tool for inference concerning the eigenvalues of a sample covariance matrix computed from an $n\times p$ data matrix $X$. We focus on the modern framework where $p/n$ is not close to 0 but…

Sparse coding--that is, modelling data vectors as sparse linear combinations of basis elements--is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the large-scale matrix factorization…

Symmetry is an important feature of many constraint programs. We show that any problem symmetry acting on a set of symmetry breaking constraints can be used to break symmetry. Different symmetries pick out different solutions in each…

Isospectral reduction is an important tool for network/matrix analysis as it reduces the dimension of a matrix/network while preserving its eigenvalues and eigenvectors. The main contribution of this manuscript is a proposed algorithmic…

In this paper, we introduce a novel algorithm for segmentation of imperfect boundary probability maps (BPM) in connectomics. Our algorithm can be a considered as an extension of spectral clustering. Instead of clustering the diffusion maps…