Factor Analysis based on multivariate $t$ distribution ($t$fa) is a useful robust tool for extracting common factors on heavy-tailed or contaminated data. However, $t$fa is only applicable to vector data. When $t$fa is applied to matrix data, it is common to first vectorize the matrix observations. This introduces two challenges for $t$fa: (i) the inherent matrix structure of the data is broken, and (ii) robustness may be lost, as vectorized matrix data typically results in a high data dimension, which could easily lead to the breakdown of $t$fa. To address these issues, starting from the intrinsic matrix structure of matrix data, a novel robust factor analysis model, namely bilinear factor analysis built on the matrix-variate $t$ distribution ($t$bfa), is proposed in this paper. The novelty is that it is capable to simultaneously extract common factors for both row and column variables of interest on heavy-tailed or contaminated matrix data. Two efficient algorithms for maximum likelihood estimation of $t$bfa are developed. Closed-form expression for the Fisher information matrix to calculate the accuracy of parameter estimates are derived. Empirical studies are conducted to understand the proposed $t$bfa model and compare with related competitors. The results demonstrate the superiority and practicality of $t$bfa. Importantly, $t$bfa exhibits a significantly higher breakdown point than $t$fa, making it more suitable for matrix data.
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We give an algorithm that given a graph $G$ with $n$ vertices and $m$ edges and an integer $k$, in time $O_k(n^{1+o(1)}) + O(m)$ either outputs a rank decomposition of $G$ of width at most $k$ or determines that the rankwidth of $G$ is larger than $k$; the $O_k(\cdot)$-notation hides factors depending on $k$. Our algorithm returns also a $(2^{k+1}-1)$-expression for cliquewidth, yielding a $(2^{k+1}-1)$-approximation algorithm for cliquewidth with the same running time. This improves upon the $O_k(n^2)$ time algorithm of Fomin and Korhonen [STOC 2022]. The main ingredient of our algorithm is a fully dynamic algorithm for maintaining rank decompositions of bounded width: We give a data structure that for a dynamic $n$-vertex graph $G$ that is updated by edge insertions and deletions maintains a rank decomposition of $G$ of width at most $4k$ under the promise that the rankwidth of $G$ never grows above $k$. The amortized running time of each update is $O_k(2^{\sqrt{\log n} \log \log n})$. The data structure furthermore can maintain whether $G$ satisfies some fixed ${\sf CMSO}_1$ property within the same running time. We also give a framework for performing ``dense'' edge updates inside a given set of vertices $X$, where the new edges inside $X$ are described by a given ${\sf CMSO}_1$ sentence and vertex labels, in amortized $O_k(|X| \cdot 2^{\sqrt{\log n} \log \log n})$ time. Our dynamic algorithm generalizes the dynamic treewidth algorithm of Korhonen, Majewski, Nadara, Pilipczuk, and Soko{\l}owski [FOCS 2023].
The main reason for query model's prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values can't be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. Finally, we study how large certificate complexity and block sensitivity can be as compared to sensitivity for symmetric functions (even up to constant factors). We show tight separations, i.e., give upper bounds on possible separations and construct functions achieving the same.
Estimating parameters from data is a fundamental problem in physics, customarily done by minimizing a loss function between a model and observed statistics. In scattering-based analysis, researchers often employ their domain expertise to select a specific range of wavevectors for analysis, a choice that can vary depending on the specific case. We introduce another paradigm that defines a probabilistic generative model from the beginning of data processing and propagates the uncertainty for parameter estimation, termed ab initio uncertainty quantification (AIUQ). As an illustrative example, we demonstrate this approach with differential dynamic microscopy (DDM) that extracts dynamical information through Fourier analysis at a selected range of wavevectors. We first show that DDM is equivalent to fitting a temporal variogram in the reciprocal space using a latent factor model as the generative model. Then we derive the maximum marginal likelihood estimator, which optimally weighs information at all wavevectors, therefore eliminating the need to select the range of wavevectors. Furthermore, we substantially reduce the computational cost by utilizing the generalized Schur algorithm for Toeplitz covariances without approximation. Simulated studies validate that AIUQ significantly improves estimation accuracy and enables model selection with automated analysis. The utility of AIUQ is also demonstrated by three distinct sets of experiments: first in an isotropic Newtonian fluid, pushing limits of optically dense systems compared to multiple particle tracking; next in a system undergoing a sol-gel transition, automating the determination of gelling points and critical exponent; and lastly, in discerning anisotropic diffusive behavior of colloids in a liquid crystal. These outcomes collectively underscore AIUQ's versatility to capture system dynamics in an efficient and automated manner.
A graph $G=(V,E)$ is a star-$k$-PCG if there exists a weight function $w: V \rightarrow R^+$ and $k$ mutually exclusive intervals $I_1, I_2, \ldots I_k$, such that there is an edge $uv \in E$ if and only if $w(u)+w(v) \in \bigcup_i I_i$. These graphs are related to two important classes of graphs: PCGs and multithreshold graphs. It is known that for any graph $G$ there exists a $k$ such that $G$ is a star-$k$-PCG. Thus, for a given graph $G$ it is interesting to know which is the minimum $k$ such that $G$ is a star-$k$-PCG. We define this minimum $k$ as the star number of the graph, denoted by $\gamma(G)$. Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of $\gamma(G)$ for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two dimensional grid graphs is 2 and that the star number of $4$-dimensional grids is at least 3. Finally, we conclude with numerous open problems.
We present an efficient preconditioner for two-by-two block system of linear equations with the coefficient matrix $ \begin{pmatrix} F & -G^H G & F \end{pmatrix}$ where $F\in\mathbb{C}^{n\times n}$ is Hermitian positive definite and $G\in\mathbb{C}^{n\times n}$ is positive semidefinite. Spectral analysis of the preconditioned matrix is analyzed. In each iteration of a Krylov subspace method, like GMRES, for solving the preconditioned system in conjunction with proposed preconditioner two subsystems with Hermitian positive definite coefficient matrix should be solved which can be accomplished exactly using the Cholesky factorization or inexactly using the conjugate gradient method. Application of the proposed preconditioner to the systems arising from finite element discretization of PDE-constrained optimization problems is presented. Numerical results are given to demonstrate the efficiency of the preconditioner.
We propose a threshold-type algorithm to the $L^2$-gradient flow of the Canham-Helfrich functional generalized to $\mathbb{R}^N$. The algorithm to the Willmore flow is derived as a special case in $\mathbb{R}^2$ or $\mathbb{R}^3$. This algorithm is constructed based on an asymptotic expansion of the solution to the initial value problem for a fourth order linear parabolic partial differential equation whose initial data is the indicator function on the compact set $\Omega_0$. The crucial points are to prove that the boundary $\partial\Omega_1$ of the new set $\Omega_1$ generated by our algorithm is included in $O(t)$-neighborhood from $\partial\Omega_0$ for small time $t>0$ and to show that the derivative of the threshold function in the normal direction for $\partial\Omega_0$ is far from zero in the small time interval. Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our threshold-type algorithm.
We prove that the values of a generalized $\psi$-estimator (introduced by Barczy and P\'ales in 2022) on samples of arbitrary length but having only two different observations uniquely determine the values of the estimator on any sample of arbitrary length without any restriction on the number of different observations. In other words, samples of arbitrary length but having only two different observations form a determining class for generalized $\psi$-estimators. We also obtain a similar statement for the comparison of generalized $\psi$-estimators using comparative functions, and, as a corollary of this result, we derive the Schweitzer's inequality (also called Kantorovich's inequality).
The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess fundamental conjugacy properties. When used as priors for the vector of parameters in general probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although such a core result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such an important gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that such a family is conjugate to the likelihood induced by general multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement substantially enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.
We analyze a Discontinuous Galerkin method for a problem with linear advection-reaction and $p$-type diffusion, with Sobolev indices $p\in (1, \infty)$. The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on local P\'eclet numbers. A set of numerical tests supports the theoretical derivations.
We consider the problem of sketching a set valuation function, which is defined as the expectation of a valuation function of independent random item values. We show that for monotone subadditive or submodular valuation functions satisfying a weak homogeneity condition, or certain other conditions, there exist discretized distributions of item values with $O(k\log(k))$ support sizes that yield a sketch valuation function which is a constant-factor approximation, for any value query for a set of items of cardinality less than or equal to $k$. The discretized distributions can be efficiently computed by an algorithm for each item's value distribution separately. Our results hold under conditions that accommodate a wide range of valuation functions arising in applications, such as the value of a team corresponding to the best performance of a team member, constant elasticity of substitution production functions exhibiting diminishing returns used in economics and consumer theory, and others. Sketch valuation functions are particularly valuable for finding approximate solutions to optimization problems such as best set selection and welfare maximization. They enable computationally efficient evaluation of approximate value oracle queries and provide an approximation guarantee for the underlying optimization problem.
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