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We examine a special case of the multilevel factor model, with covariance given by multilevel low rank (MLR) matrix~\cite{parshakova2023factor}. We develop a novel, fast implementation of the expectation-maximization (EM) algorithm, tailored for multilevel factor models, to maximize the likelihood of the observed data. This method accommodates any hierarchical structure and maintains linear time and storage complexities per iteration. This is achieved through a new efficient technique for computing the inverse of the positive definite MLR matrix. We show that the inverse of an invertible PSD MLR matrix is also an MLR matrix with the same sparsity in factors, and we use the recursive Sherman-Morrison-Woodbury matrix identity to obtain the factors of the inverse. Additionally, we present an algorithm that computes the Cholesky factorization of an expanded matrix with linear time and space complexities, yielding the covariance matrix as its Schur complement. This paper is accompanied by an open-source package that implements the proposed methods.

Given a matrix $\mathbf{A} \in \mathbb{R}^{k \times n}$, a partitioning of $[k]$ into groups $S_1,\dots,S_m$, an outer norm $p$, and a collection of inner norms such that either $p \ge 1$ and $p_1,\dots,p_m \ge 2$ or $p_1=\dots=p_m=p \ge 1/\log n$, we prove that there is a sparse weight vector $\mathbf{\beta} \in \mathbb{R}^{m}$ such that $\sum_{i=1}^m \mathbf{\beta}_i \cdot \|\mathbf{A}_{S_i}\mathbf{x}\|_{p_i}^p \approx_{1\pm\varepsilon} \sum_{i=1}^m \|\mathbf{A}_{S_i}\mathbf{x}\|_{p_i}^p$, where the number of nonzero entries of $\mathbf{\beta}$ is at most $O_{p,p_i}(\varepsilon^{-2}n^{\max(1,p/2)}(\log n)^2(\log(n/\varepsilon)))$. When $p_1\dots,p_m \ge 2$, this weight vector arises from an importance sampling procedure based on the \textit{block Lewis weights}, a recently proposed generalization of Lewis weights. Additionally, we prove that there exist efficient algorithms to find the sparse weight vector $\mathbf{\beta}$ in several important regimes of $p$ and $p_1,\dots,p_m$. Our results imply a $\widetilde{O}(\varepsilon^{-1}\sqrt{n})$-linear system solve iteration complexity for the problem of minimizing sums of Euclidean norms, improving over the previously known $\widetilde{O}(\sqrt{m}\log({1/\varepsilon}))$ iteration complexity when $m \gg n$. Our main technical contribution is a substantial generalization of the \textit{change-of-measure} method that Bourgain, Lindenstrauss, and Milman used to obtain the analogous result when every group has size $1$. Our generalization allows one to analyze change of measures beyond those implied by D. Lewis's original construction, including the measure implied by the block Lewis weights and natural approximations of this measure.

We consider a state-space model (SSM) parametrized by some parameter $\theta$, and our aim is to perform joint parameter and state inference. A simple idea to carry out this task, which almost dates back to the origin of the Kalman filter, is to replace the static parameter $\theta$ by a Markov chain $(\theta_t)_{t\geq 0}$ and then to apply a filtering algorithm to the extended, or self-organized SSM (SO-SSM). However, the practical implementation of this idea in a theoretically justified way has remained an open problem. In this paper we fill this gap by introducing various possible constructions of $(\theta_t)_{t\geq 0}$ that ensure the validity of the SO-SSM for joint parameter and state inference. Notably, we show that such SO-SSMs can be defined even if $\|\mathrm{Var}(\theta_{t}|\theta_{t-1})\|\rightarrow 0$ slowly as $t\rightarrow\infty$. This result is important since, as illustrated in our numerical experiments, these models can be efficiently approximated using particle filter algorithms. While SO-SSMs have been introduced for online inference, the development of iterated filtering (IF) algorithms has shown that they can also serve for computing the maximum likelihood estimator of a given SSM. In this work, we also derive constructions of $(\theta_t)_{t\geq 0}$ and theoretical guarantees tailored to these specific applications of SO-SSMs and, as a result, introduce new IF algorithms. From a practical point of view, the algorithms we develop have the merit of being simple to implement and only requiring minimal tuning to perform well.

Based on a theorem by Vasconcelos, we give an algorithm for equidimensional decomposition of algebraic sets using syzygy computations via Gr\"obner bases. This algorithm avoids the use of elimination, homological algebra and processing the input equations one-by-one present in previous algorithms. We experimentally demonstrate the practical interest of our algorithm compared to the state of the art.

Let $G=(V,E)$ be an undirected weighted graph on $n=|V|$ vertices and $S\subseteq V$ be a Steiner set. Steiner mincut is a well-studied concept, which provides a generalization to both (s,t)-mincut (when $|S|=2$) and global mincut (when $|S|=n$). Here, we address the problem of designing a compact data structure that can efficiently report a Steiner mincut and its capacity after the failure of any edge in $G$; such a data structure is known as a \textit{Sensitivity Oracle} for Steiner mincut. In the area of minimum cuts, although many Sensitivity Oracles have been designed in unweighted graphs, however, in weighted graphs, Sensitivity Oracles exist only for (s,t)-mincut [Annals of Operations Research 1991, NETWORKS 2019, ICALP 2024], which is just a special case of Steiner mincut. Here, we generalize this result to any arbitrary set $S\subseteq V$. 1. Sensitivity Oracle: Assuming the capacity of every edge is known, a. there is an ${\mathcal O}(n)$ space data structure that can report the capacity of Steiner mincut in ${\mathcal O}(1)$ time and b. there is an ${\mathcal O}(n(n-|S|+1))$ space data structure that can report a Steiner mincut in ${\mathcal O}(n)$ time after the failure of any edge in $G$. 2. Lower Bound: We show that any data structure that, after the failure of any edge, can report a Steiner mincut or its capacity must occupy $\Omega(n^2)$ bits of space in the worst case, irrespective of the size of the Steiner set. The lower bound in (2) shows that the assumption in (1) is essential to break the $\Omega(n^2)$ lower bound on space. For $|S|=n-k$ for any constant $k\ge 0$, it occupies only ${\mathcal O}(n)$ space. So, we also present the first Sensitivity Oracle occupying ${\mathcal O}(n)$ space for global mincut.

We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any $n \times n$ matrix pencil $(A,B)$. The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel, and Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In particular, we show that perturbing and scaling $(A,B)$ regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of $(A,B)$ in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible $S,T$ and diagonal $D$ such that $||A - SDT^{-1}||_2 \leq \varepsilon$ and $||B - ST^{-1}||_2 \leq \varepsilon$ in at most $O \left(\log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ operations, where $T_{\text{MM}}(n)$ is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact arithmetic matrix pencil diagonalization.

In a graph $G = (V,E)$, a k-ruling set $S$ is one in which all vertices $V$ \ $S$ are at most $k$ distance from $S$. Finding a minimum k-ruling set is intrinsically linked to the minimum dominating set problem and maximal independent set problem, which have been extensively studied in graph theory. This paper presents the first known algorithm for solving all k-ruling set problems in conjunction with known minimum dominating set algorithms at only additional polynomial time cost compared to a minimum dominating set. The algorithm further succeeds for $(\alpha, \alpha - 1)$ ruling sets in which $\alpha > 1$, for which constraints exist on the proximity of vertices v $\in S$. This secondary application instead works in conjunction with maximal independent set algorithms.

An additive code is an $\mathbb{F}_q$-linear subspace of $\mathbb{F}_{q^m}^n$ over $\mathbb{F}_{q^m}$, which is not a linear subspace over $\mathbb{F}_{q^m}$. Linear complementary pairs (LCP) of codes have important roles in cryptography, such as increasing the speed and capacity of digital communication and strengthening security by improving the encryption necessities to resist cryptanalytic attacks. This paper studies an algebraic structure of additive complementary pairs (ACP) of codes over $\mathbb{F}_{q^m}$. Further, we characterize an ACP of codes in analogous generator matrices and parity check matrices. Additionally, we identify a necessary condition for an ACP of codes. Besides, we present some constructions of an ACP of codes over $\mathbb{F}_{q^m}$ from LCP codes over $\mathbb{F}_{q^m}$ and also from an LCP of codes over $\mathbb{F}_{q}$. Finally, we study the constacyclic ACP of codes over $\mathbb{F}_{q^m}$ and the counting of the constacyclic ACP of codes.

We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = \Sigma^{1/2}Z_k$, for $k=1,2$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and $\Sigma$ is a $p\times p$ positive semi-definite matrix, independent of the $Z_k$'s. We assume that $Z_k$'s have independent entries with zero mean and unit variance. The skew-symmetric/skew-Hermitian matrix $S_n^{-}$ will be referred to as a random commutator matrix associated with the samples $X_1$ and $X_2$. We show that, when the dimension $p$ and sample size $n$ increase simultaneously, so that $p/n \to c \in (0,\infty)$, there exists a limiting spectral distribution (LSD) for $S_n^{-}$, supported on the imaginary axis, under the assumptions that the spectral distribution of $\Sigma$ converges weakly and the entries of $Z_k$'s have moments of sufficiently high order. This nonrandom LSD can be described through its Stieltjes transform, which satisfies a coupled Mar\v{c}enko-Pastur-type functional equations. In the special case when $\Sigma = I_p$, we show that the LSD of $S_n^{-}$ is a mixture of a degenerate distribution at zero (with positive mass if $c > 2$), and a continuous distribution with a symmetric density function supported on a compact interval on the imaginary axis. Moreover, we show that the companion matrix $S_n^{+} = \Sigma_n^\frac{1}{2}(Z_1Z_2^* + Z_2Z_1^*)\Sigma_n^\frac{1}{2}$, under identical assumptions, has an LSD supported on the real line, which can be similarly characterized.

A permutation $\pi: [n] \rightarrow [n]$ is a Baxter permutation if and only if it does not contain either of the patterns $2-41-3$ and $3-14-2$. Baxter permutations are one of the most widely studied subclasses of general permutation due to their connections with various combinatorial objects such as plane bipolar orientations and mosaic floorplans, etc. In this paper, we introduce a novel succinct representation (i.e., using $o(n)$ additional bits from their information-theoretical lower bounds) for Baxter permutations of size $n$ that supports $\pi(i)$ and $\pi^{-1}(j)$ queries for any $i \in [n]$ in $O(f_1(n))$ and $O(f_2(n))$ time, respectively. Here, $f_1(n)$ and $f_2(n)$ are arbitrary increasing functions that satisfy the conditions $\omega(\log n)$ and $\omega(\log^2 n)$, respectively. This stands out as the first succinct representation with sub-linear worst-case query times for Baxter permutations. Additionally, we consider a subclass of Baxter permutations called \textit{separable permutations}, which do not contain either of the patterns $2-4-1-3$ and $3-1-4-2$. In this paper, we provide the first succinct representation of the separable permutation $\rho: [n] \rightarrow [n]$ of size $n$ that supports both $\rho(i)$ and $\rho^{-1}(j)$ queries in $O(1)$ time. In particular, this result circumvents Golynski's [SODA 2009] lower bound result for trade-offs between redundancy and $\rho(i)$ and $\rho^{-1}(j)$ queries. Moreover, as applications of these permutations with the queries, we also introduce the first succinct representations for mosaic/slicing floorplans, and plane bipolar orientations, which can further support specific navigational queries on them efficiently.