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Consider a matrix $\mathbf{F} \in \mathbb{K}^{m \times n}$ of univariate polynomials over a field~$\mathbb{K}$. We study the problem of computing the column rank profile of $\mathbf{F}$. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of $\mathbf{F}$ with a rank-sensitive complexity of $O\tilde{~}(r^{\omega-2} n (m+D))$ operations in $\mathbb{K}$. Here, $D$ is the sum of row degrees of $\mathbf{F}$, $\omega$ is the exponent of matrix multiplication, and $O\tilde{~}(\cdot)$ hides logarithmic factors.

Sequential Monte Carlo methods are typically not straightforward to implement on parallel architectures. This is because standard resampling schemes involve communication between all particles. The $\alpha$-sequential Monte Carlo method was proposed recently as a potential solution to this which limits communication between particles. This limited communication is controlled through a sequence of stochastic matrices known as $\alpha$-matrices. We study the influence of the communication structure on the convergence and stability properties of the resulting algorithms. In particular, we quantitatively show that the mixing properties of the $\alpha$-matrices play an important role in the stability properties of the algorithm. Moreover, we prove that one can ensure good mixing properties by using randomized communication structures where each particle only communicates with a few neighboring particles. The resulting algorithms converge at the usual Monte Carlo rate. This leads to efficient versions of distributed sequential Monte Carlo.

Consider the problem of nonparametric estimation of an unknown $\beta$-H\"older smooth density $p_{XY}$ at a given point, where $X$ and $Y$ are both $d$ dimensional. An infinite sequence of i.i.d.\ samples $(X_i,Y_i)$ are generated according to this distribution, and two terminals observe $(X_i)$ and $(Y_i)$, respectively. They are allowed to exchange $k$ bits either in oneway or interactively in order for Bob to estimate the unknown density. We show that the minimax mean square risk is order $\left(\frac{k}{\log k} \right)^{-\frac{2\beta}{d+2\beta}}$ for one-way protocols and $k^{-\frac{2\beta}{d+2\beta}}$ for interactive protocols. The logarithmic improvement is nonexistent in the parametric counterparts, and therefore can be regarded as a consequence of nonparametric nature of the problem. Moreover, a few rounds of interactions achieve the interactive minimax rate: the number of rounds can grow as slowly as the super-logarithm (i.e., inverse tetration) of $k$. The proof of the upper bound is based on a novel multi-round scheme for estimating the joint distribution of a pair of biased Bernoulli variables.

We propose a model for online graph problems where algorithms are given access to an oracle that predicts (e.g., based on past data) the degrees of nodes in the graph. Within this model, we study the classic problem of online bipartite matching, and a natural greedy matching algorithm called MinPredictedDegree, which uses predictions of the degrees of offline nodes. For the bipartite version of a stochastic graph model due to Chung, Lu, and Vu where the expected values of the offline degrees are known and used as predictions, we show that MinPredictedDegree stochastically dominates any other online algorithm, i.e., it is optimal for graphs drawn from this model. Since the "symmetric" version of the model, where all online nodes are identical, is a special case of the well-studied "known i.i.d. model", it follows that the competitive ratio of MinPredictedDegree on such inputs is at least 0.7299. For the special case of graphs with power law degree distributions, we show that MinPredictedDegree frequently produces matchings almost as large as the true maximum matching on such graphs. We complement these results with an extensive empirical evaluation showing that MinPredictedDegree compares favorably to state-of-the-art online algorithms for online matching.

We introduce the $\texttt{$k$-experts}$ problem - a generalization of the classic Prediction with Expert's Advice framework. Unlike the classic version, where the learner selects exactly one expert from a pool of $N$ experts at each round, in this problem, the learner can select a subset of $k$ experts at each round $(1\leq k\leq N)$. The reward obtained by the learner at each round is assumed to be a function of the $k$ selected experts. The primary objective is to design an online learning policy with a small regret. In this pursuit, we propose $\texttt{SAGE}$ ($\textbf{Sa}$mpled Hed$\textbf{ge}$) - a framework for designing efficient online learning policies by leveraging statistical sampling techniques. For a wide class of reward functions, we show that $\texttt{SAGE}$ either achieves the first sublinear regret guarantee or improves upon the existing ones. Furthermore, going beyond the notion of regret, we fully characterize the mistake bounds achievable by online learning policies for stable loss functions. We conclude the paper by establishing a tight regret lower bound for a variant of the $\texttt{$k$-experts}$ problem and carrying out experiments with standard datasets.

We study the online scheduling problem where the machines need to be calibrated before processing any jobs. To calibrate a machine, it will take $\lambda$ time steps as the activation time, and then the machine will remain calibrated status for $T$ time steps. The job can only be processed by the machine that is in calibrated status. Given a set of jobs arriving online, each of the jobs is characterized by a release time, a processing time, and a deadline. We assume that there is an infinite number of machines for usage. The objective is to minimize the total number of calibrations while feasibly scheduling all jobs. For the case that all jobs have unit processing times, we propose an $\mathcal{O}(\lambda)$-competitive algorithm, which is asymptotically optimal. When $\lambda=0$, the problem is degraded to rent minimization, where our algorithm achieves a competitive ratio of $3e+7(\approx 15.16)$ which improves upon the previous results for such problems.

Center-based clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is undoubtedly the k-means problem, which, given a set $P$ of points from a metric space and a parameter $k<|P|$, requires to determine a subset $S$ of $k$ centers minimizing the sum of all squared distances of points in $P$ from their closest center. A more general formulation, known as k-means with $z$ outliers, introduced to deal with noisy datasets, features a further parameter $z$ and allows up to $z$ points of $P$ (outliers) to be disregarded when computing the aforementioned sum. We present a distributed coreset-based 3-round approximation algorithm for k-means with $z$ outliers for general metric spaces, using MapReduce as a computational model. Our distributed algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term $O(\gamma)$ away from the one achievable by the best known sequential (possibly bicriteria) algorithm, where $\gamma$ can be made arbitrarily small. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by the doubling dimension $D$ of the metric space. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for general metrics.

We introduce the problem of finding a set $B$ of $k$ points in $[0,1]^n$ such that the expected cost of the cheapest point in $B$ that dominates a random point from $[0,1]^n$ is minimized. We study the case where the coordinates of the random points are independently distributed and the cost function is linear. This problem arises naturally in various application areas where customers' requests are satisfied based on predefined products, each corresponding to a subset of features. We show that the problem is NP-hard already for $k=2$ when each coordinate is drawn from $\{0,1\}$, and obtain an FPTAS for general fixed $k$ under mild assumptions on the distributions.

Computing the top eigenvectors of a matrix is a problem of fundamental interest to various fields. While the majority of the literature has focused on analyzing the reconstruction error of low-rank matrices associated with the retrieved eigenvectors, in many applications one is interested in finding one vector with high Rayleigh quotient. In this paper we study the problem of approximating the top-eigenvector. Given a symmetric matrix $\mathbf{A}$ with largest eigenvalue $\lambda_1$, our goal is to find a vector \hu that approximates the leading eigenvector $\mathbf{u}_1$ with high accuracy, as measured by the ratio $R(\hat{\mathbf{u}})=\lambda_1^{-1}{\hat{\mathbf{u}}^T\mathbf{A}\hat{\mathbf{u}}}/{\hat{\mathbf{u}}^T\hat{\mathbf{u}}}$. We present a novel analysis of the randomized SVD algorithm of \citet{halko2011finding} and derive tight bounds in many cases of interest. Notably, this is the first work that provides non-trivial bounds of $R(\hat{\mathbf{u}})$ for randomized SVD with any number of iterations. Our theoretical analysis is complemented with a thorough experimental study that confirms the efficiency and accuracy of the method.

The compression of highly repetitive strings (i.e., strings with many repetitions) has been a central research topic in string processing, and quite a few compression methods for these strings have been proposed thus far. Among them, an efficient compression format gathering increasing attention is the run-length Burrows--Wheeler transform (RLBWT), which is a run-length encoded BWT as a reversible permutation of an input string on the lexicographical order of suffixes. State-of-the-art construction algorithms of RLBWT have a serious issue with respect to (i) non-optimal computation time or (ii) a working space that is linearly proportional to the length of an input string. In this paper, we present \emph{r-comp}, the first optimal-time construction algorithm of RLBWT in BWT-runs bounded space. That is, the computational complexity of r-comp is $O(n + r \log{r})$ time and $O(r\log{n})$ bits of working space for the length $n$ of an input string and the number $r$ of equal-letter runs in BWT. The computation time is optimal (i.e., $O(n)$) for strings with the property $r=O(n/\log{n})$, which holds for most highly repetitive strings. Experiments using a real-world dataset of highly repetitive strings show the effectiveness of r-comp with respect to computation time and space.