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Motivated by applications in single-cell biology and metagenomics, we consider matrix reordering based on the noisy disordered matrix model. We first establish the fundamental statistical limit for the matrix reordering problem in a decision-theoretic framework and show that a constrained least square estimator is rate-optimal. Given the computational hardness of the optimal procedure, we analyze a popular polynomial-time algorithm, spectral seriation, and show that it is suboptimal. We then propose a novel polynomial-time adaptive sorting algorithm with guaranteed improvement on the performance. The superiority of the adaptive sorting algorithm over the existing methods is demonstrated in simulation studies and in the analysis of two real single-cell RNA sequencing datasets.

Motivated by applications to the theory of rank-metric codes, we study the problem of estimating the number of common complements of a family of subspaces over a finite field in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We then use these bounds to describe the general behaviour of common complements with respect to sparseness and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. By specializing our results to matrix spaces, we obtain upper and lower bounds for the number of MRD codes in the rank metric. In particular, we answer an open question in coding theory, proving that MRD codes are sparse for all parameter sets as the field size grows, with only very few exceptions. We also investigate the density of MRD codes as their number of columns tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing general structural properties of the density function of rank-metric codes.

Recently, random walks on dynamic graphs have been studied because of their adaptivity to the time-varying structure of real-world networks. In general, there is a tremendous gap between static and dynamic graph settings for the lazy simple random walk: Although $O(n^3)$ cover time was shown for any static graphs of $n$ vertices, there is an edge-changing dynamic graph with an exponential hitting time. On the other hand, previous works indicate that the random walk on a dynamic graph with a time-homogeneous stationary distribution behaves almost identically to that on a static graph. In this paper, we strengthen this insight by obtaining general and improved bounds. Specifically, we consider a random walk according to a sequence $(P_t)_{t\geq 1}$ of irreducible and reversible transition matrices such that all $P_t$ have the same stationary distribution. We bound the mixing, hitting, and cover times in terms of the hitting and relaxation times of the random walk according to the worst fixed $P_t$. Moreover, we obtain the first bounds of the hitting and cover times of multiple random walks and the coalescing time on dynamic graphs. These bounds can be seen as an extension of the well-known bounds of random walks on static graphs. Our results generalize the previous upper bounds for specific random walks on dynamic graphs, e.g., lazy simple random walks and $d_{\max}$-lazy walks, and give improved and tight upper bounds in various cases. As an interesting consequence of our generalization, we obtain tight bounds for the lazy Metropolis walk [Nonaka, Ono, Sadakane, and Yamashita, TCS10] on any dynamic graph: $O(n^2)$ mixing time, $O(n^2)$ hitting time, and $O(n^2\log n)$ cover time. Additionally, our coalescing time bound implies the consensus time bound of the pull voting on a dynamic graph.

Maximum likelihood estimates (MLEs) are asymptotically normally distributed, and this property is used in meta-analyses to test the heterogeneity of estimates, either for a single cluster or for several sub-groups. More recently, MLEs for associations between risk factors and diseases have been hierarchically clustered to search for diseases with shared underlying causes, but the approach needs an objective statistical criterion to determine the optimum number and composition of clusters. Conventional statistical tests are briefly reviewed, before considering the posterior distribution associated with partitioning data into clusters. The posterior distribution is calculated by marginalising out the unknown cluster centres, and is different to the likelihood associated with mixture models. The calculation is equivalent to that used to obtain the Bayesian Information Criterion (BIC), but is exact, without a Laplace approximation. The result includes a sum of squares term, and terms that depend on the number and composition of clusters, that penalise the number of free parameters in the model. The usual BIC is shown to be unsuitable for clustering applications unless the number of items in all clusters are sufficiently large.

The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence is $O(n^{-1}\log(n)^r)$ with $r=d$ for extensible sequences and $r=d-1$ otherwise. Such rates hold uniformly over all $d$ dimensional integrands of Hardy-Krause variation one when using $n$ evaluation points. Implicit in those bounds is that for any sequence of QMC points, the integrand can be chosen to depend on $n$. In this paper we show that rates with any $r<(d-1)/2$ can hold when $f$ is held fixed as $n\to\infty$. This is accomplished following a suggestion of Erich Novak to use some unpublished results of Trojan from the 1980s as given in the information based complexity monograph of Traub, Wasilkowski and Wo\'zniakowski. The proof is made by applying a technique of Roth with the theorem of Trojan. The proof is non constructive and we do not know of any integrand of bounded variation in the sense of Hardy and Krause for which the QMC error exceeds $(\log n)^{1+\epsilon}/n$ for infinitely many $n$ when using a digital sequence such as one of Sobol's. An empirical search when $d=2$ for integrands designed to exploit known weaknesses in certain point sets showed no evidence that $r>1$ is needed. An example with $d=3$ and $n$ up to $2^{100}$ might possibly require $r>1$.

We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed $\alpha$ between zero and one we are given a graph and two numbers $k \in \mathbb{N}$ and $t \in \mathbb{Q}$. The task is to find a vertex subset $S$ of exactly $k$ vertices that has value at least (resp. at most for minimization) $t$. Here, the value of a vertex set computes as $\alpha$ times the number of edges with exactly one endpoint in $S$ plus $1-\alpha$ times the number of edges with both endpoints in $S$. These two problems generalize many prominent graph problems, such as Densest $k$-Subgraph, Sparsest $k$-Subgraph, Partial Vertex Cover, and Max ($k$,$n-k$)-Cut. In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max $(k,n-k)$-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms.

When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist) obtained from partial data valid? This paper answers this question by offering a new asymptotic theory about inference with missing data that is more general than existing theories. By using more powerful tools from real analysis and probability theory than those used in previous research, it proves that as the sample size increases and the extent of missingness decreases, the mean-loglikelihood function generated by partial data and that ignores the missingness mechanism will almost surely converge uniformly to that which would have been generated by complete data; and if the data are Missing at Random, this convergence depends only on sample size. Thus, inferences from partial data, such as posterior modes, uncertainty estimates, confidence intervals, likelihood ratios, test statistics, and indeed, all quantities or features derived from the partial-data loglikelihood function, will be consistently estimated. They will approximate their complete-data analogues. This adds to previous research which has only proved the consistency and asymptotic normality of the posterior mode, and developed separate theories for Direct-Likelihood, Bayesian, and Frequentist inference. Practical implications of this result are discussed, and the theory is verified using a previous study of International Human Rights Law.

We study constrained reinforcement learning (CRL) from a novel perspective by setting constraints directly on state density functions, rather than the value functions considered by previous works. State density has a clear physical and mathematical interpretation, and is able to express a wide variety of constraints such as resource limits and safety requirements. Density constraints can also avoid the time-consuming process of designing and tuning cost functions required by value function-based constraints to encode system specifications. We leverage the duality between density functions and Q functions to develop an effective algorithm to solve the density constrained RL problem optimally and the constrains are guaranteed to be satisfied. We prove that the proposed algorithm converges to a near-optimal solution with a bounded error even when the policy update is imperfect. We use a set of comprehensive experiments to demonstrate the advantages of our approach over state-of-the-art CRL methods, with a wide range of density constrained tasks as well as standard CRL benchmarks such as Safety-Gym.

In this paper, we propose a one-stage online clustering method called Contrastive Clustering (CC) which explicitly performs the instance- and cluster-level contrastive learning. To be specific, for a given dataset, the positive and negative instance pairs are constructed through data augmentations and then projected into a feature space. Therein, the instance- and cluster-level contrastive learning are respectively conducted in the row and column space by maximizing the similarities of positive pairs while minimizing those of negative ones. Our key observation is that the rows of the feature matrix could be regarded as soft labels of instances, and accordingly the columns could be further regarded as cluster representations. By simultaneously optimizing the instance- and cluster-level contrastive loss, the model jointly learns representations and cluster assignments in an end-to-end manner. Extensive experimental results show that CC remarkably outperforms 17 competitive clustering methods on six challenging image benchmarks. In particular, CC achieves an NMI of 0.705 (0.431) on the CIFAR-10 (CIFAR-100) dataset, which is an up to 19\% (39\%) performance improvement compared with the best baseline.

In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.