We construct an estimator $\widehat{\Sigma}$ for covariance matrices of unknown, centred random vectors X, with the given data consisting of N independent measurements $X_1,...,X_N$ of X and the wanted confidence level. We show under minimal assumptions on X, the estimator performs with the optimal accuracy with respect to the operator norm. In addition, the estimator is also optimal with respect to direction dependence accuracy: $\langle \widehat{\Sigma}u,u\rangle$ is an optimal estimator for $\sigma^2(u)=\mathbb{E}\langle X,u\rangle^2$ when $\sigma^2(u)$ is ``large".
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Quantifying the heterogeneity is an important issue in meta-analysis, and among the existing measures, the $I^2$ statistic is most commonly used. In this paper, we first illustrate with a simple example that the $I^2$ statistic is heavily dependent on the study sample sizes, mainly because it is used to quantify the heterogeneity between the observed effect sizes. To reduce the influence of sample sizes, we introduce an alternative measure that aims to directly measure the heterogeneity between the study populations involved in the meta-analysis. We further propose a new estimator, namely the $I_A^2$ statistic, to estimate the newly defined measure of heterogeneity. For practical implementation, the exact formulas of the $I_A^2$ statistic are also derived under two common scenarios with the effect size as the mean difference (MD) or the standardized mean difference (SMD). Simulations and real data analysis demonstrate that the $I_A^2$ statistic provides an asymptotically unbiased estimator for the absolute heterogeneity between the study populations, and it is also independent of the study sample sizes as expected. To conclude, our newly defined $I_A^2$ statistic can be used as a supplemental measure of heterogeneity to monitor the situations where the study effect sizes are indeed similar with little biological difference. In such scenario, the fixed-effect model can be appropriate; nevertheless, when the sample sizes are sufficiently large, the $I^2$ statistic may still increase to 1 and subsequently suggest the random-effects model for meta-analysis.
We consider a discrete best approximation problem formulated in the framework of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. Given a set of samples of input and output of an unknown function, the problem is to construct a generalized tropical Puiseux polynomial that best approximates the function in the sense of a tropical distance function. The construction of an approximate polynomial involves the evaluation of both unknown coefficient and exponent of each monomial in the polynomial. To solve the approximation problem, we first reduce the problem to an equation in unknown vector of coefficients, which is given by a matrix with entries parameterized by unknown exponents. We derive a best approximate solution of the equation, which yields both vector of coefficients and approximation error parameterized by the exponents. Optimal values of exponents are found by minimization of the approximation error, which is reduced to a minimization of a function of exponents over all partitions of a finite set. We solve this minimization problem in terms of max-plus algebra (where addition is defined as maximum and multiplication as arithmetic addition) by using a computational procedure based on the agglomerative clustering technique. This solution is extended to the minimization problem of finding optimal exponents in the polynomial in terms of max-algebra (where addition is defined as maximum). The results obtained are applied to develop new solutions for conventional problems of discrete best approximation of real functions by piecewise linear functions and piecewise Puiseux polynomials. We discuss computational complexity of the proposed solution and estimate upper bounds on the computational time. We demonstrate examples of approximation problems solved in terms of max-plus and max-algebra, and give graphical illustrations.
We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.
In this article, we propose a new classification of $\Sigma^0_2$ formulas under the realizability interpretation of many-one reducibility (i.e., Levin reducibility). For example, ${\sf Fin}$, the decision of being eventually zero for sequences, is many-one/Levin complete among $\Sigma^0_2$ formulas of the form $\exists n\forall m\geq n.\varphi(m,x)$, where $\varphi$ is decidable. The decision of boundedness for sequences ${\sf BddSeq}$ and posets ${\sf PO}_{\sf top}$ are many-one/Levin complete among $\Sigma^0_2$ formulas of the form $\exists n\forall m\geq n\forall k.\varphi(m,k,x)$, where $\varphi$ is decidable. However, unlike the classical many-one reducibility, none of the above is $\Sigma^0_2$-complete. The decision of non-density of linear order ${\sf NonDense}$ is truly $\Sigma^0_2$-complete.
Near-Optimal differentially private low-rank trace regression with guaranteed private initialization
We study differentially private (DP) estimation of a rank-$r$ matrix $M \in \RR^{d_1\times d_2}$ under the trace regression model with Gaussian measurement matrices. Theoretically, the sensitivity of non-private spectral initialization is precisely characterized, and the differential-privacy-constrained minimax lower bound for estimating $M$ under the Schatten-$q$ norm is established. Methodologically, the paper introduces a computationally efficient algorithm for DP-initialization with a sample size of $n \geq \wt O (r^2 (d_1\vee d_2))$. Under certain regularity conditions, the DP-initialization falls within a local ball surrounding $M$. We also propose a differentially private algorithm for estimating $M$ based on Riemannian optimization (DP-RGrad), which achieves a near-optimal convergence rate with the DP-initialization and sample size of $n \geq \wt O(r (d_1 + d_2))$. Finally, the paper discusses the non-trivial gap between the minimax lower bound and the upper bound of low-rank matrix estimation under the trace regression model. It is shown that the estimator given by DP-RGrad attains the optimal convergence rate in a weaker notion of differential privacy. Our powerful technique for analyzing the sensitivity of initialization requires no eigengap condition between $r$ non-zero singular values.
For multivariate data, tandem clustering is a well-known technique aiming to improve cluster identification through initial dimension reduction. Nevertheless, the usual approach using principal component analysis (PCA) has been criticized for focusing solely on inertia so that the first components do not necessarily retain the structure of interest for clustering. To address this limitation, a new tandem clustering approach based on invariant coordinate selection (ICS) is proposed. By jointly diagonalizing two scatter matrices, ICS is designed to find structure in the data while providing affine invariant components. Certain theoretical results have been previously derived and guarantee that under some elliptical mixture models, the group structure can be highlighted on a subset of the first and/or last components. However, ICS has garnered minimal attention within the context of clustering. Two challenges associated with ICS include choosing the pair of scatter matrices and selecting the components to retain. For effective clustering purposes, it is demonstrated that the best scatter pairs consist of one scatter matrix capturing the within-cluster structure and another capturing the global structure. For the former, local shape or pairwise scatters are of great interest, as is the minimum covariance determinant (MCD) estimator based on a carefully chosen subset size that is smaller than usual. The performance of ICS as a dimension reduction method is evaluated in terms of preserving the cluster structure in the data. In an extensive simulation study and empirical applications with benchmark data sets, various combinations of scatter matrices as well as component selection criteria are compared in situations with and without outliers. Overall, the new approach of tandem clustering with ICS shows promising results and clearly outperforms the PCA-based approach.
In 2017, Aharoni proposed the following generalization of the Caccetta-H\"{a}ggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$. In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed $r \geq 1$, there exists a constant $c_r$ such that if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $n/r + c_r$.
We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. A proof is given that no finite element approximation can converge at a better rate than that given by the definition, justifying the concept. A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of our definition.
We give an almost complete characterization of the hardness of $c$-coloring $\chi$-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing. In particular, we show that these problems do not admit any distributed quantum advantage. To do that: 1) We give a new distributed algorithm that finds a $c$-coloring in $\chi$-chromatic graphs in $\tilde{\mathcal{O}}(n^{\frac{1}{\alpha}})$ rounds, with $\alpha = \bigl\lfloor\frac{c-1}{\chi - 1}\bigr\rfloor$. 2) We prove that any distributed algorithm for this problem requires $\Omega(n^{\frac{1}{\alpha}})$ rounds. Our upper bound holds in the classical, deterministic LOCAL model, while the near-matching lower bound holds in the non-signaling model. This model, introduced by Arfaoui and Fraigniaud in 2014, captures all models of distributed graph algorithms that obey physical causality; this includes not only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL, even with a pre-shared quantum state. We also show that similar arguments can be used to prove that, e.g., 3-coloring 2-dimensional grids or $c$-coloring trees remain hard problems even for the non-signaling model, and in particular do not admit any quantum advantage. Our lower-bound arguments are purely graph-theoretic at heart; no background on quantum information theory is needed to establish the proofs.
We introduce the concept of memoryless concretization relation (MCR) to describe abstraction within the context of controller synthesis. This relation is a specific instance of alternating simulation relation (ASR), where it is possible to simplify the controller architecture. In the case of ASR, the concretized controller needs to simulate the concurrent evolution of two systems, the original and abstract systems, while for MCR, the designed controllers only need knowledge of the current concrete state. We demonstrate that the distinction between ASR and MCR becomes significant only when a non-deterministic quantizer is involved, such as in cases where the state space discretization consists of overlapping cells. We also show that any abstraction of a system that alternatingly simulates a system can be completed to satisfy MCR at the expense of increasing the non-determinism in the abstraction. We clarify the difference between the MCR and the feedback refinement relation (FRR), showing in particular that the former allows for non-constant controllers within cells. This provides greater flexibility in constructing a practical abstraction, for instance, by reducing non-determinism in the abstraction. Finally, we prove that this relation is not only sufficient, but also necessary, for ensuring the above properties.
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