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This work studies an experimental design problem where {the values of a predictor variable, denoted by $x$}, are to be determined with the goal of estimating a function $m(x)$, which is observed with noise. A linear model is fitted to $m(x)$ but it is not assumed that the model is correctly specified. It follows that the quantity of interest is the best linear approximation of $m(x)$, which is denoted by $\ell(x)$. It is shown that in this framework the ordinary least squares estimator typically leads to an inconsistent estimation of $\ell(x)$, and rather weighted least squares should be considered. An asymptotic minimax criterion is formulated for this estimator, and a design that minimizes the criterion is constructed. An important feature of this problem is that the $x$'s should be random, rather than fixed. Otherwise, the minimax risk is infinite. It is shown that the optimal random minimax design is different from its deterministic counterpart, which was studied previously, and a simulation study indicates that it generally performs better when $m(x)$ is a quadratic or a cubic function. Another finding is that when the variance of the noise goes to infinity, the random and deterministic minimax designs coincide. The results are illustrated for polynomial regression models and the general case is also discussed.

The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. The acyclic chromatic index is the analogous graph parameter for edge colorings. We first show that the acyclic chromatic index is at most $2\Delta-1$, where $\Delta$ is the maximum degree of the graph. We then show that for all $\epsilon >0$ and for $\Delta$ large enough (depending on $\epsilon$), the acyclic chromatic number of the graph is at most $\lceil(2^{-1/3} +\epsilon) {\Delta}^{4/3} \rceil +\Delta+ 1$. Both results improve long chains of previous successive advances. Both are algorithmic, in the sense that the colorings are generated by randomized algorithms. However, in contrast with extant approaches, where the randomized algorithms assume the availability of enough colors to guarantee properness deterministically, and use additional colors for randomization in dealing with the bichromatic cycles, our algorithms may initially generate colorings that are not necessarily proper; they only aim at avoiding cycles where all pairs of edges, or vertices, that are one edge, or vertex, apart in a traversal of the cycle are homochromatic (of the same color). When this goal is reached, they check for properness and if necessary they repeat until properness is attained.

Necessary and sufficient conditions of uniform consistency are explored. Nonparametric sets of alternatives are bounded convex sets in $\mathbb{L}_p$ with "small" balls deleted. The "small" balls have the center at the point of hypothesis and radii of balls tend to zero as sample size increases. For problem of hypothesis testing on a density, we show that, for the sets of alternatives, there are uniformly consistent tests for some sequence of radii of the balls, if and only if, convex set is compact. The results are established for problem of hypothesis testing on a density, for signal detection in Gaussian white noise, for linear ill-posed problems with random Gaussian noise and so on.

Two-sample tests utilizing a similarity graph on observations are useful for high-dimensional and non-Euclidean data due to their flexibility and good performance under a wide range of alternatives. Existing works mainly focused on sparse graphs, such as graphs with the number of edges in the order of the number of observations, and their asymptotic results imposed strong conditions on the graph that can easily be violated by commonly constructed graphs they suggested. Moreover, the graph-based tests have better performance with denser graphs under many settings. In this work, we establish the theoretical ground for graph-based tests with graphs ranging from those recommended in current literature to much denser ones.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size $n$ is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in $\mathbb{R}^K$, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a $\ell_2$ distance of at most $\varepsilon$ from the true simplex (for any $\varepsilon>0$). Also, we theoretically show that in order to achieve this bound, it is sufficient to have $n\ge\left(K^2/\varepsilon^2\right)e^{\Omega\left(K/\mathrm{SNR}^2\right)}$ samples, where $\mathrm{SNR}$ stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as $\mathrm{SNR}\ge\Omega\left(K^{1/2}\right)$, the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in \citep{ashtiani2018nearly}, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

We study the problems of sequential nonparametric two-sample and independence testing. Sequential tests process data online and allow using observed data to decide whether to stop and reject the null hypothesis or to collect more data while maintaining type I error control. We build upon the principle of (nonparametric) testing by betting, where a gambler places bets on future observations and their wealth measures evidence against the null hypothesis. While recently developed kernel-based betting strategies often work well on simple distributions, selecting a suitable kernel for high-dimensional or structured data, such as text and images, is often nontrivial. To address this drawback, we design prediction-based betting strategies that rely on the following fact: if a sequentially updated predictor starts to consistently determine (a) which distribution an instance is drawn from, or (b) whether an instance is drawn from the joint distribution or the product of the marginal distributions (the latter produced by external randomization), it provides evidence against the two-sample or independence nulls respectively. We empirically demonstrate the superiority of our tests over kernel-based approaches under structured settings. Our tests can be applied beyond the case of independent and identically distributed data, remaining valid and powerful even when the data distribution drifts over time.

The Kolmogorov $N$-width describes the best possible error one can achieve by elements of an $N$-dimensional linear space. Its decay has extensively been studied in Approximation Theory and for the solution of Partial Differential Equations (PDEs). Particular interest has occurred within Model Order Reduction (MOR) of parameterized PDEs e.g.\ by the Reduced Basis Method (RBM). While it is known that the $N$-width decays exponentially fast (and thus admits efficient MOR) for certain problems, there are examples of the linear transport and the wave equation, where the decay rate deteriorates to $N^{-1/2}$. On the other hand, it is widely accepted that a smooth parameter dependence admits a fast decay of the $N$-width. However, a detailed analysis of the influence of properties of the data (such as regularity or slope) on the rate of the $N$-width seems to lack. In this paper, we use techniques from Fourier Analysis to derive exact representations of the $N$-width in terms of initial and boundary conditions of the linear transport equation modeled by some function $g$ for half-wave symmetric data. For arbitrary functions $g$, we derive bounds and prove that these bounds are sharp. In particular, we prove that the $N$-width decays as $c_r N^{-(r+1/2)}$ for functions in the Sobolev space, $g\in H^r$. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insight.

We consider dependent clustering of observations in groups. The proposed model, called the plaid atoms model (PAM), estimates a set of clusters for each group and allows some clusters to be either shared with other groups or uniquely possessed by the group. PAM is based on an extension to the well-known stick-breaking process by adding zero as a possible value for the cluster weights, resulting in a zero-augmented beta (ZAB) distribution in the model. As a result, ZAB allows some cluster weights to be exactly zero in multiple groups, thereby enabling shared and unique atoms across groups. We explore theoretical properties of PAM and show its connection to known Bayesian nonparametric models. We propose an efficient slice sampler for posterior inference. Minor extensions of the proposed model for multivariate or count data are presented. Simulation studies and applications using real-world datasets illustrate the model's desirable performance.

Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of some of its partial derivatives in $u$. DDEs occur frequently in combinatorics, especially in map enumeration. If a DDE is of fixed-point type then its solution $F(t,u)$ is unique, and a general result by Popescu (1986) implies that $F(t,u)$ is an algebraic power series. Constructive proofs of algebraicity for solutions of fixed-point type DDEs were proposed by Bousquet-M\'elou and Jehanne (2006). Bostan et. al (2022) initiated a systematic algorithmic study of such DDEs of order 1. We generalize this study to DDEs of arbitrary order. First, we propose nontrivial extensions of algorithms based on polynomial elimination and on the guess-and-prove paradigm. Second, we design two brand-new algorithms that exploit the special structure of the underlying polynomial systems. Last, but not least, we report on implementations that are able to solve highly challenging DDEs with a combinatorial origin.

Autonomous Nano Aerial Vehicles have been increasingly popular in surveillance and monitoring operations due to their efficiency and maneuverability. Once a target location has been reached, drones do not have to remain active during the mission. It is possible for the vehicle to perch and stop its motors in such situations to conserve energy, as well as maintain a static position in unfavorable flying conditions. In the perching target estimation phase, the steady and accuracy of a visual camera with markers is a significant challenge. It is rapidly detectable from afar when using a large marker, but when the drone approaches, it quickly disappears as out of camera view. In this paper, a vision-based target poses estimation method using multiple markers is proposed to deal with the above-mentioned problems. First, a perching target with a small marker inside a larger one is designed to improve detection capability at wide and close ranges. Second, the relative poses of the flying vehicle are calculated from detected markers using a monocular camera. Next, a Kalman filter is applied to provide a more stable and reliable pose estimation, especially when the measurement data is missing due to unexpected reasons. Finally, we introduced an algorithm for merging the poses data from multi markers. The poses are then sent to the position controller to align the drone and the marker's center and steer it to perch on the target. The experimental results demonstrated the effectiveness and feasibility of the adopted approach. The drone can perch successfully onto the center of the markers with the attached 25mm-diameter rounded magnet.