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Privacy protection methods, such as differentially private mechanisms, introduce noise into resulting statistics which often results in complex and intractable sampling distributions. In this paper, we propose to use the simulation-based "repro sample" approach to produce statistically valid confidence intervals and hypothesis tests based on privatized statistics. We show that this methodology is applicable to a wide variety of private inference problems, appropriately accounts for biases introduced by privacy mechanisms (such as by clamping), and improves over other state-of-the-art inference methods such as the parametric bootstrap in terms of the coverage and type I error of the private inference. We also develop significant improvements and extensions for the repro sample methodology for general models (not necessarily related to privacy), including 1) modifying the procedure to ensure guaranteed coverage and type I errors, even accounting for Monte Carlo error, and 2) proposing efficient numerical algorithms to implement the confidence intervals and $p$-values.

A finite element discretization is developed for the Cai-Hu model, describing the formation of biological networks. The model consists of a non linear elliptic equation for the pressure $p$ and a non linear reaction-diffusion equation for the conductivity tensor $\mathbb{C}$. The problem requires high resolution due to the presence of multiple scales, the stiffness in all its components and the non linearities. We propose a low order finite element discretization in space coupled with a semi-implicit time advancing scheme. The code is validated with several numerical tests performed with various choices for the parameters involved in the system. In absence of the exact solution, we apply Richardson extrapolation technique to estimate the order of the method.

The existing randomized algorithms need an initial estimation of the tubal rank to compute a tensor singular value decomposition. This paper proposes a new randomized fixedprecision algorithm which for a given third-order tensor and a prescribed approximation error bound, automatically finds an optimal tubal rank and the corresponding low tubal rank approximation. The algorithm is based on the random projection technique and equipped with the power iteration method for achieving a better accuracy. We conduct simulations on synthetic and real-world datasets to show the efficiency and performance of the proposed algorithm.

Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than two decades. One of the most well-known and widely studied problems is the estimation of the quadratic variation of the continuous component of an It\^o semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed in the literature when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of the truncated moments of a locally stable L\'evy process, we construct a new rate- and variance-efficient volatility estimator for a class of It\^o semimartingales whose jumps behave locally like those of a stable L\'evy process with Blumenthal-Getoor index $Y\in (1,8/5)$ (hence, of unbounded variation). The proposed method is based on a two-step debiasing procedure for the truncated realized quadratic variation of the process. Our Monte Carlo experiments indicate that the method outperforms other efficient alternatives in the literature in the setting covered by our theoretical framework.

Body weight, as an essential physiological trait, is of considerable significance in many applications like body management, rehabilitation, and drug dosing for patient-specific treatments. Previous works on the body weight estimation task are mainly vision-based, using 2D/3D, depth, or infrared images, facing problems in illumination, occlusions, and especially privacy issues. The pressure mapping mattress is a non-invasive and privacy-preserving tool to obtain the pressure distribution image over the bed surface, which strongly correlates with the body weight of the lying person. To extract the body weight from this image, we propose a deep learning-based model, including a dual-branch network to extract the deep features and pose features respectively. A contrastive learning module is also combined with the deep-feature branch to help mine the mutual factors across different postures of every single subject. The two groups of features are then concatenated for the body weight regression task. To test the model's performance over different hardware and posture settings, we create a pressure image dataset of 10 subjects and 23 postures, using a self-made pressure-sensing bedsheet. This dataset, which is made public together with this paper, together with a public dataset, are used for the validation. The results show that our model outperforms the state-of-the-art algorithms over both 2 datasets. Our research constitutes an important step toward fully automatic weight estimation in both clinical and at-home practice. Our dataset is available for research purposes at: //github.com/USTCWzy/MassEstimation.

In this paper, we will show the $L^p$-resolvent estimate for the finite element approximation of the Stokes operator for $p \in \left( \frac{2N}{N+2}, \frac{2N}{N-2} \right)$, where $N \ge 2$ is the dimension of the domain. It is expected that this estimate can be applied to error estimates for finite element approximation of the non-stationary Navier--Stokes equations, since studies in this direction are successful in numerical analysis of nonlinear parabolic equations. To derive the resolvent estimate, we introduce the solution of the Stokes resolvent problem with a discrete external force. We then obtain local energy error estimate according to a novel localization technique and establish global $L^p$-type error estimates. The restriction for $p$ is caused by the treatment of lower-order terms appearing in the local energy error estimate. Our result may be a breakthrough in the $L^p$-theory of finite element methods for the non-stationary Navier--Stokes equations.

A novel stochastic geometry framework is proposed in this paper to study the downlink coverage performance in a millimeter wave (mmWave) cellular network by jointly considering the polar coordinates of the Base Stations (BSs) with respect to the typical user located at the origin. Specifically, both the Euclidean and the angular distances of the BSs in a maximum power-based association policy for the UE are considered to account for realistic beam management considerations, which have been largely ignored in the literature, especially in the cell association phase. For completeness, two other association schemes are considered and exact-form expressions for the coverage probability are derived. Subsequently, the key role of angular distances is highlighted by defining the dominant interferer using angular distance-based criteria instead of Euclidean distance-based, and conducting a dominant interferer-based coverage probability analysis. Among others, the numerical results revealed that considering angular distance-based criteria for determining both the serving and the dominant interfering BS, can approximate the coverage performance more accurately as compared to utilizing Euclidean distance-based criteria. To the best of the authors$'$ knowledge, this is the first work that rigorously explores the role of angular distances in the association policy and analysis of cellular networks.

The generalized coloring numbers of Kierstead and Yang (Order 2003) offer an algorithmically-useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we complete the work of Grohe et al. (WG 2015) by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining if a graph has weak $r$-coloring number at most $k$ is para-NP-hard when parameterized by $k$ for all $r \geq 2$. We adapt this to determining if a graph has $r$-coloring number at most $k$ as well, proving para-NP-hardness for all $r \geq 2$. Para-NP-hardness implies that no XP algorithm (runtime $O(n^{f(k)})$) exists for testing if a generalized coloring number is at most $k$. Moreover, there exists a constant $c$ such that it is NP-hard to approximate the generalized coloring numbers within a factor of $c$. To complement these results, we give an approximation algorithm for the generalized coloring numbers, improving both the runtime and approximation factor of the existing approach of Dvo\v{r}\'{a}k (EuJC 2013). We prove that greedily ordering vertices with small estimated backconnectivity achieves a $(k-1)^{r-1}$-approximation for the $r$-coloring number and an $O(k^{r-1})$-approximation for the weak $r$-coloring number.

Structured recursion schemes such as folds and unfolds have been widely used for structuring both functional programs and program semantics. In this context, it has been customary to implement denotational semantics as folds over an inductive data type to ensure termination and compositionality. Separately, operational models can be given by unfolds, and naturally not all operational models coincide with a given denotational semantics in a meaningful way. To ensure these semantics are coherent it is important to consider the property of full abstraction which relates the denotational and the operational model. In this paper, we show how to engineer a compositional semantics such that full abstraction comes for free. We do this by using distributive laws from which we generate both the operational and the denotational model. The distributive laws ensure the semantics are fully abstract at the type level, thus relieving the programmer from the burden of the proofs.

Recent years have witnessed a remarkable success of large deep learning models. However, training these models is challenging due to high computational costs, painfully slow convergence, and overfitting issues. In this paper, we present Deep Incubation, a novel approach that enables the efficient and effective training of large models by dividing them into smaller sub-modules that can be trained separately and assembled seamlessly. A key challenge for implementing this idea is to ensure the compatibility of the independently trained sub-modules. To address this issue, we first introduce a global, shared meta model, which is leveraged to implicitly link all the modules together, and can be designed as an extremely small network with negligible computational overhead. Then we propose a module incubation algorithm, which trains each sub-module to replace the corresponding component of the meta model and accomplish a given learning task. Despite the simplicity, our approach effectively encourages each sub-module to be aware of its role in the target large model, such that the finally-learned sub-modules can collaborate with each other smoothly after being assembled. Empirically, our method outperforms end-to-end (E2E) training in terms of both final accuracy and training efficiency. For example, on top of ViT-Huge, it improves the accuracy by 2.7% on ImageNet or achieves similar performance with 4x less training time. Notably, the gains are significant for downstream tasks as well (e.g., object detection and image segmentation on COCO and ADE20K). Code is available at //github.com/LeapLabTHU/Deep-Incubation.