In this article, the authors give a survey on the recent developments of both the John--Nirenberg space $JN_p$ and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO, $VJN_p$, and $CJN_p$ on $mathbb{R}^n$ or a given cube $Q_0sub
setmathbb{R}^n$ with finite side length. In addition, some related open questions are also presented.
The fast developments of radio astronomy open a new window to explore the properties of Dark Matter (DM). The recent direct imaging of the supermassive black hole (SMBH) at the center of M87 radio galaxy by the Event Horizon Telescope (EHT) collabora
tion is expected to be very useful to search for possible new physics. In this work, we illustrate that such results can be used to detect the possible synchrotron radiation signature produced by DM annihilation from the innermost region of the SMBH. Assuming the existence of a spiky DM density profile, we obtain the flux density due to DM annihilation induced electrons and positrons, and derive new limits on the DM annihilation cross section via the comparison with the EHT integral flux density at 230 GHz. Our results show that the parameter space can be probed by the EHT observations is largely complementary to other experiments. For DM with typical mass regions of being weakly interacting massive particles, the annihilation cross section several orders of magnitude below the thermal production level can be excluded by the EHT observations under the density spike assumption. Future EHT observations may further improve the sensitivity on the DM searches, and may also provide a unique opportunity to test the interplay between DM and the SMBH.
Most autonomous vehicles (AVs) rely on LiDAR and RGB camera sensors for perception. Using these point cloud and image data, perception models based on deep neural nets (DNNs) have achieved state-of-the-art performance in 3D detection. The vulnerabili
ty of DNNs to adversarial attacks have been heavily investigated in the RGB image domain and more recently in the point cloud domain, but rarely in both domains simultaneously. Multi-modal perception systems used in AVs can be divided into two broad types: cascaded models which use each modality independently, and fusion models which learn from different modalities simultaneously. We propose a universal and physically realizable adversarial attack for each type, and study and contrast their respective vulnerabilities to attacks. We place a single adversarial object with specific shape and texture on top of a car with the objective of making this car evade detection. Evaluating on the popular KITTI benchmark, our adversarial object made the host vehicle escape detection by each model type nearly 50% of the time. The dense RGB input contributed more to the success of the adversarial attacks on both cascaded and fusion models. We found that the fusion model was relatively more robust to adversarial attacks than the cascaded model.
We propose a universal and physically realizable adversarial attack on a cascaded multi-modal deep learning network (DNN), in the context of self-driving cars. DNNs have achieved high performance in 3D object detection, but they are known to be vulne
rable to adversarial attacks. These attacks have been heavily investigated in the RGB image domain and more recently in the point cloud domain, but rarely in both domains simultaneously - a gap to be filled in this paper. We use a single 3D mesh and differentiable rendering to explore how perturbing the meshs geometry and texture can reduce the robustness of DNNs to adversarial attacks. We attack a prominent cascaded multi-modal DNN, the Frustum-Pointnet model. Using the popular KITTI benchmark, we showed that the proposed universal multi-modal attack was successful in reducing the models ability to detect a car by nearly 73%. This work can aid in the understanding of what the cascaded RGB-point cloud DNN learns and its vulnerability to adversarial attacks.
We study the problem of learning similarity by using nonlinear embedding models (e.g., neural networks) from all possible pairs. This problem is well-known for its difficulty of training with the extreme number of pairs. For the special case of using
linear embeddings, many studies have addressed this issue of handling all pairs by considering certain loss functions and developing efficient optimization algorithms. This paper aims to extend results for general nonlinear embeddings. First, we finish detailed derivations and provide clean formulations for efficiently calculating some building blocks of optimization algorithms such as function, gradient evaluation, and Hessian-vector product. The result enables the use of many optimization methods for extreme similarity learning with nonlinear embeddings. Second, we study some optimization methods in detail. Due to the use of nonlinear embeddings, implementation issues different from linear cases are addressed. In the end, some methods are shown to be highly efficient for extreme similarity learning with nonlinear embeddings.
Some Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$ can be characterized in terms of the behavior of the Fourier--Haar coefficients. In this article, the authors discuss some necessary restrictions for the parameters $s$, $tau$, $p$, $q$ and $n$ of
this characterization. Therefore, the authors measure the regularity of the characteristic function $mathcal X$ of the unit cube in $mathbb{R}^n$ via the Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$. Furthermore, the authors study necessary and sufficient conditions such that the operation $langle f, mathcal{X} rangle$ generates a continuous linear functional on $B^{s,tau}_{p,q}(mathbb{R}^n)$.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good an
d bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first introduce the variable weak Hardy space on $mathbb R^n$, $W!H^{p(cdot)}(mathbb R^n)$, via
the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W!H^{p(cdot)}(mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_{lambda}^ast$-function. As an application, the authors establish the boundedness of convolutional $delta$-type and non-convolutional $gamma$-order Calderon-Zygmund operators from $H^{p(cdot)}(mathbb R^n)$ to $W!H^{p(cdot)}(mathbb R^n)$ including the critical case $p_-={n}/{(n+delta)}$, where $p_-:=mathopmathrm{ess,inf}_{xin rn}p(x).$
In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $varphi$-transforms in the sense of Frazier and Jawerth, smooth atoms o
r Peetre maximal functions, as well as a Sobolev-type embedding. As an application of their atomic characterization, the authors obtain a trace theorem of these variable Besov-type spaces.
