arXiv subjects
Publications and source records attributed to Ruixiang Zhang.
We prove that if $f\in L^p(\mathbb{R}^k)$ with $p<(k^2+k+2)/2$ satisfies that $\widehat{f}$ is supported on a small perturbation of the moment curve in $\mathbb{R}^k$, then $f$ is identically zero. This improves the more general result of Agranovsky and Narayanan, and the exponents are sharp in all dimensions. In the process, we develop a mechanism that should lead to further progress on related problems.
This paper focuses on the scale imbalance problem of semi-supervised object detection(SSOD) in aerial images. Compared to natural images, objects in aerial images show smaller sizes and larger quantities per image, increasing the difficulty of manual annotation. Meanwhile, the advanced SSOD technique can train superior detectors by leveraging limited labeled data and massive unlabeled data, saving annotation costs. However, as an understudied task in aerial images, SSOD suffers from a drastic performance drop when facing a large proportion of small objects. By analyzing the predictions between small and large objects, we identify three imbalance issues caused by the scale bias, i.e., pseudo-label imbalance, label assignment imbalance, and negative learning imbalance. To tackle these issues, we propose a novel Scale-discriminative Semi-Supervised Object Detection (S^3OD) learning pipeline for aerial images. In our S^3OD, three key components, Size-aware Adaptive Thresholding (SAT), Size-rebalanced Label Assignment (SLA), and Teacher-guided Negative Learning (TNL), are proposed to warrant scale unbiased learning. Specifically, SAT adaptively selects appropriate thresholds to filter pseudo-labels for objects at different scales. SLA balances positive samples of objects at different scales through resampling and reweighting. TNL alleviates the imbalance in negative samples by leveraging information generated by a teacher model. Extensive experiments conducted on the DOTA-v1.5 benchmark demonstrate the superiority of our proposed methods over state-of-the-art competitors. Codes will be released soon.
We prove some weighted refined decoupling estimates. As an application, we give an alternative proof of the following result on Falconer's distance set problem by the authors in a companion work: if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 4$, then there is a point $x\in E$ such that the pinned distance set $Δ_x(E)$ has positive Lebesgue measure. Aside from this application, the weighted refined decoupling estimates may be of independent interest.
Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick.
Let $\mathrm{SO}(3,\mathbb{R})$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $μ$. Resolving a problem by Breuillard and Green, we show that if $A \subseteq \mathrm{SO}(3,\mathbb{R})$ is an open subset with sufficiently small measure, then $$ μ(A^2) > 3.99 μ(A).$$ We also show a more general result for the product of two sets, which can be seen as a Brunn-Minkowski-type inequality for sets with small measure in $\mathrm{SO}(3,\mathbb{R})$.
We present a free Schrödinger solution studied by Barceló--Bennett--Carbery--Ruiz--Vilela and show why it can be viewed as a sharp example for the recently discovered refined decoupling theorem.
We present Bit Diffusion: a simple and generic approach for generating discrete data with continuous state and continuous time diffusion models. The main idea behind our approach is to first represent the discrete data as binary bits, and then train a continuous diffusion model to model these bits as real numbers which we call analog bits. To generate samples, the model first generates the analog bits, which are then thresholded to obtain the bits that represent the discrete variables. We further propose two simple techniques, namely Self-Conditioning and Asymmetric Time Intervals, which lead to a significant improvement in sample quality. Despite its simplicity, the proposed approach can achieve strong performance in both discrete image generation and image captioning tasks. For discrete image generation, we significantly improve previous state-of-the-art on both CIFAR-10 (which has 3K discrete 8-bit tokens) and ImageNet-64x64 (which has 12K discrete 8-bit tokens), outperforming the best autoregressive model in both sample quality (measured by FID) and efficiency. For image captioning on MS-COCO dataset, our approach achieves competitive results compared to autoregressive models.
In this paper, we first generalize the work of Bourgain and state a curvature condition for Hörmander-type oscillatory integral operators, which we call Bourgain's condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for Hörmander-type oscillatory integral operators satisfying Bourgain's condition, they satisfy the same $L^p$ bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain's condition fails, then the $L^{\infty} \to L^q$ boundedness always fails for some $q= q(n) > \frac{2n}{n-1}$, extending Bourgain's three-dimensional result. On the other hand, if Bourgain's condition holds, then we prove $L^p$ bounds for Hörmander-type oscillatory integral operators for a range of $p$ that extends the currently best-known range for the Fourier restriction conjecture in high dimensions, given by Hickman and Zahl. This gives new progress on the Fourier restriction problem and the Bochner-Riesz problem.
Deep Neural Network (DNN) based point cloud semantic segmentation has presented significant achievements on large-scale labeled aerial laser point cloud datasets. However, annotating such large-scaled point clouds is time-consuming. Due to density variations and spatial heterogeneity of the Airborne Laser Scanning (ALS) point clouds, DNNs lack generalization capability and thus lead to unpromising semantic segmentation, as the DNN trained in one region underperform when directly utilized in other regions. However, Self-Supervised Learning (SSL) is a promising way to solve this problem by pre-training a DNN model utilizing unlabeled samples followed by a fine-tuned downstream task involving very limited labels. Hence, this work proposes a hard-negative sample aware self-supervised contrastive learning method to pre-train the model for semantic segmentation. The traditional contrastive learning for point clouds selects the hardest negative samples by solely relying on the distance between the embedded features derived from the learning process, potentially evolving some negative samples from the same classes to reduce the contrastive learning effectiveness. Therefore, we design an AbsPAN (Absolute Positive And Negative samples) strategy based on k-means clustering to filter the possible false-negative samples. Experiments on two typical ALS benchmark datasets demonstrate that the proposed method is more appealing than supervised training schemes without pre-training. Especially when the labels are severely inadequate (10% of the ISPRS training set), the results obtained by the proposed HAVANA method still exceed 94% of the supervised paradigm performance with full training set.
Humans are remarkably good at understanding and reasoning about complex visual scenes. The capability to decompose low-level observations into discrete objects allows us to build a grounded abstract representation and identify the compositional structure of the world. Accordingly, it is a crucial step for machine learning models to be capable of inferring objects and their properties from visual scenes without explicit supervision. However, existing works on object-centric representation learning either rely on tailor-made neural network modules or strong probabilistic assumptions in the underlying generative and inference processes. In this work, we present \ours, a conceptually simple and general approach to learning object-centric representations through an energy-based model. By forming a permutation-invariant energy function using vanilla attention blocks readily available in Transformers, we can infer object-centric latent variables via gradient-based MCMC methods where permutation equivariance is automatically guaranteed. We show that \ours can be easily integrated into existing architectures and can effectively extract high-quality object-centric representations, leading to better segmentation accuracy and competitive downstream task performance. Further, empirical evaluations show that \ours's learned representations are robust against distribution shift. Finally, we demonstrate the effectiveness of \ours in systematic compositional generalization, by re-composing learned energy functions for novel scene generation and manipulation.
LiDAR sensor is essential to the perception system in autonomous vehicles and intelligent robots. To fulfill the real-time requirements in real-world applications, it is necessary to efficiently segment the LiDAR scans. Most of previous approaches directly project 3D point cloud onto the 2D spherical range image so that they can make use of the efficient 2D convolutional operations for image segmentation. Although having achieved the encouraging results, the neighborhood information is not well-preserved in the spherical projection. Moreover, the temporal information is not taken into consideration in the single scan segmentation task. To tackle these problems, we propose a novel approach to semantic segmentation for LiDAR sequences named Meta-RangeSeg, where a new range residual image representation is introduced to capture the spatial-temporal information. Specifically, Meta-Kernel is employed to extract the meta features, which reduces the inconsistency between the 2D range image coordinates input and 3D Cartesian coordinates output. An efficient U-Net backbone is used to obtain the multi-scale features. Furthermore, Feature Aggregation Module (FAM) strengthens the role of range channel and aggregates features at different levels. We have conducted extensive experiments for performance evaluation on SemanticKITTI and SemanticPOSS. The promising results show that our proposed Meta-RangeSeg method is more efficient and effective than the existing approaches. Our full implementation is publicly available at https://github.com/songw-zju/Meta-RangeSeg .
Brascamp-Lieb inequalities have been important in analysis, mathematical physics and neighboring areas. Recently, these inequalities have had a deep influence on Fourier analysis and, in particular, on Fourier restriction theory. In this article we motivate and explain this connection. A lot of our examples are taken from a rapidly developing subarea called "decoupling". It is the author's hope that this article will be accessible to graduate students in fields broadly related to analysis.
In this work, we focus on the high-dimensional trace regression model with a low-rank coefficient matrix. We establish a nearly optimal in-sample prediction risk bound for the rank-constrained least-squares estimator under no assumptions on the design matrix. Lying at the heart of the proof is a covering number bound for the family of projection operators corresponding to the subspaces spanned by the design. By leveraging this complexity result, we perform a power analysis for a permutation test on the existence of a low-rank signal under the high-dimensional trace regression model. We show that the permutation test based on the rank-constrained least-squares estimator achieves non-trivial power with no assumptions on the minimum (restricted) eigenvalue of the covariance matrix of the design. Finally, we use alternating minimization to approximately solve the rank-constrained least-squares problem to evaluate its empirical in-sample prediction risk and power of the resulting permutation test in our numerical study.
We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell\geq 1$ and $\mathbb{R}^{d}=\mathbb{R}^{d_1}\times\cdots \times\mathbb{R}^{d_\ell}$, $d_i\geq 2$. For any compact set $E\subset \mathbb{R}^{d}$ with Hausdorff dimension larger than $d-\frac{\min(d_i)}{2}+\frac{1}{4}$ if $\min(d_i) $ is even, $d-\frac{\min(d_i)}{2}+\frac{1}{4}+\frac{1}{4\min(d_i)}$ if $\min(d_i) $ is odd, we prove that the multiparameter distance set of $E$ has positive $\ell$-dimensional Lebesgue measure. A key ingredient in the proof is a new multiparameter radial projection theorem for fractal measures.
We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.
In this paper, we study the representation of neural networks from the view of kernels. We first define the Neural Fisher Kernel (NFK), which is the Fisher Kernel applied to neural networks. We show that NFK can be computed for both supervised and unsupervised learning models, which can serve as a unified tool for representation extraction. Furthermore, we show that practical NFKs exhibit low-rank structures. We then propose an efficient algorithm that computes a low rank approximation of NFK, which scales to large datasets and networks. We show that the low-rank approximation of NFKs derived from unsupervised generative models and supervised learning models gives rise to high-quality compact representations of data, achieving competitive results on a variety of machine learning tasks.
Suppose $G$ is a connected noncompact locally compact group, $A,B$ are nonempty and compact subsets of $G$, $μ$ is a left Haar measure on $G$. Assuming that $G$ is unimodular, and $ μ(A^2) < K μ(A) $ with $K>1$ a fixed constant, our first result shows that there is a continuous surjective group homomorphism $χ: G\to L$ with compact kernel, where $L$ is a Lie group with $$\dim(L) \leq \lfloor\log K\rfloor(\lfloor\log K\rfloor+1)/2.$$ We also demonstrate that this dimension bound is sharp, establish the relationship between $A$ and its image under the quotient map, and obtain a more general version of this result for the product set $AB$ without assuming unimodularity. Our second result classifies $G,A,B$ where $A,B$ have nearly minimal expansions (when $G$ is unimodular, this just means $μ(AB)$ is close to $μ(A)+μ(B)$). This answers a question suggested by Griesmer and Tao, and completes the last open case of the inverse Kemperman problem. The proofs of both results involve a new analysis of locally compact group $G$ with bounded $n-h$, where $n-h$ is an invariant of $G$ appearing in the recently developed nonabelian Brunn-Minkowski inequality. We also generalize Ruzsa's distance and related results to possibly nonunimodular locally compact groups.
We propose a new method of estimating oscillatory integrals, which we call a stationary set method. We use it to obtain the sharp convergence exponents of Tarry's problems in dimension two for every degree $k\ge 2$. As a consequence, we obtain sharp Fourier extension estimates for a family of monomial surfaces.