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Fine-Grained Object Detection (FGOD) is a critical task in high-resolution aerial image analysis. This letter introduces Orthogonal Mapping (OM), a simple yet effective method aimed at addressing the challenge of semantic confusion inherent in FGOD. OM introduces orthogonal constraints in the feature space by decoupling features from the last layer of the classification branch with a class-wise orthogonal vector basis. This effectively mitigates semantic confusion and enhances classification accuracy. Moreover, OM can be seamlessly integrated into mainstream object detectors. Extensive experiments conducted on three FGOD datasets (FAIR1M, ShipRSImageNet, and MAR20) demonstrate the effectiveness and superiority of the proposed approach. Notably, with just one line of code, OM achieves a 4.08% improvement in mean Average Precision (mAP) over FCOS on the ShipRSImageNet dataset. Codes are released at https://github.com/ZhuHaoranEIS/Orthogonal-FGOD.

Large language models have revolutionized natural language processing by leveraging self-supervised pretraining on vast textual data. Inspired by this success, researchers have investigated complicated speech tokenization methods to discretize continuous speech signals so that language modeling techniques can be applied to speech data. However, existing approaches either model semantic tokens, potentially losing acoustic information, or model acoustic tokens, risking the loss of semantic information. Having multiple token types also complicates the architecture and requires additional pretraining. Here we show that discretizing mel-filterbank channels into discrete intensity bins produces a simple representation (dMel), that performs better than other existing speech tokenization methods. Using a transformer decoder-only architecture for speech-text modeling, we comprehensively evaluate different speech tokenization methods on speech recognition (ASR), speech synthesis (TTS). Our results demonstrate the effectiveness of dMel in achieving high performance on both tasks within a unified framework, paving the way for efficient and effective joint modeling of speech and text., Comment: under review

Diffusion models have become the de-facto approach for generating visual data, which are trained to match the distribution of the training dataset. In addition, we also want to control generation to fulfill desired properties such as alignment to a text description, which can be specified with a black-box reward function. Prior works fine-tune pretrained diffusion models to achieve this goal through reinforcement learning-based algorithms. Nonetheless, they suffer from issues including slow credit assignment as well as low quality in their generated samples. In this work, we explore techniques that do not directly maximize the reward but rather generate high-reward images with relatively high probability -- a natural scenario for the framework of generative flow networks (GFlowNets). To this end, we propose the Diffusion Alignment with GFlowNet (DAG) algorithm to post-train diffusion models with black-box property functions. Extensive experiments on Stable Diffusion and various reward specifications corroborate that our method could effectively align large-scale text-to-image diffusion models with given reward information.

We obtain some sharp $L^p$ weighted Fourier restriction estimates of the form $\|Ef\|_{L^p(B^{n+1}(0,R),Hdx)} \lessapprox R^{\beta}\|f\|_2$, where $E$ is the Fourier extension operator over the truncated paraboloid, and $H$ is a weight function on $\mathbb R^{n+1}$ which is $n$-dimensional up to scale $\sqrt R$., Comment: 15 pages, 2 figures

We explore variants of the following open question: Split $[0,1]^2$ into $N^2$ squares with side length $1/N$. Is there a way to select $N$ such squares such that each line intersects only $O(1)$ of them?, Comment: Updated the bibliography and added Section 5

Vision-Language Models (VLMs) such as GPT-4V have recently demonstrated incredible strides on diverse vision language tasks. We dig into vision-based deductive reasoning, a more sophisticated but less explored realm, and find previously unexposed blindspots in the current SOTA VLMs. Specifically, we leverage Raven's Progressive Matrices (RPMs), to assess VLMs' abilities to perform multi-hop relational and deductive reasoning relying solely on visual clues. We perform comprehensive evaluations of several popular VLMs employing standard strategies such as in-context learning, self-consistency, and Chain-of-thoughts (CoT) on three diverse datasets, including the Mensa IQ test, IntelligenceTest, and RAVEN. The results reveal that despite the impressive capabilities of LLMs in text-based reasoning, we are still far from achieving comparable proficiency in visual deductive reasoning. We found that certain standard strategies that are effective when applied to LLMs do not seamlessly translate to the challenges presented by visual reasoning tasks. Moreover, a detailed analysis reveals that VLMs struggle to solve these tasks mainly because they are unable to perceive and comprehend multiple, confounding abstract patterns in RPM examples., Comment: ICLR 2024 AGI workshop. https://github.com/apple/ml-rpm-bench

For several weights based on lattice point constructions in $\mathbb{R}^d (d \geq 2)$, we prove that the sharp $L^2$ weighted restriction inequality for the sphere is very different than the corresponding result for the paraboloid. The proof uses Poisson summation, linear algebra, and lattice counting. We conjecture that the $L^2$ weighted restriction is generally better for the circle for a wide variety of general sparse weights.

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.

We consider Carleson-Sj\"{o}lin operators on Riemannian manifolds that arise naturally from the study of Bochner-Riesz problems on manifolds. They are special cases of H\"{o}rmander-type oscillatory integral operators. We obtain improved $L^p$ bounds of Carleson-Sj\"{o}lin operators in two cases: The case where the underlying manifold has constant sectional curvature and the case where the manifold satisfies Sogge's chaotic curvature condition. The two results rely on very different methods: To prove the former result, we show that on a Riemannian manifold, the distance function satisfies Bourgain's condition if and only if the manifold has constant sectional curvature. To obtain the second result, we introduce the notion of "contact orders" to H\"{o}rmander-type oscillatory integral operators, prove that if a H\"{o}rmander-type oscillatory integral operator is of a finite contact order, then it always has better $L^p$ bounds than "worst cases" (in spirit of Bourgain and Guth, and Guth, Hickman and Iliopoulou), and eventually verify that for Riemannian manifolds that satisfy Sogge's chaotic curvature condition, their distance functions alway have finite contact orders. As byproducts, we obtain new bounds for Nikodym maximal functions on manifolds of constant sectional curvatures., Comment: fixed typos

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 $\Delta_x(E)$ has positive Lebesgue measure. Aside from this application, the weighted refined decoupling estimates may be of independent interest., Comment: 28 pages. arXiv admin note: text overlap with arXiv:2309.04103

We show that 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 $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\dim_H (E) \in (\frac{d}{2} - \frac{1}{4} - \frac{3}{8d+4}, \frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimensions $d \ge 3$., Comment: 36 pages

Let $\mathrm{SO}(3,\mathbb{R})$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $\mu$. 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 $$ \mu(A^2) > 3.99 \mu(A).$$ 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 $\mathrm{SO}(3,\mathbb{R})$., Comment: 40 pages. Comments are welcome

We present a free Schr\"{o}dinger solution studied by Barcel\'{o}--Bennett--Carbery--Ruiz--Vilela and show why it can be viewed as a sharp example for the recently discovered refined decoupling theorem., Comment: To appear in a Contemporary Mathematics volume, based on a talk in a virtual AMS Special Session on Harmonic Analysis in March 2022

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.

In this paper, we first generalize the work of Bourgain and state a curvature condition for H\"ormander-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 H\"ormander-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 H\"ormander-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., Comment: Introduction expanded; this version submitted to journal

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., Comment: ICLR'23

Let $f$ be the germ of a real analytic function at the origin in $\mathbb{R}^n $ for $n \geq 2$, and suppose the codimension of the zero set of $f$ at $\mathbf{0}$ is at least $2$. We show that $\log |f|$ is $W^{1,1}_{\operatorname{loc}}$ near $\mathbf{0}$. In particular, this implies the differential inequality $|\nabla f |\leq V |f|$ holds with $V \in L^1_{\operatorname{loc}}$. As an application, we derive an inequality relating the {\L}ojasiewicz exponent and singularity exponent for such functions., Comment: 18 pages, comments welcome!

We study the number of degree $n$ number fields with discriminant bounded by $X$. In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for $6 \leq n \leq 94$., Comment: 19 pages; now includes section 3, clarifying relationships between polynomial coefficients, polynomial roots, and etale algebras

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., Comment: 38 pages

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 ., Comment: Accepted by RA-L with IROS 2022

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.

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.

Suppose $G$ is a connected noncompact locally compact group, $A,B$ are nonempty and compact subsets of $G$, $\mu$ is a left Haar measure on $G$. Assuming that $G$ is unimodular, and $ \mu(A^2) < K \mu(A) $ with $K>1$ a fixed constant, our first result shows that there is a continuous surjective group homomorphism $\chi: G\to L$ 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 $\mu(AB)$ is close to $\mu(A)+\mu(B)$). 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., Comment: 25 pages

We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$ with $\mathrm{Gal}(f) \subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants., Comment: Minor revisions

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., Comment: 31 pages; final version to appear in Transactions of the AMS

We introduce Attention Free Transformer (AFT), an efficient variant of Transformers that eliminates the need for dot product self attention. In an AFT layer, the key and value are first combined with a set of learned position biases, the result of which is multiplied with the query in an element-wise fashion. This new operation has a memory complexity linear w.r.t. both the context size and the dimension of features, making it compatible to both large input and model sizes. We also introduce AFT-local and AFT-conv, two model variants that take advantage of the idea of locality and spatial weight sharing while maintaining global connectivity. We conduct extensive experiments on two autoregressive modeling tasks (CIFAR10 and Enwik8) as well as an image recognition task (ImageNet-1K classification). We show that AFT demonstrates competitive performance on all the benchmarks, while providing excellent efficiency at the same time.

We show that the recent techniques developed to study the Fourier restriction problem apply equally well to the Bochner-Riesz problem. This is achieved via applying a pseudo-conformal transformation and a two-parameter induction-on-scales argument. As a consequence, we improve the Bochner-Riesz problem to the best known range of the Fourier restriction problem in all high dimensions., Comment: 51 pages

Point clouds can be represented in many forms (views), typically, point-based sets, voxel-based cells or range-based images(i.e., panoramic view). The point-based view is geometrically accurate, but it is disordered, which makes it difficult to find local neighbors efficiently. The voxel-based view is regular, but sparse, and computation grows cubically when voxel resolution increases. The range-based view is regular and generally dense, however spherical projection makes physical dimensions distorted. Both voxel- and range-based views suffer from quantization loss, especially for voxels when facing large-scale scenes. In order to utilize different view's advantages and alleviate their own shortcomings in fine-grained segmentation task, we propose a novel range-point-voxel fusion network, namely RPVNet. In this network, we devise a deep fusion framework with multiple and mutual information interactions among these three views and propose a gated fusion module (termed as GFM), which can adaptively merge the three features based on concurrent inputs. Moreover, the proposed RPV interaction mechanism is highly efficient, and we summarize it into a more general formulation. By leveraging this efficient interaction and relatively lower voxel resolution, our method is also proved to be more efficient. Finally, we evaluated the proposed model on two large-scale datasets, i.e., SemanticKITTI and nuScenes, and it shows state-of-the-art performance on both of them. Note that, our method currently ranks 1st on SemanticKITTI leaderboard without any extra tricks.

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., Comment: Added an argument showing sharpness of the main result in an average sense; shared to us by Wright

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., Comment: 56 pages; incorporated referee comments, to appear in GAFA

Autoencoder and its variants have been widely applicated in anomaly detection.The previous work memory-augmented deep autoencoder proposed memorizing normality to detect anomaly, however it neglects the feature discrepancy between different resolution scales, therefore we introduce multi-scale memories to record scale-specific features and multi-scale attention fuser between the encoding and decoding module of the autoencoder for anomaly detection, namely MMAE.MMAE updates slots at corresponding resolution scale as prototype features during unsupervised learning. For anomaly detection, we accomplish anomaly removal by replacing the original encoded image features at each scale with most relevant prototype features,and fuse these features before feeding to the decoding module to reconstruct image. Experimental results on various datasets testify that our MMAE successfully removes anomalies at different scales and performs favorably on several datasets compared to similar reconstruction-based methods.

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., Comment: v2: corrected following referee reports, 37 pages

Learning controllable and generalizable representation of multivariate data with desired structural properties remains a fundamental problem in machine learning. In this paper, we present a novel framework for learning generative models with various underlying structures in the latent space. We represent the inductive bias in the form of mask variables to model the dependency structure in the graphical model and extend the theory of multivariate information bottleneck to enforce it. Our model provides a principled approach to learn a set of semantically meaningful latent factors that reflect various types of desired structures like capturing correlation or encoding invariance, while also offering the flexibility to automatically estimate the dependency structure from data. We show that our framework unifies many existing generative models and can be applied to a variety of tasks including multi-modal data modeling, algorithmic fairness, and invariant risk minimization., Comment: ICML2020 accepted paper. Author name fixed

We show that if compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}$, where $d\geq 4$ is an even integer, then the distance set of $E$ has positive Lebesgue measure. This improves the previously best known result towards Falconer's distance set conjecture in even dimensions., Comment: 15 pages. To appear in Math. Ann

For general gravitational collapse, inside the black-hole region, singularities $(r=0)$ may arise. In this article, we aim to answer how strong these singularities could be. We analyse the behaviours of various geometric quantities. In particular, we show that in the most singular scenario, the Kretschmann scalar obeys polynomial blow-up upper bounds $O(1/r^N)$. This improves previously best-known double-exponential upper bounds $O\big(\exp\exp(1/r)\big)$. Our result is sharp in the sense that there are known examples showing that no sub-polynomial upper bound could hold. Finally we do a case study on perturbations of the Schwarzschild solution., Comment: 34 pages, published in Commun. Math. Phys. online on Jan 21, 2020

We show that the sum of the implicit generator log-density $\log p_g$ of a GAN with the logit score of the discriminator defines an energy function which yields the true data density when the generator is imperfect but the discriminator is optimal, thus making it possible to improve on the typical generator (with implicit density $p_g$). To make that practical, we show that sampling from this modified density can be achieved by sampling in latent space according to an energy-based model induced by the sum of the latent prior log-density and the discriminator output score. This can be achieved by running a Langevin MCMC in latent space and then applying the generator function, which we call Discriminator Driven Latent Sampling~(DDLS). We show that DDLS is highly efficient compared to previous methods which work in the high-dimensional pixel space and can be applied to improve on previously trained GANs of many types. We evaluate DDLS on both synthetic and real-world datasets qualitatively and quantitatively. On CIFAR-10, DDLS substantially improves the Inception Score of an off-the-shelf pre-trained SN-GAN~\citep{sngan} from $8.22$ to $9.09$ which is even comparable to the class-conditional BigGAN~\citep{biggan} model. This achieves a new state-of-the-art in unconditional image synthesis setting without introducing extra parameters or additional training.

AI Safety is a major concern in many deep learning applications such as autonomous driving. Given a trained deep learning model, an important natural problem is how to reliably verify the model's prediction. In this paper, we propose a novel framework -- deep verifier networks (DVN) to verify the inputs and outputs of deep discriminative models with deep generative models. Our proposed model is based on conditional variational auto-encoders with disentanglement constraints. We give both intuitive and theoretical justifications of the model. Our verifier network is trained independently with the prediction model, which eliminates the need of retraining the verifier network for a new model. We test the verifier network on out-of-distribution detection and adversarial example detection problems, as well as anomaly detection problems in structured prediction tasks such as image caption generation. We achieve state-of-the-art results in all of these problems., Comment: Accepted to AAAI 2021

We prove a sharp square function estimate for the cone in $\mathbb{R}^3$ and consequently the local smoothing conjecture for the wave equation in $2+1$ dimensions., Comment: Referee's suggestions incorporated

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions., Comment: 43 pages, 6 figures. This article supersedes arXiv:1901.01802