We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $\Sigma,\Gamma$ and $\Phi$ are alphabets of constant size, and $\mu$ is a pairwise connected distribution over $\Sigma\times\Gamma\times\Phi$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is $\Omega(1)$, then the following holds. Any triplets of $1$-bounded functions $f\colon \Sigma^n\to\mathbb{C}$, $g\colon \Gamma^n\to\mathbb{C}$, $h\colon \Phi^n\to\mathbb{C}$ satisfying \[ \left|\mathbb{E}_{(x,y,z)\sim \mu^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \] must arise from an Abelian group associated with the distribution $\mu$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $\tilde{f}(x) = \chi(\sigma(x_1),\ldots,\sigma(x_n)) L (x)$, where $\sigma\colon \Sigma \to H$ is some map, $\chi\in \hat{H}^{\otimes n}$ is a character, and $L\colon \Sigma^n\to\mathbb{C}$ is a low-degree function with bounded $2$-norm. En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.
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Many real-world datasets are represented as tensors, i.e., multi-dimensional arrays of numerical values. Storing them without compression often requires substantial space, which grows exponentially with the order. While many tensor compression algorithms are available, many of them rely on strong data assumptions regarding its order, sparsity, rank, and smoothness. In this work, we propose TENSORCODEC, a lossy compression algorithm for general tensors that do not necessarily adhere to strong input data assumptions. TENSORCODEC incorporates three key ideas. The first idea is Neural Tensor-Train Decomposition (NTTD) where we integrate a recurrent neural network into Tensor-Train Decomposition to enhance its expressive power and alleviate the limitations imposed by the low-rank assumption. Another idea is to fold the input tensor into a higher-order tensor to reduce the space required by NTTD. Finally, the mode indices of the input tensor are reordered to reveal patterns that can be exploited by NTTD for improved approximation. Our analysis and experiments on 8 real-world datasets demonstrate that TENSORCODEC is (a) Concise: it gives up to 7.38x more compact compression than the best competitor with similar reconstruction error, (b) Accurate: given the same budget for compressed size, it yields up to 3.33x more accurate reconstruction than the best competitor, (c) Scalable: its empirical compression time is linear in the number of tensor entries, and it reconstructs each entry in logarithmic time. Our code and datasets are available at //github.com/kbrother/TensorCodec.
Ordered sequences of data, specified with a join operation to combine sequences, serve as a foundation for the implementation of parallel functional algorithms. This abstract data type can be elegantly and efficiently implemented using balanced binary trees, where a join operation is provided to combine two trees and rebalance as necessary. In this work, we present a verified implementation and cost analysis of joinable red-black trees in $\textbf{calf}$, a dependent type theory for cost analysis. We implement red-black trees and auxiliary intermediate data structures in such a way that all correctness invariants are intrinsically maintained. Then, we describe and verify precise cost bounds on the operations, making use of the red-black tree invariants. Finally, we implement standard algorithms on sequences using the simple join-based signature and bound their cost in the case that red-black trees are used as the underlying implementation. All proofs are formally mechanized using the embedding of $\textbf{calf}$ in the Agda theorem prover.
Single-view depth estimation can be remarkably effective if there is enough ground-truth depth data for supervised training. However, there are scenarios, especially in medicine in the case of endoscopies, where such data cannot be obtained. In such cases, multi-view self-supervision and synthetic-to-real transfer serve as alternative approaches, however, with a considerable performance reduction in comparison to supervised case. Instead, we propose a single-view self-supervised method that achieves a performance similar to the supervised case. In some medical devices, such as endoscopes, the camera and light sources are co-located at a small distance from the target surfaces. Thus, we can exploit that, for any given albedo and surface orientation, pixel brightness is inversely proportional to the square of the distance to the surface, providing a strong single-view self-supervisory signal. In our experiments, our self-supervised models deliver accuracies comparable to those of fully supervised ones, while being applicable without depth ground-truth data.
The number and arrangement of sensors on an autonomous mobile robot dramatically influence its perception capabilities. Ensuring that sensors are mounted in a manner that enables accurate detection, localization, and mapping is essential for the success of downstream control tasks. However, when designing a new robotic platform, researchers and practitioners alike usually mimic standard configurations or maximize simple heuristics like field-of-view (FOV) coverage to decide where to place exteroceptive sensors. In this work, we conduct an information-theoretic investigation of this overlooked element of mobile robotic perception in the context of simultaneous localization and mapping (SLAM). We show how to formalize the sensor arrangement problem as a form of subset selection under the E-optimality performance criterion. While this formulation is NP-hard in general, we further show that a combination of greedy sensor selection and fast convex relaxation-based post-hoc verification enables the efficient recovery of certifiably optimal sensor designs in practice. Results from synthetic experiments reveal that sensors placed with OASIS outperform benchmarks in terms of mean squared error of visual SLAM estimates.
Out-of-distribution (OOD) detection is critical to ensuring the reliability and safety of machine learning systems. For instance, in autonomous driving, we would like the driving system to issue an alert and hand over the control to humans when it detects unusual scenes or objects that it has never seen before and cannot make a safe decision. This problem first emerged in 2017 and since then has received increasing attention from the research community, leading to a plethora of methods developed, ranging from classification-based to density-based to distance-based ones. Meanwhile, several other problems are closely related to OOD detection in terms of motivation and methodology. These include anomaly detection (AD), novelty detection (ND), open set recognition (OSR), and outlier detection (OD). Despite having different definitions and problem settings, these problems often confuse readers and practitioners, and as a result, some existing studies misuse terms. In this survey, we first present a generic framework called generalized OOD detection, which encompasses the five aforementioned problems, i.e., AD, ND, OSR, OOD detection, and OD. Under our framework, these five problems can be seen as special cases or sub-tasks, and are easier to distinguish. Then, we conduct a thorough review of each of the five areas by summarizing their recent technical developments. We conclude this survey with open challenges and potential research directions.
Generalization to out-of-distribution (OOD) data is a capability natural to humans yet challenging for machines to reproduce. This is because most learning algorithms strongly rely on the i.i.d.~assumption on source/target data, which is often violated in practice due to domain shift. Domain generalization (DG) aims to achieve OOD generalization by using only source data for model learning. Since first introduced in 2011, research in DG has made great progresses. In particular, intensive research in this topic has led to a broad spectrum of methodologies, e.g., those based on domain alignment, meta-learning, data augmentation, or ensemble learning, just to name a few; and has covered various vision applications such as object recognition, segmentation, action recognition, and person re-identification. In this paper, for the first time a comprehensive literature review is provided to summarize the developments in DG for computer vision over the past decade. Specifically, we first cover the background by formally defining DG and relating it to other research fields like domain adaptation and transfer learning. Second, we conduct a thorough review into existing methods and present a categorization based on their methodologies and motivations. Finally, we conclude this survey with insights and discussions on future research directions.
Recently, Mutual Information (MI) has attracted attention in bounding the generalization error of Deep Neural Networks (DNNs). However, it is intractable to accurately estimate the MI in DNNs, thus most previous works have to relax the MI bound, which in turn weakens the information theoretic explanation for generalization. To address the limitation, this paper introduces a probabilistic representation of DNNs for accurately estimating the MI. Leveraging the proposed MI estimator, we validate the information theoretic explanation for generalization, and derive a tighter generalization bound than the state-of-the-art relaxations.
Domain generalization (DG), i.e., out-of-distribution generalization, has attracted increased interests in recent years. Domain generalization deals with a challenging setting where one or several different but related domain(s) are given, and the goal is to learn a model that can generalize to an unseen test domain. For years, great progress has been achieved. This paper presents the first review for recent advances in domain generalization. First, we provide a formal definition of domain generalization and discuss several related fields. Next, we thoroughly review the theories related to domain generalization and carefully analyze the theory behind generalization. Then, we categorize recent algorithms into three classes and present them in detail: data manipulation, representation learning, and learning strategy, each of which contains several popular algorithms. Third, we introduce the commonly used datasets and applications. Finally, we summarize existing literature and present some potential research topics for the future.
Deep Learning (DL) is vulnerable to out-of-distribution and adversarial examples resulting in incorrect outputs. To make DL more robust, several posthoc anomaly detection techniques to detect (and discard) these anomalous samples have been proposed in the recent past. This survey tries to provide a structured and comprehensive overview of the research on anomaly detection for DL based applications. We provide a taxonomy for existing techniques based on their underlying assumptions and adopted approaches. We discuss various techniques in each of the categories and provide the relative strengths and weaknesses of the approaches. Our goal in this survey is to provide an easier yet better understanding of the techniques belonging to different categories in which research has been done on this topic. Finally, we highlight the unsolved research challenges while applying anomaly detection techniques in DL systems and present some high-impact future research directions.
Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.
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