Dimensionality reduction is a crucial technique in data analysis, as it allows for the efficient visualization and understanding of high-dimensional datasets. The circular coordinate is one of the topological data analysis techniques associated with dimensionality reduction but can be sensitive to variations in density. To address this issue, we propose new circular coordinates to extract robust and density-independent features. Our new methods generate a new coordinate system that depends on a shape of an underlying manifold preserving topological structures. We demonstrate the effectiveness of our methods through extensive experiments on synthetic and real-world datasets.
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Deep neural networks are susceptible to generating overconfident yet erroneous predictions when presented with data beyond known concepts. This challenge underscores the importance of detecting out-of-distribution (OOD) samples in the open world. In this work, we propose a novel feature-space OOD detection score that jointly reasons with both class-specific and class-agnostic information. Specifically, our approach utilizes Whitened Linear Discriminant Analysis to project features into two subspaces - the discriminative and residual subspaces - in which the ID classes are maximally separated and closely clustered, respectively. The OOD score is then determined by combining the deviation from the input data to the ID distribution in both subspaces. The efficacy of our method, named WDiscOOD, is verified on the large-scale ImageNet-1k benchmark, with six OOD datasets that covers a variety of distribution shifts. WDiscOOD demonstrates superior performance on deep classifiers with diverse backbone architectures, including CNN and vision transformer. Furthermore, we also show that our method can more effectively detect novel concepts in representation space trained with contrastive objectives, including supervised contrastive loss and multi-modality contrastive loss.
Satisfiability Modulo the Theory of Nonlinear Real Arithmetic, SMT(NRA) for short, concerns the satisfiability of polynomial formulas, which are quantifier-free Boolean combinations of polynomial equations and inequalities with integer coefficients and real variables. In this paper, we propose a local search algorithm for a special subclass of SMT(NRA), where all constraints are strict inequalities. An important fact is that, given a polynomial formula with $n$ variables, the zero level set of the polynomials in the formula decomposes the $n$-dimensional real space into finitely many components (cells) and every polynomial has constant sign in each cell. The key point of our algorithm is a new operation based on real root isolation, called cell-jump, which updates the current assignment along a given direction such that the assignment can `jump' from one cell to another. One cell-jump may adjust the values of several variables while traditional local search operations, such as flip for SAT and critical move for SMT(LIA), only change that of one variable. We also design a two-level operation selection to balance the success rate and efficiency. Furthermore, our algorithm can be easily generalized to a wider subclass of SMT(NRA) where polynomial equations linear with respect to some variable are allowed. Experiments show the algorithm is competitive with state-of-the-art SMT solvers, and performs particularly well on those formulas with high-degree polynomials.
We propose new tools for the geometric exploration of data objects taking values in a general separable metric space $(\Omega, d)$. Given a probability measure on $\Omega$, we introduce depth profiles, where the depth profile of an element $\omega\in\Omega$ refers to the distribution of the distances between $\omega$ and the other elements of $\Omega$. Depth profiles can be harnessed to define transport ranks, which capture the centrality of each element in $\Omega$ with respect to the entire data cloud based on optimal transport maps between depth profiles. We study the properties of transport ranks and show that they provide an effective device for detecting and visualizing patterns in samples of random objects and also entail notions of transport medians, modes, level sets and quantiles for data in general separable metric spaces. Specifically, we study estimates of depth profiles and transport ranks based on samples of random objects and establish the convergence of the empirical estimates to the population targets using empirical process theory. We demonstrate the usefulness of depth profiles and associated transport ranks and visualizations for distributional data through a sample of age-at-death distributions for various countries, for compositional data through energy usage for U.S. states and for network data through New York taxi trips.
Deep neural networks are susceptible to generating overconfident yet erroneous predictions when presented with data beyond known concepts. This challenge underscores the importance of detecting out-of-distribution (OOD) samples in the open world. In this work, we propose a novel feature-space OOD detection score that jointly reasons with both class-specific and class-agnostic information. Specifically, our approach utilizes Whitened Linear Discriminative Analysis to project features into two subspaces - the discriminative and residual subspaces - in which the ID classes are maximally separated and closely clustered, respectively. The OOD score is then determined by combining the deviation from the input data to the ID distribution in both subspaces. The efficacy of our method, named WDiscOOD, is verified on the large-scale ImageNet-1k benchmark, with six OOD datasets that covers a variety of distribution shifts. WDiscOOD demonstrates superior performance on deep classifiers with diverse backbone architectures, including CNN and vision transformer. Furthermore, we also show that our method can more effectively detect novel concepts in representation space trained with contrastive objectives, including supervised contrastive loss and multi-modality contrastive loss.
With the explosive growth in the number of fine-grained images in the Internet era, it has become a challenging problem to perform fast and efficient retrieval from large-scale fine-grained images. Among the many retrieval methods, hashing methods are widely used due to their high efficiency and small storage space occupation. Fine-grained hashing is more challenging than traditional hashing problems due to the difficulties such as low inter-class variances and high intra-class variances caused by the characteristics of fine-grained images. To improve the retrieval accuracy of fine-grained hashing, we propose a cascaded network to learn compact and highly semantic hash codes, and introduce an attention-guided data augmentation method. We refer to this network as a cascaded hierarchical data augmentation network. We also propose a novel approach to coordinately balance the loss of multi-task learning. We do extensive experiments on some common fine-grained visual classification datasets. The experimental results demonstrate that our proposed method outperforms several state-of-art hashing methods and can effectively improve the accuracy of fine-grained retrieval. The source code is publicly available: //github.com/kaiba007/FG-CNET.
Robust semantic segmentation of intraoperative image data could pave the way for automatic surgical scene understanding and autonomous robotic surgery. Geometric domain shifts, however, although common in real-world open surgeries due to variations in surgical procedures or situs occlusions, remain a topic largely unaddressed in the field. To address this gap in the literature, we (1) present the first analysis of state-of-the-art (SOA) semantic segmentation networks in the presence of geometric out-of-distribution (OOD) data, and (2) address generalizability with a dedicated augmentation technique termed "Organ Transplantation" that we adapted from the general computer vision community. According to a comprehensive validation on six different OOD data sets comprising 600 RGB and hyperspectral imaging (HSI) cubes from 33 pigs semantically annotated with 19 classes, we demonstrate a large performance drop of SOA organ segmentation networks applied to geometric OOD data. Surprisingly, this holds true not only for conventional RGB data (drop of Dice similarity coefficient (DSC) by 46 %) but also for HSI data (drop by 45 %), despite the latter's rich information content per pixel. Using our augmentation scheme improves on the SOA DSC by up to 67 % (RGB) and 90 % (HSI) and renders performance on par with in-distribution performance on real OOD test data. The simplicity and effectiveness of our augmentation scheme makes it a valuable network-independent tool for addressing geometric domain shifts in semantic scene segmentation of intraoperative data. Our code and pre-trained models will be made publicly available.
In computational biology, $k$-mers and edit distance are fundamental concepts. However, little is known about the metric space of all $k$-mers equipped with the edit distance. In this work, we explore the structure of the $k$-mer space by studying its maximal independent sets (MISs). An MIS is a sparse sketch of all $k$-mers with nice theoretical properties, and therefore admits critical applications in clustering, indexing, hashing, and sketching large-scale sequencing data, particularly those with high error-rates. Finding an MIS is a challenging problem, as the size of a $k$-mer space grows geometrically with respect to $k$. We propose three algorithms for this problem. The first and the most intuitive one uses a greedy strategy. The second method implements two techniques to avoid redundant comparisons by taking advantage of the locality-property of the $k$-mer space and the estimated bounds on the edit distance. The last algorithm avoids expensive calculations of the edit distance by translating the edit distance into the shortest path in a specifically designed graph. These algorithms are implemented and the calculated MISs of $k$-mer spaces and their statistical properties are reported and analyzed for $k$ up to 15. Source code is freely available at //github.com/Shao-Group/kmerspace .
The use of expectiles in risk management has recently gathered remarkable momentum due to their excellent axiomatic and probabilistic properties. In particular, the class of elicitable law-invariant coherent risk measures only consists of expectiles. While the theory of expectile estimation at central levels is substantial, tail estimation at extreme levels has so far only been considered when the tail of the underlying distribution is heavy. This article is the first work to handle the short-tailed setting where the loss (e.g. negative log-returns) distribution of interest is bounded to the right and the corresponding extreme value index is negative. We derive an asymptotic expansion of tail expectiles in this challenging context under a general second-order extreme value condition, which allows to come up with two semiparametric estimators of extreme expectiles, and with their asymptotic properties in a general model of strictly stationary but weakly dependent observations. A simulation study and a real data analysis from a forecasting perspective are performed to verify and compare the proposed competing estimation procedures.
Blind source separation (BSS) aims to recover an unobserved signal $S$ from its mixture $X=f(S)$ under the condition that the effecting transformation $f$ is invertible but unknown. As this is a basic problem with many practical applications, a fundamental issue is to understand how the solutions to this problem behave when their supporting statistical prior assumptions are violated. In the classical context of linear mixtures, we present a general framework for analysing such violations and quantifying their impact on the blind recovery of $S$ from $X$. Modelling $S$ as a multidimensional stochastic process, we introduce an informative topology on the space of possible causes underlying a mixture $X$, and show that the behaviour of a generic BSS-solution in response to general deviations from its defining structural assumptions can be profitably analysed in the form of explicit continuity guarantees with respect to this topology. This allows for a flexible and convenient quantification of general model uncertainty scenarios and amounts to the first comprehensive robustness framework for BSS. Our approach is entirely constructive, and we demonstrate its utility with novel theoretical guarantees for a number of statistical applications.
Edge intelligence refers to a set of connected systems and devices for data collection, caching, processing, and analysis in locations close to where data is captured based on artificial intelligence. The aim of edge intelligence is to enhance the quality and speed of data processing and protect the privacy and security of the data. Although recently emerged, spanning the period from 2011 to now, this field of research has shown explosive growth over the past five years. In this paper, we present a thorough and comprehensive survey on the literature surrounding edge intelligence. We first identify four fundamental components of edge intelligence, namely edge caching, edge training, edge inference, and edge offloading, based on theoretical and practical results pertaining to proposed and deployed systems. We then aim for a systematic classification of the state of the solutions by examining research results and observations for each of the four components and present a taxonomy that includes practical problems, adopted techniques, and application goals. For each category, we elaborate, compare and analyse the literature from the perspectives of adopted techniques, objectives, performance, advantages and drawbacks, etc. This survey article provides a comprehensive introduction to edge intelligence and its application areas. In addition, we summarise the development of the emerging research field and the current state-of-the-art and discuss the important open issues and possible theoretical and technical solutions.
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